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A study of the condensed phase in phosphorus doped silicon Halliwell, Robin Ernest 1973

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A STUDY OF THE CONDENSED PHASE IN PHOSPHORUS DOPED SILICON  by  ROBIN ERNEST HALLIWELL B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1967 M . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  In t h e Department  of Physics  We a c c e p t t h i s required  t h e s i s as conforming t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s representatives.  be granted by the Head of my Department or I t i s understood that copying or p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  ABSTRACT  Photoluminescent s t u d i e s give evidence phase o f e l e c t r o n s and  h o l e s can be  t h a t a condensed  formed i n s i l i c o n  phorus donor c o n c e n t r a t i o n s from 9.0  x 10  1 5  cm""  3  c o n t a i n i n g phos-  to 4.3  x lO ^  cm .  1  -3  When the i m p u r i t y c o n c e n t r a t i o n i s l e s s than t h a t r e q u i r e d f o r d e r e a l i z a t i o n o f the donor e l e c t r o n s , the condensed phase takes  the form of *!  e l e c t r o n - h o l e drops w i t h c a r r i e r c o n c e n t r a t i o n s of 3.0 For Impurity  x 10 ° cm  p r e t e d as e v i d e n c e  observations are  the photoluminescence and  p a r a m a g n e t i c resonance f o r samples c o n t a i n i n g i m p u r i t y cm- ^ 3  N  inter-  of s i l i c o n doped w i t h phosphorus i s found  produce marked changes i n both  1 8  .  -  f o r a condensed phase i n v o l v i n g o n l y h o l e s .  Heat treatment  x 10  3  c o n c e n t r a t i o n s g r e a t e r than t h a t r e q u i r e d f o r d e r e a l i z a -  t i o n o f t h e donor e l e c t r o n s the e x p e r i m e n t a l  2.2  ft  D  < 1.3  x 10  1 9  cm" . 3  to  the e l e c t r o n  concentrations  These changes a r e i n t e r p r e t e d  i n terms of t h e p r o d u c t i o n of n e u t r a l t r a p s by r e s i d u a l i m p u r i t i e s such as c a r b o n and  oxygen.  The  luminescence lends support for  high impurity  e f f e c t o f t h i s heat t o the h y p o t h e s i s  concentrations.  treatment  on the  photo-  of a condensed h o l e phase  TABLE OF CONTENTS Page Abstract  i  Table of Contents  .  .  L i s t of Tables  v  L i s t of Figures  vi  Acknowledgements Chapter 1  i l  .  viii  INTRODUCTION  1.1  General Introduction  1  1.2  Purpose and Outline of this Thesis  2  Chapter 2  PHOTOLUMINESCENCE OF INTRINSIC SILICON 'AND GERMANIUM  2.1  Introduction  3  2.2  Free Excitons  4  2.3  Excitonic Molecule  8  2.4  Bose-Einstein Condensation  2.5  Electron-Hole Drops (EHD)  2.6  Experimental Evidence for Biexcitdns and Electron-Hole Drops  Chapter 3  10 10  16  EXPERIMENTAL  3.1  Sample Preparation  19  3.2  Photoluminescent Spectrometer  20  3.3  Electron Paramagnetic Resonance Spectrometer  22  3.4  Temperature Measurements  22  iii Page Chapter  4  CONCENTRATION DEPENDENCE OF THE PHOTOLUMINESCENCE OF S i : P  4.1  Introduction  4.2  C o n c e n t r a t i o n Dependence o f the  29 Photoluminescent  S p e c t r a o f S i : P a t 2K 4.3  Low C o n c e n t r a t i o n  (9 x 1 0 4.4  15  cm  1 N  -3  < 3.6 x 1 0  D  17  30  cm" ) 3  T r a n s i t i o n Range  (1.1 x 1 0  Chapter  30  18  cm  -3  1 N  D  < 5.5 x 1 0  (N^ > I O  18  34  cm" ) 3  4.5  High C o n c e n t r a t i o n  5  TEMPERATURE DEPENDENCE OF THE PHOTOLUMINESCENCE OF S i : P  5.1  Introduction  5.2  Temperature Dependence o f t h e Photoluminescence  2 9  cm )  36  -3  ,  37  38  of S i : P 5.3  Low C o n c e n t r a t i o n  1)  9 x 10  2)  3.6 x 1 0  15  cm" 17  38  3  cm"  41  3  !  i 5.4  T r a n s i t i o n Range  (1.1 x 1 0 5.5  Chapter  18  cm"  3  < N  High  Concentration  1)  1.3 x 1 0  19  cm  - 3  2)  4.3 x 1 0  19  cm  -3  D  < 5.5 x 1 0  6  DISCUSSION OF RESULTS  6.1  Screening of Excitons  6.2  I n t e r p r e t a t i o n of Spectra  19  41  cm ) - 3  43 43  \  44 48  iv  Page 6.3  Low C o n c e n t r a t i o n  (9 x 1 0 6.4  1 5  cm  - 3  < N  D  < 1.1 x 1 0  1 8  49  cm ) -3  T r a n s i t i o n Range  (2.2 x 1 0  1 8  cm"  3  < N  D  <  5.5 x 1 0  1 8  54  cm ) -3  6.5  High C o n c e n t r a t i o n (N^ > 1.3 x 1 0  6.6  I n t e n s i t y o f Photoluminescence  58  6.7  Temperature  60  Chapter 7  1 9  cm ) -3  .......  Dependence o f Photoluminescence  56  EFFECT OF HEAT TREATMENT ON S i ; P  7.1  Introduction  63  7.2  P h o t o l u m i n e s c e n c e from Heat T r e a t e d S i : P  64  7.3  EPR o f Heat T r e a t e d S i : P  67  7.4  Discussion of Results  72  Chapter 8  ...;  CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY  8.1  Conclusions  78  8.2  S u g g e s t i o n s f o r F u r t h e r Study  79  Appendix A  C o r r e c t i o n f o r I n s t r u m e n t a l Broadening  81  Appendix  C a l c u l a t i o n o f L i n e s h a p e f o r E l e c t r o n - H o l e Drops  84  B  Bibliography  88  V  LIST OF TABLES Table  2.1  Page  Intrinsic  Recombination R a d i a t i o n  Bands o f  S i l i c o n a t 26K  2.2  Band Parameters f o r S i l i c o n  2.3  T h e o r e t i c a l E q u i l i b r i u m D e n s i t y and Ground  7  12  State  Energy o f EHD  2.4  E x p e r i m e n t a l V a l u e s of E q u i l i b r i u m D e n s i t y and Ground S t a t e Energy o f EHD  14  vi  LIST OF FIGURES  Figure  Page  2.1  Band S t r u c t u r e of I n t r i n s i c  S i l i c o n a t 2K  6  2.2  Ground S t a t e of EHD  3.1  O p t i c a l C o n f i g u r a t i o n o f P h o t o l u m i n e s c e n t Spectrometer . . .  21  3.2  D e t a i l s of Temperature  24  3.3  Temperature  C a l i b r a t i o n Procedure  26  3.4  Temperature  D i f f e r e n c e Between Sample and Sample Mount . . .  28  4.1  Photoluminescence f o r S i : P w i t h N D  = 9 x 1015  4.2  Photoluminescence f o r Si:P w i t h  = 3.6 x 1 0 1 7  4.3  C o n c e n t r a t i o n Dependence o f t h e Photoluminescence  in Silicon  14  C o n t r o l System  cm - 3  a t 2K .  31  a t 2K  32  cm - 3  of S i : P a t 2K 5.1  35  Temperature Dependence of the Photoluminescence f o r i  N 5.2  D  = 9 x 10  15  cm  -3  '  Temperature Dependence o f the Photoluminescence f o r  N = 3.6 x 1 0 1 7 cm - 3 D  5.3  40  Temperature Dependence of the Photoluminescene f o r  N - 1.3 x 1 0 1 9 cm - 3  42  C o n c e n t r a t i o n Dependence o f t h e T h r e s h o l d Energy E  50  D  6.1  39  vii  Figure  Page  6.2  Half-Width  6.3  Comparison of C a l c u l a t e d and E x p e r i m e n t a l  o f TO A s s i s t e d Peak  v s . Impurity  Concentration  L i n e Shapes  a t 2K  52  6.4  I n t e n s i t y o f TO A s s i s t e d Peak  v s . Impurity  7.1  P h o t o l u m i n e s c e n c e v s . Impurity  C o n c e n t r a t i o n f o r Heat-  Concentration  T r e a t e d S i : P a t 13K  7.2  D  = 3.0  x 10  59.  65  P h o t o l u m i n e s c e n c e v s . Temperature f o r H e a t - T r e a t e d with N  51  1 8  cm  •  -3  7.3  EPR  Spectra f o r  = 2.0  x 10  1 8  cm  7.4  EPR  Spectra f o r N  = 6.2  x 10  1 8  cm-  -3  3  Si:P 66  69  70  viii  ACKNOWLEDGEMENTS  I w i s h t o e x p r e s s my thanks t o my s u p e r v i s o r Dr. R.R. Parsons f o r h i s s u p p o r t , guidance and f r i e n d s h i p d u r i n g t h e s e s t u d i e s . To D r . R. B a r r i e , I am g r a t e f u l f o r h i s c r i t i c i s m , and many i n v a l u a b l e  advice,  discussions.  I would l i k e t o thank Dr. J . Marko f o r p r o v i d i n g .the o r i g i n a l impetus  f o r t h i s work and g u i d i n g i t through t h e e a r l y s t a g e s , a l s o  Dr. J . Q u i r t  f o r a d v i c e and s u g g e s t i o n s on some o f t h e e x p e r i m e n t a l  aspects. A s p e c i a l thanks must go t o Dr. C.F. Schwerdtfeger f o r h i s s u p p o r t and encouragement d u r i n g my e a r l y y e a r s as a graduate s t u d e n t . G r a t e f u l acknowledgement i s g i v e n t o t h e N a t i o n a l Research C o u n c i l and t h e H.R. M a c M i l l a n f a m i l y f o r f i n a n c i a l s u p p o r t . T h i s r e s e a r c h was s u p p o r t e d by g r a n t s t o Dr. C.F. Schwerdtfeger CNRC 67-2228) and Dr. R.R. Parsons  (NRC 67-6714).  F i n a l l y I w i s h t o thank my w i f e J a n e t , w i t h o u t whose, h e l p t h i s t h e s i s may n e v e r have been  completed.  - 1 CHAPTER 1  INTRODUCTION  1.1  General Introduction A s u f f i c i e n t l y h i g h d e n s i t y o f p h o t o c r e a t e d e l e c t r o n s and  h o l e s i n a semiconductor  can condense to form what i s known as  an  (1 2) "electron-hole droplet"  '  As a r e s u l t o f t h i s c o n d e n s a t i o n some  r e g i o n s o f the c r y s t a l c o n t a i n two plasmas, one due moving i n a u n i f o r m p o s i t i v e background and one, i n a u n i f o r m n e g a t i v e background. n e u t r a l and  to the h o l e s moving  These two plasmas a r e  can be t r e a t e d as b e i n g n o n - i n t e r a c t i n g .  are macroscopic  t o the e l e c t r o n s  electrically  These r e g i o n s  and i n q u a s i - t h e r m a l e q u i l i b r i u m w i t h the  Only semiconductors  lattice.  h a v i n g an i n d i r e c t band gap,  s i l i c o n and germanium, are expected  t o have f r e e c a r r i e r  l o n g enough f o r such a c o n d e n s a t i o n  to o c c u r .  been c o n s i d e r a b l e e x p e r i m e n t a l and  such  as  lifetimes  Consequently  t h e r e has  t h e o r e t i c a l work done on these  m a t e r i a l s , b o t h t o e s t a b l i s h the e x i s t e n c e o f s u c h a condensate,  two and  (3) to s t u d y i t s p r o p e r t i e s  .  Some work has been done on t h e d i r e c t band  (4 5) gap  semiconductor  GaAs but the r e s u l t s a r e i n c o n c l u s i v e  '  .  To  date,  work i n t h i s f i e l d has been r e s t r i c t e d t o i n t r i n s i c m a t e r i a l s or t o m a t e r i a l s w i t h i m p u r i t y c o n c e n t r a t i o n s low enough t h a t i m p u r i t y — i m p u r i t y i n t e r a c t i o n s may One  be n e g l e c t e d .  interesting  f e a t u r e t h a t has come from t h e s e s t u d i e s i s  t h a t the e q u i l i b r i u m d e n s i t y o f the condensed phase i s a p p r o x i m a t e l y same as the i m p u r i t y c o n c e n t r a t i o n r e q u i r e d f o r the d e r e a l i z a t i o n o f the i m p u r i t y s t a t e s .  T h i s immediately  r a i s e s the q u e s t i o n o f what  the  - 2-  happens t o the condensed  phase when t h e i m p u r i t y c o n c e n t r a t i o n I s c l o s e  to o r above t h i s c r i t i c a l c o n c e n t r a t i o n f o r d e r e a l i z a t i o n .  1.2  Purpose  and O u t l i n e o f t h i s T h e s i s  The main purpose of t h i s t h e s i s i s t o i n v e s t i g a t e t h e photol u m i n e s c e n c e o f s i l i c o n h e a v i l y doped w i t h phosphorus.  In p a r t i c u l a r  the  n a t u r e o f the condensed  phase i s s t u d i e d as the m a t e r i a l  the  t r a n s i t i o n from " s e m i c o n d u c t i n g " t o " m e t a l l i c " b e h a v i o r .  undergoes This i s  c o r r e l a t e d w i t h a s u b s i d i a r y study o f the e f f e c t s o f h e a t - t r e a t m e n t on the  photoluminescence The  and e l e c t r o n paramagnetic  resonance.  t h e s i s i s d i v i d e d i n t o seven c h a p t e r s .  The t h e o r e t i c a l  and e x p e r i m e n t a l i n v e s t i g a t i o n s o f e x c i t o n s and v a r i o u s condensed in intrinsic  s i l i c o n and germanium a r e d i s c u s s e d i n Chapter 2.  3 o u t l i n e s the e x p e r i m e n t a l p r o c e d u r e s and p r e c a u t i o n s .  phases  Chapter  The p h o t o l u m i n -  e s c e n t d a t a as f u n c t i o n s o f i m p u r i t y c o n c e n t r a t i o n and temperature a r e p r e s e n t e d i n Chapters 4 and 5, t h e s e r e s u l t s a r e then a n a l y s e d and d i s c u s s e d i n Chapter 6.  A s t u d y o f the e f f e c t s o f h e a t - t r e a t m e n t on t h e  p h o t o l u m i n e s c e n c e and e l e c t r o n paramagnetic Chapter 7 and an attempt the  previous chapters.  and  conclusions.  resonance i s p r e s e n t e d i n  i s made t o c o r r e l a t e t h i s w i t h t h e r e s u l t s o f A f i n a l c h a p t e r p r o v i d e s a summary o f the r e s u l t s  - 3-  CHAPTER 2 PHOTOLUMINESCENCE OF INTRINSIC SILICON AND GERMANIUM  2.1  Introduction I n r e c e n t y e a r s photoluminescence  has developed  f u l t o o l f o r the i n v e s t i g a t i o n of semiconductors. absorbs  a photon w i t h energy  This photo-created  When a c r y s t a l  g r e a t e r than t h e band gap energy  t r o n i s e j e c t e d i n t o the c o n d u c t i o n band, l e a v i n g band.  i n t o a power-  an e l e c -  a hole i n the valence  e l e c t r o n and h o l e r a p i d l y l o s e energy  and f a l l  t o t h e bottom o f t h e c o n d u c t i o n band and t h e top o f t h e v a l e n c e band r e s p e c t i v e l y , where they recombine w i t h t h e e m i s s i o n o f a photon.  Direct  Band Gap  I n d i r e c t Band Gap  - 4 -  T h i s r e c o m b i n a t i o n may  be d i r e c t band to band or i t may  involve  an  electron-hole  complex, such as an e x c i t o n or the more r e c e n t l y observed  electron-hole  droplet.  B e f o r e i t i s p o s s i b l e to u n d e r s t a n d the e f f e c t s o f  specific  i m p u r i t i e s on the p r o p e r t i e s of a m a t e r i a l i t i s f i r s t n e c e s s a r y understand i t s i n t r i n s i c p r o p e r t i e s . easy t o p r e p a r e i n a pure s t a t e , and is  e x c i t o n s and  the v a r i o u s  to dope by p r e s c r i b e d amounts, i t  the two  t h e o r e t i c a l models f o r f r e e  e x c i t o n condensate models t h a t have been p r o -  posed t o e x p l a i n the r e c e n t  2.2  Because s i l i c o n i s r e l a t i v e l y  a n e a r l y i d e a l m a t e r i a l w i t h which t o work. T h i s chapter w i l l present  and  p h o t o l u m i n e s c e n t d a t a from i n t r i n s i c  silicon  germanium.  Free  Excitons E x c i t o n s were f i r s t d e s c r i b e d  by F r e n k e l  r e s u l t o f i n v e s t i g a t i o n s i n t o the a b s o r p t i o n in  to  insulating solids.  c r y s t a l c o u l d be l a t e d atoms.  i n 1 9 3 1 ^ ^ as  processes that take  He proposed t h a t t r i a l e x c i t e d s t a t e s of  constructed  a  from the ground and  place the  e x c i t e d s t a t e s of  iso-  In t h i s model, the ground s t a t e of the c r y s t a l i s w r i t t e n !  as an a n t i s y m m e t r i z e d p r o d u c t o f n o n - o v e r l a p p i n g o r t h o n o r m a l e l e c t r o n wave f u n c t i o n s . constructed and  by  In an analogous manner an e x c i t e d s t a t e can  assuming t h a t an atom a t one  s u b s t i t u t i n g the a p p r o p r i a t e  The  be  s i t e i s i n an e x c i t e d s t a t e ,  one-electron  i n t o the ground s t a t e wave f u n c t i o n . o f the  one-  e x c i t e d s t a t e wave f u n c t i o n  true stationary excited  c r y s t a l can be e x p r e s s e d as l i n e a r combinations of these  states  localized  e x c i t a t i o n s , each h a v i n g a wave v e c t o r K i n the r e c i p r o c a l space.  - 5 -  The F r e n k e l model, which i s a c t u a l l y a t i g h t b i n d i n g  model,  (7 8) .has been u s e d e x t e n s i v e l y i n the s t u d y o f m o l e c u l a r c r y s t a l s  '  where e x c i t a t i o n o n l y i n v o l v e s a rearrangement o f atomic o r b i t a l s t h e m o l e c u l e and does not a p p r e c i a b l y cules.  increase the overlap  within  between mole-  The model has been found t o have o n l y l i m i t e d u s e f u l n e s s  i n rare  (9) gas  solids  d e s p i t e the r e l a t i v e l y  and  i n general  i s n o t a p p l i c a b l e t o most o t h e r  Of more i n t e r e s t adopted by Wannier  i n the present  work on s i l i c o n i s the approach  The p a i r would move t h r o u g h the l a t t i c e ,  related, just  representing  band e l e c -  considerable  t h e i r centre of  The F r e n k e l and Wannier models a r e  the two extremes o f e x c i t o n r a d i u s r :  F r e n k e l model, r < a, t h e i n t e r a t o m i c »  crystalline insulators.  band h o l e , a l t h o u g h p o s s i b l y w i t h  mass h a v i n g t o t a l wave v e c t o r K.  r  i n t h e i r ground s t a t e s ,  He viewed t h e e x c i t o n as a c o n d u c t i o n  t r o n bound t o a v a l e n c e separation.  tight binding  f o r the  d i s t a n c e ; f o r the Wannier model,  a. The H a m i l t o n i a n  f o r the Wannier e x c i t o n i s t h a t f o r a two  p a r t i c l e system, i n t e r a c t i n g v i a the coulomb i n t e r a c t i o n m o d i f i e d dielectric  c o n s t a n t of the medium: "& V Ti V = £. k - £2 2  K  2  2  by the  2  2  * 2m  * 2^  e  (2.2-11  icr  T h i s has h y d r o g e n i c energy l e v e l s which can be w r i t t e n a s :  E  (1) -Ji£— 2n2 2 2 K  * where  n  f \ ( m * + n^)  (2.2-2)  K2  m  n  2  * and m^  a r e the e f f e c t i v e masses o f t h e e l e c t r o n and  hole  - 6 -  E i  •  Conduction  band  i 1  E = 1.1698 eV n  \  r Valence  bands >  (0,0,0) FIGURE 2 . 1  ! (1,0,0) Band Structure of Intrinsic Silicon at 2 K .  T a b l e 2.1  Intrinsic  Recombination R a d i a t i o n Bands o f S i l i c o n a t 26K  Threshold  Phonon  Energy (eV)  Phonon  Energy (meV)  (12,14)  Relative  Assignment  Intensity  1.1550  ^0  1.1365  18.3  TA  0.035  1.0998  55.3  LO  -  1.0978  57.3  TO  1.074  80.8  TO + I V  a  1.051  103.8  TO + I V  b  1.0315  122.3  TO + 0  F  1.013  142  TO + 0  r  0.968  187  TO + 2 0  No phonon  ^0.004  1.00 0.016 M3.008 0.07 + IV  3  ^0.01  F  TA:  T r a n s v e r s e a c o u s t i c a l momentum c o n s e r v i n g  TO:  Transverse  iv  a  0 : F  'k;  o p t i c a l momentum c o n s e r v i n g  Phonons s e l e c t e d f o r i n t e r v a l l e y C e n t r e o f reduced  -vO.0025  scattering  zone - z e r o wave v e c t o r  - 8 -  r e s p e c t i v e l y , K i s the d i e l e c t r i c c o n s t a n t , and y i s the reduced mass of  the system.  brillouin energy is  S i n c e the wave v e c t o r K can range over the  first  zone t h e r e are e x c i t o n "bands" r a t h e r than s t a t e s .  The  r e q u i r e d t o d i s s o c i a t e the e l e c t r o n and h o l e when the e x c i t o n  i n i t s ground s t a t e , more c o r r e c t l y , when t h e c r y s t a l i s i n i t s  l o w e s t e x c i t e d s t a t e , i s c a l l e d the e x c i t o n " r y d b e r g " and  yel  G=  *  2h K 2  i s given  *"  by:  (2.2-3)  2  E x c i t o n s were f i r s t  observed  i n s i l i c o n by M a c f a r l a n e  w h i l e d o i n g a b s o r p t i o n s t u d i e s a t the band edge.  et a l .  They observed s e v e r a l  knees i n the a b s o r p t i o n c o e f f i c i e n t o f the band edge which were e x p l a i n ed, i n terms o f Wannier e x c i t o n s w i t h an e x c i t o n r y d b e r g o f 10 meV.  More  (12) r e c e n t l y Shaklee  and Nahory  , u s i n g wave l e n g t h d e r i v a t i v e  have r e f i n e d t h i s v a l u e t o G = 14.7 n - 2 e x c i t o n e x c i t e d s t a t e and  nons was  first  observed  by d i r e c t o b s e r v a t i o n o f  the  the onset of the e x c i t o n continuum.  The photoluminescence e x c i t o n s w i t h the simultaneous  meV  techniques,  produced  by the r e c o m b i n a t i o n  e m i s s i o n of a photon and  ( F i g u r e 2.1)  free  one o r more (13)  i n s i l i c o n by Haynes e t a l . i n 1960  s i l i c o n has an i n d i r e c t band gap  of  .  pho-  Because  the r e c o m b i n a t i o n must be  phonon a s s i s t e d , w i t h a t l e a s t one phonon r e q u i r e d i n o r d e r t h a t momentum be c o n s e r v e d .  T a b l e 2.1  t i o n r a d i a t i o n bands and 2.3  Excitonic  lists  the e n e r g i e s of the i n t r i n s i c  the phonons i n v o l v e d w i t h  1  f o r two  each.  Molecule  Lampert^ " ^, i n 1 9 5 8 , ' f i r s t suggested ble  recombina-  5  t h a t i t s h o u l d be  possi~  e x c i t o n s to b i n d t o g e t h e r t o form an e x c i t o n i c m o l e c u l e .  His  - 9 -  contention  was based upon the e a r l i e r  W h e e l e r o f  the existence  t h e o r e t i c a l p r e d i c t i o n of  o f p o s i t r o n i u m and a c o r r e s p o n d i n g  posi-  tronium molecule. Conservation considerations when two e x c i t o n s  bind  t h a t a phonon be e m i t t e d  t o form an e x c i t o n i c m o l e c u l e o r what i s u s u a l l y  r e f e r r e d t o as a b i e x c i t o n . tion  require  For two e x c i t o n s  a t r e s t , energy c o n s e r v a -  requires:  AE  where  = E  b  - Tua  (2.3-1)  i  i s t h e d i s s o c i a t i o n energy o f the b i e x c i t o n , Tiw^ i s the energy  o f t h e e m i t t e d phonon, and AE i s the k i n e t i c energy o f the b i e x c i t o n acquired  by the e m i s s i o n of the phonon.  l y studied  by A s n i n e t a l .  who  biexcitons,  extensivequasi-  and t h e e m i t t e d phonons.  can be e x p r e s s e d by:  n  L f C n  + 1 )  where  nnfe  =  f  Y =  and  have been  show t h a t t h e r e s h o u l d be a  e q u i l i b r i u m between f r e e e x c i t o n s , This  Biexcitons  where g ^  exciton  ex  -AE/kT  (2.3-2)  > m  ex  *  \ 3/z  2 urn kT ex h  2  and g^ a r e the degeneracy f a c t o r s o f the e x c i t o n  and b i -  s t a t e s ; n ^ , n^, and n^ r e f e r t o the number o f e x c i t o n s , b i -  excitons,  and phonons i n a g i v e n s t a t e r e s p e c t i v e l y .  F o r low e x c i t a t i o n  l e v e l s a t low temperatures t h e number of phonons i s g i v e n by t h e e q u i l i brium  number:  - 10  -nto nf  » n  -x. e  -  /kT *  C2.3-3)  o and  s u b s t i t u t i n g t h i s i n t o equation  (2.3-2) g i v e s n 2 ^  a n ^ , the number  of b i e x c i t o n s v a r i e s as the square of the number o f f r e e e x c i t o n s . h i g h e x c i t a t i o n l e v e l s the number of phonons c r e a t e d by the  formation  of b i e x c i t o n s exceeds the e q u i l i b r i u m v a l u e , t h e r e f o r e n^ ^ n^ s u b s t i t u t i o n i n t o equation  (2.3-2) g i v e s n g x  experimental  that E^  0.1  - 1.5  2.4  meV  and  a n ^ , the number of b i -  e x c i t o n s v a r i e s l i n e a r l y as the number o f f r e e e x c i t o n s . of  At  d a t a r e l a t i n g t o bound e x c i t o n s , Haynes  G, where G i s the e x c i t o n r y d b e r g .  On  the b a s i s estimates  T h i s would g i v e  as the b i n d i n g energy of a b i e x c i t o n i n s i l i c o n .  Bose-Einstein  Condensation (19)  I t was  p o s t u l a t e d by Moskalenko  i n 1961  that there  could  e x i s t a condensed phase i n a semiconductor c o n s i s t i n g o f e x c i t o n s bound by Van  der waals a t t r a c t i o n .  condensation t i o n , and 2.5  I I possess  E l e c t r o n - H o l e Drops  condensa-  superfluidity.  (EHD)  e x i s t e n c e o f a condensed phase c o n s i s t i n g o f a degenerate  e l e c t r o n - h o l e plasma was  first  suggested by K e l d y s h ^  p h o t o c o n d u c t i v i t y s t u d i e s made by R o g a c h e v ^ ^ . found i n germanium at low  temperatures and  which were q u a l i t a t i v e l y e x p l a i n e d plasmas.  i n t e g r a l s p i n , such a  would have the c h a r a c t e r i s t i c s of a B o s e - E i n s t e i n  l i k e He  The  As the e x c i t o n has  on the b a s i s o f  H i g h m o b i l i t i e s were  h i g h photo e x c i t a t i o n  levels  i n terms of degenerate e l e c t r o n - h o l e  - 11 -  The problem o f m o b i l e c a r r i e r s o f one s i g n moving i n a homogeneous background  formed by c a r r i e r s of t h e o t h e r s i g n was s o l v e d by  (21) Pines the  .  The problem o f two m o b i l e c a r r i e r s i s approached by summing  s o l u t i o n s f o r an e l e c t r o n plasma and a h o l e plasma.  has t h e o b v i o u s d e f e c t  o f n o t i n c l u d i n g any e l e c t r o n - h o l e  Such a s o l u t i o n correlations.  A t p r e s e n t t h e r e i s no method f o r c a l c u l a t i n g t h i s term, however i t i s believed  that  the electron-hole  the e l e c t r o n - e l e c t r o n  c o r r e l a t i o n i s o f the same magnitude as  and h o l e - h o l e c o r r e l a t i o n s  (3)  .  I n a degenerate system the average energy p e r e l e c t r o n - h o l e (21) p a i r c a n be w r i t t e n  as  :  E = 3/5 (F + F J + E e S . + E h h . + E 6 6 + E™ 1 e h exch exch corr corr  F E  g  and F ^ a r e t h e F e r m i e n e r g i e s o f t h e e l e c t r o n s  6 6  corr  and h o l e s ,  (2.5-1)  ee ^ i » e x c  1  , and E*1*1 E*1*1 a r e t h e exchange and c o r r e l a t i o n e n e r g i e s o f the exch' corr  electrons  and h o l e s ,  made t o i n c l u d e  respectively.  electron-hole  interactions.  S i l i c o n i s characterized minima which a r e l o c a t e d  As n o t e d above, no attempt has been  by s i x p a r a b o l i c  c o n d u c t i o n band  a l o n g the (100) d i r e c t i o n s , about t w o - t h i r d s o f  the way to t h e zone edge, and a t w o - f o l d degenerate v a l e n c e band i t s maximum a t K = 0. the e x c i t o n  The e l e c t r o n and h o l e Fermi e n e r g i e s ,  with  i n units of  r y d b e r g , may be w r i t t e n a s :  3/5  F  -  ^1  e  r  2 z  s  ,2/3  6  (2.5-2) m, de  - 12 -  3/5 F. =  2.21  (2.5-3)  3/2 1 +  where r  g  i s t h e mean s p a c i n g between e l e c t r o n s o r h o l e s i n u n i t s o f t h e  e x c i t o n Bohr r a d i u s , a e* x u  m  m , oh  oe  (2.5-4)  K  a = e x c i t o n Bohr r a d i u s x  m  m  1 2  oh  1  m lh  1 3  oe  _  2m  2/3  m, = m ' de t  +  1  (2.5-5)  'V  (2.5-6)  ^  t  1/3  (2.5-7)  m. / 1  The a p p r o p r i a t e v a l u e s f o r the d i e l e c t r i c c o n s t a n t K and t h e e f f e c t i v e masses a r e g i v e n i n T a b l e 2.2.  Table  2.2  Band Parameters f o r S i l i c o n  C o n d u c t i o n Band  V a l e n c e Band  ni = 0.1905 m t o m. = 0.9163 m 1 o m, = 0.32 m de o m = 0.26 m oe o  m,,. In  p = 0.14 m,.  H  0.154 m  = 0.523  o m Q  m . = 0.237 m oh o  K  » 11.4  - 13 -  The exchange e n e r g i e s have been c a l c u l a t e d by Brinkman and Rice  (4)  u s i n g t h e d e r i v a t i o n o f the exchange energy i n an e l l i p s o i d a l  band by Combescot and N o z i e r e s  (22)  .  T h e i r r e s u l t can be w r i t t e n a s :  Eee . + Ehh = - 1^51 exch exch r  (2.5-8)  u  The c o r r e l a t i o n e n e r g i e s do n o t l e n d t h e m s e l v e s t o s t r a i g h t (23) forward c a l c u l a t i o n s .  Brinkman e t a l .  have used a  generalization  o f a scheme s u g g e s t e d by Hubbard based on t h e random-phase a p p r o x i m a tion.  A l t h o u g h t h i s method g i v e s q u i t e a c c u r a t e r e s u l t s f o r a s i m p l e  band s t r u c t u r e i t cannot be e a s i l y e x t e n d e d t o i n c l u d e band s t r u c t u r e s  complicated  as a r e found i n s i l i c o n and germanium.  Similar results  (22) were o b t a i n e d by Combescot and N o z i e r e s  u s i n g an i n t e r p o l a t i o n  method w h i c h i m p l i c i t l y i n c l u d e s t h e d e t a i l s o f t h e band These terms may be added and t h e e n e r g y p e r  electron-hole  as i s done i n F i g u r e 2.2.  p a i r p l o t t e d as a f u n c t i o n o f r  structure.  Here t h e  s  energy i s e x p r e s s e d i n terms o f t h e e x c i t o n r y d b e r g G. The ground s t a t e e n e r g y o f t h e condensed phase i s o b t a i n e d by m i n i m i z i n g t h e energy p e r e l e c t r o n - h o l e corresponding value of r  g  i  p a i r with respect to r . s  The  i s then r e l a t e d to the e q u i l i b r i u m density  nQ  by — n  o  = | 3  3  3  7rr a s x  These r e s u l t s a r e summarized i n T a b l e 2.3.  C2.5-9]  - 14 -  -2.0 (4) FIGURE 2.2  Ground State of EHD in Silicon.  From Brinkman and Rice  Table 2.3 Theoretical Equilibrium Density and Ground State Energy of EHD  Equilibrium Concentration  Ground State Energy o'  Ge (a)  Cb) S i (a)  '  (b)  Ca) r e f . 4 (b) r e f . 22  1.8 x 1 0  17  cm"  5.3 meV  2.0 x 1 0  17  cm"  6.1 meV  3.4 x 1 0  18  cm"  3  20.4 meV  3.1 x 1 0  18  cm  -3  21.0 meV  3  3  - 15 -  (2) P o k r o v s k i i and S v i s t u n o v a  have used a s i m p l e model t o  o b t a i n t h e dependence o f t h e r e c o m b i n a t i o n r a d i a t i o n from t h e EHD on e x c i t a t i o n l e v e l and t e m p e r a t u r e . They c o n s i d e r e d t h e condensed phase as a c l o u d o f s p h e r i c a l e l e c t r o n - h o l e drops o f r a d i u s r o c c u p y i n g a s m a l l f r a c t i o n o f t h e t o t a l volume o f t h e s a m p l e .  T h i s a s s u m p t i o n was  n e c e s s a r y so t h a t g e n e r a t i o n o f c a r r i e r s w i t h i n t h e drops c o u l d be ignored.  Under s t e a d y s t a t e c o n d i t i o n s t h e f l u x o f e x c i t o n s i n t o t h e  d r o p must be e q u a l t o t h e sum o f t h e r e c o m b i n a t i o n r a t e i n s i d e t h e drop and t h e l o s s o f c a r r i e r s o u t o f t h e drop due t o t h e r m a l e m i s s i o n .  When  t h i s c o n d i t i o n i s combined w i t h t h e s t e a d y s t a t e g e n e r a t i o n r a t e o f c a r r i e r s , g , i t i s found t h a t a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of  t h e EHD i s t h a t : 8 -  >0  (2.5-10)  '  where <}> i s t h e work f u n c t i o n f o r e m i s s i o n o f c a r r i e r s from t h e EHD, T i s t h e e x c i t o n l i f e t i m e , v i s t h e t h e r m a l v e l o c i t y and A i s t h e R i c h a r d s o n c o n s t a n t (A = 7.5 x 1 0 2 0 c m - 2 s e c - 1 d e g - 2 ) .  T h i s i m m e d i a t e l y shows t h a t  t h e r e i s a c r i t i c a l e x c i t a t i o n l e v e l , g c r > and a c r i t i c a l  temperature,  T » which a r e r e l a t e d through: cr  -<|>/kT 4 AT g p  m  cr  2  -  (2.5-11)  vx  On t h e b a s i s o f t h i s model i t i s a l s o p o s s i b l e t o show t h a t f o r l o w e x c i t a t i o n l e v e l s , b u t g r e a t e r than gcr> t h e i n t e n s i t y o f t h e recombinat i o n r a d i a t i o n from w i t h i n t h e EHD, I d , i s p r o p o r t i o n a l t o g 3 . A t h i g h  - 16  -  e x c i t a t i o n l e v e l s 1^ i s l i n e a r l y p r o p o r t i o n a l to g.  These c a l c u l a t i o n s  (3) have been extended somewhat by P o k r o v s k i i o f the EHD  to show t h a t the i n t e n s i t y  r e c o m b i n a t i o n i s r e l a t e d t o t h a t o f the f r e e e x c i t o n , I , ' ex'  through:  I, a I d ex  (2.5-12)  3  T h i s i s independent o f temperature and  e x c i t a t i o n l e v e l i f the  t r a t i o n of drops i s c o n t r o l l e d by n u c l e a t i o n The an EHD  can be  obtained  o f the e l e c t r o n and  n(E  d  4  o  where hv and  centres.  t h e o r e t i c a l l i n e shape f o r r e c o m b i n a t i o n r a d i a t i o n w i t h i n i n a s t r a i g h t f o r w a r d manner.  o n l y t o assume t h a t the m a t r i x  I (hv) a  concen-  e  element f o r r e c o m b i n a t i o n i s independent  hole energies,  ) n(Ej h  I t i s necessary  giving:  f (E ) f ( E j e h  6(hv-E -E -E g e h  ) dE dE^ e h  (2.5-13)  4  o  i s the energy o f the e m i t t e d  photon; and n ( E ) , n ( E ^ ) , e  f(E ), e  f ( E ^ ) are r e s p e c t i v e l y , the d e n s i t y o f s t a t e s i n the c o n d u c t i o n  valence  bands, and  the Fermi d i s t r i b u t i o n f u n c t i o n s f o r e l e c t r o n s  and and  holes.  2.6  E x p e r i m e n t a l E v i d e n c e f o r B i e x c i t o n s and At present  densation  of e x c i t o n s  Electron-Hole  there i s strong experimental can occur  k i n e t i c s of i t s formation  in intrinsic  Drops  evidence that a  germanium and  con-  silicon.  however, have not y e t been agreed upon.  The  - 17  In 1966 photoluminescent levels.  On  Haynes  (18)  observed  -  a new  spectrum of i n t r i n s i c  s e r i e s of peaks i n the  s i l i c o n at high  the b a s i s of p o s i t i o n , w i d t h ,  and  excitation  e x c i t a t i o n l e v e l depen-  dence t h e s e l i n e s were a t t r i b u t e d to r a d i a t i v e a n n i h i l a t i o n of a biexciton. ium  The  analogous s e r i e s of l i n e s were a l s o observed  i n german-  (24) These s p e c t r a , p a r t i c u l a r l y  t h a t o f germanium, have been (25—28  e x t e n s i v e l y s t u d i e d by a number of workers.  Asnin et a l .  '  done a number o f e x p e r i m e n t s s t u d y i n g the p h o t o l u m i n e s c e n c e , and p h o t o c o n d u c t i v i t y which suggest  On  at h i g h  (2) Svistunova  t h a t the EHD  Both these  notably,  (29) and  Benoit a l a Guillaume  c l u d e d t h a t the b i e x c i t o n phase does not e x i s t a t any but  lifetimes,  excitation  the b a s i s of s i m i l a r experiments o t h e r w o r k e r s ,  P o k r o v s k i i and  have  that the s p e c t r a are a s s o c i a t e d with  b i e x c i t o n s a t low e x c i t a t i o n l e v e l s and w i t h EHD levels.  17)  phase does e x i s t  , have c o n excitation  f o r e x c i t a t i o n above a c r i t i c a l  c o n c l u s i o n s a r e based upon the d e t a i l e d dependence o f  level, value. the  l u m i n e s c e n t i n t e n s i t y and e x c i t o n c o n c e n t r a t i o n on the e x c i t a t i o n l e v e l c l o s e t o the c r i t i c a l v a l u e . I t i s u n l i k e l y , however, t h a t the a v a i l a b l e d a t a i s r e l i a b l e enough t o f a v o u r one i s ample e v i d e n c e  i  p o i n t of v i e w or the o t h e r .  from p h o t o c o n d u c t i v i t y , d i f f u s i o n , and  light  There  scattering  (3) studies  (see f o r example P o k r o v s k i i  ) which c l e a r l y demonstrates  e x i s t e n c e o f m a c r o s c o p i c r e g i o n s of the c r y s t a l h a v i n g i b l e w i t h the assumption o f EHD. w h i c h a r e not  gas o r a b i e x c i t o n gas,  p r o p e r t i e s compat-  I t i s o n l y the k i n e t i c s o f  c l e a r , whether the EHD  the  formation  i s i n e q u i l i b r i u m w i t h an e x c i t o n  o r perhaps b o t h .  There i s as y e t no  the e x i s t e n c e o f a B o s e - E i n s t e i n c o n d e n s a t i o n  as suggested  evidence  for  by Moskalenko.  - 18 -  Because the l i f e t i m e and  quantum e f f i c i e n c y of the condensed  (2) phase a r e much l a r g e r i n germanium, T silicon, T  0  = 0.18  ys and n = 5 x 1 0  a A  - 1 +  20 ys and n  Consequently  0.8  , than i n  ^ ^ \ i t i s p o s s i b l e to c r e a t e a  much h i g h e r average d e n s i t y of c a r r i e r s and data.  =  to o b t a i n much more r e l i a b l e ^  t h e r e i s c o n s i d e r a b l y more e x p e r i m e n t a l  a b l e f o r germanium than s i l i c o n . t i o n o f e l e c t r o n s and  Experimental  h o l e s w i t h i n the EHD  a r e g i v e n i n T a b l e 2.4.  data  v a l u e s of the  and  avail-  concentra-  i t s ground s t a t e energy  These appear to be i n good agreement w i t h  the  p r e d i c t e d v a l u e s i n T a b l e 2.3.  '** Table  Experimental Values  2.4  of E q u i l i b r i u m Concentration  and Ground S t a t e Energy o f  Equilibrium Concentration  Ge  Si  EHD  Ground S t a t e Energy  (a)  1:95 X 10 . cm _3  6.0  meV  (b)  2.6  X 10  6.3  meV  Cc)  1.0  X 10*6 c m  -3  (d)  2.0  X IO* c m  -3  Ce)  3.7  X IO * c m  21.5  meV  17  0  17  cm  7  1  3  -3  -3  (a) r e f .  29  (d) r e f .  31  (b) r e f .  2  (e) r e f .  32  (c) r e f . 27  -  19 -  CHAPTER 3 EXPERIMENTAL 3.1  Sample Preparation The samples were cut from single crystal ingots of phosphorus  doped silicon purchased from Ventron and General Diode Corporations. The impurity concentration was determined by resistivity measurements using a four-point probe on the face of the crystal before and after cutting the s l i c e .  The face was polished and etched with hot concen-  trated KOH before measurements were made. A series of ten or more measurements were made at different positions on the crystal face. These were averaged to give the resistivity of the face.  The f i n a l  value was taken to be the average of the r e s i s t i v i t i e s obtained before and after cutting the s l i c e .  The impurity concentrations were deter(33)  mined from the measured r e s i s t i v i t i e s using the Irvin chart  and  were estimated to have an accuracy of ± 5%, subject to the validity of the Irvin chart. Samples to be used in photoluminescent studies were polished with 40 um abrasive on a napped cloth and then etched for 10 seconds in a mixture of HNO^ and HF (20:1) before being mounted i n the spectrometer.  For the heat treatment studies the samples were heated to 1150 C  i n a helium atmosphere for 30 minutes.  At the end of this time the  samples were quenched in acetone and then etched.  If i t was not possible  to perform the measurements immediately the samples were stored i n liquid nitrogen.  The samples could be stored at this temperature for several  weeks with no noticeable annealing.  - 20 -  To a v o i d the d i f f i c u l t i e s encountered when s t u d y i n g the EPR s p e c t r a o f samples  l a r g e r than the microwave s k i n d e p t h i t was n e c e s s a r y  t h a t a t l e a s t one dimension of the sample be s m a l l compared t o the s k i n depth.  To a c h i e v e t h i s the samples were ground  m o r t a r and p e s t l e .  t o a f i n e powder w i t h a  At each c o n c e n t r a t i o n s t u d i e d two samples were p r e -  p a r e d , one was h e a t t r e a t e d a t 1150 C i n h e l i u m f o r 30 minutes b e f o r e b e i n g ground,  the o t h e r was n o t heat t r e a t e d and was used as a s t a n d a r d  f o r t h e purpose o f comparison. a m i x t u r e o f HNO^  A f t e r g r i n d i n g , the powder was e t c h e d i n  and HF (20:1).  A f l o a t a t i o n s e p a r a t i o n p r o c e d u r e was  t h e n u s e d to grade the powder i n t o t h r e e l o t s w i t h average p a r t i c l e 1-5  um, 5 - 1 5  um, and g r e a t e r than 15 ym.  by e x a m i n a t i o n w i t h an o p t i c a l m i c r o s c o p e .  The s i z e s were determined The samples were t h e n weighed,  s e a l e d i n l u c i t e c a p s u l e s and s t o r e d i n l i q u i d n i t r o g e n u n t i l  3.2  Photoluminescent  size  used.  Spectrometer  A S p e c t r a P h y s i c s 165 Argon l a s e r w i t h a c o n t i n u o u s maximum o  a v a i l a b l e power o f 1.8 watts a t 5150 A was used as an e x c i t a t i o n source.  light  The p h o t o l u m i n e s c e n c e was a n a l y s e d w i t h a P e r k i n Elmer 98G g r a t -  i n g s p e c t r o m e t e r f i t t e d w i t h a Bausch i n the f i r s t  and Lomb g r a t i n g b l a z e d a t 1.0 ym  o r d e r and h a v i n g 600 lines/mm.  A PbS d e t e c t o r c o o l e d w i t h  d r y i c e and a c e t o n e , t o g e t h e r w i t h a P r i n c e t o n A p p l i e d R e s e a r c h  (PAR)  113 p r e a m p l i f i e r and a PAR JB-5 l o c k - i n a m p l i f i e r o p e r a t i n g a t ^ 80 Hz gave a minimum d e t e c t a b l e i n t e n s i t y o f 1 0 u r a t i o n used i s shown i n F i g u r e 3.1.  - 1 3  watts.  Corning f i l t e r s  The o p t i c a l  config-  4-96 and 9-92 were  used on the l a s e r output t o e l i m i n a t e i n f r a r e d l a s e r e m i s s i o n , and filters  7-56 and 2-64 were used b e f o r e the s p e c t r o m e t e r e n t r a n c e  slits  A  laser chart filters  recorder  4-96,4-92  lock-in amplifier  1  10  sample i n cryostat  FIGURE 3.1  chopper  filters 7-56,2-64  Optical Configuration of Photoluminescent Spectrometer.  - 22 -  to prevent any laser light from appearing i n the second order. Care had to be exercised i n both the selection and order of f i l t e r s , as many of the dyes used i n their manufacture are photoluminescent. 3.3 Electron Paramagnetic Resonance Spectrometer (34) A standard X-band homodyne EPR spectrometer obtain the EPR results.  was used to  Through the use of a microwave bias system and  automatic frequency controller which "locked" the microwave frequency to that of the resonant cavity, the spectrometer was tuned to measure the absorption due to the sample.  A pair of Helmholtz c o i l s mounted on  the magnet pole faces were used to provide magnetic f i e l d modulation at 500 Hz. The resultant audio frequency signal was then detected and amplified by a PAR HR-8 lock-in amplifier.  The spectrometer was f i t t e d  with a double-sample modulation-switched cavity so that control samples which  had not been heat-treated could be compared d i r e c t l y with the  heat-treated samples.  Details of this cavity design and i t s operation (35) are presented i n the thesis of J.D. Quirt .  3.4  Temperature Measurements Electron paramagnetic resonance studies were made at three  fixed temperatures, 77K, 4.2K, and 1.1K; Because of high d i e l e c t r i c losses and excessive bubbling i t was not p r a c t i c a l to obtain 77K by immersion of the microwave cavity i n l i q u i d nitrogen.  Instead, a helium  exchange gas system employing an outer dewar f i l l e d with l i q u i d nitrogen and an inner dewar f i l l e d with helium gas was used.  I t was found that  1 1/2 hours was sufficient for such a system to equilibrate and that the resultant temperature was within 1 degree of l i q u i d nitrogen temperature.  - 23 -  A temperature o f 4.2K  was  o b t a i n e d by immersion o f the sample and  c a v i t y i n l i q u i d h e l i u m , and 1.1K t o r e d u c e i t s vapour p r e s s u r e . c i e n t l y low t h a t t h e r e was Because d i f f i c u l t y was  helium  o b t a i n e d by pumping on the h e l i u m  The microwave power i n p u t was heating of the  o f the need  suffi-  sample.  f o r h i g h e x c i t a t i o n power l e v e l s some  e x p e r i e n c e d i n o b t a i n i n g sample temperatures f o r the  photoluminescent s t u d i e s . immersing  no  was  A nominal 2K temperature was  o b t a i n e d by  the sample i n a l i q u i d h e l i u m b a t h kept a t 1.9K.  Superfluid  ( i . e . h e l i u m below the A p o i n t ) does n o t p r o v i d e a good heat  s i n k i f the sample must d i s s i p a t e l a r g e amounts of h e a t .  To o b t a i n an  e s t i m a t e o f the sample temperature the p o s i t i o n and w i d t h o f the photol u m i n e s c e n t peaks were s t u d i e d as a f u n c t i o n o f the e x c i t a t i o n power. I t was  found t h a t f o r e x c i t a t i o n l e v e l s g r e a t e r than 0.2  broadened  and s h i f t e d  w a t t s the peaks  to h i g h e r e n e r g i e s i n d i c a t i n g t h a t the sample  b e i n g h e a t e d . Thus a l e v e l o f 0.15  w a t t s was  used t o ensure t h a t as  was little  h e a t i n g as p o s s i b l e o c c u r r e d w i t h o u t r e d u c i n g the power l e v e l too s e r i o u s ly.  T h e r e f o r e i t i s b e l i e v e d t h a t T = 2K i s n o t s e r i o u s l y i n e r r o r , a l -  though the temperature c o u l d be as h i g h as 3K. For h i g h e r temperatures a d i f f e r e n t system employing h e l i u m gas was  used to c o o l the sample.  e x p e r i m e n t a l d e t a i l s are shown i n F i g u r e 3.2. t u r e c o n t r o l l e r w i t h a GaAs heat s e n s o r was a t u r e o f the sample mounting was  tempera-  used t o c o n t r o l the temper-  b l o c k to w i t h i n ± .5 d e g r e e s .  mounted w i t h a screw clamp  i n c r e a s e the t h e r m a l c o n t a c t .  A PAR model 152  The  The  sample  a t one end w i t h indium pads used t o Mounted i n t h i s way  i t was  found t h a t t h e  - 24 -  liquid  helium  excitation light molecular sieve  heater  GaAs sample  helium gas flow  FIGURE 3.2  D e t a i l s o f Temperature C o n t r o l System.  heat sensor  - 25 -  i n t e n s i t y o f t h e EHD photoluminescent peak, w h i c h i s q u i t e  temperature  dependent i n t h e case o f i n t r i n s i c s i l i c o n , c o r r e l a t e d v e r y  closely  w i t h t h e heat s e n s o r output w i t h a time c o n s t a n t o f much l e s s than 1 second.  T h i s was taken as an i n d i c a t i o n t h a t the sample mount tempera-  t u r e as measured by t h e GaAs d i o d e was c l o s e l y c o u p l e d t o t h e temperat u r e o f t h e sample and responded q u i c k l y t o any change i n sample temperature. Temperature c a l i b r a t i o n was c a r r i e d out i n two s t e p s .  First  the GaAs d i o d e was c a l i b r a t e d by mounting a 33 ohm A l l e n - B r a d l e y c a r b o n r e s i s t o r on t h e sample mount as shown i n F i g u r e 3.3 ( a ) .  G.C. E l e c t r o n -  i c DC-Z9 s i l i c o n e compound was used t o ensure good t h e r m a l c o n t a c t . r e s i s t a n c e o f t h e carbon r e s i s t o r was c a l i b r a t e d a t f o u r f i x e d  The  points;  l i q u i d h e l i u m , l i q u i d n i t r o g e n , d r y - i c e a c e t o n e , and room t e m p e r a t u r e . I n t e r m e d i a t e temperatures were o b t a i n e d by f i t t i n g  t h e r e s i s t a n c e t o the  semi-empirical e q u a t i o n ^ ^ :  l o 8  10R  +  isf^R  =  2  (3.4-1)  f  b +  where R i s the r e s i s t a n c e i n ohms and K, b and a a r e determined by f i t t i n g a t known t e m p e r a t u r e s . o v e r t h e range 4.2K - 200K.  T h i s gave t h e temperature t o w i t h i n 10%  Secondly the sample temperature was measured  w i t h t h e c a r b o n r e s i s t o r as shown i n F i g u r e 3.3 (b) w i t h t h e l a s e r  oper-  a t i n g a t 1.8 w a t t s , and t h i s temperature compared t o t h a t o f the sample mount.  The d i f f e r e n c e between t h e sample temperature and t h a t o f t h e  sample mount i s shown i n F i g u r e 3.4.  T h i s i n d i c a t e s that although the  - 26 -  laser light  resistor  FIGURE 3 . 3  Temperature C a l i b r a t i o n P r o c e d u r e .  of GaAs Heat Sensor.  (b)  (a)  Calibration  C a l i b r a t i o n o f Sample Temperature.  - 27 -  sample i s s i g n i f i c a n t l y heated f o r T < 20K, s y s t e m was  able  t o d i s s i p a t e the heat q u i t e w e l l .  l i n e w x d t h and peak p o s i t i o n i n d i c a t e t h a t ducible  to ^  at h i g h e r temperatures  IK.  Reproducibility  the temperature was  the of  repro-  - 28 -  10  20  30  40  TEMPERATURE  FIGURE 3.4  50  60  70  80  90  OF SAMPLE MOUNT ( K)  Temperature Difference between Sample and Sample Mount  -  29  -  CHAPTER 4  CONCENTRATION DEPENDENCE OF THE PHOTOLUMINESCENCE OF S i ; P  • I 4.1  Intro duction L a m p e r t ^ ^ ^ p r e d i c t e d i n 1958 t h a t i t s h o u l d be p o s s i b l e f o r  an e x c i t o n t o be bound t o an i m p u r i t y i n a semiconductor.  T h i s was  (37) c o n f i r m e d by Haynes photoluminescent  , who i d e n t i f i e d a bound e x c i t o n l i n e  i n the  s p e c t r a o f s i l i c o n doped w i t h donor i m p u r i t i e s .  The  o b s e r v e d dependence o f t h e b i n d i n g energy on t h e donor i o n i z a t i o n was found  energy  t o be i n c o n s i s t e n t w i t h the p i c t u r e o f an e x c i t o n bound t o a  n e u t r a l d o n o r , b u t r e a d i l y e x p l a i n e d i f t h e n e u t r a l donor f i r s t t u r e d a s e c o n d e l e c t r o n and t h e n bound a h o l e .  cap-  More r e c e n t l y , Dean e t  C4) al. h a v e done e x t e n s i y e s t u d i e s on t h e - r e c o m b i n a t i o n p r o c e s s e s a s s o c i a t e d w i t h e x c i t o n s bound t o i m p u r i t i e s i n s i l i c o n and germanium. (38) Alekseev et a l .  have s t u d i e d germanium doped w i t h donor  i m p u r i t i e s u s i n g h i g h photo e x c i t a t i o n l e v e l s . photoluminescent seen i n i n t r i n s i c cm  - 3  spectrum  a t t r i b u t e d t o r e c o m b i n a t i o n w i t h i n an EHD as  germanium.  F o r donor c o n c e n t r a t i o n s N^ < 5 x 1 0  the EHD observed i n i n t r i n s i c  not involve residual impurities.  Recombination  energy a s s o c i a t e d w i t h t h e EHD i n i n t r i n s i c for at  1 5  no change was o b s e r v e d i n t h e p o s i t i o n o f t h e p h o t o l u m i n e s c e n t  peaks, hence i t was c o n c l u d e d t h a t did  They o b s e r v e d t h e same  donor c o n c e n t r a t i o n s o f  = 2 x 10  1 6  r a d i a t i o n a t the  germanium was n o t observed  cm . - 3  R a d i a t i o n was observed  l o w e r energy and was a t t r i b u t e d t o t r a n s i t i o n s o f excess  i n i m p u r i t y l e v e l s i n t o t h e v a l e n c e band.  crystals  electrons  -  Gbbel et a l .  30  -  have done photoluminescence studies on  s i l i c o n doped with phosphorus which show a similar trend, with no change i n the position of the spectra associated with the EHD for donor concentrations  < 6.5 x 10  cm .  17  -3  For unspecified higher  concentrations they observed very broad emission bands at lower energy, which were attributed to band tailing. The results of extrinsic photoluminescence studies of silicon doped with phosphorus are presented i n this chapter. These samples have impurity concentrations i n the range 9 x 1 0  15  cm ^  i 4.3 x 1 0  -3  19  cm . -3  4.2  Concentration Dependence of the Photoluminescent Spectra of Si;P at 2K There are two reasons why i t is'convenient to consider the  photoluminescent spectra at low temperatures.  First, at sufficiently  low temperatures the broadening of the lines due to thermal excitation of electrons and holes i s reduced.  Secondly, the condensed phase, i n  which this work i s primarily interested, i s only observed below 20K i n intrinsic silicon.  Thus i f the EHD i s to be considered, i t i s logical  to start with temperatures below 20K.  To make comparisons more meaning-  f u l a l l samples were excited with the same laser power of 0.15 watt. This i s a factor of 5 greater than that used by Gobel et a l . 4.3  Low Concentration Samples with phosphorus concentrations of 9 x 10  3.6 x 1 0 al.^.  1 7  cm  -3  15  cm  -3  and  were studied to confirm the previous work of GObel et  The photoluminescent spectrum observed from the sample  •  FIGURE 4.1 refer  Photoluminescene  f o r Si:P with N  D  «* 9.0 x 1 0  1 5  cm at - 3  2K.  The s u p e r s c r i p t  t o t h e phonon a s s i s t i n g the r e c o m b i n a t i o n ; T O - t r a n s v e r s e o p t i c a l , T A - t r a n s v e r s e  acoustical,  NP-no phonon.  The peaks l a b e l e d BE are bound e x c i t o n  peaks.  N =3.6 D  X  10  17  PHOTON  FIGURE 4.2  ENERGY  Photoluminescence for Si:P with  ( eV)  » 3.6 x 1 0  17  ND i s due to the capture of a free hole by a neutral donor.  cm~ at 2K. The peak labeled 3  - 33 -  containing 9 x 1 0  1 5  cm  - 3  phosphorus i m p u r i t i e s e x h i b i t e d f o u r l i n e s  a t 2K as shown i n F i g u r e 4.1. i d e n t i f i e d by Dean e t a l .  .  Two o f t h e s e l i n e s have been p r e v i o u s l y The l i n e a t hv = 1.092 eV i s a t t r i b u t e d  t o t h e r e c o m b i n a t i o n o f a bound e x c i t o n w i t h t h e simultaneous of a transverse o p t i c a l  (TO) momentum c o n s e r v i n g phonon.  hv SB 1.151 eV i s a t t r i b u t e d exciton.  t o t h e phononless  The l i n e  at  r e c o m b i n a t i o n o f a bound  There i s no phonon i n v o l v e d i n t h i s t r a n s i t i o n ;  atom t a k e s up t h e e x t r a momentum.  emission  the impurity  The two r e m a i n i n g l i n e s a t hv = 1.082  eV and hv = 1.12 eV a r e i d e n t i c a l i n p o s i t i o n , w i d t h and r e l a t i v e amplitude  t o t h e l i n e s observed  i n pure s i l i c o n  (39)  and a t t r i b u t e d t o  TO and TA ( t r a n s v e r s e a c o u s t i c a l ) phonon a s s i s t e d r e c o m b i n a t i o n w i t h i n (12) t h e EHD r e s p e c t i v e l y .  I t s h o u l d be noted  t h a t Shaklee and Nahory  have shown t h a t t h e l i n e s a t t r i b u t e d t o TO phonon r e p l i c a s a r e a c t u a l l y u n r e s o l v e d d o u b l e t s due t o TO and LO ( l o n g i t u d i n a l o p t i c a l ) phonon r e p l i c a s s e p a r a t e d by 1.8 meV. l i n e s w i l l be r e f e r r e d  F o r t h e purpose  spectrum  these l i n e s , observed  such  t o as TO phonon r e p l i c a s .  A t t h e h i g h e r c o n c e n t r a t i o n (N luminescent  of this thesis  i s comprised  = 3.6 x 1 0  of four l i n e s  1 7  c m ) the photo- 3  ( F i g u r e 4.2).  Three o f  at hv = 1.08 eV, 1.12 eV, and 1.14 eV, a r e i d e n t i -  f i e d w i t h TO phonon, TA phonon, and no phonon r e p l i c a s o f t h e recombinat i o n r a d i a t i o n a s s o c i a t e d w i t h t h e EHD.  These were f i r s t observed and  i d e n t i f i e d by Gobel e t a l . f o r t h i s i m p u r i t y c o n c e n t r a t i o n .  A fourth  l i n e a t 1.065 eV was n o t observed by Gb*bel e t a l . , but has been observed by o t h e r workers a t lower c o n c e n t r a t i o n s and i s i d e n t i f i e d w i t h t h e r a d i a t i v e c a p t u r e o f f r e e h o l e s a t n e u t r a l donors  .  - 34 -  The observed spectra are consistent with the conclusions of Gobel et a l . that recombination radiation from the EHD i s observable for impurity concentrations up to 3.5 x 1 0 4.4  Transition Range (1.1 x 1 0  18  cm  -3  cm .  17  -3  i Np < 5.5 x 1 0  18  cm ) -3  As the concentration of phosphorus impurities i s increased above  = 3 x 10  17  cm  3  states begin to overlap.  the wave functions of the excited impurity This results i n a merging of the higher  excited states with the conduction band, producing a conduction band tail.  When the impurity concentration exceeds  = 3 x 10  18  cm  -3  the  impurity atoms are s u f f i c i e n t l y close that the ground state wave functions overlap and the donor electron can no longer be considered to (41) be localized at a particular l a t t i c e s i t e  . For a l l higher impurity  concentrations the material i s characterized by metallic behavior, e.g. there i s no activation energy for conductivity. The photoluminescent spectra for a number of different impurity concentrations i n the region of this semiconductor-metal trans i t i o n are shown i n Figure 4.3.  This figure i l l u s t r a t e s the most  important feature of the photoluminescent spectra i n this range; that the l i n e width and shape, except for some t a i l i n g at low energy, remains essentially unchanged with only a monotonic s h i f t to lower energy. As the recombination radiation intensity i s dependent upon the surface of the sample i n a manner which i s not f u l l y understood, i t i s not possible to quantitatively monitor the intensity as a function of impurity concentration.  I t i s possible to say, at least qualitatively,  that the intensity of the luminescence decreases with increasing concentration.  - 35 -  4.3 x IO  19  cm  I.3xl0 cm" l 9  - 3  3  5 . 5 x | 0 '8 ' ° cm- 3 r m  rm-3 3 x | 0 •8 cm  '8  1.03  1.06  1.09  PHOTON ENERGY  r m  -3  1.12 (eV)  FIGURE 4.3 Concentration Dependence of the Photoluminescence of Si:P at 2K. The dashed portion of the spectrum for = 4.3 x 1 0 cm was inferred from the spectra at higher temperatures. The p a r a l l e l bars give the spectrometer s l i t width. -3  l a  - 36 -  4.5  High Concentration The spectra observed for samples with phosphorus concentra-  tions of N = 1.3 x 1 0  19  Q  cm and Np = 4.3 x 1 0 -3  19  cm" are shown i n 3  Figure 4.3. The lines are found to be very much broadened on the low energy side with the width increasing with concentration.  There i s a  continuing s h i f t of the peak position to lower energy as the impurity concentration increases.  CHAPTER 5 TEMPERATURE DEPENDENCE OF THE PHOTOLUMINESCENCE OF Si:P  • I 5.1 Introduction An investigation of the temperature dependence of the photoluminescence i s of interest for several reasons.  In particular, the  c r i t i c a l temperature above which a peak i s not observed, and the temperature variations of the linewidth are of considerable aid i n identifying the recombination mechanism responsible for the observed spectra. In the case of s i l i c o n such studies may make i t possible to determine whether biexcitons are formed, as suggested by Asnin et al.^\  Other  reasons for such studies include the understanding of the effects of impurities on the c r i t i c a l temperature and excitation level, and perhaps further insight into the semiconductor-metal transition i t s e l f . Recent work by Gobel et a l .  (c\  (38)  and Alekseev et a l .  indicates at  least f o r low impurity concentrations, that the c r i t i c a l temperature i s raised and the c r i t i c a l excitation level lowered as the impurity concentration i s increased. This chapter w i l l present the results of an investigation of the photoluminescent spectra of phosphorus doped silicon i n the temperature range 2K < T < 140K. Some of the important features w i l l be discussed briefly, although any detailed discussion w i l l be l e f t to a later chapter.  - 38 -  5.2  Temperature Dependence of the Photoluminescence of Si:P As was done when studying the concentration dependence of  the photoluminescence, a l l experiments were performed with the same incident power.  For these experiments the maximum power level, 1.8  watts, was used in order to obtain the largest possible signal.  Unfor-  tunately, this power level could not be used to study samples at 2K as the poor thermal contact with the superfluid helium resulted in s i g n i f i cant sample heating.  The poor signal obtained at low incident power  levels also ruled out any meaningful attempt to determine the excitation threshold levels or the variation of photoluminescent intensity with excitation level. 5-3 1)  Low  Concentration  9 x 10  15  cm"  3  This was the only concentration studied for which recombination radiation from free excitons and bound excitons could be clearly identified.  As shown in Figure 5.1 the free exciton peak at hv = 1.10  eV was not observed at very low temperatures, T = 2K, but increased in relative magnitude such that for T £ 30K, i t was the dominant peak. The two peaks associated with the bound exciton, TO phonon assisted at hv = 1.092  eV and no phonon at hv = 1.15 eV were only observed for (14)  T £ 25K.  Other workers  concentrations of  have considered samples with phosphorus  = 8 x 10  16  cm  -3  and found that with the higher  impurity concentration i t was possible to observe the hound exciton peaks up to I = 80K.  The peaks at hv - 1.082  eV and hv = 1.12 eV, iden-  t i f i e d with recombination within an EHD, were observed only for T < 20K.  - 39 -  32 K  25 K  21 K  13.5 K  2 K —i  1  1.03  '.  1.06  PHOTON  FIGURE 5.1 N  =  9  x  10  Temperature 15  cm  - 3  .  i  1.09  ENERGY  i  1.12  i  1.15  (eV)  Dependence o f the Photoluminescence f o r  -  40  -  - 41 -  The d i s a p p e a r a n c e o f t h e s e l i n e s , and the simultaneous i n c r e a s e i n s t r e n g t h o f t h e f r e e e x c i t o n l i n e was t h e EHD  i n t o an e x c i t o n gas  2)  x 10  3.6  cm  1 7  (38)  .  N  D  x 10  = 9 x 10  cm  1 7  1 5  ,  -3  As shown i n F i g u r e 5.2 - 3.6  a t t r i b u t e d t o the e v a p o r a t i o n of  -3  cm .  was  the p h o t o l u m i n e s c e n t  q u i t e d i f f e r e n t from t h a t observed f o r  There were no l i n e s which c o u l d be a t t r i b u t e d to  -3  f r e e o r bound e x c i t o n s , and t h e r e was T h i s l i n e was  a new  i d e n t i f i e d i n S e c t i o n 4.3  and broaden  l i n e a t hv - 1.065  as a r i s i n g from the  t i o n of" a f r e e h o l e w i t h a n e u t r a l donor. b u t e d t o t h e EHD,  spectrum f o r  eV.  recombina-  The r e m a i n i n g peaks,  attri-  x>irere found to s h i f t m o n o t o n i c a l l y t o h i g h e r e n e r g i e s  a s the temperature  i n c r e a s e s ; b e h a v i o r which i s i n a c c o r d  w i t h t h e p i c t u r e o f r e c o m b i n a t i o n between-two bands.  For T > 45K  TO a s s i s t e d p e a k d e v e l o p e d a s h o u l d e r on t h e h i g h photon energy  the  side.  U n f o r t u n a t e l y , due t o t h e w i d t h o f the l i n e s a t these temperatures i t i s not p o s s i b l e t o determine whether t h e r e a r e two  photoluminescent  I t i s p o s s i b l e to s e t a lower bound o f T -  peaks p r e s e n t .  t h r e s h o l d temperature  t o be a s s o c i a t e d w i t h t h e EHD  44K  f o r any  for this impurity  concentration.  .5.4  T r a n s i t i o n Range (1.1 x 1 0 The  temperature  cm  1 8  <  -3  x 10  cm )  1 8  -3  dependence of the p h o t o l u m i n e s c e n t  the c o n c e n t r a t i o n range 1.1 x 1 0  1 8  cm  -3  to be v e r y s i m i l a r t o t h a t observed f o r due  < 5.5  <,  <; 5.5  x 10  = 3.6 x l O  1 7  1 8  cm  cm" . 3  spectra i n -3  was  found  The  line  t o r e c o m b i n a t i o n o f a f r e e h o l e w i t h a n e u t r a l donor was n o t  a l t h o u g h i t may  observe  c o n t r i b u t e t o the t a i l observed on the low energy s i d e o  - 42 -  —  «  1.06  1-03  PHOTON FIGURE 5.3  i  1  1.12  1.15  ENERGY (eV)  Temperature Dependence of the Photoluminescence  for N = 1.3 x 10 D  1  1.09  19  cm" . 3  - 43 -  of the TO assisted recombination peak. A l i n e identified as due to phononless recombination was present i n this concentration range, with i t s relative intensity increasing with concentration.  For a l l samples  studied i n this concentration range a broad shoulder developed on the high photon energy side of the TO assisted peak for T £ 40K. There appears to be an increase i n the temperature at which the shoulder appears, however due to the uncertainty i n the l i n e shapes, i t i s impossible to make any quantitative v e r i f i c a t i o n of this trend. 5.5' High Concentration 1)  1.3 x 1 0  1 9  phosphorus/cm  3  The photoluminescent spectrum, shown i n Figure 5.3, i s comprised of two broad lines whose energy separation suggests that they are TO phonon and no phonon replicas of the same recombination mechanism. The lines broaden and s h i f t to higher energy with increasing temperature, with the broadening taking place primarily on the high energy side. 2)  4.3 x 1 0  1 9  phosphorus/cm  3  The spectrum i s composed of two broad bands at hv = 1.06 eV i  and hv = 1.11 eV which broaden with increasing temperature u n t i l by T = 125K they can no longer be resolved.  The peak positions appear to  be independent of temperature, although the broadening and overlap make i t impossible to determine this with any degree of r e l i a b i l i t y .  - 44 -  CHAPTER 6 DISCUSSION OF RESULTS 6.1  Screening of Excitons As a f i r s t step towards identifying the recombination process  responsible for the observed photoluminescence, i t i s necessary to consider what role, i f any, i s played by excitons.  In this section i t w i l l  be demonstrated that free and bound excitons do not contribute s i g n i f i cantly to the photoluminescence for N^ > 3.6 x 1 0  17  cm . -3  (42) Asnin and Rogachev  noted that the energy of the free  exciton peak i n the absorption spectrum of germanium was independent of impurity concentration, although the peak was found to broaden with increasing impurity, concentration. ^ 5 meV.  The maximum width attained was  I t was found that the excitons disappeared when the decrease  i n the band gap was equal to the exciton ionization energy. For T = 77K, the impurity concentration at which the exciton l i n e disappeared agreed well with that calculated using the simple Debye expression: '  \l/2  KkT  4ire N  (6.1-1)  2  D  The impurity concentration at which there are sufficient free carriers to screen the coulomb interaction between an electron and hole, and prevent the formation of excitons can be estimated i n several ways. Using the c r i t e r i o n of equation (6.1-1) suggested by Asnin et a l . , or a (43) s l i g h t l y more sophisticated approach adopted by Albers  to calculate  - 45 -  the exciton binding energy, i t i s found that there can be no exciton states i n silicon for free carrier concentrations greater than n = 2.2 x 1 0  17  cm .  Both approaches suffer from only being applicable  3  at high temperature, however since screening increases with decreasing temperature, the results w i l l at least constitute an upper bound for (44) the existence of excitons. Mott  , using the Thomas-Fermi approxima-  tion arrived at the criterion n  l/3  For silicon a cm  -3  x  a > 0.25 x  (6.1-2)  -' 4.29 x 10  -7  cm, which gives a value of n = 2 x  10  17  as the concentration of free carriers required to prevent the for-  mation of a bound state. This approach i s applicable at low temperature and i t s close agreement with the Debye model indicates that i t i s reasonable to accept n ^ 2 x 10  17  cm"  3  free carriers as the maximum  allowed for the existence of free excitons. The impurity concentration required to give 2 x 10 free carriers at 2K can be calculated quite readily.  17  cm""  3  If the Fermi level  is at least 4 kT below the bottom of the conduction band then the occupation probability of the conduction band can be described by Boltzmann statistics, and the number of electrons in the conduction band given by N n - N  •  — . N n  1  (6.1-3)  - 46 -  where m, kT 3/2 de -  N' = 6 c  exp  2  2TTR  (6.1-4)  ~ kT  The activation energy, E^, of phosphorus donors in silicon has been shown by Kuwahara E  A  =  E  (45)  -  O  to obey the empirical expression  < > A  l/3 N  (6.1-5)  D  where E Q = 45 meV and <a> = 3.27 x I O i t i s found that to have 2 x 1 0  17  necessary to have  18  = 2.5 x I O  cm  meV-cm. From equation (6.1-3)  -5  free carriers at 2K i t i s  -3  cm" . 3  This calculation can be extended to bound excitons i n an (43) approximate manner. Albers  has shown that the free carrier concen-  tration required to screen excitons i s proportional to the square of the reduced mass. n  g  a y  (6.1-6)  2  The exciton binding energy i s proportional to the reduced mass, 1  Gay  (6.1-7)  and therefore h a G  (6.1-8)  2  The binding energy of a bound exciton i s ^ 1/2 G (.0065 eV) which means that the free carrier concentration required to screen bound excitons i s ^ 5 x 10  le  cm . -3  In terms of impurity concentration, a phosphorus  - 47 -  concentration of  = 1.0  1Q  x  cm""  18  3  do n o t b i n d to i m p u r i t y atoms at 2K. reasonable to  i s r e q u i r e d to ensure t h a t  These r e s u l t s show t h a t i t i s  to assume t h a t t h e r e w i l l be no r e c o m b i n a t i o n  f r e e or bound  excitons for  excitons  > 2.5  x 10  cm  1 8  emission  and  -3  1.0  x 10  due cm"  i 8  3  respectively. The by  two  orders  screening.  b i n d i n g energy f o r a b i e x c i t o n i s ^ 0.1  G.  T h i s reduces  o f magnitude the number o f f r e e c a r r i e r s r e q u i r e d f o r  T h i s e f f e c t i v e l y r u l e s out  i n the i m p u r i t y  c o n c e n t r a t i o n range  I t i s not n e c e s s a r y  the e x i s t e n c e o f the b i e x c i t o n > 10  1 6  cm . -3  to consider screening of excitons  e x p l a i n the p h o t o l u m i n e s c e n t s p e c t r a f o r N  = 9 x 10^  Q  t h i s i m p u r i t y c o n c e n t r a t i o n t h e r e are i n s u f f i c i e n t shown i n F i g u r e 5.1  5  cm" , 3  since at  free c a r r i e r s .  the TO phonon a s s i s t e d f r e e e x c i t o n l i n e  hv = 1 . 0 9 8 eV i s observed f o r T > 13.5K.  to  As  at  T h i s l i n e i s not p r e s e n t  T = 2K because the time r e q u i r e d f o r the c a p t u r e  o f an e x c i t o n by  the  condensed phase i s s h o r t compared to the e x c i t o n l i f e t i m e at t h i s perature. is  The  t h a t a t hv  p r i n c i p a l l i n e due = 1.092  eV and  impurity concentration l i n e and  difficult  f o r T ^ 80K  to the r e c o m b i n a t i o n  i s observed a t T < 32K.  when the t h e r m a l energy i s i n s u f f i c i e n t  low  with  = 3.6  x 10  1 7  eV due  to bound e x c i t o n s .  be  i n d i c a t e t h a t bound cm"  3  present  t o cause d i s s o c i a t i o n excitons  i m p u r i t i e s , the p h o t o -  spectrum f o r t h i s i m p u r i t y c o n c e n t r a t i o n does not  l i n e a t hv = 1.092  excitons  Because o f the  however i t s h o u l d  A l t h o u g h s c r e e n i n g arguments a l o n e  luminescent  tem-  t h i s l i n e i s weak r e l a t i v e to the f r e e e x c i t o n  to i d e n t i f y at T = 32K,  c o u l d e x i s t f o r T 1 50K  of bound  at  Were i t p r e s e n t  contain a this  line  - 48  c o u l d be r e a d i l y r e s o l v e d . a t t h i s c o n c e n t r a t i o n , and  Free e x c i t o n s a r e p o s s i b l e f o r T < t h e i r p o s s i b l e c o n t r i b u t i o n to the  w i l l be d i s c u s s e d i n a l a t e r T h e r e i s no hv = 1.092. eV 1019cm-3  e m i s s i o n peak observed a t hv = 1.098  T i 2K.  T < 50K when  x 1017  cm - 3  £  cm - 3  <  or x  the not  t h a t f r e e e x c i t o n s s h o u l d not e x i s t except f o r  x 1018  = 1.1  spectrum  < 4.3  This observation i s i n accord with  I t i s concluded 3.6  x 1018  eV  c a l c u l a t i o n s which show t h a t bound e x c i t o n s should  e x i s t i n t h i s range and  100K  section.  i n the c o n c e n t r a t i o n range 1.1  f o r any  preceeding  -  cm-3.  t h a t f o r the i m p u r i t y  < 4.3  x IO19  cm - 3  o n c e n t r a t i o n range  t h e r e i s no  c o n t r i b u t i o n due  to  the recombination  of e x c i t o n s t o the observed p h o t o l u m i n e s c e n c e except  p o s s i b l y t h a t due  t o f r e e e x c i t o n s at  6.2  I n t e r p r e t a t i o n of  = 3.6  x 10  cm  .  Spectra  I n the p r e v i o u s s e c t i o n ' i t was  shown t h a t the  experimental  r e s u l t s cannot be e x p l a i n e d i n terms of the r e c o m b i n a t i o n bound e x c i t o n s . ved  T h i s l e a v e s two  s p e c t r a c o u l d be due  of f r e e or  o t h e r p o s s i b l e mechanisms.  The  to s i m p l e band-to-band r e c o m b i n a t i o n  Obser-  of  e l e c t r o n s i n the c o n d u c t i o n band w i t h h o l e s i n the v a l e n c e band w i t h o u t t h e f o r m a t i o n o f any  form of c o n d e n s a t e , or i t c o u l d be due  to r e c o m b i n -  a t i o n w i t h i n some form of condensed p h a s e . I f the p h o t o l u m i n e s c e n c e i s due  to band-to-band  t h e o b s e r v e d peak must have photon e n e r g i e s g r e a t e r t h a n E  S  i s t h e band gap  the r e c o m b i n a t i o n .  energy and At low  hv  P  recombination, - hv^,  where  i s the energy o f the phonon a s s i s t i n g  temperature the w i d t h o f s u c h a  recombination  - 49 -  l i n e depends on the d e n s i t y o f p h o t o c r e a t e d c a r r i e r s .  F o r t h e maximum  e x c i t a t i o n power used i n these e x p e r i m e n t s , 1.8 w a t t s , t h e r e would be *v> 5 x 10  cm  carriers.  -J  i n a narrow l i n e the e f f e c t i v e  T h i s c o n c e n t r a t i o n o f c a r r i e r s would  result  (^2.0 meV) a t T = 2K whose w i d t h would be l i m i t e d by  s l i t width of the spectrometer  (y 3 meV).  A n e c e s s a r y , b u t n o t s u f f i c i e n t , c o n d i t i o n f o r the e x i s t e n c e o f a condensed  phase i s t h a t t h e b i n d i n g energy  E = E o a  -  be l e s s t h a n z e r o . and E  g i v e n by  (E ) + hv g opt p  v  In t h i s equation ( g ) p E  Q  i s t h e t h r e s h o l d energy.  t  I  s t n  (6.2-1) '  e o p t i c a l band gap  The t h r e s h o l d energy i s o b t a i n e d by  3. e x t r a p o l a t i n g t h e h i g h photon energy s i d e o f t h e p h o t o l u m i n e s c e n t peak t o the b a s e l i n e .  The w i d t h o f the peak would depend on the e q u i l i b r i u m  c o n c e n t r a t i o n o f c a r r i e r s i n t h e condensed tion  phase and n o t on t h e e x c i t a -  level. In t h e f o l l o w i n g a n a l y s i s i t w i l l be shown t h a t t h e observed  photoluminescence mechanisms.  c a n be e x p l a i n e d i n terms o f b o t h o f t h e above two  F o r convenience o n l y t h e TO a s s i s t e d r e c o m b i n a t i o n l i n e  w i l l be d i s c u s s e d , t h e o t h e r s b e i n g j u s t phonon r e p l i c a s . .6.3  Low C o n c e n t r a t i o n C9 x 1 0 The  2 5  cm"" .< N 3  D  .< 1.1 x 1 0  1 8  cm" ) 3  t h r e s h o l d energy and w i d t h a t h a l f maximum a t 2K f o r t h e  TO a s s i s t e d peak a r e shown i n F i g u r e s 6.1 and 6.2 r e s p e c t i v e l y .  The  e x p e r i m e n t a l d a t a shown i n these f i g u r e s have been c o r r e c t e d f o r i n s t r u m e n t a l b r o a d e n i n g i n t h e manner d e s c r i b e d i n Appendix  A.  The  v a l u e s used f o r t h e band gap and o p t i c a l band gap were o b t a i n e d from  (Eg)  l.l  o p  -hI/p  1.10(Eg)  0 p  -hvp  > >-  1.09  e> cr. in 2 UJ  1.08  1I  O  g  Threshold Energy J  1.07  CL  o 1.06  1.05  J  16 10  L  I I Mill  ^7 10"  j  I  IMPURITY  FIGURE 6.1  '  I I I I 11 J8 10'°  10  CONCENTRATION  '  ' i i *i 111  19  20 10  ( / m ) P  3  C  C o n c e n t r a t i o n Dependence o f t h e T h r e s h o l d Energy, E a . F o r  Eg * (Eg) ^ and was o b t a i n e d from K u w a h a r a ^ \ opt were o b t a i n e d from B a l k a n s k i e t a l . . 5  6  ' i i i i 11  6  (  4  6  )  F o r N_, > 6.0 x 1 0 D  1 8  cm  < 1.0 x 1 0 - 3 e  1 8  Eg and (Eg) opt 6 /  cm  - 3  FIGURE 6.2  Half-width of TO Assisted Peak vs. Impurity Concentration.  are calculated assuming parabolic bands and constant effective masses.  Solid curves  9 X 10  FIGURE 6.3  1 5  3.6 X 1 0  1.1 X 10,18  1 7  Comparison of Calculated and Experimental Line Shapes at 2K,  curves represent the profile for EHD with n » 3.0 x 10  1 8  cm" . 3  The dashed  - 53 -  Kuwahara^ ^ f o r  < 1.1 x 1 0  45  for  N  1.1  x 10  D  > 6 x 10 cm  1 8  a condensed  cm .  1 8  1 8  cm"  and from B a l k a n s k i et  3  It i s clear that f o r 9 x 1 0  -3  cm  1 5  al.^ ^ 6  £  -3  <  t h e c r i t e r i o n o f e q u a t i o n (6.2-1) f o r the e x i s t e n c e of  -3  phase i s s a t i s f i e d .  The requirement  of band-to-band recom-  • b i n a t i o n , t h a t the e m i t t e d photons have energy g r e a t e r t h a n the band i s not s a t i s f i e d t i o n power.  and the l i n e w i d t h i s found to be independent  gap,  of e x c i t a -  These f i n d i n g s i n d i c a t e t h a t the mechanism r e s p o n s i b l e f o r  t h i s peak i s not s i m p l e band-to-band r e c o m b i n a t i o n , but must i n v o l v e a condensed  phase. It  of  i s found t h a t the t h r e s h o l d energy, E , and  the l i n e i n t h i s c o n c e n t r a t i o n range a r e e s s e n t i a l l y t h e same as  t h o s e observed i n i n t r i n s i c  s i l i c o n f o r the l i n e a t t r i b u t e d  phonon a s s i s t e d r e c o m b i n a t i o n w i t h i n an EHD. c l o s e agreement  x 10  n = 3.0  cm"  1 8  x 10  Appendix B. ent  1 8  3  F i g u r e 6.3  for  = 9 x- 1 0  1 5  cm , -3  3.6  cm . -3  1 7  cm  and  -3  with  T h i s v a l u e f o r the e q u i l i b r i u m d e n s i t y i s somewhat d i f f e r x 10  1 8  cm  -3  found by P o k r o v s k i i e t a l . ^ " ^  in  l a r g e l y to the c h o i c e  v a l u e s f o r the e f f e c t i v e masses.  luminescence  ing  shows the  The d e t a i l s of t h i s c a l c u l a t i o n can be found i n  from the v a l u e o f 3.8  On  to  x 10  and the c a l c u l a t e d l i n e shape f o r an EHD  i n t r i n s i c s i l i c o n , however the d i s c r e p a n c y i s due of  to TO  (except i n t h e wings) found between the e x p e r i m e n t a l l y  o b s e r v e d l i n e shapes 1.1  the h a l f w i d t h  the b a s i s o f the above e v i d e n c e i t i s c l e a r t h a t the observed f o r 9.0  r e c o m b i n a t i o n w i t h i n an EHD phonons.  x IO  1 5  cm"  3  H  1 1.1  x 10  1 8  cm  -3  photoi s due  a s s i s t e d by a p p r o p r i a t e momentum c o n s e r v -  - 54 -  6.4  Transition Range (2.2 x 10 18 . cnT3.s.N  x 1Q18  <.5.5  cm" ) 3  Unfortunately there are no data available on either the band gap or the optical band gap in the concentration range 2.2 x 10 18 <  < 5.5 x 10  8  cm . -3  - hv  -3  It is possible, though, to make a reasonable From Figure 6.1  estimate of E - hv in this range. § P E  cm  i t appears that  can be extrapolated over this range by connecting the data of  Kuwahara with that of Balkanski et a l . with a smooth curve.  If this  i s done, then for a l l samples studied in this concentration range, the criterion of equation (6.2-1) i s satisfied, indicating that the recombination may involve a condensed phase.  In the next section i t i s suggested  -that the data of Balkanski et a l . i s misleading and this would mean that the above estimate of E^ - hv^ i s not a good one.  Since the donor ground  state is 45 meV below the intrinsic conduction band edge this constitutes 'a lower limit for the band edge when N  D  ^ 3,0 x l Q  cm" ,  i 8  3  tion at which the ground state be.comes delocalized. case, i t i s s t i l l found that for  = 3.0  x 10 18  the concentra*-  For this extreme  cm"  3  an appreciable  fraction of the observed photoluminescent line has photon energies less than this value of E g  - hv . On the basis of these estimates i t i s P |  unlikely that the observed photoluminescence  i s due to simple band-to-  band recombination. The linewidth data shown in Figure 6.2  i s more informative.  If the data i s to be analysed i n terms of a condensed phase, the simple picture of an EHD must be modified somewhat. When the impurity concentration exceeds the equilibrium density (3.0 x 1Q  18  able to consider a condensation of electrons.  cm" ) 3  i t i s unreason-  Maintenance of an electron  - 55 -  d e n s i t y o f n = 3.0 x 1 0  c n f ^ would c o n s t i t u t e a r a r e f a c t i o n o f  1 8  e l e c t r o n s which i s p h y s i c a l l y u n r e a l i s t i c . have an e l e c t r o n d e n s i t y o f n = 3.0 x I O cm" , and an e l e c t r o n d e n s i t y o f n =  cm"  3  when  3  If  1 8  The m o d i f i e d when  EHD would 1 3.0 x 1 0  £ 3.0 x 1 0  1 8  cm  _ 3 +  t h e p r o b l e m o f a h o l e plasma moving i n a homogeneous n e g a t i v e  1 8  .  back-  ground i s s o l v e d , i . e . by c o n s i d e r i n g o n l y t h e terms i n v o l v i n g h o l e s i n the equation  (2.5-1),  then i t i s found t h a t i n s i l i c o n  t h e lowest  energy c o n f i g u r a t i o n i s a h o l e plasma w i t h d e n s i t y n^ ^ 3.0 x 1 0 Thus i t i s r e a s o n a b l e holes w i l l  continue  t o c o n s i d e r as a f i r s t  cm" . 3  a p p r o x i m a t i o n t h a t the  t o condense t o t h i s d e n s i t y i r r e s p e c t i v e o f t h e  electron concentration.  This i s i n accord with  t h e model f o r t h e EHD  w h i c h c o n t a i n s no e l e c t r o n - h o l e i n t e r a c t i o n s b u t c o n s i d e r s t r o n s and h o l e s t o be independent.  i s g i v e n by curve  and h o l e d e n s i t y n = 3.0 x 1 0  ( i ) o f F i g u r e 6.2.  p a r a b o l i c bands and c o n s t a n t  the e l e c -  The l i n e w i d t h e x p e c t e d o f a c o n -  densed phase w i t h e l e c t r o n d e n s i t y NQ cm  1 8  1 8  T h i s c a l c u l a t i o n assumes  e f f e c t i v e masses.  The l i n e w i d t h s  obtained  I n t h i s manner a r e l a r g e r t h a n t h e measured l i n e w i d t h s i n t h e i m p u r i t y c o n c e n t r a t i o n range 2.2 x 1 0  1 8  cm"  3  d e n s i t y of s t a t e s f o r the conduction  <  < 5.5 x 1 0  1 8  cm" . 3  Using a  band o f t h e form suggested by  (47) Mott  i s e x p e c t e d t o g i v e even l a r g e r c a l c u l a t e d l i n e w i d t h s r a t h e r  t h a n improve t h e agreement. in  terms o f band-to-band r e c o m b i n a t i o n  The to  I f , on the o t h e r hand, the d a t a i s a n a l y s e d  a d d i t i o n o f 5.0 x 1 0  1 6  cm  without  photocreated  a condensed phase then  e l e c t r o n s i s n o t expected  cause any a p p r e c i a b l e r e d i s t r i b u t i o n o f donor e l e c t r o n s when n » N  - 56  -  the l i n e w i d t h would be g i v e n by curve  ( i i ) of F i g u r e 6.2.  c u l a t i o n a l s o assumes p a r a b o l i c bands and The  cal-  c o n s t a n t e f f e c t i v e masses.  l i n e w i d t h s c a l c u l a t e d w i t h t h i s model a r e s m a l l e r than the  l i n e w i d t h s , but may  This  observed  the use of a more s u i t a b l e d e n s i t y o f s t a t e s f u n c t i o n  improve the agreement. The  concept  recombination  of a c o n s t a n t h o l e d e n s i t y used  i n v o l v i n g a condensed phase may  e l e c t r o n d e n s i t y o f n > 3.0  x 10  1 8  cm"  be a l i t t l e  i s sufficient  3  i n the model f o r naive.  to s c r e e n  any  bound i m p u r i t y s t a t e s , hence the condensed phase w i l l c o n t a i n tive i o n cores.  Although  p o s i t i v e f i e l d w i l l reduce  x 10  1 8  predicted and to  cm"  3  < N  D  f o r no  condensed phase  the l i n e w i d t h o f the EHD  of  6.5  x 10  1 8  cm"  t h a t i n the c o n c e n t r a t i o n range the l i n e w i d t h l i e s between those  3  (curve ( i i ) ) ,  and  recombination  a c t u a l l y decreases  line i n Intrinsic  band—to-band r e c o m b i n a t i o n  H i g h C o n c e n t r a t i o n '.N  cannot  and  the p o s s i b i l i t y  be d e f i n i t e l y r u l e d  > .1.0 .x .10  1 9  . cm"  out.  3  p r i n c i p l e i t s h o u l d be p o s s i b l e t o draw d e f i n i t e c o n c l u s i o n s  about t h e e x i s t e n c e o f a condensed phase f o r N band gap  silicon.  o f a condensate w i t h a d e c r e a s i n g h o l e  However, w i t h o u t knowledge about the band gap,  In  relative  o f a condensate w i t h a c o n s t a n t h o l e d e n s i t y but  c o m p a t i b l e w i t h t h e concept  density.  that t h e i r  f o r a condensed phase w i t h c o n s t a n t h o l e d e n s i t y (curve ( i ) ) ,  T h i s i s not expected is  < 5.5  these  the h o l e d e n s i t y .  E x p e r i m e n t a l l y i t i s observed 2.2  posi-  the d e t a i l e d n a t u r e of the e f f e c t s of  p o s i t i v e c o r e s i s not known, i t i s r e a s o n a b l e to expect  The  > 6'x  o p t i c a l band gap have been determined  10  1 8  f o r 6.0  cm .  The  -3  x 10  1 8  cm"  3  - 57  < N  D  < 4 . 9 x 10 20  cm""  a t 35K  3  absorption techniques. ments t h i s s h o u l d be a condensed phase.  coefficient be f i t t e d  2  1  To o b t a i n v a l u e s f o r the band gap  first  two  t h a t the f r e e a b s o r p t i o n c o e f f i c i e n t  dependence, and  secondly  t h a t the r e m a i n i n g  had  absorption  a f t e r s u b t r a c t i o n o f the f r e e c a r r i e r c o n c e n t r a t i o n c o u l d  to a t h e o r e t i c a l e x p r e s s i o n assuming a p a r a b o l i c d e n s i t y of assumption of X  The  t i o n may  be r e a s o n a b l e ,  2  dependence f o r the f r e e c a r r i e r  The  not v a l i d  f o r i m p u r i t y c o n c e n t r a t i o n s j u s t above the  transition. conduction  absorp-  s i n c e i t does g i v e good agreement a t l o n g wave-  lengths.  a s s u m p t i o n o f a p a r a b o l i c d e n s i t y of s t a t e s i s c e r t a i n l y  At  semiconductor-metal  these concentrations there i s s i g n i f i c a n t  tailing  band, a f e a t u r e w e l l documented by o t h e r workers.  of  the  Further  of the e x i s t e n c e o f such a t a i l on the c o n d u c t i o n band i s p r o -  v i d e d i n F i g u r e 4.3. s i d e o f t h e TO  Here l o n g t a i l s a r e observed  a s s i s t e d peaks, p a r t i c u l a r l y f o r  Balkanski's value of E has  infrared  U n f o r t u n a t e l y the r e s u l t s o f B a l k a n s k i et a l . must  states.  evidence  using  s u f f i c i e n t to determine whether o r n o t t h e r e i s  assumptions were made: A  by B a l k a n s k i e t al,  Together w i t h the p h o t o l u m i n e s c e n c e e x p e r i -  be v i e w e d w i t h some c a r e .  the c l a s s i c a l  -  = 1.15  would suggest  o n l y been l o w e r e d by 20 meV,  on the low < 3.0  B a l k a n s k i e t a l . may  in  t h e c o n c e n t r a t i o n range 1.3  band  below the  intrinsic  With the p r o v i s o t h a t the d a t a  n o t be c o r r e c t , F i g u r e 6.1  equation  3  even though f o r t h i s c o n c e n t r a t i o n i t  delocalized.  of  cm"" .  1 8  t h a t the c o n d u c t i o n  i s known t h a t the donor ground s t a t e , which i s 45 meV band edge, i s c o m p l e t e l y  x 10  energy  shows t h a t the  of  criterion  (6.2-1) f o r the e x i s t e n c e of a condensed phase i s s a t i s f i e d x 10  1 9  cm  -3  & N  4 4.3  x 10  1 9  cm . -3  - 58  -  The measured l i n e w i d t h s a r e l e s s c o n c l u s i v e . a p p e a r s t o be  some agreement between c u r v e  Although  ( i ) of F i g u r e 6.2  measured v a l u e s , c o r r e c t i o n s f o r non p a r a b o l i c bands may b e t t e r f i t w i t h curve t h e r e i s no  I t s h o u l d be noted  o f a p a r a b o l i c band i s based on a one c o r r e l a t i o n s between e l e c t r o n s . i s expected N  D  -v 1 0  1 9  3  t o draw any  The  8  t h a t the  if  assumption ignores  i n c l u s i o n of c o r r e l a t i o n  effects  o f the peak, e s p e c i a l l y f o r  Thus the l i n e w i d t h d a t a by i t s e l f  conclusions with regard  the  result in a  e l e c t r o n p i c t u r e which  t o produce a f u r t h e r broadening  cm" ^ ^.  and  ( i i ) which r e p r e s e n t s the w i d t h expected  condensed phase.  there  i s not  sufficient  to the e x i s t e n c e of a condensed  phase.  6.6  I n t e n s i t y o f Photoluminescence The  r e l a t i v e i n t e g r a t e d i n t e n s i t y of the TO a s s i s t e d peak i s  shown a s a f u n c t i o n o f i m p u r i t y c o n c e n t r a t i o n a t 2K i n F i g u r e Although should be  t h e i n t e n s i t i e s a r e shown w i t h an u n c e r t a i n t y o f 10% c o n s i d e r e d o n l y to show a q u a l i t a t i v e t r e n d .  The  o f p h o t o l u m i n e s c e n c e depends i n an unknown manner upon the c o n d i t i o n o f t h e sample s u r f a c e .  Care was  i n the p r e p a r a t i o n o f the samples, but i n t e n s i t y of t» 10%  occurred  t i o n were compared. spectrometer  i t was  i f different  be o p t i m i s t i c .  data  efficiency detailed  to maintain u n i f o r m i t y  found  t h a t changes i n  samples o f the same  between s u c c e s s i v e experiments can a l s o cause  may  the  concentra-  S m a l l v a r i a t i o n s i n the o p t i c a l alignment  v a r i a t i o n s i n the s i g n a l i n t e n s i t y . i n F i g u r e 6.4  taken  6.4.  of  the  significant  Thus the quoted u n c e r t a i n t y o f  10%  However, t h e r e i s an o r d e r of magnitude  change i n t h e i n t e g r a t e d i n t e n s i t y o f the T O - a s s i s t e d peak over  the  -  IMPURITY  FIGURE 6.4  59  -  CONCENTRATION  (P/cm ) 3  Intensity of TO Assisted Peak vs. Impurity Concentration.  - 60 -  c o n c e n t r a t i o n range  s t u d i e d which must be c o n s i d e r e d r e a l .  Such a  change i s p l a u s i b l e as t h e i n c r e a s e d number o f s c a t t e r i n g c e n t r e s would enhance non r a d i a t i v e r e c o m b i n a t i o n p r o c e s s e s . (38) The  f i n d i n g of A l e e k s e v et a l .  t i o n l e v e l i s lowered  t h a t the c r i t i c a l  as the c o n c e n t r a t i o n i s i n c r e a s e d would appear  suggest t h a t an i n c r e a s e i n photoluminescence expected. set  excita-  Such an i n c r e a s e may  i n t e n s i t y might  be  o c c u r , however i t c o u l d be e a s i l y  i f t h e l i f e t i m e w i t h i n the EHD  to  off-  decreased as the i m p u r i t y c o n c e n t r a -  tion increased. 6.7  Temperature Dependence o f The  temperature  Photoluminescence  dependence o f the photoluminescence  of  (39) intrinsic observed  s i l i c o n has been s t u d i e d by K a m i n s k i i e t a l . t h a t at low temperature  from the EHD  was  present.  The  As the temperature was  t h e r e was  temperature  radiation  r a i s e d , peaks due  to  photoluminescent  i n c r e a s e d w i t h i n c r e a s i n g temperature no r a d i a t i o n observed from the EHD.  f o r t h i s " e v a p o r a t i o n " o f the EHD  w h i c h t h e r e l a t i v e i n t e n s i t i e s o f t h e two cal  i n the  was  i n t e n s i t y o f t h i s r e c o m b i n a t i o n r a d i a t i o n , r e l a t i v e to  t h a t from the EHD, T > 20K  It  (T i 2K) o n l y r e c o m b i n a t i o n  t h e r e c o m b i n a t i o n o f f r e e e x c i t o n s appeared spectra.  .  temperature, T  was  until  The  by  critical  taken t o be t h a t a t  l i n e s were e q u a l .  The  criti-  » i s dependent upon the e x c i t a t i o n l e v e l i n the  manner shown by e q u a t i o n (2.5-11), and f o r a l l r e a l i s t i c power l e v e l s i s about  15K. As F i g u r e 5.1  samples w i t h N  = 9 x 10  shows, a s i m i l a r b e h a v i o r i s observed f o r 1 5  cm"" , except t h a t i n t h i s case t h e r e a r e a l s o 3  - 61 -  bound e x c i t o n s p r e s e n t .  A l t h o u g h t h e exact  employed f o r d e t e r m i n i n g  c r i t e r i o n t h a t s h o u l d be  i s no l o n g e r c l e a r , i t i s apparent from  F i g u r e 5.1 t h a t 13.5 < T  < 21K.  T h i s i n d i c a t e s t h a t t h e EHD-exciton  cr e q u i l i b r i u m i s n o t s i g n i f i c a n t l y a f f e c t e d by the p r e s e n c e o f cm"  phosphorus i m p u r i t i e s .  3  = 9 x 10  I t i s c l e a r from F i g u r e 5.2 t h a t the e f f e c t  o f t h e i m p u r i t i e s c a n no l o n g e r be i g n o r e d t h i s c o n c e n t r a t i o n t h e lowest  for  = 3.6 x 1 0  1 7  cm"" .  temperature f o r w h i c h t h e r e i s any  evidence  o f f r e e e x c i t o n r a d i a t i o n i s T ^ 45K.  shoulder  has begun to d e v e l o p on the h i g h photon energy s i d e o f the  TO a s s i s t e d peak.  T h i s shoulder  At t h i s temperature a  i n c r e a s e s i n r e l a t i v e s i z e u n t i l by  T  100K t h e r e i s a d e f i n i t e s h i f t  eV,  t h e energy e x p e c t e d f o r r e c o m b i n a t i o n  Unfortunately  At  3  o f t h e peak p o s i t i o n t o hv = 1.098 o f a TO a s s i s t e d f r e e e x c i t o n .  the t h e r m a l b r o a d e n i n g o f t h e l i n e s i s such t h a t a t no  temperature i s i t p o s s i b l e t o r e s o l v e two l i n e s .  Thus t h e p o s s i b i l i t y  t h a t t h e o b s e r v e d peak i s due t o t h e same r e c o m b i n a t i o n temperatures i s not r u l e d out.  process  at a l l  C a l c u l a t i o n s of the e f f e c t s of screen-  i n g i n d i c a t e t h a t f r e e e x c i t o n s can e x i s t up t o a t l e a s t T ^ 100K f o r samples o f t h i s i m p u r i t y c o n c e n t r a t i o n . peak a t T = 100K w i t h The 5,  free excitons i s conceptually  valid.  temperature dependences o f samples w i t h 1.1 x 10"  5 1.3 x 1 0  sample w i t h  Thus t h e i d e n t i f i c a t i o n o f t h e  1 9  cm  - 3  18  cm"  3  i m p u r i t i e s a r e much t h e same as t h a t o f the  = 3.6 x 1 0  1 7  cm . -3  I n t h i s c o n c e n t r a t i o n range i t i s  p o s s i b l e t o r u l e out any r e c o m b i n a t i o n r a d i a t i o n due t o e x c i t o n s . The (43) IR c a l c u l a t i o n s of Albers can be u s e d t o show t h a t f o r = 1.0 x 10 cm"* f r e e e x c i t o n s cannot e x i s t a t T > 30K, and f o r N „ > 3 x 1 0 3  1 8  cm  - 3  1 5  - 62 -  they cannot  e x i s t at a l l .  On t h i s b a s i s , i f a condensed phase does  e x i s t , , t h e work f u n c t i o n f o r an e l e c t r o n - h o l e p a i r would not be measured r e l a t i v e t o t h e f r e e e x c i t o n , but r e l a t i v e to f r e e e l e c t r o n s and  holes  •I i n t h e c o n d u c t i o n and v a l e n c e bands r e s p e c t i v e l y . broad it  :  l i n e s and  A g a i n , because o f  l a c k of i n f o r m a t i o n on w h i c h to p r e d i c t the l i n e  shapes,  i s n o t p o s s i b l e t o say w i t h any c e r t a i n t y i f t h e r e i s any change i n  the r e c o m b i n a t i o n mechanism r e s p o n s i b l e f o r the TO a s s i s t e d l i n e as the temperature  i s raised.  the h i g h e n e r g y range.  The peak does appear to develop a s h o u l d e r on  s i d e f o r T >• 45K  T h i s may  f o r a l l samples i n t h i s c o n c e n t r a t i o n  be c o n s t r u e d as the o n s e t of e v a p o r a t i o n o f a condensed  phase. The  temperature  dependence o f the  = 4.3  x 10  1 9  i s s t r a i g h t f o r w a r d . W i t h i n the e x p e r i m e n t a l e r r o r , the two peaks b r o a d e n w i t h temperature  cm  -3  sample  observed  but do not s h i f t a p p r e c i a b l y i n energy.  Based on t h e d a t a of B a l k a n s k i e t a l . t h e s e peaks l i e e n t i r e l y above t h e i r r e s p e c t i v e values of E - hv which suggests t h a t the p h o t o l u m i n i 6 P escence  i s due  t o r e c o m b i n a t i o n o f donor e l e c t r o n s and  out t h e p r e s e n c e o f a condensed phase.  The  temperature  free holes withdependence o f  the p e a k p o s i t i o n would depend p r i m a r i l y on the e l e c t r o n d e n s i t y and t h i s d o n o r c o n c e n t r a t i o n would be i n s e n s i t i v e to temperature i t would be s u b j e c t t o  broadening.  although  at  - 63  -  CHAPTER 7  ''EFFECT OF HEAT-TREATMENT ON  Si : P •  7.1  Introduction I t i s known t h a t c o m m e r c i a l l y  produced s i l i c o n c r y s t a l s  c o n t a i n h i g h c o n c e n t r a t i o n s of oxygen and c a u t i o n s a r e t a k e n d u r i n g manufacture. i t i e s present  iq x 10  and vacuum f l o a t  -3  x 10 ' 1  been f o u n d  pre-  number o f oxygen impur-  c o n c e n t r a t i o n s as h i g h  crystal, as  cm  .  t o be ^ 5 x 1 0  of p r o d u c t i o n  zoned c r y s t a l s u s u a l l y h a v i n g  less  ^ (50)  17  t h a n 1.0  The  may  ^(49) cm  3  carbon u n l e s s s p e c i a l  depends s t r o n g l y on the method used to grow the  w i t h C z o c h r a l s k i grown c r y s t a l s h a v i n g  1.5  !  .  The 1 8  cm  c o n c e n t r a t i o n of carbon i m p u r i t i e s has -3  r e l a t i v e l y independent of the manner  Both carbon and  as n e u t r a l i m p u r i t i e s , and  oxygen e n t e r the s i l i c o n  as s u c h have no  lattice  a p p r e c i a b l e e f f e c t on  the  e l e c t r o n i c p r o p e r t i e s of s i l i c o n .  T h e i r p r e s e n c e has been d e t e c t e d by (52 53) s t u d i e s o f t h e f a r i n f r a r e d a b s o r p t i o n due t o l a t t i c e v i b r a t i o n s ' (54) A r e p o r t by 2 x 10 300  1 6  donors-cm""  - 500C sparked  ment on s i l i c o n .  3  F u l l e r et a l . in intrinsic  on the f o r m a t i o n of more than  s i l i c o n a f t e r heat  considerable interest Kaiser  found  treatment  i n the e f f e c t s of heat  that prolonged  heat  known t o be o f two  precipi-  A c o n s i d e r a b l e amount of work has been done  on i d e n t i f y i n g t h e s e p r e c i p i t a t e s and (see f o r example, B u l l o u g h  treat-  treatment  ( g r e a t e r t h a n 10 h o u r s ) a t 1000C r e s u l t e d i n the f o r m a t i o n of t a t e s w i t h i n the c r y s t a l .  at  and Newman  t h e k i n e t i c s of t h e i r ).  The  t y p e s , l a r g e ones (10 - 40 um)  formation  p r e c i p i t a t e s are c o n s i s t i n g of a  now Si02  - 64 -  core surrounded by a layer of SiC, and smaller ones (5 - 10 ym) cons i s t i n g only of S I C ^ \ 5 7  I t i s also known that heat treatment for  periods as short as 15 minutes at 1000C can produce considerable structural changes within the c r y s t a l , corresponding to the early stages of p r e c i p i t a t i o n O n e of these early stages i s l i k e l y to be the ejection of carbon from i t s usual substitutional sites into i n t e r s t i t i a l sites.  In this configuration carbon i s highly mobile and (Kg)  can diffuse rapidly, even at 300K  '.  In this chapter the effects of heat treatment at 1150C for 30 minutes on the photoluminescence and the electron paramagnetic resonance are presented for the phosphorus impurity range 9 x 1 0 < N  7.2  D  < 4.3 x 1 0  19  15  cm  -3  cm . -3  Photoluminescence from Heat Treated Si:P Heat treatment at 1150C for 30 minutes was carried out on  samples of s i l i c o n with phosphorus impurity concentration i n the range 9 x 10  15  cm  c  -3  < 4.3 x 1 0  19  cm" .  The photoluminescence of heat  3  treated samples was found to d i f f e r markedly from that of samples which had not undergone heat treatment i n the impurity concentration range 2.2 x 1 0  18  cm"  3  < N  D  < 1.3 x 1 0  19  cm" . 3  Samples with impurity concen-  tration outside this range were found to be essentially unaffected by heat treatment. Within this concentration range heat treatment produced two major changes i n the photoluminescent spectra. developed at hv ^ 1.1 eV.  F i r s t , a broad l i n e  This l i n e was about 200 meV wide and  appeared to be independent of both concentration and temperature.  - 65 -  13 K  —  J  1.03  FIGURE 7.1  1  '  1  1.06  1.09  1.12  PHOTON  ENERGY  ( eV)  Photoluminescence  a t 13K f o r H e a t - T r e a t e d S i : P .  :  v s . Impurity Concentration  L  _  1.15  - 66 -  N D = 3 X 1018  125 K UJ (J  z  UJ  80 K  u <> / Ul 2  2  58 K  r>  _j  O r-  o  36 K  I  CL U.  28 K  o  >(/)  z  UJ  19  I-  K  •13 K 103  1.06 PHOTON  1.09 ENERGY  1.12  1.15  (eV)  FIGURE 7.2 Photoluminescence vs. Temperature for Heat-Treated Si:P with N - 3.0 x 1 0 cm . 18  D  -3  - 67 -  S e c o n d l y , a new l i n e d e v e l o p e d a t hv = 1.05 eV which, was o n l y observed f o r T < 20K and whose p o s i t i o n was o n l y s l i g h t l y c o n c e n t r a t i o n dependent. This: e m i s s i o n l i n e i s somewhat broader than t h a t observed due t o recomb i n a t i o n w i t h i n t h e EHD.  The photoluminescence  a t 13K i s shown as a  f u n c t i o n o f c o n c e n t r a t i o n i n F i g u r e 7.1 and t h e photoluminescence = 3»0 x 1 0  1 8  cm  3  i s shown as a f u n c t i o n o f temperature  i n F i g u r e 7.2.  A l l o w i n g t h e samples t o a n n e a l a t 300K f o r s e v e r a l days was f o r t h e photoluminescent  of  sufficient  s p e c t r a t o r e v e r t t o t h a t observed b e f o r e h e a t -  treatment .  7.3  EPR o f H e a t - T r e a t e d S i : P E l e c t r o n paramagnetic  resonance  (EPR) i s the s t u d y o f magnetic  d i p o l e t r a n s i t i o n s between s p i n s t a t e s o f a system. donors  F o r phosphorus  i n s i l i c o n o n l y t h e o r d i n a r y Zeeman i n t e r a c t i o n need be c o n s i d e r e d  for impurity concentrations o f t h e d o n o r ground  > 2 x 10  1 8  cm . -3  This gives a s p l i t t i n g  state  AE = g u H  (7.3-1)  g  where g i s t h e e f f e c t i v e g - v a l u e , u m a g n i t u d e o f t h e a p p l i e d magnetic  p  i s t h e Bohr magneton and H i s t h e  field.  A l l o t h e r terms o f the s p i n  Hamiltonian, i n c l u d i n g the hyperfine i n t e r a c t i o n with P  3 1  nuclei,  average  t o z e r o due t o t h e d e r e a l i z a t i o n o f the donor e l e c t r o n s . S i l i c o n samples from t h e same i m p u r i t y c o n c e n t r a t i o n range as that used  f o r t h e photoluminescent  f o r 30 m i n u t e s  s t u d i e s were heat t r e a t e d a t 1150C  and t h e i r EPR s p e c t r a o b s e r v e d a t 77K, 4.2K and 1.1K.  w i t h t h e photoluminescent  s t u d i e s , i t was observed t h a t t h e heat  As  treat-  - 68  ment had  no  N  x 10  D  < 1.1  -  e f f e c t upon the spectrum f o r i m p u r i t y cm" ,  18  The  3  w i t h i n the e x p e r i m e n t a l  concentrations  accuracy.  s p e c t r u m f o r s i l i c o n c o n t a i n i n g 2.0  x 10  1 8  cm  -3  phos-  phorus c o n s i s t s o f a s i n g l e symmetric l o r e n t z i a n l i n e as a r e s u l t the  p a r t i a l d e r e a l i z a t i o n o f the e l e c t r o n s a t t h i s  A f t e r h e a t t r e a t m e n t , the spectrum ( F i g u r e 7.3) s l i g h t l y broadened s i s t s of field  (y  1 0 % ) , w h i l e at 4.2K  and  (higher  shows the EPR 1 8  spectrum observed a f t e r heat cm  i s t i c of  cm  s  -3  peak a t g 'v 2.0  to  and  1.1K  $ 1.3  w h i c h may  components has  conduction  treated.  x 10 be  1 9  cm .  The  -3  separated  t h e same w i d t h and  character-  components.  range  One  of  g - v a l u e as the s i n g l e l i n e  due  e l e c t r o n s o b s e r v e d i n samples t h a t have not been heat  A f t e r s u b t r a c t i o n o f t h i s component t h e r e remains a broad The w i d t h , asymmetry and  of  t h i s l i n e a l l increase with decreasing  of  knowledge as to the shape o f t h i s l i n e makes p r o p e r l i n e s i m p o s s i b l e , one  s i t i v e to  The  spectrum i s a composite  i n t o two  asymmetric l i n e w i t h a h i g h e r g - v a l u e .  two  are  t h e changes o b s e r v e d i n the i m p u r i t y c o n c e n t r a t i o n la  treat-  phosphorus i m p u r i t i e s .  -3  changes i n the s p e c t r a of t h i s sample at 4.2K  these  the spectrum c o n -  g-value).  ment o f samples c o n t a i n i n g 6 x 1 0  x 10  is  a s i n g l e g r e a t l y broadened l i n e s h i f t e d to lower magnetic  F i g u r e 7.4  3.0  concentration.  observed at 77K  1.1K  of  impurity  r a p i d l y with  may  temperature.  Although  g-value lack  r e s o l u t i o n of  say t h a t the g-value i s r e l a t i v e l y  insen-  c o n c e n t r a t i o n , whereas the l i n e w i d t h i n c r e a s e s  concentration.  A l l of the samples i n t h i s  r a n g e showed a s m a l l b r o a d e n i n g of ^ 10%  o f the l i n e a t  concentration 77K.  the  FIGURE 7.3  EPR  curves represent curves, a f t e r  S p e c t r a f o r Np = 2.0  x 10  1 8  cm- . 3  the s p e c t r a b e f o r e h e a t - t r e a t m e n t ,  treatment.  The  dashed  the  solid  - 70  Nr»= 6.2 X  -  10  1  \/  "> MAGNETIC FIGURE 7.4  EPR  Spectra for  FIELD  = 6.2  x 10  1 8  cm . -3  c u r v e s r e p r e s e n t the s p e c t r a b e f o r e h e a t - t r e a t m e n t ; curves, a f t e r  treatment.  The  dashed  the  solid  - 71 -  The number o f e l e c t r o n s r e s p o n s i b l e f o r an EPR  l i n e i s given  by N  where x"  e  a  x"(H)dH  (7.4-1)  I s t h e complex s u s c e p t i b i l i t y .  Due  t o the m o d u l a t i o n t e c h n i q u e 9Y"  u s e d , the o b s e r v e d l i n e shape i s p r o p o r t i o n a l t o —  on.  ,  hence i t i s  n e c e s s a r y t o i n t e g r a t e the observed l i n e t w i c e t o o b t a i n the a r e a under the  absorption curve.  A comparison  technique using a  double-sample  (35) modulation-switched sample c a v i t y developed by Q u i r t d e t e r m i n e i f Nfi was  changed by the heat t r e a t m e n t .  was  used t o  A l t h o u g h t h i s method  i s a c c u r a t e t o ^ 3% i f the l i n e shapes a r e known, the u n c e r t a i n t y i n the c o r r e c t l i n e shape due t o the p r e s e n c e o f l o n g a b s o r p t i o n t a i l s made i t i m p o s s i b l e to determine the a r e a under the curve t o b e t t e r than 15%. t h i s a c c u r a c y and assuming no change i n s p i n s u s c e p t i b i l i t y , t h e r e was no change i n the number o f e l e c t r o n s r e s p o n s i b l e f o r the observed  EPR  s p e c t r a b e f o r e and a f t e r heat t r e a t m e n t . As w i t h the p h o t o l u m i n e s c e n t work, the changes due to heat treatment d i s a p p e a r and the EPR  s p e c t r a r e v e r t t o t h a t observed b e f o r e  h e a t treatment i f the sample i s a l l o w e d t o a n n e a l at 300K f o r s e v e r a l days.  To a v o i d problems a s s o c i a t e d w i t h the microwave s k i n d e p t h i n  t h i c k s a m p l e s , the samples used f o r the EPR powder w i t h average s i z e < 50 ym.  I t was  s t u d i e s were ground t o a  found t h a t i f the sample  ground b e f o r e h e a t treatment t h e r e were no changes observed i n the  was  To  - 72 -  spectra.  S i m i l a r l y i f the heat treatment was  l o n g as 24 hours t h e r e was The EPR trations all  - 4.3  temperatures.  o f a second  no change i n t h e s p e c t r a from the u n t r e a t e d .  l i n e f o r heat t r e a t e d samples w i t h i m p u r i t y concenx 10  1 9  cm"  3  was  observed t o be broadened  by  10% a t  T h i s b r o a d e n i n g would be c o m p a t i b l e w i t h the presence  l i n e , however attempts  w i d t h o f the o b s e r v e d  7.4  c a r r i e d out f o r times as  to r e s o l v e t h i s were hampered by the  line.  D i s c u s s i o n of R e s u l t s Any attempt  t o d e v e l o p a model which can e x p l a i n the e f f e c t s  o f h e a t t r e a t m e n t must a l s o d e a l w i t h the q u e s t i o n o f why a n n e a l s out a t 300K and why  powdered samples  the e f f e c t  are- u n a f f e c t e d .  I f the  k i n e t i c s o f the heat treatment are as o u t l i n e d i n s e c t i o n 7.1, t h e s e two  a s p e c t s can be e x p l a i n e d q u i t e r e a d i l y .  When the sample i s  h e a t e d f o r o n l y s h o r t p e r i o d s o f time and then quenched, c a r b o n ities  can be t r a p p e d a t i n t e r s t i t i a l s i t e s .  then  impur-  I f allowed to anneal at  300K t h e s e c a r b o n i n t e r s t i t i a l s w i l l d i f f u s e t o and be t r a p p e d by v a c a n c i e s , o r as i n the case o f powdered samples, d i f f u s i o n l e n g t h o f carbon at 1150C  a t the s u r f a c e .  The  i n s i l i c o n i s s u c h t h a t the carbon (59)  i m p u r i t i e s would m i g r a t e to the s u r f a c e o f the powdered  samples  d u r i n g the p e r i o d o f heat treatment where they would be t r a p p e d and l a t e r removed by e t c h i n g .  S i m i l a r l y , heat treatment f o r p r o l o n g e d times  r e s u l t s i n the f o r m a t i o n o f S i C p r e c i p i t a t e s which e f f e c t i v e l y remove t h e c a r b o n i m p u r i t y from the c r y s t a l .  Thus the heat treatment  and  a n n e a l i n g p r o p e r t i e s c o u l d be e x p l a i n e d i f the change's to the p h o t o l u m i n e s c e n c e and EPR were i n some way carbon  impurities.  related  t o the presence o f  interstitial  - 73 -  The powder samples used f o r the EPR a c c o r d i n g to p a r t i c l e  s i z e , and the e f f e c t  s t u d i e s were graded  of heat treatment on the  o b s e r v e d s p e c t r a was  found t o be a f u n c t i o n o f t h i s s i z e .  be due t o the e f f e c t  o f heat treatment on the s u r f a c e o f the  s i n c e the sample i s ground  after  from those o f i n t e r e s t .  s e p a r a t i n g the two  l i n e s a t g ^ 2.00  under the two a b s o r p t i o n c u r v e s make statement,  There i s  and i s q u i t e  A l t h o u g h the problems i n v o l v e d i n and d e t e r m i n i n g the r e l a t i v e i t difficult  to make a  areas  definitive  t h e r e i s not s u f f i c i e n t e v i d e n c e f o r b e l i e v i n g t h a t the s u r -  f a c e p l a y s a major r o l e  i n the observed EPR  heat t r e a t m e n t produces  local distortions  throughout  the c r y s t a l  bution i n p a r t i c l e  spectra.  More l i k e l y ,  the  whose d i s t r i b u t i o n w i l l v a r y  and which w i l l i n f l u e n c e the manner i n which the  c r y s t a l f r a c t u r e s when ground  tortions  Nor i s  of untreated Si:P.  l i n e due t o the s u r f a c e b u t t h i s has g = 2.06  w e l l resolved  particles,  treatment and then e t c h e d .  t h e r e any such s i z e dependence i n the EPR an EPR  T h i s cannot  t o a powder.  s i z e which i s r e l a t e d  i n the heat t r e a t e d  T h i s would l e a d to a  t o the d e n s i t y o f t h e s e  distridis-  samples.  There a r e a number of complexes t h a t i n t e r s t i t i a l c a r b o n  can  (52) form w i t h oxygen distortions  and s i l i c o n .  These would produce  a variety  and accompanying s t r a i n f i e l d s i n t h e l a t t i c e .  of  The work of  (57) Shul'pina et a l .  suggests that such d i s t o r t i o n s  p o s s i b l y extending only s e v e r a l l a t t i c e spacings. b o t h the EPR  and photoluminescence  c e n t r a t i o n and  temperature  sets of  results.  It i s unlikely  that  would show changes i n the same con-  r a n g e s , and t h a t t h e s e changes would have the  same a n n e a l i n g c h a r a c t e r i s t i c crystal defect.  would be s h o r t range,  u n l e s s b o t h a r e a s s o c i a t e d w i t h the same  On t h i s b a s i s any model put forward must e x p l a i n  both  - 74 -  The enumerated.  p r o p e r t i e s t h a t these complexes must p o s s e s s can be They must be a b l e t o t r a p an e l e c t r o n and form a p a r a -  magnetic s p e c i e s .  The time r e q u i r e d f o r the c a p t u r e o f t h e e l e c t r o n  must be s h o r t compared t o any e x p e r i m e n t a l in equilibrium. f o r capture  times so t h a t t h e system i s  I t must, however, be l o n g e r than the time r e q u i r e d  o f an e l e c t r o n by an i o n i z e d donor o r f o r r e c o m b i n a t i o n o f  e l e c t r o n s and h o l e s .  I f t h i s were n o t t r u e than t h e e f f e c t s o f the  complexes would be o b s e r v e d a t a l l i m p u r i t y c o n c e n t r a t i o n s . captured  The  e l e c t r o n must be c a p a b l e o f t u n n e l l i n g out o f t h e " t r a p " by  20K,  s i n c e t h e e f f e c t s o f the h e a t treatment a r e o n l y o b s e r v e d below  this  temperature.  The paramagnetic c e n t r e i n v o l v i n g a complex and a  t r a p p e d e l e c t r o n w i l l c o n t r i b u t e i n some way t o t h e p h o t o l u m i n e s c e n c e . T h i s f o l l o w s from t h e o b s e r v a t i o n t h a t t h e r e i s a new l i n e i n t h e p h o t o luminescent  s p e c t r a o n l y when the new paramagnetic s p e c i e s , as measured  by EPR, i s p r e s e n t .  I n t h e v i c i n i t y o f t h i s paramagnetic s p e c i e s  e l e c t r o n - h o l e recomination  o c c u r s w i t h the e m i s s i o n o f a photon w i t h  energy hv ='1.05 eV independent o f i m p u r i t y c o n c e n t r a t i o n .  Any model  s h o u l d a l s o g i v e some e x p l a n a t i o n f o r t h e broad background p h o t o l u m i n e s cence o b s e r v e d . The any  L  q u e s t i o n i s , can these v a r i e d p r o p e r t i e s be r e c o n c i l e d t o  one mechanism o r a r e the e x p e r i m e n t a l  r e s u l t s being  interpreted i n  too r e s t r i c t i v e a f a s h i o n ? One such mechanism t h a t does appear t o s a t i s f y a l l t h e r e q u i r e ments i s t h e f o r m a t i o n o f n e u t r a l e l e c t r o n t r a p s h a v i n g a bound s t a t e j u s t s l i g h t l y below t h e i m p u r i t y ground s t a t e , i . e . about 47 meV below  - 75 -  the; bottom o f t h e i n t r i n s i c c o n d u c t i o n band.  The p r o c e s s  responsible  for. t h e p h o t o l u m i n e s c e n t l i n e a t hv = 1.05 eV would be the recombination, o f a trapped e l e c t r o n with a of the previous  photocreated hole.  chapter i n d i c a t e s that t h i s hole  condensed p h a s e .  The d i s c u s s i o n  i s probably i n a  The o n l y assumption n e c e s s a r y t o e x p l a i n t h e observed  p r o p e r t i e s i s t h a t t h e time r e q u i r e d t o t r a p an e l e c t r o n be l o n g compared t o t h e l i f e t i m e o f p h o t o c r e a t e d c a r r i e r s .  The time r e q u i r e d  f o r am e l e c t r o n t o i n t e r a c t and recombine w i t h a h o l e s e c o n d s ^  a  Q  (  j  m  a  y  ^  e  u  s  e  <  j  a  s  a  i  o  w  e  r  limit  i s^ 2 x 10  - 6  f o r t h e c a p t u r e time f o r  the t r a p s , s i n c e t h e c a p t u r e o f e l e c t r o n s by i o n i z e d donors i s much faster  C— 1 0  - 9  seconds).  The upper l i m i t  i s g i v e n by the s h o r t e s t  c h a r a c t e r i s t i c t i m e o f t h e experiments and i s c e r t a i n l y l e s s t h a n 1  secondAs  long as the impurity  o f an e l e c t r o n b e i n g  captured  s t a t e s aire l o c a l i z e d , t h e p r o b a b i l i t y  by a n e u t r a l t r a p i s low, s i n c e b o t h  c a p t u r e by i o n i z e d donor and r e c o m b i n a t i o n w i t h i n the EHD a r e much f a s t e r mechanisms. that  there  bility  I t i s o n l y when t h e r e a r e s u f f i c i e n t  a r e d e l o c a l i z e d e l e c t r o n s present  impurities  f o r T < 20K t h a t t h e p r o b a -  o f e l e c t r o n c a p t u r e by a t r a p w i l l become a p p r e c i a b l e .  l o c a l i z a t i o n which occurs f o r  £ 2.2 x 10-  18  cm  - 3  The de-  r e s u l t s i n an excess  o f f r e e e l e c t r o n s w i t h i n f i n i t e l i f e t i m e which can now be c a p t u r e d . t r a p becomes n e g a t i v e l y  c h a r g e d and r e p e l s other  The  e l e c t r o n s , thus t h e  t r a p w i l l n o t be s c r e e n e d by e i t h e r t h e e x i s t i n g donor e l e c t r o n s o r by any  e l e c t r o n c o n d e n s a t e which may form.  This negatively  w i l l b e p a r a m a g n e t i c and c o n t r i b u t e t o t h e EPR.  Since  charged  trap  t h e r e w i l l be a  -  76  -  range of trap energies there w i l l be a variety of contributions, resulting i n an asymmetric resonance l i n e .  Being negatively charged,  the trapped electron w i l l attract holes and recombine resulting i n radiative emission.  There are two types of holes with which recombin-  ation can occur, free holes and holes i n a condensed phase. processes are l i k e l y to occur, giving photoluminescence  Both  at hv = 1.065  eV and hv = 1.05 eV respectively, the condensed holes having a binding energy of ^ 15 meV.^  The width of.the.observed l i n e would be controlled  by the variation i n trap energies. Experimentally there i s a photoluminescent peak at hv =1.05  eV  indicating that the recombination occurs v i a a condensed phase. This i s not surprising, since i t i s known that the time for capture by the condensed phase i s short compared to the free carrier l i f e t i m e .  This i s  the reason recombination from free excitons i s not observed at 2K i n (3) intrinsic silicon  . The electron would only remain trapped for T < 20K,  since above this temperature the probability i s high that i t w i l l tunnel to the donor ground state, which i s only a few meV higher i n energy. When the impurity concentration i s high enough,i.e. for cm , -3  > 4.3 x 1 0  19  the conduction band w i l l be lower than the trap level so that there  w i l l be no bound state.  The broad background luminescence i s not  explained by this model, but could be attributed to a low density of traps with a wide range of energies. Recombination with free hole at hv *=> Eg  *- hv ~ 47 meV ~ 1,065 eV. o p Recombination with hole i n condensed phase at hv ~ Eg - hv - 47 meV - o p - 15 meV =1.05 eV. Eg = 1.170 eV i n t r i n s i c band gap, hv = 57.3 meV for TO phonon. r  Q  p  - l i -  lt  I s p o s s i b l e t o o b t a i n an e s t i m a t e o f t h e number o f such  traps i n the c r y s t a l .  The photoluminescence s p e c t r a a t 13K and t h e EPR  s p e c t r a a t 1.1K i n d i c a t e t h a t the m a j o r i t y o f t h e e l e c t r o n s a r e a s s o c i a t e d w i t h t r a p s when  = 2.2 x 1 0  1 8  cm .  c o n t r i b u t i o n t o t h e EPR from c o n d u c t i o n the p h o t o l u m i n e s c e n t  line attributed  -3  There i s not a s i g n i f i c a n t  e l e c t r o n s , and the i n t e n s i t y o f  t o recombination  t r o n s I s much g r e a t e r than t h a t a t t r i b u t e d t o f r e e trons.  of trapped  Cor condensed) e l e c -  F o r h i g h e r impurity concentrations there are c l e a r l y  b u t i o n s t o t h e EPR and photoluminescence, traps I s taken  t o be ^ 2 x 10  t r a t i o n o f carbon  expected  cm  J  elec-  two c o n t r i -  therefore the concentration of  which i s i n a c c o r d w i t h t h e concen-  to be p r e s e n t .  T h i s c o n c e n t r a t i o n i s expected  t o v a r y f r o m sample t o sample. The a c t u a l p h y s i c a l s i z e o f the l a t t i c e d i s t o r t i o n g i v i n g r i s e t o t h e s e t r a p s may be s e v e r a l l a t t i c e  spacings  even though t h e e f f e c t i v e c a p t u r e c r o s s - s e c t i o n as determined  by the  c a p t u r e p r o b a b i l i t y may be q u i t e s m a l l . One outcome o f t h i s model i s t h e i d e n t i f i c a t i o n o f a condensed phase o f h o l e s f o r i m p u r i t y c o n c e n t r a t i o n s condensed phase may have a lower  < 1.3 x 1 0  i 9  cm . -3  d e n s i t y than t h e EHD observed  i n intrin-  s i c s i l i c o n , i n f a c t t h e r e s u l t s o f the p r e v i o u s c h a p t e r suggest must.  that i t  In a s i m p l e p i c t u r e , t h e b i n d i n g energy o f a h o l e condensate s h o u l d  decrease  i f t h e d e n s i t y changes s i g n i f i c a n t l y from n, = 3 x 1 0 n.  however t h i s p i c t u r e does n o t take i n t o account donors.  This  1 8  the e f f e c t of Ionized  These may r e s u l t i n a s h i f t o f the e q u i l i b r i u m t o lower  w h i l e m a i n t a i n i n g the b i n d i n g energy.  cm~^  density  - 78 -  CHAPTER 8  CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY  • I 8.1  Conclusions The p h o t o l u m i n e s c e n c e  o f phosphorus  doped s i l i c o n has been  s t u d i e d f o r 2K < T < 125K over t h e c o n c e n t r a t i o n range 9 x 1 0 N„ < 4.3 x 1 0  1 9  cm  .  cence o f t h e samples s t u d i e s on the e f f e c t  1 5  cm  -  -3  S i m i l a r s t u d i e s were made o f t h e p h o t o l u m i n e s -  a f t e r heat treatment a t 1150C and c o r r e l a t e d w i t h o f t h i s heat treatment on the e l e c t r o n  paramagnetic  r e s o n a n c e o f S i : P i n t h i s c o n c e n t r a t i o n range. On the b a s i s o f t h i s work i t i s c o n c l u d e d t h a t e l e c t r o n h o l e (39) d r o p l e t s , as o b s e r v e d i n i n t r i n s i c s i l i c o n phosphorus  , can be c r e a t e d i n  doped s i l i c o n w i t h i m p u r i t y c o n c e n t r a t i o n N^ < 2.2 x 1 0  A l s o t h e EHD r e t a i n s t h e same e q u i l i b r i u m d e n s i t y o f n = 3.0 x 1 0 e l e c t r o n s and h o l e s throughout t h i s range.  cm .  1 8  - 3  cm  1 8  -3  The t h r e s h o l d energy and  d e t a i l e d l i n e shape o f the p h o t o l u m i n e s c e n t peaks a s s o c i a t e d w i t h t h e EHD a r e m o d i f i e d by changes i n the c o n d u c t i o n band d e n s i t y o f s t a t e s as the m a t e r i a l approaches 10  1 8  cm"  3  cm" . 3  t h e semiconductor-metal  transition at  - 3.0 x  When t h e i m p u r i t y c o n c e n t r a t i o n i s g r e a t e r than 2.2 x 1 0  1 8  i t i s p h y s i c a l l y u n r e a l i s t i c f o r an e l e c t r o n condensate t o o c c u r .  There i s e v i d e n c e , however, t h a t a h o l e condensate  can be formed.  The  e q u i l i b r i u m c o n c e n t r a t i o n o f t h i s h o l e condensate d e c r e a s e s w i t h i n c r e a s i n g i m p u r i t y c o n c e n t r a t i o n such t h a t f o r N^ > 4.3 x 1 0 l o n g e r makes a s i g n i f i c a n t  1 9  cm  -3  c o n t r i b u t i o n t o the l u m i n e s c e n c e .  i t no  - 79 -  The  c r i t i c a l temperature f o r " e v a p o r a t i o n " o f t h e > 3.6 x 1 0 1 7  phase i s found t o i n c r e a s e to 1 « 45K f o r compared t o T = 15K i n i n t r i n s i c  silicon.  condensed  cm" 3 ,  as  This increase i s interpreted  as b e i n g a r e s u l t o f the s c r e e n i n g o f e x c i t o n s and the replacement of the  EHD-exciton e q u i l i b r i u m w i t h an EHD-free  electron-free hole e q u i l i b -  rium. Any  c o n c l u s i o n s drawn f o r  > 2.2  x 1018  cm - 3  must be  con-  s i d e r e d as t e n t a t i v e pending more a c c u r a t e knowledge -of t h e c o n c e n t r a t i o n dependence o f the c o n d u c t i o n band d e n s i t y of s t a t e s i n t h i s tration  concen-  range. Heat treatment o f phosphorus  doped s i l i c o n a t 1150C  f o r 30  minutes i s e x p l a i n e d i n terms of n e u t r a l e l e c t r o n t r a p s w i t h a range of e n e r g i e s c l o s e to 47 meV  below the i n t r i n s i c  c o n d u c t i o n band.  These  t r a p s a r e a s s o c i a t e d w i t h i n t e r s t i t i a l carbon b e i n g e j e c t e d from i t s substitutional sites. silicon  The  carbon i s p r e s e n t as a n e u t r a l i m p u r i t y i n  c r y s t a l s u n l e s s s p e c i a l p r e c a u t i o n s are taken to remove i t .  The  i n t e r s t i t i a l carbon i s q u i t e m o b i l e at 300K, d i f f u s i n g r e a d i l y to v a c a n c i e s on the c r y s t a l s u r f a c e w i t h i n s e v e r a l d a y s .  8.2  S u g g e s t i o n s f o r F u r t h e r Work A l t h o u g h d e t a i l e d i n f o r m a t i o n on the c o n c e n t r a t i o n dependence  of  the s i l i c o n  c o n d u c t i o n band would be v e r y u s e f u l , t h e r e a r e s e v e r a l  experiments which may  h e l p c l a r i f y the p r e s e n t work w i t h o u t t h i s  infor-  mation. L i g h t s c a t t e r i n g e x p e r i m e n t s , s i m i l a r t o t h o s e done i n i n t r i n s i c (3) germanium  , would c o n f i r m o r r e f u t e the e x i s t e n c e o f m a c r o s c o p i c EHD f o r  - 80 -  <. 1.1 x 1Q  1B  cm" . 3  The proposal of the existence of a condensed  hole phase could also be tested By this technique. A pulsed laser could be used to measure the lifetime of the condensed phase as a function of both concentration and temperature. This may prove to be the only way to settle the question of evaporation of the condensed phase, since i t should be possible to distinguish between different recombination mechanisms. Two other experiments which may be interesting to consider are the replacement of donor impurities with acceptor impurities to check the possible formation of an electron condensate, and replacement of phosphorus with a donor of different ionization energy.  I f the donor  ground state i s different from that of phosphorus then the photoluminescence and electron paramagnetic resonance lines produced by heat treatment should have a different temperature dependence. This information could then be used to gain more insight into the nature of the traps produced.  - 81 -  APPENDIX A  The an i n t e g r a l of  o u t p u t of any s p e c t r o m e t e r which u s e s a monochromator as  p a r t i s o f n e c e s s i t y broadened  the i n s t r u m e n t .  necessary that t h i s The  band w i d t h  In o r d e r to study l i n e s h a p e s o r w i d t h s i t i s b r o a d e n i n g be removed.  observed  spectra with a s l i t  by the e f f e c t i v e  s p e c t r a are a c t u a l l y a c o n v o l u t i o n o f the t r u e  function.  The  slit  f u n c t i o n f o r the monochromators  used f o r t h i s work a r e g a u s s i a n and g i v e n by:  (E-E  F  s  the o b s e r v e d  narrower  - i -  r- a  o  exp  *  or i s the w i d t h a t h a l f maximum o f the s l i t p r o f i l e .  where 2flnl is  CE - E ) = o  This  l i n e shape when i t i s known t h a t the t r u e l i n e i s much  than the observed l i n e .  The observed p r o f i l e L  (hv) i s g i v e n  by:  L (hv)  F ( E - hv)  p  s  where hv i s t h e photon A fairly Savitzky^^.  L (E)dE T  energy and L^CE)  i s the t r u e l i n e  profile.  simple t e c h n i q u e f o r o b t a i n i n g L^,(E) i s g i v e n by  T h i s t e c h n i q u e i s a p p l i c a b l e i n cases where the  w i d t h i s more t h a n t w i c e the s p e c t r a l w i d t h cr, and employs the procedure  line iterative  - 82 -  L (hv) « L (hv) + L (hv) n n«~l c  F (E - hv)L ,(E)dE s n-l  LpChv) i s used as the f i r s t t r i a l function L (hv). Q  The following i s a sample computer program written i n FORTRAN G which converts the spectrometer wave drive readings to photon energies, smooths the data with a nine point smoothing routine, and extracts the true line profile.  The inputs required are:  WD - the starting value of the wave drive STPSZ - the wavedrive interval between successive data points N - the number of data points SLTW - the effective s l i t width given in eV LF(I) - the data to be analysed. The constants A, B, C, and D are obtained from a calibration of the particular grating used.  On output the true profile i s contained i n  OLDOM(I) and the corresponding photon energies in HV(I).  - 83 -  REAL LF(300),HV CiOO) jNOMEGA(30 READ (S,l,F.NP=i?) WD,STPSZ,N FORMA T (2F10,0,13) Re*n—«T2->-3tT-* FORMAT (F10.0) READ (5,3) ( L F ( I ) , I = l , N ) FORMAT M6F5,1 ) SMOOTH OAT A M=N-b  7 1 2 3 C  po  io  I=O,H  0),OLDOMC500},F(300,300),SPECTR(300) 1 '• ! • —  —  — :  — :  :—  L F C I ) = ( - 3 6 , * ( L F ( I - 5 ) * L K I + 5 ) )+9, * (IF (I •»«)+LF (I •<!) ) , * U F ( I ^3) t*LFCI*3)W69,*(LF(I»2)*LF(1+2))+84,*(LF(I-I)+LF(1*1))*B9,* 2LF(I>)/429 CONTINUE CONVtRTSPfcCTRUM TO PHOTON ENfcHGIES P s 10,/b A = 0,23620 S 3 0.783S2E-02 e—Qr,-u±jaiE-Qb 00 20 1=1,N THETA s A+B*W0*C*W0*N0 H V ( l ) 8 l g 3 9 , a 5 a i / t . ' , y O / 3 I N ( T H C T A ) / l ,0E»03 mo = wO*STPSZ OLOOM(I) = L F ( I ) CONTINUE '• < CALCULATE SLIT FUNCTION F SIGMAsSLTw/l,665 00 30 1»1,N 00 30 J=1,N F C I r J ) = ,b6il2/SIGMA*EXP(-(CHV(J)-HV(I))/SIGMA)**2) CONT I NUfc ; PERFORM CONVOLUTION AND ITERATE NOMEGA(1 ) = L F ( 1 ) 0LD1FF a ^ E + gb 00 40 M=l,10 DIFF s o , t  10 C  %  59 C :  -JO— C  :  DO so—I »2 rN  SUMso, OLOVAL = F ( I , 1 ) » 0 L D 0 M ( 1 ) 00 60 J=2,N VAL = F(I,J)»OLDOHtJi '• : — ~ SUM=SUM*,5*(VAL*0LDVAL)*(HV(J)-.HV(Jwl)) OLOVAL = VAL CONTINUE NOMEGA(I)=OLOOM(I)+LF(I)*SUH DIFF=DIFF*(SUM-.LF (I ) )*(SUM-PLF ( I ) ) CONTIfWE : : IF (OLOIFF ,LT, OIFF) GO TO 11 ' •_, PRINT 8, M, DIFF FORMAT—(-•—AT- ITER A T ION-«+rI 3 r« SOU ARE—OF—DIFFERENCES - S U M M E D — — 1 F15.3) OLOIFF a OIFF DO 70 1 = 1,N OLDOM(I)sNOMEGAU) CONTINUE CONTINUE:  60 -SO —9  70 40 — «  - 84 -  APPENDIX K  The l i n e p r o f i l e expected f o r r a d i a t i v e recombination between two bands can be calculated i n a straightforward manner. of photoluminescence  The  intensity  I (hv) i s given by: i-i  ILChv)  N (E ) N ( E ) f ( E ) f ( E ) 5(hv-E - E - E ) d E d E e  0  h  e  h  e  h  e  h  0  where A i s a constant depending on the matrix elements connecting the states and i s assumed to be independent N(E ) g  and NCE^)  a r e  of energy.  the density of states f o r the conduction  band and valence band respectively and f o r parabolic bands are given by:  N(E) =  2m  *V3/2  •1/2  f ( E ) and f(E^) are the electron and hole p r o b a b i l i t y d i s t r i b u t i o n e  functions r e s p e c t i v e l y . The Fermi energy f o r a parabolic band i s r e a d i l y obtained :  from N(E):  2m  s u b s t i t u t i n g i n the appropriate values f o r s i l i c o n , allowing for the s i x conduction band minima and doubly degenerate valence band one obtains  - 85 -  I* -  1.14  E** = 0.697 Jc  I0~  x  lh  (|)  x lO-  2/3  eV  (n) /  1 4  2  eV  3  f o r e l e c t r o n s and h o l e s , where n i s the c o n c e n t r a t i o n  o f e l e c t r o n s and  holes. At f i n i t e temperatures t h e e l e c t r o n and h o l e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s a r e g i v e n by:  fCE) exp  where cfT) - E  p  (E - )'/kT ?  - j- ( k T )  9  2  3E  i  l n  + 1  r  \  N(E) E = E,  f o r p a r a b o l i c bands N(E) a E  •  r  m  - F  —  C  K  1/2  T  )  2  For an e l e c t r o n - h o l e drop w i t h b i n d i n g energy EA the i n t e n s i t y o f p h o t o l u m i n e s c e n c e i s g i v e n by OO  0  QO  f  E1/2E1/2  I (hv) d  exp 0  ^ e ' ^ / k T +1  0  x S(hv-E -E -E +EA+E^+E^+tUo)dE dE g e h F F ' e n L  exp  ° W / k T  +1  - 86  -  where hco i s the energy of the phonon a s s i s t i n g the r e c o m b i n a t i o n . I n t e g r a l can be  s o l v e d q u i t e r e a d i l y -using n u m e r i c a l  techniques.  f o l l o w i n g i s a computer program w r i t t e n i n FORTRAN G t o s o l v e i n t e g r a l and  p l o t the l i n e p r o f i l e .  hu = .0578 eV  i s used.  The  This The  this  In t h i s program a phonon energy  program i n p u t r e q u i r e d i s  T - temperature f o r which the l i n e shape i s r e q u i r e d EGAP - band  gap  ESTP - the number of e l e c t r o n and  hole energies  N - the number of p o i n t s i n the output EA CON  The  used  profile  - the b i n d i n g energy i n eV - e q u i l i b r i u m concentration of  l i n e p r o f i l e i s s c a l e d to 5 inches  energies  to be  a r e p r i n t e d on  EHD.  and  the e l e c t r o n and  output.  I  hole  Fermi  - 87 -  IMEGEW ESTP REAL K , K T » t E L ( 2 0 0 ) » F D E ( 2 0 0 ) » H V ( 5 0 0 ) # I N T ( S O O ) R E A D C S . l ) T i EGAP RE-AO-fS ^ - K W T N  :  READ(5,2,EN0=8) EA.COM FO*MAT(2F10.0) FOHMAT(F10,0,E7,1) — : FORHATC2I3) CALCULATE FERMI ENERGIES E-F-E-s-l-r-Hli-4*4C-0N At>-r-)-«*42« / 3 « ) • HFEc0.6O7E-lO*C0N**(2,/3.) PRINT U , E F E , M F E FQk'MAT ( i ' , T 1 0 . IEFE • , T 2 5 , > H F E ' V / ( 2 F 1 5 . 5 ) J CALCULATE CHEMICAL POTENTIALS K=8.62E-05 K^f-K*4 :— FAC = V.8h96>/12.E-0«*KT*KT CPE = EFE-FAC*ALOG( CEFE + 1 . OE-OU)/EFE) CPM = HFfc-FAC*ALOG( ( H F t * l ,0E-0tt)/HFE) PRINT b, CHE,CPH FORMAT( I » T 1 0i CPE #T25»'CPH'//(2F15.5)) fcU4^=EFE*S.«KT HUL=HFE*S.**T ESS=EUL/(ESTP-1) : — E E L U ) = 0.0 : — 00 10 I = 2 » E S T P  9 1 —g 3 C  —H C  5  1  ,  ,  :  EELCI)=EELCI-n+ESS  CONTINUE — CALCULATE FERMI OIRAC FUNCTION DO 20 1 = 1 , t S T P E00M=(EELCI ) - C P E ) / K T . : '. FDECI)=1.0/CEXP(tDOMj*I.O) CONTINUE PBOBsFDE (ESTP) PRINT 6, EULfPKOB F 0RMAT ( ' •,T10,'EUL',T25, PROR'//(2F15,5n _eALCULATE-HHaiOLU.MlNESCE^CE-IHTENSI-IY HV(1 ) = E G A P - 0 . 0 b 7 c i - E A - E F E - H F E DELTA=(EUL*HUL)/(N-U  W C _ 20  6 C  ,  o o 30 i=2v» 30  ,  UO  21  :  HV (I ) = H V ( I - l ) + D E L T A CONTINUE  D o-jw-j=I,N  :  :  :  INT(J)=0.0 OLDVAL=0,0 00 UP I=2,ESTP EH=HV(J)-HV(l)-EELCn IF CEH.LT , 0 . 0 ) fcNsO.O HDOMs.tE.H-CeM.J-/K.T. 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