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The effect of strain on the exciton spectrum of germanium Glass, Alastair Malcolm 1964

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THE EFFECT OF STRAIN ON THE EXCITON SPECTRUM OF GERMANIUM by ALASTAIR MALCOLM GLASS B.Sc.(special)  The U n i v e r s i t y of London,England. 1961  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the department of physics  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1964  In the  r e q u i r e m e n t s f o r an  British  mission  for reference  for extensive  p u r p o s e s may  be  advanced  of  and  written  Department  of  for  JO  the  I further  Head o f my  ' tuple?  /h*g^  Columbia,  agree for  that  of •  per-  scholarly or  c o p y i n g or  shall  of  make i t f r e e l y  Department  that  f i n a n c i a l gain  fulfilment  University  shall  this thesis  permission*  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date  study,  the  Library  I t i s understood  this thesis  w i t h o u t my  by  in partial  degree at  the  copying of  granted  representatives.  cation  this thesis  Columbia, I agree that  available  his  presenting  not  be  by publi-  allowed  The U n i v e r s i t y  of B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE  DEGREE OF  DOCTOR OF PHILOSOPHY  of  ALASTAIR MALCOLM GLASS  B.Sc.  (Special),  University  MONDAY, AUGUST 10,  of London, 1961  1964, a t 9:30 A.M.  IN ROOM 301, HENNINGS BUILDING (PHYSICS)  COMMITTEE IN CHARGE Chairman: I . McT. Cowan R. B a r r i e J.W. B i c h a r d L.C. Brown External  F.W. Dalby K.B. Harvey H. Schmidt Examiner:  W.H. K l e i n e r  L i n c o l n Laboratory Massachusetts I n s t i t u t e o f Technology.^  THE EFFECT OF STRAIN ON THE EXCITON SPECTRUM OF GERMANIUM A B S T R A C T  The i n t e n s i t y  of o p t i c a l a b s o r p t i o n , near the  a b s o r p t i o n edge i n germanium,  i s examined as a func-  t i o n of s t r a i n a p p l i e d t o the l a t t i c e at 9 0 " K . ,The r e s u l t s a r e i n t e r p r e t e d i n terms of t h e change of the band s t r u c t u r e of the l a t t i c e w i t h a p p l i e d  strain.  The a b s o r p t i o n edge i n u n s t r a i n e d germanium shows a s i n g l e sharp peak due t o e x c i t o n f o r m a t i o n , whereas the edge i n the s t r a i n e d peaks.  specimens shows two e x c i t o n  The peak p o s i t i o n s v a r y l i n e a r l y w i t h b o t h  compressional and t e n s i o n a l s t r a i n up t o the maximum strains applied  (0.1% d e f o r m a t i o n ) .  The e x c i t o n  b i n d i n g e n e r g i e s i n the s t r a i n e d germanium  lattice,  c a l c u l a t e d w i t h i n the e f f e c t i v e mass approximation, are a p p r o x i m a t e l y equal and independent tude of the s t r a i n .  of the magni-  The s e p a r a t i o n of the peaks  g i v e s f o r the shear d e f o r m a t i o n p o t e n t i a l s , |b| = (2.7 t (4.7 t  0.3)eV/unit <100)>shear and  0 . 5 ) e V / u n i t < 1 1 1 ^ shear.  peaks g i v e the s h i f t eV/unit  the v a l u e s  |d| =  The p o s i t i o n s of the  of the energy gap as - (10.3 1" 1)  dilatation.  L a t t i c e i m p e r f e c t i o n s a r e found t o have no e f f e c t on the i n t e r p r e t a t i o n of t h e ' - r e s u l t s . a b s o r p t i o n edge observed manium i s accounted  The broad  i n evaporated f i l m s of ger-  f o r i n terms of i n t e r n a l  strains.  GRADUATE STUDIES  Field  of Study:  Physics  Elementary Quantum Mechanics  F.A. Kaempffer  Waves  R.W.  Stewart  E l e c t r o m a g n e t i c Theory  G.M.  Volkoff  Quantum Theory o f S o l i d s Noise i n P h y s i c a l Advanced  Related  Systems  Magnetism  R. B a r r i e R.E. Burgess M. Bloom  Studies:  A p p l i e d E l e c t r o m a g n e t i c Theory  G. Walker  S t r u c t u r e of M e t a l s  E. T e g h t s o o n i a n  Quantum Chemistry  R, H o c h s t r a s s e r  Spectroscopy and M o l e c u l a r Structure  K.B.: Harvey A.V. Bree L.W. Reeves  ABSTRACT The i n t e n s i t y o f o p t i c a l a b s o r p t i o n , n e a r the a b s o r p t i o n edge i n germanium,  i s examined as a f u n c t i o n o f s t r a i n a p p l i e d  to the l a t t i c e a t 90°K. The r e s u l t s are i n t e r p r e t e d i n terms of the change o f the band s t r u c t u r e o f the l a t t i c e w i t h a p p l i e d s t r a i n . The a b s o r p t i o n edge i n u n s t r a i n e d germanium shows a s i n g l e sharp peak due to e x c i t o n formation, whereas the edge i n the s t r a i n e d specimens shows two e x c i t o n peaks. The peak p o s i t i o n s vary l i n e a r l y w i t h both compressional  and t e n s i o n a l  s t r a i n up to the maximum s t r a i n s a p p l i e d ( 0 . 1 % deformation). The e x c i t o n b i n d i n g e n e r g i e s i n the s t r a i n e d germanium  lattice,  c a l c u l a t e d w i t h i n the e f f e c t i v e mass approximation, are approximately  equal and independent  o f the magnitude o f the  s t r a i n . The s e p a r a t i o n o f the peaks g i v e s f o r theshear deformation potentials, and  the values  I b| = ( 2. 7 ± 0. 3) eV/unit <100> shear  |d| = (4. 7 ± 0. 5) eV/unit (111) shear. The p o s i t i o n s o f the  peaks g i v e the s h i f t o f the energy gap a s — ( 10. 3 ± 1) e V / u n i t dilatation. L a t t i c e i m p e r f e c t i o n s are found to have no e f f e c t on the i n t e r p r e t a t i o n o f the r e s u l t s . The broad a b s o r p t i o n edge observed i n evaporated terms o f i n t e r n a l  f i l m s o f germanium i s accounted  strains.  fori n  iii  TABLE OF CONTENTS page no. CHAPTER I .  Introduction  CHAPTER I I .  Experimental  1 methods and r e s u l t s  4  A.  C r y s t a l growth  4  B.  Specimen p r e p a r a t i o n  6  C.  The monochromator and d e t e c t o r system  10  D.  Experimental  procedure  13  E.  R e s u l t s o f a b s o r p t i o n measurements  18  F.  Test f o r p l a s t i c  21  CHAPTER I I I .  flow  T h e o r e t i c a l background and a n a l y s i s of r e s u l t s  ;  23  A.  The band s t r u c t u r e o f u n s t r a i n e d germanium  24 25  B.  The band s t r u c t u r e o f s t r a i n e d germanium D e s c r i p t i o n o f the e x c i t o n s t a t e i n the e f f e c t i v e mass approximation f o r the s t r a i n e d and u n s t r a i n e d l a t t i c e  C.  30  The i n t e n s i t y o f o p t i c a l a b s o r p t i o n near the a b s o r p t i o n edge i n s t r a i n e d and u n s t r a i n e d germanium.  36  D.  A n a l y s i s o f experimental  E.  Discussion of results  41  The e f f e c t o f l a t t i c e d e f e c t s on the o p t i c a l a b s o r p t i o n edge.  43  A.  Phonons  43  B.  I m p u r i t i e s and vacancies  45  C.  Surface  46  D.  Dislocations  47  E.  Evaporated f i l m s o f germanium  53  F.  Discussion  55  CHAPTER IV.  results  states  38  IV  TABLE OF CONTENTS (continued) . CHAPTER V.  page no.  Conclusions  56  APPENDIX A. C a l c u l a t i o n o f the s t r a i n s on the specimens  57  APPENDIX B. T r a n s f o r m a t i o n o f e l a s t i c constants under r o t a t i o n o f axes  62  BIBLIOGRAPHY.  64 LIST OF TABLES  Table Table  I.  The expansion c o e f f i c i e n t s o f germanium and v a r i o u s s u b s t r a t e s .  I I . The p o s i t i o n s o f the a b s o r p t i o n peaks i n the a b s o r p t i o n edge o f germanium a t  Table I I I .  The e f f e c t i v e h o l e masses a t k=0 s t r a i n e d germanium.  15  90°K. 22  in 29  Table  IV.  The deformation p o t e n t i a l s o f germanium  40  Table  V.  Comparison between theory and experiment of the temperature s h i f t o f the a b s o r p t i o n edge o f germanium  44  <  V  LIST OF FIGURES to f o l l o w page Figure 1. P l a n view of the Ebert monochromator.  10  Figure 2. P l a n view of the f o r e - d i s p e r s i n g o p t i c s  10  Figure 3.  I n t e n s i t y d i s t r i b u t i o n of r a d i a t i o n entering the monochromator f o r a) l i n e source  b) extended source s l i t  p. 11  Figure 4., Block diagram of the detector e l e c t r o n i c s Figure 5.  The nitrogen dewar and specimen holder  12 p.13  The absorption spectra of germanium Figure 6.  Unstrained at 90°K and 300°K.  (<110>)  21  Figure 7.  Mounted on #0080 glass at 90*K. (U10) )  21  Figure 8.  Mounted on 7740 glass at 90°K. «110>)  21  Figure 9.  Mounted on #7900 glass at 90 K. (<110>)  21  6  The v a r i a t i o n of e x c i t o n peak p o s i t i o n s w i t h s t r a i n Figure 10. ( I l l ) o r i e n t e d specimens  22  Figure 11. (10Q) o r i e n t e d specimens  22  Figure 12. (110) o r i e n t e d specimens  22  Figure 13. The band s t r u c t u r e of unstrained germanium  p24  Figure 14. a) The band s t r u c t u r e of s t r a i n e d germanium f o r a (100) o r i e n t e d specimen  27  b) V a r i a t i o n of hole mass w i t h wave-vector k Figure 15. The absorption spectrum of evaporated f i l m s of germanium  54  ACKNOWLEDGEMENTS I would l i k e t o thank Dr.R.Barrie and Dr.J.Bichard f o r t h e i r a s s i s t a n c e i n t h i s i n v e s t i g a t i o n and f o r t h e i r advice and c r i t i c i s m i n the p r e p a r a t i o n of t h i s t h e s i s . I would a l s o l i k e t o express my thanks to A.S.Syed f o r help w i t h c r y s t a l growth, and t o E.M.Edwards and C.Pulley f o r the p r e p a r a t i o n of the epoxy, and the design and constr u c t i o n of the a m p l i f i e r used i n t h i s work. The research f o r t h i s t h e s i s was supported by the Defence Research Board o f Canada under grant no.9510-35. I would a l s o l i k e t o express my g r a t t i t u d e t o the Commonwealth Scholarship Committee f o r the award o f a s c h o l a r ship during my course o f studies .  1 CHAPTER I :  INTRODUCTION  In-the ground s t a t e o f a semiconductor, a l l s t a t e s o f valence band are are empty. The  occupied and  a l l those o f the conduction baiid  f i r s t e x c i t e d s t a t e o f the c r y s t a l i s the  of an e l e c t r o n h o l e p a i r . The gives  the  creation  coulomb i n t e r a c t i o n of the  pair  r i s e to bound s t a t e s of the e l e c t r o n and h o l e , w i t h energies  below the c o n d u c t i o n band ( r e f e r e n c e  1 ).These s t a t e s are c a l l e d  exciton  p a i r i s often  s t a t e s and  the e l e c t r o n - h o l e  r e f e r r e d to as an e x c i t o n . A band and  a f r e e h o l e i n the  loosely  free electron i n t h e c o n d u c t i o n valence band, may  thus be  considered  as an i o n i s e d s t a t e o f the e x c i t o n . L i g h t i s absorbed by the c r y s t a l i n t o e x c i t o n  exciting  states.  D i r e c t t r a n s i t i o n s , i n v o l v i n g the i n t e r a c t i o n o f  electrons  w i t h photons only, must s a t i s f y the momentum c o n s e r v a t i o n k  + g  kh  =  0  vectors),  ( where the k  e  , k  h  are  rule  the e l e c t r o n and h o l e wave  s i n c e the photon momentum i s n e g l i g i b l e . Thus t r a n s i t i o n s  to bound e x c i t o n  states give  o f continuous a b s o r p t i o n  r i s e to a l i n e spectrum. The  occurs when the photon energy i s equal  to the "forbidden"energy gap  between the  valence and  bands. S t u d i e s o f t h i s a b s o r p t i o n  edge have y i e l d e d  information  of several  on  the  onset  band s t r u c t u r e  In the case of germanium, the  conduction useful  semiconductors.  valence band maximum i s  f o l d degenerate ( i n c l u d i n g s p i n ) a t k= 0 o f the B r i l l o u i n The  fourzone.  conduction band minima are at the B r i l l o u i n zone boundaries,  w i t h an a u x i l i a r y minimum a t k= 0 ( r e f e r e n c e of e x c i t o n s  of lowest energy are  forbidden  2).Thus the  by momentum  a t i o n s u n l e s s a phonon i s s i m u l t a n e o u s l y c r e a t e d (reference  3).  Macfarlane et a l ( r e f e r e n c e  absorption  o f l i g h t due  4)  or  creation consider-  annihilated  have shown t h a t  to these i n d i r e c t t r a n s i t i o n s appears  2  appears only as a long-wavelength t a i l t o the absorption edge, since these t r a n s i t i o n s are f a r l e s s probable than the d i r e c t transitions at k=0. Using high r e s o l u t i o n and t h i n , f r e e l y hanging specimens of high p u r i t y germanium, Macfarlane  et a l (reference 5)  observed a sharp absorption peak i n the edge a t high absorption c o e f f i c i e n t s , a t temperatures below 196°K. They i n t e r p r e t e d t h i s as the lowest energy l i n e i n the l i n e spectrum due t o e x c i t o n formation. However, two peaks were observed i n the spectrum by Zwerdling,  et a l i a  (reference 6) using s i m i l a r  specimens, glued to a glass substrate. The absorption edge was a l s o s h i f t e d to s h o r t e r wavelengths. Macfarlane e t a l l a t e r showed that the d i f f e r e n c e i n the two spectra was because the specimens of Zwerdling e t a l . were s t r a i n e d by the substrate on c o o l i n g (reference 7) . This l e d K l e i n e r and Roth (reference 8) t o i n v e s t i g a t e the e f f e c t of s t r a i n on the band s t r u c t u r e of germanium from a t h e o r e t i c a l p o i n t of view. They found that when the cubic symmetry of the l a t t i c e i s removed by a homogenious s t r a i n , the valence band degeneracy a t k= 0 i s removed. There i s a l s o a change i n the energy gap corresponding  to the change i n the l a t t i c e parameters.  They then i n t e r p r e t e d the two peaks observed by Zwerdling e t a l (and l a t e r by Macfarlane)  as the e x c i t o n spectra a s s o c i a t e d w i t h  the two p a i r s of bands. Owing t o the l a c k of data, K l e i n e r and Roth were unable to evaluate the deformation p o t e n t i a l s of the germanium l a t t i c e e x p l i c i t l y , from t h e i r theory. A more complete theory on the e f f e c t of s t r a i n o n the hole energy spectrum of germanium has been given by picus and B i r (reference 9) .  3  Deformation  p o t e n t i a l s are important parameters i n a l l  t h e o r i e s a s s o c i a t e d w i t h l a t t i c e deformation. They were f i r s t i n t r o d u c e d by Bardeen and Shockley who of  used them i n the theory  e l e c t r o n and h o l e s c a t t e r i n g . ( r e f e r e n c e 10). They are a l s o  important i n the theory of the e f f e c t of i n t e r n a l and  externally  a p p l i e d s t r a i n s on the band s t r u c t u r e of a c r y s t a l . Thus i t i s d e s i r a b l e to have accurate values of these S t u d y i n g the e x c i t o n spectrum  parameters.  as a f u n c t i o n of s t r a i n  p r o v i d e s a check of the agreement between experiment theory o f K l e i n e r and Roth, and i s a p a r t i c u l a r l y  and  the  convenient  method o f determining the deformation p o t e n t i a l s of the pure germanium l a t t i c e e x p l i c i t l y . I n t h i s i n v e s t i g a t i o n the is  s t u d i e d as a f u n c t i o n of both compressional  s t r a i n . The  spectrum  and t e n s i o n a l  e x c i t o n b i n d i n g energies i n germanium are c a l c u l a t e d  w i t h i n the e f f e c t i v e mass approximation,  u s i n g the formalism o f  Dresselhaus ( r e f e r e n c e 2) i n o r d e r t h a t the p o s i t i o n s o f the band edges may of  be o b t a i n e d from the experimental  the p o s s i b i l i t y t h a t t h i s i n f o r m a t i o n may  d i r e c t l y from the experimental ( r e f e r e n c e 11)  data. I n view  be o b t a i n e d  data, the theory o f E l l i o t t  on the o p t i c a l a b s o r p t i o n near the edge, f o r two  simple s p h e r i c a l bands, was  extended  to the case o f germanium,  where the bands have a more c o m p l i c a t e d shape. Finally,  the e f f e c t o f l a t t i c e i m p e r f e c t i o n s on the  absorp-  t i o n edge of the germanium specimens i s s t u d i e d i n some d e t a i l , in  order to ensure  of  the p e r f e c t germanium l a t t i c e . The  evaporated  t h a t the observed  e f f e c t s were p r o p e r t i e s a b s o r p t i o n spectrum  of  f i l m s of i n t r i n s i c germanium i n the wavelength range -4  0.5yw to l.O^t discussion.  (y*=10  cm)  i s accounted  f o r i n the l i g h t o f t h i s  4 CHAPTER I I  :  EXPERIMENTAL  METHODS AND PROCEDURE.  Experimentally, observing the e f f e c t o f s t r a i n on the e x c i t o n spectrum o f germanium, i n v o l v e d the f o l l o w i n g problems: a) i n order to minimise the e f f e c t o f l a t t i c e i m p e r f e c t i o n s , s i n g l e c r y s t a l s of high p u r i t y and low d i s l o c a t i o n d e n s i t y , had to be prepared. Since the e l a s t i c moduli o f germanium depend on the choice o f c r y s t a l axes, the c r y s t a l o r i e n t a t i o n had t o be known. b)  The f i n e s t r u c t u r e due t o d i r e c t e x c i t o n absorption  at high absorption c o e f f i c i e n t s Thus i t was necessary  occurs  ( ~ 3500 cm" ) , ( r e f e r e n c e 5). 1  to prepare t h i n wafers o f germanium from  these s i n g l e c r y s t a l s . c)  For the examination  of the e x c i t o n spectrum o f both s t r a i n e d  and u n s t r a i n e d germanium specimens, a high r e s o l u t i o n raonochromator was r e q u i r e d , w i t h an arrangement t o c o o l the specimens to about 77°K. (reference 5). A. CRYSTAL GROWTH. C r y s t a l s of pure germanium were p u l l e d from the melt by the C z o c h r a l s k i tecnique: Zone r e f i n e d germanium of high p u r i t y was melted i n a graphite i n a graphite c r u c i b l e by r e s i s t a n c e heating and germanium seed c r y s t a l s o f (100) , (110) , ( i l l ) , o r i e n t a t i o n s were used t o p u l l the c r y s t a l s from the melt. The c r y s t a l growth was performed i n an argon atmosphere t o avoid contamination o f the germanium by r e a c t i v e m a t e r i a l s . The germanium ingots were then s l i c e d i n t o d i s c s , o f about 1mm. i n t h i c k n e s s , w i t h a diamond impregnated saw, so that the plane o f the d i s c s was perpendicular t o the growth a x i s . PURITY : The p u r i t y o f the d i s c s was checked by measuring t h e i r  5 r e s i s t a n c e s at 23°C w i t h a four-probe machine. The r e s i s t i v i t y of a l l d i s c s was found to be about 60 ohm-cm,(except at the two extreme ends of the ingot,which were discarded) , i n d i c a t i n g that the i m p u r i t y content was l e s s than 10 i m p u r i t i e s / c c . TWINNING : A check was made f o r twinning of the c r y s t a l during growth, by b o i l i n g f r e s h l y lapped d i s c s i n a 30% hydrogen peroxide etching s o l u t i o n . Since there was no change i n the r e f l e c t i v i t y of the d i s c s across t h e i r surfaces, they were assumed t o be s i n g l e c r y s t a l . DISLOCATION DENSITY : The method of Pfann (reference 12) was used to determine the d i s l o c a t i o n density. The germanium d i s c s were lapped w i t h 800 mesh (#800) Alumina and etched f o r about 90 seconds i n f a s t CP4 (25cc n i t r i c a c i d , 15cc h y d r o f l u o r i c a c i d , 15cc a c e t i c acid) . The etch p i t s were c l e a r l y v i s i b l e under a m a g n i f i c a t i o n of 100 x , f o r the d i s c s i n the [100] and [ i l l ] planes only. I f i t i s assumed that the etch p i t s are l o c a t e d at the i n t e r s e c t i o n of the d i s l o c a t i o n l i n e s w i t h the c r y s t a l surface (reference 12), then the etch p i t counts i n d i c a t e d d i s l o c a t i o n d e n s i t i e s of about 5000 l i n e s /cm;* f o r both c r y s t a l s . CRYSTAL ORIENTATION : The c r y s t a l o r i e n t a t i o n was c l e a r l y i n d i c a t e d by the symmetry of the ingots about the growth a x i s ; however an independent check was afforded by examining the shapes of the etch p i t s and comparing them w i t h those described by E l l i s (reference 13). The agreement i n d i c a t e d that the planes of the d i s c s were w i t h i n 3 degrees of the nominal planes. I n the(100) o r i e n t e d d i s c s , terraces i n the etch p i t s , corresponding of the d i s l o c a t i o n , could be seen.  t o jogs  6  B. SPECIMEN  PREPARATION.  The e x c i t o n absorption peak i n germanium occurs at a photon energy where the absorption c o e f f i c i e n t i s about 3500 cm"' (reference 5). A t higher energies the absorption c o e f f i c i e n t increases. I t i s shown l a t e r that the transmission T of a specimen of thickness d i s r e l a t e d to the absorption c o e f f i c i e n t by  Tocexp-cxd  . The optimum c o n d i t i o n f o r the observation of dT  the e x c i t o n peak i s cTw: i s a maximum at oC= 3500 cm"'. Thus the optimum specimen thickness i s about 3y* . Three p o s s i b l e methods of specimen p r e p a r a t i o n were i n v e s t i g a t e d : Chemical etching, e l e c t r o e t c h i n g , and mechanical p o l i s h i n g . The f i r s t two methods proved unsuccessful, and i t i s worthwhile  to o u t l i n e the reasons f o r f a i l u r e .  a) CHEMICAL ETCHING : The germanium d i s c s were ground to a thickness of about 50^t and etched i n slow CP4 (10 cc n i t r i c a c i d , lOcc h y d r o f l u o r i c a c i d , 20cc a c e t i c acid) . I t was found that the edges of the wafers d i s s o l v e d i n the etch very r a p i d l y , while there was only l i t t l e change i n thickness. Masking the edges w i t h epoxy d i d not r e t a r d the e f f e c t as the epoxy was undermined by the etch. Further, the p o l i s h i n g was uneven, and holes developed through the wafer. b) ELECTROETCHING The germanium wafer was used as the anode i n a KOH e l e c t r o l y t e , and platinum as the cathode. The procedure followed was that described by U h l i r (reference 14). I t was found that etching occurred,but  i t was l o c a l i s e d at the p a r t s of the wafer nearest to  the metal contact w i t h the germanium. The primary anodic  process  i n germanium i s c a r r i e d out by holes (reference 13) , so t h i s  7  r e s u l t could be accounted f o r i f the hole density were greatest near the metal contact. Hole i n j e c t i o n may take place a t a metal-germanium contact, so t h i s i s the probable explanation of the above e f f e c t . S h i n i n g l i g h t on the wafer, or i n c r e a s i n g the temperature d i d increase the anodic current density, but uneven etching s t i l l occurred. Making l a r g e area contacts or u s i n g a moving pressure contact would be t e c h n i c a l l y d i f f i c u l t f o r very t h i n wafers, so e l e c t r o e t c h i n g was abandoned, c) MECHANICAL POLISHING. This method was t r i e d l a s t o f a l l because i t has been reported (reference 14) that abrasion damage penetrates over 20^t below the c r y s t a l surface, even when a lapped surface i s subsequently h i g h l y p o l i s h e d . Photomagnetoelectric the surface recombination  measurements on  v e l o c i t y (reference 15), on the other  hand, r e v e a l that the abrasion damage extends, on the average, to a depth comparable t o the abrasive p a r t i c l e s i z e only. Specimens down t o 5yu t h i c k were s u c c e s s f u l l y prepared by mechanical p o l i s h i n g . Although  t h i s was greater than the optimum  thickness, i t was found that thicknesses up to 20yw were adequate f o r the oservation of the e x c i t o n spectrum. The sequence of abrasives and laps used was e x t e n s i v e l y studied w i t h a view to the q u a l i t y of the p o l i s h and the speed of specimen preparation. The f o l l o w i n g procedure gave best r e s u l t s : Several wafers ( u s u a l l y four) were shaped i n cross s e c t i o n so that they f i t t e d together i n a mosaic form, without any appreciable gaps between them. They were glued to a p l a t e of glass w i t h r o s i n wax,  and chips of germanium were glued around the wafers on the  glass so that a l l edges o f a wafer were i n contact with another piece of germanium. The chips prevented  the edges of the wafers  8  from rounding during polishing. The complete mosaic was ground flat with #450 Silicon Carbide (SiC) on glass ( abrasive particle size ~22yu) . Subsequent abrasives used in order were: #600 SiC on a glass plate (abrasive particle size-l^*) , 50yu removed #800 A I 2 O 3 on glass (abrasive particle size ^10^ ) , 50/A removed. The mosaic was finely ground at this stage. It was mounted in an automatic j i g fitted to a mechanical polishing wheel, and the polishing sequence was: #600 SiC on astromet cloth*(see footnote), with the wheel rotating at 600 revolutions per minute until a l l the pits due to lapping were removed and the surface was well polished. # Linde  B  Al 0a (abrasive particle size ~0.2/A ) on astromet cloth 2  with the wheel rotating at 600 revs/min. After one hour with this abrasive, a l l scratches from the previous abrasive were removed and the surface was highly polished. The polished surfaces were flat over 90% of the surface and only a very slight curvature was seen right at the wafer edges. The wafers were s t i l l at least 300/n thick. They were removed from the glass plate by melting the wax, washed in alcohol and glued to another plate with the polished surface downwards. For the specimens which were to be removed subsequently from the glass the  glue used was Dennison type A14*(see footnote). The specimens  which were not to be removed from the substrate, were glued with an epoxy, especially prepared for the purpose of obtaining thin *Astromet cloth is a Precision Scientific Instruments product. It is a durable cloth with a very short nap,for polishing flat surfaces, tDennison Type A14 was sent to me by Raytheon Co. for the purpose of obtaining very thin layers of cement.It has a very low melting point,low viscosity just above the melting point,has l i t t l e tendency to form bubbles,and is readily soluble in alcohol. It is very hard at room temperature.  9  layers of glue. The epoxy had a low viscosity and high bond strength; i t was cured for 48 hours at room temperature. A micrometer was used to measure the specimen thicknesses since the substrate thickness was known from a previous measurement. It was found from repeated experiments that the glue formed a very thin parallel layer. The wafer was lapped by hand with #600 SiC on glass until the thickness was about 100^, and then with #800 Alj.03 on glass until the thickness was about 25// . This point was the most critical in the specimen preparation and i t was essential that only a very small,even pressure be applied evenly to the wafer while lapping. The wafer thickness was constantly checked to ensure uniformity.The wafer and substrate were then mounted in the j i g and polished on astromet cloth with #600 SiC, with the wheel rotating at 300 revs/min., until a l l lapping pits were removed and the wafer was reduced to the required thickness. Since no masking of the wafer edges was used for this side of the specimen an estimate of the wafer thickness could be made, and its uniformity, by looking at the condition of the extreme edges of the wafer. Only a slight reduction in the area of the wafer occurred before i t was about 10/>i thick. A high final polish was obtained with Linde B abrasive on astromet cloth. The specimens mounted on glass with epoxy were ready for use in the monochromator at this stage, but those mounted with Dennison Type A14 were placed in alcohol until a l l the cement had dissolved. The unsupported germanium specimens were slightly flexible, though they did not curl on removal from the substrate i f both surfaces were equally well polished. The specimens were  10 washed In alcohol and placed between two sapphire windows to avoid handling them further. This method of specimen preparation proved very fast and successful. Since only the centre section of the specimens were used for optical measurements, the rounding of the edges did not matter. Interference fringes were observed in the transmission spectra of the specimens, indicating that they; were nearly uniform in thickness. The optical properties of the specimens were essentially unaffected by the polishing rate or the presence of a few pits or scratches. C. THE MONOCHROMATOR. A high resolution infra red Ebert monochromator was used in this investigation. The optical arrangement is shown in Fig 1. The entrance and exit slits, curved to minimise aberrations, are placed in the focal plane of the converging spherical mirror M|. The diffraction grating G is a Bausch and Lomb reflexion grating of 300 lines/mm. , blazed at 1.5y«in the second order. The fnumber of the system,limited by the grating, is £12.5. The whole system  is mounted inside a tank, which  can be evacuated.  The monochromator was operated at a resolution ^  of 6000  to 15000 at 1.4/4 in the second order, corresponding to mechanical s l i t widths of 0.4 and 0.18mm respectively. FORE-DISPERSING OPTICS : In order to prevent different orders of reflexion, from overlapping at the exit s l i t S , fore- dispersing optics were 2  added to the monochromator.This allowed a determined band width of radiation to enter S . To obtain only one wavelength X in the f  second order from the monochromator, the maximum bandwidth to enter S, must be 2% to •£. The Ebert system could not be used in  Fig. 1 P l a n view o f the monochromator,  showing the l i g h t p a t h .  Fig P l a n view of the  My.  M* P S3  S,  fore-dispersing  2 o p t i c s , showing the l i g h t p a t h .  s p h e r i c a l converging m i r r o r (10" f . l ) p l a n e m i r r o r at 45° to a x i s of M calcium f l u o r i d e fore-prism entrance s l i t to the f o r e - o p t i c s entrance s l i t to the main monochromator B : b a f f l e s to minimise s c a t t e r e d scattered radiation. 3  a  rt O  t-t» O  i T3 05  OP  11 the f o r e - o p t i c s because of mechanical l i m i t a t i o n s , but since the s e l e c t e d bandwidths were l a r g e , small aberrations were unimportant. The o p t i c a l system i s shown i n F i g 2. The entrance s l i t S curved so that the image of S l i n e a r w i t h S,  . M3  ?  5  is  i n the s p h e r i c a l m i r r o r M i s co3  i s about 5 degrees o f f a x i s . The system has  u n i t m a g n i f i c a t i o n so S  3  and S, were both i n the f o c a l plane o f  M . The f o r e - d i s p e r s i n g prism i s calcium f l u o r i d e , aluminised on 2  one face. The d i s p e r s i o n and r e f r a c t i v e index of CaF  2  are  r e s p e c t i v e l y 0.005 and 1.42 a t about 1.4yu , g i v i n g a l i n e a r d i s p e r s i o n at S| of O.^cm^u. . The bandwidth o f r a d i a t i o n could be v a r i e d by changing the s l i t width of S  3  . I f S was very narrow 3  - e f f e c t i v e l y a l i n e source - only those wavelengths  r-04-l S  A  mm  X  f o r which the image of S3  f e l l d i r e c t l y on S| entered the main monochromator. The expected i n t e n s i t y A  d i s t r i b u t i o n of r a d i a t i o n passing through S, i s i l l u s t r a t e d i n F i g 3a. This was the minimum band width f o r a given s l i t width of S| . For S, « 0.4mm, the bandwidth  A^=0.2/n. Using an extended  source s l i t S3, images of S wavelengths  3  for different  overlapped at S| so that the bandwidth o f r a d i a t i o n  entering S, was increased. The i n t e n s i t y d i s t r i b u t i o n f o r t h i s case i s shown i n F i g 3b. I f Sj s 0.7mm and S, = 0.4mm then A\ 0.6//. s  The p r e d i c t e d behavior o f the f o r e - o p t i c s was borne out i n p r a c t i c e . This e l i m i n a t e d the n e c e s s i t y o f f i l t e r s i n the system, and maximum i n t e n s i t y was maintained and s c a t t e r e d r a d i a t i o n reaching the e x i t s l i t S^ was kept to a minimum. The f o r e - o p t i c s  12 chamber was  evacuated w i t h the main monochromator, and the f o r e -  d i p e r s i n g p r i s m could be r o t a t e d from outside the vacuum system f o r s e l e c t i o n of the wavelength bands. SOURCE : The  source used was  a S y l v a n i a Sun Gun  tungsten  filament lamp  w i t h a c o l o u r temperature of 3400°K when run at 120 v o l t s  a.c.  This source was p a r t i c u l a r l y u s e f u l because of i t s very small s i z e and small h e l i c a l filament (3g i n c h long), so that the filament could be placed very c l o s e to S , 3  the only focussing  o p t i c s being a c y l i n d r i c a l m i r r o r behind the source. This gave the maximum i n t e n s i t y p o s s i b l e from the source and the small o p t i c a l path i n a i r minimised atmospheric absorption. The -wavelength curve was  a good approximation  intensity  to black body r a d i a t i o n  which peaks at about 1.2^ at 3400 K. The l i g h t beam was modulated at 870 cycles/second chopper p l a c e d between S  3  and the  by a d i s c  source.  DETECTOR SYSTEM : I n order to measure l a r g e absorption c o e f f i c i e n t s i t was necessary to keep the s i g n a l - t o - n o i s e r a t i o as high as p o s s i b l e , and the d r i f t of the source and detector system as low as p o s s i b l e . The e x i t o p t i c s c o n s i s t e d of an image d i s t o r t e r which a d i s t o r t e d image of about %cm  z  of the s l i t S  a  focussed  onto the specimen,  and the l i g h t transmitted through the specimen was  focussed onto  a l e a d sulphide detector. Since the s i g n a l / n o i s e r a t i o of a PbS detector i s p r o p o r t i o n a l to  ( s e n s i t i v e area ) ^ , a very short  f o c a l length m i r r o r and a detector of \ mm*" P r e l i m i n a r y experiments on various PbS I n f r a t r o n B-J-SA19 was  area were used.  detectors showed that the  the most s u i t a b l e . This detector had a  reponse time of 200yWseconds at 300 K and a dark r e s i s t a n c e of  Fig 4 Block diagram of the detector electronics.  24 Enclosed in vacuum system. j — W V V — ( D ) — Pre-amp load V/  + 24  Amplifier tuned to 870>±20cps  T 24V bias power supply  -*-24volt regulated power supply  Brown strip chart: recorder  rt O l-h O  i  rt) N3  13 1 megohm. A t r a n s i s t o r p r e a m p l i f i e r was mounted d i r e c t l y behind the detector and m i r r o r , i n s i d e the vacuum system to avoid noise pickup. The output s i g n a l of the p r e a m p l i f i e r was fed i n t o a narrow band a m p l i f i e r tuned t o 870 cycles/second. The e f f e c t i v e noise input of the a m p l i f i e r was l e s s than the shot noise of the detector. With a monochromator s l i t width of 0.4mm, maximum l i g h t i n t e n s i t y , and no specimen i n the l i g h t path, the s i g n a l - i to-noise r a t i o was about 500. The source and detector system d r i f t s were n e g l i g i b l e . The complete e l e c t r o n i c s i s shown diagrammatically i n F i g 4. The e x i t o p t i c s was evacuated separately from the main monochromator t o a pressure of about 10~^mm Hg. EXPERIMENTAL PROCEDURE The specimens and mount were clamped on the dewar block as shown i n F i g 5. The copper p l a t e was "0"ring vacuum  screwed down, s u f f i c i e n t l y t i g h t l y to j u s t h o l d the specimen i n p l a c e ,  'A  g/Ls  so as not to s t r a i n the specimen by the c o n t r a c t i o n of the r e t a i n i n g screws on c o o l i n g . The copper  refrigerant ~ZZ7%&  —  —  r a d i a t i on  shield  copper dewar block  r a d i a t i o n s h i e l d , cooled to the same temperature as the dewar block,  sapphire window  minimised heat exchange between the  specimen  specimen and the surrounding w a l l s . The e x i t o p t i c a l system was  seal  specimen r e t a i n e r p l a t e  evacuated w i t h the dewar i n p l a c e and the specimen was cooled by conduction through the copper block  Fig 5 :  The dewar and specimen holder  14 when the r e f r i g e r a n t was introduced i n t o the dewar. The germanium wafers were s t r a i n e d by the d i f f e r e n t i a l cont r a c t i o n of the germanium and substrate on c o o l i n g . The degree of s t r a i n could be v a r i e d by a s u i t a b l e choice of substrate.Since the expansion c o e f f i c i e n t s of germanium and the substrates were i s o t r o p i c , the a p p l i e d s t r a i n s were uniform i n the plane of the wafer. The choice bf substrate and glue was l i m i t e d by the requirements that both had t o be transparent i n the wavelength range 1.3?to 2.0//, both had t o be hard enough to withstand the mechanical stresses during p o l i s h i n g and had t o be unaffected by water. The substrates used i n these experiments are tabulated w i t h t h e i r expansion c o e f f i c i e n t s i n t a b l e I . The glue had not t o crack when cooled to 77°K and had t o be i n a l a y e r , t h i n enough that i t s c o n t r a c t i o n d i d not i n f l u e n c e the s t r a i n on the specimens (see Appendix A). The epoxy used s a t i s f i e d a l l these  requirements.  The temperature of the specimens could be determined by comparing the spectrum of the s t r a i n e d specimens w i t h an uns t r a i n e d specimen mounted i n a s i m i l a r manner, and using the data of Macfarlane  et a l (reference 5)*(see footnote). The specimen  0 temperature was found to be about 10 K higher than that of the r e f r i g e r a n t , but was reproducible from specimen to specimen w i t h i n 2°K. The transmission T of the specimen and substrate (and glue) was recorded on a s t r i p chart recorder while the monochromator g r a t i n g was rotated. A run without the specimen i n the l i g h t path was made w i t h the dewar lowered so that the beam from the * Macfarlane e t a l used a dewar i n which the unstrained specimens were i n d i r e c t contact w i t h the exchange gas o f the r e f r i g e r a n t which a c c u r a t e l y determined the temperature.  TABLE I The expansion c o e f f i c i e n t s of germanium and various substrates.  Expansion c o e f f 300°K t o 600K Germanium*  Expansion c o e f f 70 K t o 300"K a  Expansion c o e f f r e l a t i v e t o Ge 70'K t o 300 K #  45 x 10" /*C. 7  #0080  92 x 10~ /*C.  #8410  71  #7056  65  20 x 10" /°C.  51  39.5  -5.5  #7740  32.5  25.5  -19.5  #79 00  8  4  -41.0  Sapphire ( X r c-axis)  7  7  50  # i n d i c a t e s Corning glass no.  * S.I.Novikova ( r e f e r e n c e 16)  The 300 K t o 600 K. expansion c o e f f i c i e n t s are t a b u l a t e d f o r comparison of sapphire and #8410 w i t h the other glasses.  16  monochromator passed d i r e c t l y to the detector, through the upper p o r t i n the dewar block. The r e l a t i o n between T and the absorption c o e f f i c i e n t c< of a specimen of thickness d may be w r i t t e n T  =  Transmitted beam i n t e n s i t y I n c i d e n t beam i n t e n s i t y  ~  f( R ^ ( 1-R ) ( 1-Ra) exp-*d l-R,R exp-2<*d t  a  where R,,R , are the r e f l e c t i v i t i e s of the f i r s t and second 2  specimen surfaces and f ( R i ) i s a f u n c t i o n of the r e f l e c t i v i t i e s of a l l the other i n t e r f a c e s traversed by the beam. I n t h i s equation the R^have been assumed to be small. The denominator takes i n t o account the l i g h t transmitted a f t e r an even number of r e f l e x i o n s i n s i d e the specimen. The phase f a c t o r s of the l i g h t beam have been omitted because the observed f r i n g e s were very weak and were e a s i l y averaged. The above equation f o r T i s the averaged transmission. The r e f l e c t i v i t y at the i n t e r f a c e of two media of r e f r a c t i v e i n d i c e s n, , n , i s a  R  s  ( ; "0* ( n, + n ) * a  The v a r i a t i o n of n w i t h <x i n these experiments i s n e g l i g i b l e since i n the energy region examined the absorption c o e f f i c i e n t of germanium i s l e s s than 5000cm"*' . A l l the substrate m a t e r i a l s were transparent i n t h i s energy region and f o r t h i s small range the v a r i a t i o n of n with photon energy was n e g l i g i b l e f o r a l l m a t e r i a l s used. For the unstrained specimens both surfaces were i n contact w i t h sapphire,( n =• .1. 7 ) , so that at the germanium-sapphire interfaces (n=4,  f o r ge.)  R =• 0.16. Thus the denominator i n  the equation f o r T could c o n t r i b u t e at most 2.5% to T and f(R^) i s approximately u n i t y .  17  For the §trained specimens, one s u r f a c e was i n c o n t a c t w i t h sapphire and the other w i t h epoxy, the r e f r a c t i v e index o f which was not known. I t i s a l s o u n l i k e l y t h a t a l l s u r f a c e s made p e r f e c t o p t i c a l c o n t a c t . To a v o i d d e t a i l e d c o r r e c t i o n f a c t o r s , the t r a n s m i s s i o n data were normalised  so t h a t T n  1 i n the r e g i o n o f  the spectrum where the specimens were t r a n s p a r e n t and the e x p r e s s i o n used t o c a l c u l a t e °< was T  n  = exp-°<d  w i t h an estimated maximum e r r o r o f 5% i n exp-<*d. The  apparent t r a n s m i s s i o n o f a specimen a t h i g h photon  energies where the specimens should have been opaque ,was 0.47 . o  T h i s s i g n a l was due t o s c a t t e r e d r a d i a t i o n r e a c h i n g the d e t e c t o r , i n s p i t e o f the b a f f l e s p l a c e d i n the monochromator. F o r the measurement o f small t r a n s m i s s i o n r a t i o s t h i s e f f e c t  became  s i g n i f i c a n t s i n c e the b a s e l i n e o f t r a n s m i s s i o n was s h i f t e d by the background r a d i a t i o n . T h i s s h i f t was accounted f o r t o some extent by t a k i n g the true b a s e l i n e as the asymptote o f the t r a n s m i s s i o n curve a t h i g h photon e n e r g i e s . Finally, data,  t o o b t a i n a b s o r p t i o n c o e f f i e n t s from the t r a n s m i s s i o n  the specimen t h i c k n e s s d had to be known. F o r many o f the  specimens used, i n t e r f e r e n c e f r i n g e s were seen i n the p a r t o f the spectrum where the specimens were t r a n s p a r e n t ,  and these  f r i n g e s gave an average value o f the specimen t h i c k n e s s . The f r i n g e s were u s u a l l y q u i t e f a i n t , i n d i c a t i n g t h a t the specimens were not p e r f e c t l y uniform i n t h i c k n e s s . F o r the e v a l u a t i o n o f °C, the value o f d r e q u i r e d was approximately since T arose  n  the minimum  thickness  » exp-<*d , and not the average t h i c k n e s s so some e r r o r  here.  18 With the above approximations, the estimated maximum e r r o r i n the absorption c o e f f i c i e n t s i s 107 . The r e p r o d u c i b i l i t y of o  measurements was w i t h i n t h i s range. The measurement of high absorption c o e f f i e n t s ( 5000cm ' ) was l i m i t e d by noise i n the detector system and s c a t t e r e d r a d i a t i o n . The monochromator wavedrive was c a l i b r a t e d , using the 1.4425/* and the 1.4119/w water vapour absorption l i n e s (reference 17) . A i r was l e t i n t o the main monochromator tank and a b r i e f run over t h i s wavelength range was made. E : RESULTS OF ABSORPTION MEASUREMENTS T y p i c a l absorption spectra of the (110> o r i e n t e d specimens mounted on various substrates at 90°K are shown i n f i g u r e s 6 to 9. The spectrum of unstrained germanium was the same f o r a l l c r y s t a l o r i e n t a t i o n s and s i m i l a r to the spectrum described by Macfarlane et a l (reference 5) . The s i n g l e peak observed i n the absorption edge at an absorption c o e f f i c i e n t of about 3500cm"' , was i d e n t i f i e d o by Macfarlane as e x c i t o n absorption. The p o s i t i o n of the peak E  e x  could be l o c a t e d w i t h i n the r e s o l u t i o n of the monochromator and was conveniently used as the zero of photon energy f o r a l l other measurements at the same temperature. The spectrum of unstrained germanium at 300°K i s shown f o r comparison i n F i g 6. I t was found to be a l i t t l e broader than the edge at 90 K and the e x c i t o n absorption appears only as a knee i n the edge, i n agreement w i t h Macfarlane et a l . A t 300*K  E°  0.8040± 0.0006 eV and w i t h l i q u i d  nitrogen r e f r i g e r a n t E ° = 0.8790 ± 0.0003eV. The l a t t e r was x  reproducible w i t h i n 0.0006eV, and i n d i c a t e s a specimen temperature of 90 * 2°K. A l l specimens showed i d e n t i c a l absorption spectra at room temperature. The specimens mounted on #0080 glass were under  compression when c o o l e d t o 90 K ( s e e t a b l e I ) . The a b s o r p t i o n edge o f these energies  specimens was found t o be s h i f t e d t o h i g h e r photon  r e l a t i v e t o the u n s t r a i n e d specimens, and a  second,well  d e f i n e d a b s o r p t i o n peak appeared on the a b s o r p t i o n edge. T h i s second peak has a l s o been a s s o c i a t e d w i t h e x c i t o n  absorption  ( r e f e r e n c e ? , 8 ) . The spectrum was q u a l i t a t i v e l y s i m i l a r f o r a l l specimens and the p o s i t i o n s o f the two peaks are t a b u l a t e d i n t a b l e I I . The spectrum o f the specimen  on #8410 g l a s s was a l s o  q u a l i t a t i v e l y s i m i l a r t o f i g 7, but the s h i f t o f the a b s o r p t i o n edge and the s e p a r a t i o n o f the two peaks was s m a l l e r . The r e s u l t s i n t a b l e I I are the mean values o f r e s u l t s  from f i v e specimens  o f each o r i e n t a t i o n on #0080 g l a s s and j u s t one specimen on #8410 g l a s s . Three runs at 90°K were done f o r each specimen. The e r r o r s quoted are the estimated maximum e r r o r s , and a l l specimens were r e p r o d u c i b l e w i t h i n t h i s range. The r e p r o d u c i b i l i t y was l i m i t e d by the r e p r o d u c i b i l i t y o f specimen temperature, s i n c e f o r e  a change o f 2 K t h e a b s o r p t i o n edge s h i f t e d through about 0.9 m i l l i - e V . The peak s e p a r a t i o n was r e l a t i v e l y i n s e n s i t i v e t o temperature v a r i a t i o n s and the e r r o r s i n these  values are s m a l l e r .  I t was found t h a t i f vacuum grease was used t o p r o v i d e  good  thermal c o n t a c t between the dewar block and the specimens t h a t the r e s u l t s were i r r e p r o d u c i b l e and the s e p a r a t i o n o f the a b s o r p t i o n peaks was c o n s i d e r a b l y g r e a t e r than without The  grease.  cause f o r t h i s must be t h a t on c o o l i n g , the grease forms a  f i r m bond between the specimen and the block which s t r a i n s the specimen t o a g r e a t e r extent were thus mounted without The  than the s u b s t r a t e . The specimens  grease.  specimens mounted on #7740 and #7900 g l a s s were under  t e n s i o n when c o o l e d t o 9 0 ° K , to lower photon energies.  a n d t h e a b s o r p t i o n edge was s h i f t e d  The s p e c t r u m o f s p e c i m e n s  showed one p e a k a t a b s o r p t i o n c o e f f i c i e n t s and  a d e f i n i t e knee  be q u i t e w e l l l o c a t e d . structure  apparent  o f a b o u t 3500cm"'  i n t h e edge a t 1500cm"' . T h i s k n e e  be a n a l y s e d a s a s e c o n d a b s o r p t i o n p e a k  no f i n e  The s p e c t r u m o f s p e c i m e n s  i n the a b s o r p t i o n edge.  at absorption coefficients  t h e two a b s o r p t i o n p e a k s w e r e  h a l f w i d t h o f about 1.5 m i l l i  on#7900  o f 1100cm*' a n d a n o t h e r  broadened w i t h a broadening o f the peak  O n l y one o r two s p e c i m e n s  c o u l d n o t be l o c a t e d w i t h s u f f i c i e n t  c l e a r l y i n d i c a t e the differences  at  f o r by a s s u m i n g  e a c h o r i e n t a t i o n w e r e p r e p a r e d on #7740 a n d #7900 g l a s s , the peaks  showed  One v e r y b r o a d k n e e was  eV. A rough e s t i m a t e  p o s i t i o n s c o u l d t h e n be o b t a i n e d .  could  and i t s p o s i t i o n c o u l d  a b o u t 3 5 0 0 c m " ' . T h i s a b s o r p t i o n c o u l d be a c c o u n t e d that  on #7740  between  accuracy  different  of  since to  specimen  orientations. Slight  v a r i a t i o n s i n t h e o p t i c a l a b s o r p t i o n edge w e r e  f r o m s p e c i m e n t o s p e c i m e n on s i m i l a r s u b s t r a t e s . peak h e i g h t s  v a r i e d i n magnitude,  b r o a d e r i n some s p e c i m e n s .  s p e c i m e n s m o u n t e d on #7740 g l a s s was l e s s cases than i n others.  The a b s o r p t i o n  a n d t h e edge a p p e a r e d  In particular,  seen  slightly  the l o w energy peak o f w e l l d e f i n e d i n some  Possible explanations  o f these e f f e c t s  will  be d i s c u s s e d i n c h a p t e r IV. The v a r i a t i o n o f t h e a b s o r p t i o n p e a k p o s i t i o n s wLth s t r a i n are p l o t t e d i n f i g u r e s  1 0 , 1 1 , 1 2 . The e r r o r f l a g s  are again the  e s t i m a t e d maximum e r r o r . S i n c e e a c h p o i n t i s t h e mean r e s u l t five  to fifteen  readings,  the probable  of  e r r o r i n each p o i n t i s  s m a l l e r t h a n t h i s . The u n c e r t a i n t y i n t h e t h e r m a l e x p a n s i o n o f germanium i s i n c l u d e d i n t h e e r r o r o f t h e p o i n t f o r u n s t r a i n e d  21 germanium. F o r a l l o r i e n t a t i o n s the graphs are l i n e a r w i t h i n the e r r o r s , and the i n t e r s e c t i o n o f the two l i n e s i s a t E °  x  (approx).  The r e s u l t s o f the peak p o s i t i o n s i n the specimens mounted on #8410 and sapphire  are f i t t e d t o the graph f o r the ( i l l )  o r i e n t e d specimens. They are c o n s i s t e n t w i t h the other r e s u l t s , and t h e i r expansion c o e f f i c i e n t s (90°K t o 300°K) may be deduced from the graphs as : Sapphire : 29 x  l(fVc.  #8410 : 47 x 10~Vc.  S i n c e no broadening o f the a b s o r p t i o n was  edge on the low energy s i d e  seen i n the s p e c t r a o f the " u n s t r a i n e d " germanium, t h i s  r e s u l t i n d i c a t e s that the specimens were t r u l y u n s t r a i n e d  by the  sapphire windows. The s p e c t r a o f the specimens were a l s o examined a t 196°K, but because o f the small s t r a i n s a p p l i e d , the d i f f i c u l t y i n maintaining  a constant  temperature and the broadening o f the  a b s o r p t i o n peaks a t t h i s temperature, the data only served t o show c o n s i s t e n c y with F : PLASTIC  the above r e s u l t s .  FLOW.  The c a l c u l a t i o n o f the s t r a i n s on the specimens a t 90°K. ( s e e appendix A) assumes that the specimens d i d not deform p l a s t i c a l l y . A check f o r p l a s t i c  flow was made f o r two specimens mounted on  #7900 g l a s s s i n c e these specimens were under t e n s i o n and experienced  the l a r g e s t s t r a i n s a t 90°K. The specimens were c o o l e d  to 90°K and warmed up t o 300°K a few times, and each time the s p e c t r a a t both temperatures were u n a l t e r e d . I f p l a s t i c taking place,  flow were  f r o z e n i n s t r e s s e s would be expected a f t e r the f i r s t  c y c l e and the a b s o r p t i o n  s p e c t r a would be a l t e r e d .  Figure 6 The absorption speo-trum of unstrained germanium at 90°K and 300°K  Figure 7 The absorption spectrum of germanium mounted on #0080 glass a t 90°K  4000 1  3000 absorption coefficient o< cm -1  <110> o r i e n t e d specimens  20001  monochromator r e s o l u t i o n  = 0.3 m i H i - e V  1000  Photon energy ( m i l l i - e V )  10  -42-  Figure 8  0  Figure 9 A b s o r p t i o n spectrum of germaniijim mounted on #7900 glass a t 90°K  Table I I The p o s i t i o n s o f the a b s o r p t i o n peaks i n the edge o f the s t r a i n e d germanium s p e c t r a a t 90*K measured r e l a t i v e t o the mean p o s i t i o n o f the peak i n u n s t r a i n e d germanium a t 90°K. Substrate #0080  Specimen orientation <111> < 110 > < 100 >  S t r a i n on specimen -0.00042 II  •i  A b s o r p t i o n peak p o s i t i o n s milli-eV 5.7 4.6 4.5  8.8 8.0 8.5  0.3  1.1  ± ± +  Peak s e p a r a t i o n milli-eV  0.9 0.9 0.9  3.1 • 0.3 3.4 ± 0.3 4.0 0.4  0.9  0.8  #8410  < 111>  #7056  <111>  0.00012  -1.2  -  #7740  < 111> < 110 > < 100 >  0.00041 <i ii  -4.7 -5.1 -3.6  -7.0 -7.8 -6.4  + 0.9 0.9 0.9  2.3 + 0.4 2.7 + 0.4 2.8 0.4  #7900  < 111> < 110 > < 100 >  0.00086 •f •i  -11 -11 -9  -17 -17 -15  4  + 1.5 1.5 f 1.5  6 6 6  ± 0.9  1.5  sapphire  < 111>  -4.4  -5.9  effect of grease  < 111 >  8.2  14.8  •f-  0.3  0.9  6.6  2 ± 2 + 2 0.4  Fig 10 Diagram showing the variation of the exciton peak positions with strain on the specimens when the applisd stress is in the [111] plane. ..10 Exciton peak positions (relative to E ° ) x  milli-eV  #8410  rt O  8, xlO -4 Tensile strain Q on specimens q  Compressional strain  o  05 09.  mean shift of the exciton peaks  to to ro  sapphire  F i g 11 Diagram showing the v a r i a t i o n o f e x c i t o n peak p o s i t i o n s w i t h when the s t r e s s i s a p p l i e d i n the ClOO] p l a n e  strain  Fig  12  Diagram showing the v a r i a t i o n o f E x c i t o n peak p o s i t i o n s w i t h s t r a i n on the specimens, when the s t r a i n i s a p p l i e d i n the L"llO] p l a n e .  23 CHAPTER I I I  :  THEORETICAL BACKGROUND AND  In the l a s t chapter i t was absorption  seen how  ANALYSIS OF RESULTS. s t r a i n a f f e c t s the  edge of germanium. I t i s the purpose of t h i s  to show how  chapter  the r e s u l t s were i n t e r p r e t e d .  K l e i n e r and Roth ( r e f e r e n c e 8) have shown how  the  exciton  a b s o r p t i o n peak p o s i t i o n s i n the spectrum o f s t r a i n e d germanium may  be used to evaluate  a n a l y s i s had  the deformation p o t e n t i a l s . T h e i r  i n i t the i m p l i c i t  assumptions that the  a r i s i n g from the two p a i r s o f bands had were equal and  and B i r ( r e f e r e n c e 9)  o f the s t r a i n e d germanium l a t t i c e , not be  b i n d i n g energies  i n v a r i a n t under s t r a i n . I t i s evident  d e s c r i p t i o n by P i c u s  excitons which  from  the  o f the band s t r u c t u r e  t h a t these assumptions  may  to check the v a l i d i t y o f these assumptions,  the  valid.  I n order  band s t r u c t u r e o f u n s t r a i n e d i n s e c t i o n (a)  and  germanium w i l l  be b r i e f l y reviewed  the e f f e c t o f the s t r a i n a p p l i e d i n t h i s  experiment on the band s t r u c t u r e of the specimens, w i l l , be discussed.  Then i n s e c t i o n (b)  and u n s t r a i n e d  germanium w i l l  the e x c i t o n s t a t e s i n both s t r a i n e d be d e s c r i b e d w i t h i n the e f f e c t i v e  mass approximation. In view of the p o s s i b i l i t y t h a t the e x c i t o n b i n d i n g may  be obtained  d i r e c t l y from experimental data, ( r e f e r e n c e  the i n t e n s i t y o f o p t i c a l a b s o r p t i o n discussed  simple s p h e r i c a l bands and  the a b s o r p t i o n of the band gap treatment w i l l  shown how  edge of such a semiconductor may and  exciton binding  11),  near the band edge w i l l  i n s e c t i o n ( c ) . E l l i o t t ( r e f e r e n c e 11)  case o f two  energies  has  be  treated  analysis y i e l d the  energy. I n s e c t i o n (c)  be b r i e f l y extended to the case of s t r a i n e d  the  of values this and  24 unstrained germanium. I n s e c t i o n (d) the experimental  data o f chapter I I will be  i n t e r p r e t e d i n terms o f the e f f e c t o f s t r a i n on the valence and conduction bands of germanium and the deformation p o t e n t i a l s w i l l be evaluated,  IE  a) THE BAND STRUCTURE OF GERMANIUM, i . The unstrained  lattice.  The band s t r u c t u r e o f unstrained  <Hl>  germanium has been described by Dresselhaus  conduction band minima are at the edges of the B r i l l o u i n zone where  ofthe side of the u n i t cubic c e l l  k  A V \  (reference 2 ) . The  k - §(il *1,±1) , ( a being the length  si'  F i g . 13 The band s t r u c t u r e o f unstrained germanium.  of the germanium l a t t i c e ) . A t k= 0 there i s an a u x i l i a r y minimum, 0„16eV above the zone boundary minima; t h i s minimum transforms according to the  i r r e d u c i b l e representation of the T^ x I p o i n t  group. This minimum i s two-fold degenerate ( i n c l u d i n g spin) and approximately  s p h e r i c a l so that i t may be described by a s i n g l e  e f f e c t i v e e l e c t r o n mass. Other conduction bands are s p l i t o f f to higher energies by at l e a s t 0.5 eV. The only valence band maximum i s at k= 0 o f the B r i l l o u i n zone and transforms according to the  i r r e d u c i b l e representation  of the T j x I p o i n t group. This maximum i s f o u r f o l d degenerate, i n unstrained germanium,at k= 0, but the degeneracy i s p a r t l y removed away from k = 0 i n t o two two-fold degenerate bands. A twof o l d degenerate band i s s p l i t o f f 0,3 eV ( a t k= 0) from the valence band maximum, by s p i n - o r b i t coupling.  25 The two upper valence bands are warped spheres near k= 0 and may be described by = Ak* t ( B*k*+ C* [ k* k^ •+- k* \a) + k* k* ) = E, ( k, 0)  1  >7  where E i s the hole energy f o r zero s t r a i n . Each root occurs twice as each band i s doubly degenerate. The p o s i t i v e sign i s maintained  f o r the lower " l i g h t " hole band and the negative  sign f o r the upper "heavy" hole band. Since the warping i s small, the bands may be described approximately symmetric e f f e c t i v e hole masses  by two s p h e r i c a l l y  ^heavy ~ 0.34m  o }  m  light  =  0°043m  o  The valence bands may be conveniently l a b e l l e d by the usual atomic o r b i t a l symbols as shown on the diagram. These l a b e l s have r e a l s i g n i f i c a n c e only i n the t i g h t binding  approximation,  i i . The s t r a i n e d l a t t i c e . When a s t r a i n £ i s a p p l i e d t o a germanium c r y s t a l , the l a t t i c e , i n general, no longer has cubic symmetry and the fourdimensional i r r e d u c i b l e representation T breaks down i n t o two ?  two-fold representations. Because of the presence of a centre of i n v e r s i o n i n the c r y s t a l , the degeneracy i s not e n t i r e l y removed ( r e f e r e n c e l 8 ) . The conduction band degeneracy at k = 0 i s unaffected by the s t r a i n , though the degeneracy o f the zone boundary minima  is  p a r t i a l l y removed. The e n t i r e conduction a ' A where A i s the l a t t i c e  band i s s h i f t e d through an energy dilation  ^  E (k,I) e  = S„ + £  + 21  £33 . Thus the e l e c t r o n energy  = E (k,0) + a ' A  .....  e  2  When the degeneracy a t k = 0 o f the valence band i s removed by the s t r a i n £ a p p l i e d t o the c r y s t a l , the hole energy spectrum i s considerably a l t e r e d . P i c u s and B i r (reference 9) have shown that the bands i n the s t r a i n e d l a t t i c e may be described by E 0M> i a  =  aA  ±  Jt  k  + t  + tk  Ak*.....  3  26 where  £\  + c  = B V  f « = Bb[3(k*£  X  ( V ^  lt  +  ^  K  +  K  + k|e„ +  * <) K  k*£  + 2Dd( The  constants  are deformation  c y c l o t r o n resonance parameter  ) - k*A ]  33  k,k £, f k a k j f ^ +• k k, £ 2  2  3  p o t e n t i a l constants  3 )  )  and D i s the  D = ( C + 3B*/* E ( k , £ ) i s the 2  l 2  h o l e energy, measured from the band edge i n u n s t r a i n e d germanium, the s u b s c r i p t s r e f e r r i n g t o the p l u s and minus s i g n s . At k - 0  we see from equations  E (0,£)  =  e  Writing E valence  = \  K  E (0,0) + a'A e  [E,(0,£)  +  2 and 3  and  E (0,|) = a A u  E (0,£)]  ±  -M  t  f o r the mean s h i f t o f the  t  band edge, we f i n d the v a r i a t i o n o f the f o r b i d d e n  energy gap w i t h l a t t i c e  dilatation  ..... 4  [E (0,£) • E (0,£)] - E (0,0) = ( a ' + a) A e  and The  h  the s p l i t t i n g o f the bands a t k = 0  i s 2j£  e  .....5  same e x p r e s s i o n i s a r r i v e d a t by d i a g o n a l i s i n g the o f K l e i n e r and Roth ( r e f e r e n c e 8 ) . T h e i r n o t a t i o n  hamiltonian is  e  o b t a i n e d by p u t t i n g b = - \ D* ,  d = -|D«' ,  For the two cases  H  may be w r i t t e n i n a simple  K  a ^-D^  >  > t  t  ,  Case_l.  t  >  k  u  where the  £ <  equation 3  k  form i n a c o o r d i n a t e system a s s o c i a t e d  ;* j  tensor  where  p =  ^  Z, >h k  F o r l a r g e k equation E (k,£)  1  and  w i t h the p r i n c i p a l axes o f the s t r a i n I £'.-j =p£jj  a = Dj  =  3 becomes  aA+E^k.O)  i  I  ^g-  are components o f £\j along i t s p r i n c i p a l  axes.  27  The l a r g e r k, the l e s s important i s the l a s t  term and the  bands go t o t h e i r shapes i n the u n s t r a i n e d l a t t i c e . They are approximately s p h e r i c a l and can be d e s c r i b e d by the s i n g l e e f f e c t i v e masses m  t  , m  h t  heotV  ^  , having the same values as  i n the u n s t r a i n e d l a t t i c e . The requirement energy d i f f e r e n c e s  E , (k,0) - E ( k ' , 0 )  of germanium  implies  k  l a r g e compared w i t h  z  the s p l i t t i n g o f the bands  f >  E , (0>£) - E "(0-,£). I n the case 2  t h i s c e r t a i n l y means t h a t E , (k,£.)  where E, >  E  2  i s g r e a t e r than twice the s p l i t t i n g p f the bands a t k = 0. ( s e e f i g u r e 14). Case 2. For small k equation 3 becomes 1 A  (k,e)  =  ±Jz + Hz*  • •  L  Thus when the degeneracy i s removed the valence bands near k = 0 become e l l i p s o i d a l  and are d e s c r i b e d by e f f e c t i v e masses  The shape o f the bands near k — 0 i s independent o f the magnitude o f the s t r a i n and depends o n l y on i t s o r i e n t a t i o n . I n both the above cases the e f f e c t i v e mass t e n s o r i s d i a g o n a l . I n the r e g i o n where order of  E , (k,0) - E ^ k j O ) i s o f the  E,(0,£) - E ( 0 , £ ) , the bands cannot be d e s c r i b e d z  simply. O f f d i a g o n a l terms appear i n the e f f e c t i v e mass t e n s o r and a l l  the elements are f u n c t i o n s o f k.  The valence band s t r u c t u r e d e s c r i b e d by equation 3 i s p l o t t e d i n f i g u r e 14 f o r a s t r a i n  c r y s t a l plane.  £ -  T (1 0 0\ 0 10 \0 0 - ^  i n the [l00]  k  ( to f o l l o w page 27) Figure 14 (a)  Energy band s t r u c t u r e of germanium s t r a i n e d i n the [100] plane  ------ unstrained germanium (b) Curvature of the above E-k curves showing the v a r i a t i o n of hole mass.  \•  —-»»  20 •  10 -  -=—  !  2_  4  l i g h t hole  2  hole masses i n unstrained germanium ^  1 ^ ^  o,  heavy hole  1 1  1  1  k i cm"' For curves 1,2, k; = k, or k k' is the valve of k- /or uh,ch t  z  ; f o r curves 3,4, k^=k . E,(h,0) - F (£, 0 ) * E,(oJ) 3  t  E  t  (0,1)  ,  28  From appendix B i t can be seen that the p r i n c i p a l axes of £ and 6 c o i n c i d e f o r the s t r a i n s a p p l i e d i n the [100] , [HQ] and [ill]  c r y s t a l planes. Thus the Inverse e f f e c t i v e masses  m  i  ±  may be c a l c u l a t e d from equation 6, i ) S t r e s s i n a [lOO] plane. / 1 0 0\ E' = £ = T 0 1 0  Thus  n* IR - = A ± %B 2m, + 2m 7  21  I o o=a<  X  3nd  i i ) Stress i n a [ i l l ]  =• A +• B  plane.  I t has been shown i n appendix B that f o r the s t r a i n s applied i n a [ill] £  H  e  n  C  experiment  = 3 [ -1=1 2-X 1+X V 1 + X 1+X 2-X/  2mfT 2m>  e  plane i n t h i s  = A  ^  1  a  n  and 2m>  d  r  ±  A  +  £  = *.•<*£'  JS  i i i ) S t r e s s a p p l i e d i n a [100] plane. Again fronappendix B we have  £= H  'l-l 0 m 0 0 2 20 0 1+1 0 i - a  Hence H  e  n  C  G  and  2mr  *  +  +  _ = A ±  D -( b/d) B  * 2[l (b/d7>  = A ±  -R" 2m*  1  f  and  E"  2m>  =  ±  =  £  *  B( b/d)  [l (b/d)^ +  D + (b/d) B 2[1+ (b/dFJ'*  Using the; c y c l o t r o n resonance parameters o f Levinger and Frank! ( r e f e r e n c e 31) the e f f e c t i v e masses were c a l c u l a t e d and tabulated i n t a b l e I I I . Only the e f f e c t i v e hole masses i n specimens o f [100J and [ i l l ]  o r i e n t a t i o n s may be t a b u l a t e d  without p r i o r knowledge of the deformation p o t e n t i a l s . The l a s t two columns were completed u s i n g the values of b and d obtained later.  TABLE I I I  The e f f e c t i v e hole masses i n s t r a i n e d germanium specimens  [100]  [ 110  [111] 4  +  heavy ta/e 0.057*. 0.111 rMc /io/e"  «* m  *•  2  m^ ^/mfmjm^  A=  The  0.130*.  0.051*.  0.143*,,  0.057  0.111  0.053  0.130  0,057  0.111  0.213  0.046  0.476  0.041  0,370  0.042  0.089  0.083  0.113  0.089  0.102  0.087  0.020  0.024  0.020  0.025  0,019  0,025  0.020  0.024  0,020  0.025  0,020  0,024  0.027  0.018  0.029  0.018  0.029  0.018  0.022  0.022  0,022  0,022  0.022  0.022  13.3  * Using  0.053  B = 8.6  C = 12,4  hence  D = 19,4  b/d = 0.57 from the f i n a l r e s u l t s . are the reduced e f f e c t i v e exciton masses  =  (reference 30)  election  effecTi^  mass -  O.03\m  o  1  _ / _L  4.  _L \  The e f f e c t o f s p i n - o r b i t s p l i t o f f valence band has been n e g l e c t e d i n these c a l c u l a t i o n s s i n c e i n a l l cases the band splitting  by s t r a i n was very much l e s s than 0.3 eV.  I t i s seen i n t a b l e I I I t h a t t h e " d e n s i t y mass  of s t a t e s " hole  V r - t ? *- approximately the same f o r both bands, so we m  m  m  s  may l a b l e the bands by " l i g h t " and "heavy" h o l e s only because of t h e i r behaviour a t l a r g e k. b) DESCRIPTION OF THE EXCITON STATES IN THE EFFECTIVE MASS APPROXIMATION. I t has been shown ( r e f e r e n c e d i e l e c t r i c c o n s t a n t that  2) that i n media o f h i g h  the d e s c r i p t i o n o f the e x c i t o n  state  may be c a r r i e d out i n the e f f e c t i v e mass approximation., I n t h i s s e c t i o n the d e s c r i p t i o n o f the e x c i t o n w i l l  be d i r e c t e d t o the  p a r t i c u l a r cases o f the s t r a i n e d and u n s t r a i n e d  germanium  lattice. I n the e f f e c t i v e mass approximation the h a m i l t o n i a n o f an electron-hole  pair, i s  H = H + H j, +- V( r)  .....  e  7  where H , H/,, are the h a m i l t o n i a n s o f the e l e c t r o n i n the e  conduction band and h o l e i n the valence band r e s p e c t i v e l y and V("r) i s the coulomb i n t e r a c t i o n o f the e l e c t r o n and h o l e .  The use o f the e f f e c t i v e mass approximation i s l i m i t e d by the condition  t h a t V(r) be s l o w l y  k describes  the m o d i f i c a t i o n  varying  over the l a t t i c e  spacing.  o f the coulomb f i e l d o f the  e l e c t r o n and h o l e by the p o l a r i s a t i o n o f the l a t t i c e ,  and i n  germanium i s taken t o be the s t a t i c d i e l e c t r i c c o n s t a n t . I n  31 germanium X i s l a r g e  so the i n t e r a c t i o n i s weak and the  e f f e c t i v e mass approximation i s expected t o be good. I f |k ,j> and |k>,, j')are the B l o c h f u n c t i o n s  o f an e l e c t r o n  e  i n the conduction band j and a h o l e i n the valence band j r e s p e c t i v e l y , then the s t a t e o f the e l e c t r o n - h o l e written  1  p a i r may be  as an expansion i n terms o f |k , j >( k , j') , e  i.e.  lK\r>,-£  "^" jk ,j>  |k,,j>  n  e  T h i s i s c a l l e d an e x c i t o n  h  ,..,,..8  s t a t e and the e l e c t r o n - h o l e  pair i s  u s u a l l y r e f e r r e d t o as an e x c i t o n , K i s the e x c i t o n wave-number vector  and the sum i n equation 8 i s r e s t r i c t e d by the r e l a t i o n K = k +- kj, e  When a c r y s t a l i s i l l u m i n a t e d , o f an e x c i t o n i n the n  state  the p r o b a b i l i t y o f c r e a t i o n  |K,r > , p e r u n i t time, f o r a n  direct transition i s l*V)  z  =  e  ^  A  ° |<0 I f ( e x p  iJ[.r)7 |K,r > ^(hv e  A  e  - E  +  E„)  e  and E  h  a r e the h o l e and  e l e c t r o n e n e r g i e s and A , V , f , 1 are the amplitude, 0  p o l a r i s a t i o n vector  £  -r7 i s a momentum  where I0> i s the c r y s t a l ground s t a t e , o p e r a t o r a c t i n g on e l e c t r o n s t a t e s , E  j y  and p r o p a g a t i o n v e c t o r  frequency,  o f the i n c i d e n t  r a d i a t i o n . F o r the l i n e a r combination 8, o f e l e c t r o n - h o l e states, This  |H /vanishes u n l e s s  k  0)  i s a statement o f c o n s e r v a t i o n  photon momentum i s small  - k  e  A  pair  -^ ~ 0  o f c r y s t a l momentum. The  compared w i t h the e l e c t r o n and h o l e  momenta. For germanium value o f ( ' E - ET^ A  occurs a t the c e n t r e  e  = AE, f o r which k  - k^- 0  o f the B r i l l o u i n zone, so t h a t the onset  o f d i r e c t band t o band t r a n s i t i o n s occurs when At  e  the band edge, the h o l e p a r t  hv  -AE^ . Q  o f the h a m i l t o n i a n may be w r i t t e n  32 i n Shockley  representation  H  =  A  H2  JE  \  f  HJ  a f t e r the u s u a l s e p a r a t i o n of the s p i n = o r b i t s p l i t o f f band terms. The  l a b e l s 1,2,  r e f e r to the two p  2  valence bands and  H. l2  i s an exchange term. Thus the e x c i t o n h a m i l t o n i a n i s H  /H  =  H  e +  + V(r)  M  H,I2. 1  Hia. Kane ( r e f e r e n c e 19)  has  found  H  I t was  + H^  + V(r).  t h a t exchange terms from other  bands i n the germanium l a t t i c e n e g l e c t e d here.(see  e  appear to be small and w i l l  be  footnote).  seen i n the l a s t s e c t i o n t h a t the e l e c t r o n and  e f f e c t i v e mass tensors i n u n s t r a i n e d germanium are and a l s o i n s t r a i n e d germanium, except r e g i o n where  E,(k,0) = E ( k , 0 ) z  ~  t h i s energy r e g i o n i s not important experimental  results i t w i l l  hole  diagonal,  f o r the h o l e energy  E,(0,£) - E ( 0 , £ ) . Since t  i n the a n a l y s i s o f the  be d i s r e g a r d e d i n the f o l l o w i n g  a n a l y s i s , and only diagonal e f f e c t i v e mass tensors w i l l  be  considered. Now First  H = e  2m  and e  of a l l i t w i l l  H,,, = ~ V 2  '>  be assumed t h a t H,  z  n e g l e c t e d . L a t e r the e f f e c t o f H ,  z  each pair o f bands, the Schrodinger  (.-  -1 s;Vv* -  where |K,r*)> i s the n  *j  will  \ 7* 2m •  (1 = 1,2,3)  ±  i s small and may  be d i s c u s s e d . Thus f o r  equation has  m)^>  be  -  E  *  the  form  i > -> r  ?  e x c i t o n s t a t e a s s o c i a t e d w i t h one  •••  1  0  pair  The e f f e c t i v e mass formalism would be u n a l t e r e d i f exchange terms were i n c l u d e d , but the e f f e c t i v e masses used would not be the c y c l o t r o n resonance masses.  33 of bands. I n the expansion bands  8 o f the state, the sum over the  may be omitted s i n c e only a small range o f Bloch  f u n c t i o n s i s important i n the sum, ( T h i s i s analogous t o neglecting H ) . a  Making the " c e n t r e o f mass t r a n s f o r m a t i o n " X; =  +m X + m-*  m Xei  t  &  ra*  1  At  x  / ' =  _  x  x h L  where x - ,x - are the e l e c t r o n and h o l e c o o r d i n a t e s , the et  h  equation 8 becomes I where  f2M^X>  t2 u dx'>~ /  M = m* + mf  A s o l u t i o n o f t h i s equation i s T h i s i s the F o u r i e r transform o f  where  E' =  E  n  JA. =^  and  C  |K,r >  ;  +- I  IK,^) =  =.E |K,r,> n  j F ( r ) exp iK.X  ^_,±n the equation 8.  - J  n  T h i s equation can o n l y be s o l v e d e x a c t l y f o r the case o f two s p h e r i c a l bands y K r / u = ^ 1  l  J o  For e l l i p s o i d a l  bands, v a r i a t i o n a l  methods may i n p r i n c i p l e be used t o s o l v e equation 11, Three cases w i l l be s e p a r a t e l y considered: i ) S p h e r i c a l , non-degenerate bands. T h i s i s approximately the case i n  s t r a i n e d germanium when the h o l e energy i s g r e a t e r than  twice the s p l i t t i n g o f the p  3  bands a t k = 0, ( c a s e 1 o f l a s t  s e c t i o n ) , and i n u n s t r a i n e d germanium when E ^ (k,0) > 0, ii)  S p h e r o i d a l , non-degenerate valence bands. T h i s i s the case  i n s t r a i n e d germanium near k = 0 o f the B r i l l o u i n  zone ( s e e  table I I I ) . iii)  S p h e r i c a l degenerate  bands. T h i s i s approximately  i n u n s t r a i n e d germanium at k = 0.  the case  34 i ) Simple s p h e r i c a l bands. E q u a t i o n 11 has two  been s o l v e d by E l l i o t t ( r e f e r e n c e  s p h e r i c a l non-degenerate bands. The  perturbation  11)  V(r)  gives  r i s e to a band o f s t a t e s below the conduction band. With optical selection rule k *  = 0  e  for  the  a l i n e spectrum appears from  the K = 0 s t a t e s of the e x c i t o n bands°, £'„  and  = AE  a continuum of s t a t e s  binding  energy  G -  -  0  |  for  z  f o r hv > ZiE . The 0  !~ £  2  =  -~—  <AE  hv  C  ground s t a t e  where a i s the  ground  s t a t e "Bohr r a d i u s " . The  eigenfunctions  F (r) n  f o r the bound s t a t e s  hydrogen atom f u n c t i o n s w i t h a n d the i o n i s e d s t a t e s F ( r ) n  — i n place  i s a coulomb  Thus i n germanium the e x c i t o n arising  from the two  of t h i s  type.  ii)  0  and  e.  For  function.  s t a t e s i n the  s p h e r i c a l bands, w i l l  Simple s p h e r o i d a l  of m  are  continuum,  be coulomb  functions  bands yu, -JA  V  I n t h i s case the equation 11 i s s i m i l a r to the equation f o r donor i m p u r i t i e s i n germanium. T h i s equation has approximately f o r ^n, <r« ( r e f e r e n c e  20)  / 3  function  F  i / ^  1  =  e  (na^-b)'^  k T )  X  D  6 X P  =  by u s i n g + y  / I  a*-  F (r) n  of the e x c i t o n  were used. So spectrum i s very  to o b t a i n  solved  the v a r i a t i o n a l z- \''  2  z  +  f o r the ground s t a t e s o l u t i o n . For o t h e r e x c i t o n functions  been  bM  , •  states  a complete d e s c r i p t i o n  tedious.  In germanium, the reduced e f f e c t i v e masses JA. are L  p r i m a r i l y by the small the jt  t  other  e l e c t r o n mass (see  t a b l e I I I ) so  determined that  are approximately equal f o r a l l s t r a i n s a p p l i e d . U s i n g  the above f u n c t i o n F'(r)  i t i s found that the ground s t a t e  35  binding energy i s approximately  the same as that f o r an  e x c i t o n formed from two s p h e r i c a l bands w i t h t h i s case. I f the same i s true when  - sJf^i^Jj  >  i  n  then the e x c i t o n  'i  i  binding energies i n s t r a i n e d germanium are both ~1.2 m i l l i - e V . (The v a l i d i t y of the v a r i a t i o n a l functions has not been t e s t e d for i i i ) Degenerate s p h e r i c a l bands. In equation 11 the exchange term H  l4>  was neglected. When  the two valence bands are degenerate, H . w i l l be of the same l?  order o f magnitude as  and H/, so that equation does not 2  describe the e x c i t o n states at k - 0 of unstrained germanium. For degenerate bands an exact s o l u t i o n cannot be obtained, however, i t may be expected that a minimum and maximum estimate o f the e x c i t o n binding energies may be obtained by using the l i g h t and heavy hole masses i n the expression f o r G. 0.9 < G  ^ 1.4  milli-eV.  The e f f e c t o f exchange, The separation,of the  valence bands i n germanium i s  afunction o f s t r a i n a p p l i e d t o the c r y s t a l . Thus the e f f e c t o f H  (2  on the e x c i t o n s t a t e w i l l a l s o be a f u n c t i o n of s t r a i n . P r i c e  (reference 21) has discussed the s t r a i n dependence of acceptor binding energy by i n c l u d i n g H i n the hamiltonian. Because H i r  appears i n the e x c i t o n hamiltonian, the e f f e c t of H  {%  on the  ,exciton binding energy would be much smaller than on the acceptor binding energy, so that i t may be neglected. supports  the conclusion o f p a r t ( i i i ) .  This  e  36  c) THE INTENSITY OF OPTICAL ABSORPTION NEAR THE ABSORPTION EDGE. The  i n t e n s i t y of o p t i c a l absorption  creation i s described  by an a b s o r p t i o n  associated with  coefficient  where two bands j.j. j ' only are c o n s i d e r e d , yo (o>) d e n s i t y o f i n c i d e n t r a d i a t i o n o f frequency density of exciton states  IK,r>  i s the energy  and N(E) i s the  p e r u n i t energy range a t E,  a s s o c i a t e d w i t h the bands  -  Because o f the o p t i c a l density of exciton  exciton  selection rule  k  e  4 k = 0. the h  s t a t e s i n k-space i s the same as t h a t o f  the e l e c t r o n s t a t e s ; i . e .  p e r u n i t volume. Thus the d e n s i t y  of e x c i t o n s t a t e s p e r u n i t energy range at E i s the s u r f a c e i n t e g r a l i n k=space:  '»  •  In the continuum o f s t a t e s the e x c i t o n k i n e t i c energy i s  To  evaluate  equation  12 the s u b s t i t u t i o n i/  yielding  ^  (2^,y^ ) 3  2nft  1  Following E l l i o t t (reference  where again  i s made,  ^  E  K e  3  Remembering />(w) oc co A* we have l  K J. -  *7u>) oc frWj) ' f l ~ 0 |<o|f e ' ^ %\*, y  11), we have, u s i n g  the sum over j , j ' has been n e g l e c t e d .  small range o f k  e  A  the expansion 8  Since  only a  , k^ i s important i n t h i s sum, i t may be  assumed t h a t the i n t e g r a l s are independent o f k , k^ . Thus e  \<o\f  7,lK r>l >  z  =  \<o,j'ie^ fVe\o >l lL¥°^l r  z  >J  2  37  where the f o u r i e r transform  o f ty" " has  p j / w i l l be w r i t t e n f o r | < 0 , j ' / e ' i t was  shown how  the F ( r ) may  been used. For convenience  f V*\0,j)\, In the l a s t s e c t i o n  r  be found f o r the case o f  n  germanium, so the ot(cJ) can i n p r i n c i p l e be  evaluated.  iv  i ) Simple s p h e r i c a l bands. o i h a s case o f two  been e v a l u a t e d  f o r the  simple s p h e r i c a l bands. For t r a n s i t i o n s i n t o the (oo)  bound e x c i t o n s t a t e s and  by E l l i o t t ( r e f e r e n c e 11)  oC  oj AA p^^>  f o r t r a n s i t i o n s i n t o the continuum o f s t a t e s ,\ „  N f l  ,  For u n s t r a i n e d  i 4  exp  x  where  germanium, when  k > 0,  band to band t r a n s i t i o n s i s '  x ~  i s slowly  crystal,  <^/w)  and  r  to  Xz.  > 4,/jf^ the  the same as i n the  o<j (u>) axe  due  varying)  ' *jisinh  For the s t r a i n e d germanium when (hv - AE) components  \  the a b s o r p t i o n  s i n h x,  J  —___f—  absorption  unstrained  but they are d i s p l a c e d r e l a t i v e to eachother V 3k* - k " •«  along  2  the energy a x i s through and  s h i f t e d together  — ^=—^.v (see s e c t i o n a ) L  ^  through (a' + a)A.(see equation  I n the above cases, energies,  8b  G, ,G ,  are not  2  the e x c i t o n  4). binding  but numbers c a l c u l a t e d from the reduced e f f e c t i v e  e x c i t o n masses  ( s i n c e jtA i s d i f f e r e n t at k - 0) .  i i ) Simple s p h e r o i d a l valence O ^ O J ) may  bands  be c a l c u l a t e d u s i n g the v a r i a t i o n a l f u n c t i o n s  given by Kohn and L u t t i n g e r ( r e f e r e n c e 20) . F ( 0 ) n  zero only f o r s - s t a t e s where  IF (0)I^ n  i s non-  ( TTa b n ) ' . The 1  3  v a r i a t i o n a l parameters a,b must be minim i s e d w.r.t. energy and  the p r o d u c t a b z  w i l l be o f the same order  f o r both bands  38 in  the specimens used. As  i n the case o f s p h e r i c a l bands, a  s e r i e s of l i n e s i s p r e d i c t e d from the K = 0 s t a t e s o f the bands, w i t h i n t e n s i t y f a l l i n g o f f as n" . The  absorption  3  icient  Since  exciton  coeff-  f o r t r a n s i t i o n s i n t o the bound s t a t e s i s  the  factor  (J**f ) t  i s approximately the same f o r the  the two p a i r s o f bands at k = 0 f o r the s t r a i n e d specimens used, the r a t i o of i n t e n s i t i e s o f the two  iii)  exciton absorption  peaks  Degenerate valence bands In t h i s case the  expansion o f |K,r>  sum  and  over j ' must be I n c l u d e d  i n the  e v a l u a t i o n o f the i n t e n s i t y o f absorp-  t i o n a s s o c i a t e d w i t h e x c i t o n bound s t a t e s In germanium becomes very  unstrained  complicated.  From the above c o n s i d e r a t i o n s , e x c i t o n b i n d i n g energies may  not  i t i s evident  be obtained  that  the  by the a n a l y s i s o f P•  absorption estimated  curves of germanium. However, the r a t i o u s i n g equation  14, and  e f f e c t i v e masses i n equation e m p i r i c a l values  then the reduced  13 may  be obtained,  r i ' may  be  exciton using  the  o f «(u>).  d) ANALYSIS OF EXPERIMENTAL RESULTS. I) A b s o r p t i o n  peaks.  I n the a b s o r p t i o n ( f i g u r e 6)  o n l y one  spectrum o f u n s t r a i n e d  e x c i t o n peak i s seen, whereas i n the  o f s t r a i n e d germanium ( f i g u r e s 7,8,9) two knees) or 2) the  germanium  are seen. The  r a t i o of i n t e n s i t i e s  spectra  e x c i t o n peaks ( o r <*,<i( > w  °JKV)  o f the two peaks Is seen to be about 3. S i n c e  bound e x c i t o n s t a t e s , t h i s r a t i o i n d i c a t e s t h a t  (&,ir=.l  "j/,^*^'  the  for  39  observed peaks are ground s t a t e e x c i t o n peaks associated w i t h the two p a i r s o f bands. From equation 14 i t i s seen that p and p.^are o f comparable magnitude when  3yuf)~^"  j(  ( The l a b e l s  ik ir are used i n s t e a d of 1,2 since the bands were not i d e n t i f i e d ) y  i i ) Continuum, An attempt t o f i t equation 13 t o the experimental r e s u l t s was made as follows: The value of fftcj,  - 4E)  a p o i n t where  ^(UJ,)at  3x( e x c i t o n peak separation) was found. This was then  equated t o  i n equation 13, Since the separation  <*j/tjj *• ^(w,)  of the e x c i t o n peaks was small, no c o r r e c t i o n was made f o r the r e l a t i v e s h i f t o f PJI  'Pj2 J * / i  /  ,  K  w  e  r  e  and  °$J<<o)  . Then the four parameters  v a r i e d so that equation 13 gave a good f i t  to the experimental  curves f o r a l l  UJ^CJ,.  I t was found that the f i t was not unique. A good f i t of Pj^— 0 and w.-=0,024m  the data could be obtained by assuming i n agreement with Macfarlane  y  4  o  et a l (reference 5) who considered  only one c o n t r i b u t i o n t o o<(V)in unstrained germanium. However, t h i s i s not suggested by equation 14 and the observed spectra. By t a k i n g values of ju ^A_ c a l c u l a t e d from the e l e c t r o n and 4  )  hole c y c l o t r o n resonance masses, i t was found that a good f i t of the curves was obtained by t a k i n g p^ = p.^ . With other choice (  of the parameters e q u a l l y good f i t s could be obtained. Though experimental  r e s u l t s were c o n s i s t e n t w i t h the theory  no conclusive information on the e x c i t o n e f f e c t i v e masses was obtained from the data. i i i ) Values o f deformation p o t e n t i a l s Since the e x c i t o n binding energies are approximately equal and i n v a r i a n t under s t r a i n (see s e c t i o n b), the e x c i t o n  TABLE IV The deformation p o t e n t i a l s o f germanium.  Plane of specimen  [100] /  T  -AT  £  12-X 2-X 2-X 1 +X  \  T  T  0  3  T  0  1 2-X  = -(11.3 ± l ) e V shear deformation potentials  2  ° J  \ 0/  \-l-X  0.748 E" - E  1-A  T  1+1  \0 /  ( a ' + a)  [110J  [111]  2( c„ +• 2c,-_~ c + 2 c + 4c*,) a  M  E  -  E  T  J 6 J  1  b |  | d l  ^  c„ + 3CB- 2c,, c,i + c,i+ ^Qn. E" -  2-X  E  1  ITi 2(l+A)  = (4.7 + 0.5) eV  0  =0  1  1^1  T  = -(9.7 ± 1 )eV M l .  l ~ TTI 2(I+ = (2.7 ± 0.3) eV  0  *  - -(10.0 ± l ) e V S i A'+d - - JYI (\+X) 1  = (5.6 ± 0.6) eV  *The e l a s t i c constants data are taken from the data of Fine ( r e f e r e n c e 21)  41 peak separation i s equal to the s p l i t t i n g of the p  valence  bands, and the change i n the mean e x c i t o n p o s i t i o n w i t h s t r a i n i s equal to the change i n the forbidden energy gap w i t h s t r a i n . I n f i g u r e s 10,11,12, i t i s seen that f o r a l l specimen o r i e n t a t i o n s the e x c i t o n peak p o s i t i o n s s h i f t l i n e a r l y w i t h the s t r a i n a p p l i e d to the c r y s t a l . Thus the valence band s p l i t t i n g and the change of the band gap are l i n e a r w i t h s t r a i n . This behaviour was t h e o r e t i c a l l y p r e d i c t e d by Picus and B i r and K l e i n e r and Roth (references 8,9 ) . Using equations 3,4 the deformation p o t e n t i a l s were c a l c u l a t e d from the experimental E - E r e s u l t s and are tabulated i n t a b l e IV. The value of was obtained from f i g u r e s 10,11,12  —^—  P  and the values of /b_l and fd|  were c a l c u l a t e d using the r e s u l t s of specimens mounted on #0080 glass only, since these readings were much more r e l i a b l e than the others. The s t r a i n s  1 a p p l i e d to the specimens, r e f e r r e d  to the p r i n c i p a l c r y s t a l axes are evaluated i n appendix B. e) DISCUSSION OF RESULTS. I t i s seen from t a b l e IV that the mean value of ( a ' + a) i s -(10.3 ± 1 )eV per u n i t d i l a t a t i o n of the l a t t i c e . The e r r o r has been estimated as the maximum probable e r r o r . This deformation p o t e n t i a l may be w r i t t e n as a pressure c o e f f i c i e n t of  -(13.7 ± 1.5) x 10  eVcm'7dyne. This value i s i n e x c e l l e n t  agreement w i t h other workers on the pressure v a r i a t i o n of the band gap. Cardona and Paul ( r e f e r e n c e 23 ) found  ( a ' + a) was  -(13,3 ± 1.5) x 10" eVcm /dyne using t h i n specimens and low r e s o l u t i o n . Fan (reference21) only examined the absorption edge up to absorption c o e f f i c i e n t s of about 400 cm" , however the N  s h i f t at t h i s c o e f f i c i e n t was  1  -14.1 x lO'^eVcmVdyne.  42  E l e c t r i c a l and o p t i c a l measurements of the pressure v a r i a t i o n of the i n d i r e c t band gap (reference 24) show that the s h i f t of the zone boundary minima of the conduction band i s much smaller than at the centre of the B r i l l o u i n zone. Thus the e n t i r e band does not s h i f t uniformly w i t h pure l a t t i c e dilatation. The shear deformation p o t e n t i a l s , | b | and l d | were found to be (2.7 ± 0.3)eV and (4.7 + 0.5)eV per u n i t shear. The value of V'b*'+ d* i s i n e x c e l l e n t agreement w i t h these values and a f f o r d s a  a check as to the consistency of the method. Good agreement between lb I and Id I found i n t h i s work and those quoted r e c e n t l y by H a l l (reference 25) i s obtained. H a l l s t u d i e d the e f f e c t of s t r a i n on the acceptor binding energy, and found b =  - (2.4 i 0,4)eV  and  d=- = (6.0 t 1.5) eV  per u n i t shear.  I n t h i s work the signs of b and d could not be determined since the e x c i t o n peaks could not be i d e n t i f i e d w i t h the valence bands w i t h which they were associated.  43 CHAPTER IV : THE EFFECT OF LATTICE DEFECTS ON THE OPTICAL ABSORPTION SPECTRUM OF GERMANIUM. In order to draw conclusions about the pure germanium lattice from the optical absorption spectrum, i t is necessary to eliminate the effect of lattice defects in the specimens used. Probable defects in the single crystal germanium lattice are phonons, impurities and vacancies, surface states and dislocations. In this chapter, the effect of these defects on the absorption edge of the specimens will be discussed, a) PHONONS. In the foregoing experiments the lowest specimen temperature was about 90*K, so the effect of phonons will be of importance. Two major effects will be separately considered. i . Violation of the optical selection rule  t  k = 0 results,when fc  phonons are simultaneously emmitted or absorbed during an electronic transition. The selection rule for such "indirect" transitions is k  e  + k^ ± k = 0 where k p  p  is the wave-vector of the absorbed  or emmitted phonon (reference 3). In germanium, the conduction band minima are at the Brillouin zone boundary at an energy AE from the valence band maximum ( AE*< AE ) 0  t  1  SO that the onset of  band to band transitions is by these indirect transitions when hv =r AE'  and k = £ (±1,±1,±1) . These transitions are much less P  probable than the direct transitions when hv > AE  0  and Macfarlane  et al (reference 4) have shown that they do not contribute more than 10% of the total absorption. For AE' < hv <AE  0  a long-  wavelength tail to the absorption edge is seen with absorption coeffients o<<200cm"' . i i . The change of lattice vibrations with temperature produces a change in the mean potential of an electron in the lattice. The  TABLE V Comparison of the t h e o r e t i c a l value of f^^f-y w i t h experiment.  T°K  c* xl0 cm/°C <5  Novikova reference 16  cT(a' f a) milli-eV/r t h i s work  \dT  (—)  IP  Macfarlane reference 5  Antoncik reference 25  4  0  0  0.01 »*v/t 0.01/"e//t  40  0  0  0.08  0.08  0.09  80  1.3  0.04  0.18  0.14  0.11  100  2.3  0.07  0.23  0.16  0.13  200  4.8  0.15  0.37  0.22  0.14  300  5.8  0.18  0.42  0.24  0.16  0.01 m V/°c £  c< i s the l i n e a r expansion c o e f f i c i e n t and X i s the volume c o e f f i c i e n t .  45 e f f e c t causes a s h i f t of the e n t i r e absorption edge w i t h temperature. The magnitude of the s h i f t i n germanium has been t h e o r e t i c a l l y evaluated by Antoncik (reference 25). I t i s i n t e r e s t i n g to compare the r e s u l t s of Antonfcik w i t h the experimental r e s u l t s of t h i s work and the r e s u l t s of Macfarlane et a l (reference 5) edge.  on the temperature s h i f t of the absorption  Now  to 300°K. The are  t a b u l a t e d i n t a b l e V.  results  I t i s seen t h a t there i s q u i t e good  agreement between theory and experiment at a l l temperatures, i f i t i s assumed that ( a b) IMPURITIES AND  1  4 a) i s not temperature dependent,  VACANCIES.  The r e s i s t i v i t y of the germanium c r y s t a l s at 23°C. before specimen p r e p a r a t i o n was  about 60 ohm-cm. The  resistivity  increased by a f a c t o r of about 100 when the c r y s t a l was cooled to 90°K. This i n d i c a t e d that the i m p u r i t y and vacancy content  was  (2.  l e s s than 10  defects/cc. ( I t i s expected that the e f f e c t of  vacancies would be s i m i l a r to the e f f e c t of acceptor i m p u r i t i e s ) . I t i s u n l i k e l y that any i m p u r i t i e s were introduced i n t o the c r y s t a l s during specimen p r e p a r a t i o n , since at no time during the p r e p a r a t i o n was  the c r y s t a l temperature r a i s e d above 80°C.  Copper i s the only common element that would d i f f u s e i n t o germanium below 80°C. (even t h i s process i s slow) so care  was  taken that the specimens d i d not come i n t o contact w i t h copper. I t i s a l s o u n l i k e l y that the vacancy concentration increased  46 appreciably during specimen p r e p a r a t i o n and during the s t r a i n i n g of the specimen at these low temperatures. The absorption c r o s s - s e c t i o n of i m p u r i t i e s i n germanium i s t y p i c a l l y 10~' to IO ' cm (reference 26). I f the same c r o s s - s e c t i o n 4  -  6  2  -12  i s taken f o r vacancies then w i t h only 10  defects/cc the i m p u r i t y  and vacancy absorption would be n e g l i g i b l e , even i f i t were i n the s p e c t r a l region examined. The spectra of three unstrained specimens from three d i f f e r e n t c r y s t a l s were a l l i d e n t i c a l , so the f i n e s t r u c t u r e i n the edge was not a property of one c r y s t a l only. One specimen was prepared (10  ,S  from 2 ohm-cm N-type germanium  i m p u r i t i e s / c c ) . The absorption edge was somewhat broader,  but the e x c i t o n peak was s t i l l v i s i b l e at the same photon energy as i n the purer m a t e r i a l , though l e s s intense. The broadening may be a t t r i b u t e d to the decrease i n l i f e t i m e of the e x c i t o n i n the impure l a t t i c e , c) SURFACE STATES At the surface of a c r y s t a l the p e r i o d i c l a t t i c e i s terminated and band theory i s no longer v a l i d . The surface may be regarded as a plane of atoms w i t h unpaired bonds i n t o which an e l e c t r o n may be e x c i t e d . I f an absorption c r o s s - s e c t i o n assigned to a surface s t a t e then f o r N states/cm  2  o- i s  the t o t a l  absorption at the surface i s crN. Since there are only two surfaces of the c r y s t a l traversed by the l i g h t beam, the normalised transmission through a specimen i s T  n  =  ( 1 - crN ) exp-ad a  x  C l e a r l y crN < 1. I t i s apparent that the f a c t o r  ( 1 - crN)  which  takes i n t o account the surface states, i s independent of the specimen thickness.  47 When o< = 0 i n the bulk of the specimen, the transmission i s a function of cr only. I t has been shown by Dexter ( reference  27)  that such unpaired bonds lead to a long-wavelength t a i l to the absorption edge*. The value of cr would not be s i g n i f i c a n t l y a f f e c t e d by s t r a i n so that no change i n the "surface s t a t e spectrum" would be expected on s t r a i n i n g the specimens. Thus i t appears from f i g u r e s 6 to 9 that surface states d i d not a f f e c t the o p t i c a l absorption of the specimens. Thus crN  1. I t i s f o r t h i s  reason  that the surface states have not n o t i c a b l y a f f e c t e d previous absorption measurements, d) DISLOCATIONS. I n i t i a l l y , the bulk germanium c r y s t a l s had a d i s l o c a t i o n density of about 5000 l i n e s / c m . During the specimen p r e p a r a t i o n 2  i t i s p o s s i b l e that d i s l o c a t i o n s may have been introduced i n t o the c r y s t a l as a r e s u l t of abrasion damage. A few of the specimens were etched by p l a c i n g them face-downwards i n a bubble of f a s t CP4 on a polyethylene sheet. This avoided etching the glass substrates. The specimens could only be etched f o r about 20 seconds before they were completely d i s s o l v e d . These c o n d i t i o n s were not adequate f o r the observation of etch p i t s , but even w i t h t h i s b r i e f etch,dots appeared randomly on the specimen at a density of about 10  s  to 1 0  6  per cm Cwhen viewed w i t h a microscope of 500x magnific2  ation) I t i s p o s s i b l e that these marks were an i n d i c a t i o n of the d i s l o c a t i o n density. The e f f e c t of d i s l o c a t i o n s i s two-fold, ( i ) d i s l o c a t i o n s act * Dexter appears to i n t e r p r e t the magnitude of the absorption associated With surface states i n c o r r e c t l y . I t seems that he considers the volume concentration of surface s t a t e s and assigns to them an absorption c o e f f i c i e n t . The absorption by these states w i l l thus depend on specimen thickness.  48 as acceptors since along a d i s l o c a t i o n l i n e there i s a l i n e o f atoms w i t h unpaired bonds. Dexter (reference 27) has made an estimate o f the absorption by e l e c t r o n s e x c i t e d i n t o these bonds. The absorption appears as a long-wavelength t a i l t o the edge, w i t h a maximum absorption c o e f f i c i e n t of 100cm"' f o r a d i s l o c a t i o n density o f 10  lines/cm ". For lower d i s l o c a t i o n d e n s i t i e s , 2  this  e f f e c t would be n e g l i g i b l e i n the s p e c t r a l region examined i n t h i s experiment. ( i i ) The second e f f e c t o f d i s l o c a t i o n s i s due t o the d i s t o r t i o n of the c r y s t a l l a t t i c e around a d i s l o c a t i o n . This e f f e c t w i l l be discussed i n d e t a i l i n the f o l l o w i n g  section:  INTERNAL STRAINS. The d i s t o r t i o n o f a c r y s t a l l a t t i c e associated w i t h a d i s l o c a t i o n c o n s i s t s o f severe atomic m i s f i t around the d i s l o c a t i o n (bad material) and e l a s t i c s t r a i n s i n the surrounding good material. I n the germanium specimens i n t e r n a l s t r a i n s w i l l be frozen i n as a r e s u l t of d i s l o c a t i o n s introduced during c r y s t a l growth and specimen p r e p a r a t i o n . A t any p o i n t i n the c r y s t a l there w i l l be an i n t e r n a l s t r a i n £T i n a d d i t i o n t o the e x t e r n a l a p p l i e d s t r a i n , £  w i t h which there i s associated a s h i f t i n the absorption edge and a s p l i t t i n g o f the valence bands. I n t h i s s e c t i o n an order of magnitude estimate w i l l be made o f the e f f e c t o f the i n t e r n a l s t r a i n s on the absorption edge of a specimen. The normalised transmission o f l i g h t o f photon energy hy through an area A o f specimen o f thickness d i s T  where crystal.  n  =  e  x  p  "Uo^ ^ i h  dz  i s the absorption c o e f f i c i e n t at a p o i n t (xyz) i n the  49 The e f f e c t of  i s to s h i f t the two absorption edges,  corresponding to t r a n s i t i o n s from the two p. valence bands to the conduction band, through energies AE +  AE  =  +  aZiu  ± h  given by (see chapter I I I )  [b Z(£n-tJ+2d £ Ehl' X  2  i,i  L  1  J  /Z  iJ  where ( a ' + a) has been abbreviated to a and the f i j are i n t e r n a l s t r a i n components. The displacement  u- of a p o i n t i n an i s o t r o p i c c r y s t a l at t  a distance r from the d i s l o c a t i o n has been c a l c u l a t e d by Read (reference 28). From these equations the s t r a i n s were r e a d i l y calculated.  For an estimate of the e f f e c t of f- on the absorption  edge of germanium these equations are used and b to ^ d .(see t a b l e IV ) . I t was 2  AE±(r,fi) = where and  ±  bf (0)] 2  f 19 cos* £ - 21cos 6 + 3cos 0 *• 4Xcos*0 +1 + 4-X* - 4 a 4  9  1  and y3  found that  [af^B)  ^(0) _ f (Q) =  i s set equal  l  i s the burghers vector of the d i s l o c a t i o n ,  x  y  9 i s the angle  between r~ and the s l i p plane of the d i s l o c a t i o n . The absorption edges a s s o c i a t e d w i t h the two p a i r s of bands w i l l be represented by step functions: 0  for  hv < E  0  - ^E+.  oe, f o r hv > E  e  - AE +  and «*= 0  for  hv < E  0  - AE_  * - <* f o r hv ) E  8  - 4E_  oc =  E  0  a  i s the forbidden energy gap i n u n s t r a i n e d germanium at k= 0  ( E = E (0,0) ) 0  e  The two cases of low and high d i s l o c a t i o n d e n s i t i e s i n the t h i n germanium specimens w i l l be t r e a t e d separately, i ) Low d i s l o c a t i o n density. ( < 10 * l i n e s /cm*") . For low d i s l o c a t i o n d e n s i t i e s the number of d i s l o c a t i o n s i n the plane of the specimens i s s m a l l , f o r specimens 10>«  thick.  The  50 model assumed f o r t h i s c a l c u l a t i o n i s a r e g u l a r l a t t i c e o f s t r a i g h t d i s l o c a t i o n s with equal numbers o f p o s i t i v e and negative dislocations perpendicular being  separated  edge  t o the plane o f the specimen, each  from i t s n e a r e s t  neighbour by a d i s t a n c e h. With  t h i s model there i s no v a r i a t i o n o f s t r a i n through the specimen thickness The  so  T  n ~  £ /  r d  fl  # d r exp-<* 0(hv) d r  c r y s t a l i s t r e a t e d as an e l a s t i c continuum, r a t h e r than  d i s c r e t e l a t t i c e p o i n t s so t h a t sums over elemental volumes may be r e p l a c e d by i n t e g r a l s . When  (E„- hv) > 0  a b s o r p t i o n w i l l only occur i n the f r a c t i o n a l  volume o f the c r y s t a l where ( t r e a t i n g the a b s o r p t i o n from the two bands s e p a r a t e l y ) ( E - hv) < and e  [af (B)+  1  f af^B)  ( E - hv) < 0  ...equations 1 , 2.  bf (0)] 2  then  exp-« (hv) d = exp-*,d  - bf (0)]  then  exp-o^(hy) d = exp-« d  1  _  2  For both c o n t r i b u t i o n s , when ( E - hv) > 0  then T  rff  2  - j ^ . [ af^S)  ±  bf (G)] 2  exp-<v (hv) d — 1 . Thus the t r a n s m i s s i o n o f the specimen i s rN f N f N f' - J d©f £ exp-(«, + <*,) d r d r + £ exp-<*,d J r d r + 7 J r d r rg  x  n  contributions  n  r  + N J rdr + N J rdr exp-^d J o  where r , , r , z  are d e f i n e d by equations 1 and 2 above, r ^ i s d e f i n e d  by the requirement that  NTrr£ =-1,  r  b  i s the r a d i u s o f bad m a t e r i a l  around a d i s l o c a t i o n and cv i s the a b s o r p t i o n  coefficient  t  associated with first  the bad m a t e r i a l . The f a c t o r % appears i n the  three i n t e g r a l s s i n c e f o r o n l y one h a l f o f the c r o s s -  s e c t i o n a l area o f the specimens i s the product p o s i t i v e . The c h o i c e  r, >• r , has been made 2  For low d i s l o c a t i o n d e n s i t i e s Now Using  f  d e f rdr ^ = 0.6  =  ^  ^[af-^(t9) ±  bf ((9)J 2  arbitrarily.  r <gr r ^ , so we can l e t r — » 0 . b  b  j^jaf^)  this i n t e g r a l gives  - bf (9)f 2  approximately  d«>  51  8TT*(E - hv)'  L  6  For germanium  T  a ~ lOeV, b ~ 3eV  so the term i n ab i s small  compared w i t h the other terms, thus a n d T  T  n=L  1  N//(  j dd j_ r d r  2  J d© J r d r  ^  Q  a - + 18b ") f , 64TT( E. - h v f L  '  J  2  °  1  The procedure f o r (hv - E ) ^ 0  ^vN^lT  exp-(*. + « O d J J  i s s i m i l a r . I n t h i s case  e  absorption occurs i n the whole c r y s t a l except that f o r which (hv - E. ) <|4E |. I t i s found that ±  This f u n c t i o n i s not defined near ( E - hv) = 0  since i n t h i s  0  treatment there i s no region of the c r y s t a l unaffected by d i s l o c a t i o n s . Because of the symmetry of the f u n c t i o n about E , e  f o r c o n t i n u i t y at E„ For  hv^-Eo ,  T — n  >  T  exp-(*. + O  Q  =  \ [ 1 - exp-(<*, + <*,) d  d  and f or hv « E  0  ;  ~\  T -* 1  as  n  expected. The e f f e c t of d i s l o c a t i o n s i s to broaden the  absorption  edge w i t h no nett s h i f t of the edge. A h a l f - w i d t h of the broadening may  be defined by the value of ( E t  i.e. where we have used  ( E.- h v ) . . %  w  d t h  d =• 10,* , ("x'. + O i ) =  hv ) when  -5  T= n  %  x lO'V*  4000cm* , ,/3-<100} l a t t i c e 1  o  vector  - 4 A.  To obtain a broadening h a l f width equal to the monochromator r e s o l u t i o n of 0.2 m i l l i - e V , the specimens would need a d i s l o c a t i o n density of ^10  lines/cm . (This i s outside the l i m i t of the model)  I t i s p o s s i b l e that the observed v a r i a t i o n i n peak height was  due  to t h i s broadening mechanism. The p o s i t i o n of the e x c i t o n peak i n the absorption spectrum of the specimens would not be a f f e c t e d by these i n t e r n a l s t r a i n s .  52 i i . High d i s l o c a t i o n density ( > 10 lines/cm*) . With a high d i s l o c a t i o n density there w i l l be a considerable v a r i a t i o n of s t r a i n through the specimen thickness. A random d i s t r i b u t i o n of s t r a i g h t d i s l o c a t i o n s w i l l be assumed, no other r e s t r i c t i o n s being p l a c e d on the model. Dexter (reference 27) has shown how d i s l o c a t i o n s may account f o r a long wavelength t a i l , or apparent s h i f t o f the absorption edge, using t h i s model, and h i s treatment w i l l be extended f o r the case o f germanium. For a random d i s t r i b u t i o n of d i s l o c a t i o n s , the absorption of a c r y s t a l may be described by a s i n g l e absorption c o e f f i c i e n t , equal to the mean absorption c o e f f i c i e n t o f the c r y s t a l , averaged over the volume.  T  where  *^ x  ^ j  =  n  z  -  dA exp- \_ Jtf,^ hv) dzj = exp-?*,id  \  f & ^dV y  x  When ( E ~ hv) > 0 absorption w i l l only occur i n the f r a c t i o n a l 0  ( E - hv) < ^E+  volume of the c r y s t a l f o r which  0  - again the two  bands being considered separately. Now we have = %Ny«,j d© J r d r + %N <x \ d$ J^rdr + Nj d& j o/ rdr ft  r  v  a  q  0  Q  fc  where N = 3N i s the d i s l o c a t i o n density i n l i n e s / u n i t volume, and /  r , r , were defined by equations 1 and 2 above. The i n t e g r a l s have (  2  already been evaluated and i t i s found that 64TT( Ef-hvf 3  ( a  %  ^ H * ^ )  '  N  [ 2  («• ^ )  - £"d$ J « r d r ] 0  4  S i m i l a r l y f o r (hy - E„) > 0 5. .(-,.«0 w  [ l-  Zfvl^f  (  a  Z  +  18fe,  >  +  N  ]  i i*l'<«*  The l a s t terms have been i n c l u d e d here since the e f f e c t o f bad m a t e r i a l i s not known. For very high d i s l o c a t i o n d e n s i t i e s the volume o f bad m a t e r i a l may not be n e g l i g i b l e compared w i t h the good m a t e r i a l . For a s o l i d t o e x h i b i t c r y s t a l l i n e character l a t t i c e periodicity,-then  2 r must be l e s s than r ^ . For r ~ 1 0 k b  t  53  '2 t h i s requirement i s N < 10  2.  lines/cm.  For an estimate o f o^,*. the l a s t terms i n the above equations, i n v o l v i n g the bad m a t e r i a l w i l l  be n e g l e c t e d . A g a i n i t i s c l e a r  t h a t the equations are not d e f i n e d a t E ^ s r h v , but f o r c o n t i n u i t y at E<s , o( ^= x  +  a t E . T h i s i s expected s i n c e h a l f the 0  c r y s t a l i s under s t r a i n  and the other h a l f under s t r a i n -£ .  Again i t i s c l e a r t h a t the e f f e c t o f the d i s l o c a t i o n s i s t o broaden  the a b s o r p t i o n edge. A h a l f - w i d t h o f the broadening may  be d e f i n e d as the value o f ( E - h v )  when  0  i.e.  ( E„- h v ) . . %  w  d t h  =  < * i + <*».)  .  ~ 10-V*  u s i n g the same c o n s t a n t s as b e f o r e . When T = \ , ( E - hv ) ~ 10 n  6  N  which i s approximately the same r e s u l t as w i t h the p r e v i o u s model. From t h i s a n a l y s i s i t i s seen t h a t the a b s o r p t i o n s p e c t r a o f the specimens used i n these experiments were not a f f e c t e d by d i s l o c a t i o n s s i n c e no measureable  broadening o f the edge was  o b t a i n e d . I n o r d e r t o observe the e f f e c t o f d i s l o c a t i o n s e x p e r i m e n t a l l y , i t would be n e c e s s a r y t o i n t r o d u c e more than 10 lines/cm  i n t o the c r y s t a l . The e a s i e s t way o f p r o d u c i n g h i g h  d i s l o c a t i o n d e n s i t i e s i s t o evaporate germanium f i l m s onto a s u b s t r a t e i n a p o l y c r y s t a l l i n e s t a t e . The g r a i n boundaries may be r e p r e s e n t e d by d i s l o c a t i o n s ( r e f e r e n c e 28). e) EVAPORATED FILMS OF GERMANIUM Three specimens o f germanium were p r e p a r e d by e v a p o r a t i n g i n t r i n s i c germanium from a s m a l l g r a p h i t e boat, heated by j o u l e h e a t i n g w i t h a heavy c u r r e n t . The b e l l - j a r was evacuated t o a p r e s s u r e o f about 10 mm Hg. The g l a s s s u b s t r a t e s were cleaned, 5  p r i o r t o e v a p o r a t i o n , by i o n bombardment. Two specimens  5.Lvtand  9 . L/t t h i c k were evaporated onto c o l d s u b s t r a t e s and one specimen 6.4/A t h i c k was evaporated onto a s u b s t r a t e , heated from the back  54  w i t h a heating c o i l . "Thick" f i l m s were prepared i n order t o make comparison with the s i n g l e c r y s t a l specimens. The transmission of the f i l m s was examined from 0.5eV to l.OeV. A very broad absorption edge was seen i n a l l the specimens, i n d i c a t i n g that the f i l m s were c r y s t a l l i n e , but no f i n e s t r u c t u r e was seen. The f i l m thicknesses were found by by measuring the i n t e r f e r e n c e f r i n g e s where the f i l m s were transparent, and using the r e f a c t i v e index data of B r a t t a i n and Briggs (reference 29). The f r i n g e s were averaged f o r the e v a l u a t i o n of the absorption data. The two specimens prepared on c o l d substrates showed i d e n t i c a l absorption, yet the specimen prepared on a hot substrate was n o t i c a b l y different,as seen i n F i g 15. The s i n g l e c r y s t a l absorption edge i s shown f o r comparison. The two absorption  curves  of the evaporated specimens i n t e r s e c t the s i n g l e c r y s t a l edge at about 1800cm~' . The observations are i n e x c e l l e n t q u a l i t a t i v e agreement w i t h the theory of c r y s t a l s with a high d i s l o c a t i o n density. Using ii  the value  N = 9x10 v  «  lines/cm , the curve of  vs ( E - hv) e  p r e d i c t e d i n the l a s t s e c t i o n i s a l s o p l o t t e d i n f i g 15. On the high energy side of E  0  the experimental curves are steeper than  p r e d i c t e d as would be expected since the theory assumed a step f u n c t i o n absorption edge f o r the unstrained c r y s t a l . I t was n o t i c e d that apart from the broad absorption edge, the evaporated f i l m s had the same o p t i c a l p r o p e r t i e s as the s i n g l e c r y s t a l s . The r e f l e c t i v i t y was the same, the r e f r a c t i v e index increased 2% from 300°K to 90°K (deduced from the phase s h i f t of the i n t e r f e r e n c e fringes) and the s h i f t of the absorpt i o n edge between these temperatures was the same as the s i n g l e crystal.  Figure 15. The absorption spectrum of evaporated fims o f germanium a) F u l l l i n e ; germanium evaporated onto a c o l d substrate  -•  /4000  b) Broken l i n e : germanium evaporated onto a hot substrate c) S i n g l e c r y s t a l absorption f o r comparison. *  P l o t o f the t h e o r e t i c a l curve f o r N = 9x10" l i n e s / c m . 1  v  -3000  Temperature = 300° K  Ml O  i  •a  Absorption coefficient  OQ  ro  ••1000  -,0.2  Photon energy (referred t o milli-eV  M  ,  JL42  55 Thus i t appears that the absorption edge can give an i n d i c a t i o n of the q u a l i t y of an evaporated f i l m . From the curves of f i g u r e 15, i t i s seen that the f i l m prepared on a hot substrate was l e s s s t r a i n e d than the others as might be expected, f) DISCUSSION. From the discussions of t h i s chapter i t i s seen that the l a t t i c e defects i n the specimens d i d not s i g n i f i c a n t l y a f f e c t the absorption spectra. The small v a r i a t i o n s i n the e x c i t o n peak heights may be accounted f o r by l a t t i c e imperfections, yet the large broadening of the absorption peaks i n the specimens mounted on #7900 glass could not be accounted f o r .  56 CHAPTER V  : CONCLUSIONS.  The study of the e f f e c t of s t r a i n on the e x c i t o n spectrum of germanium afforded a very d i r e c t method of obtaining the deformation p o t e n t i a l s of germanium: |b| = ( 2.7 ± 0.3 ) eV/unit <100>shear, |d|=(  4.7 ± 0.5 ) eV/unit <111>shear,  ( a ' - a) = -(10.3 ± 1.0) eV/unit d i l a t a t i o n of the l a t t i c e . The experiment i s i n h e r e n t l y quite accurate because o f the sharpness of the e x c i t o n absorption peaks and the f a c t that the e x c i t o n binding energy i s determined p r i m a r i l y by the small e l e c t r o n mass. An experimental check of the e x c i t o n binding energies i n s t r a i n e d and unstrained germanium could i n p r i n c i p l e be made by i n v e s t i g a t i n g the Stark e f f e c t of the e x c i t o n spectrum. The Stark e f f e c t depends only on the e x c i t o n r a d i u s , which i n turn depends only on the reduced e f f e c t i v e e x c i t o n mass. The study of the e f f e c t of l a t t i c e defects on the absorption spectrum i n d i c a t e d that the experimental r e s u l t s of t h i s work were p r o p e r t i e s of the p e r f e c t germanium l a t t i c e , and were not a f f e c t e d by the l a t t i c e imperfections. On the other hand the broad absorption edge observed i n evaporated f i l m s of germanium could be accounted f o r i n terms of i n t e r n a l s t r a i n s due to d i s l o c a t i o n s . Measurements of the absorption edge could prove a u s e f u l t o o l i n i n v e s t i g a t i n g the s t r u c t u r e of f i l m s . Further experiments,  c o r r e l a t i n g i n t e r n a l s t r a i n s w i t h the c r y s t a l l i t e  s i z e of the f i l m s are c l e a r l y  necessary.  I t i s a l s o i n t e r e s t i n g that the theory of Antoncik on the change of the energy gap w i t h temperature i s i n agreement w i t h experimental  results.  57 APPENDIX A:  CALCULATION OF STRAINS ON THE SPECIMENS  Basic e l a s t i c i t y theory. To avoid confusion w i t h n o t a t i o n , a r i s i n g from the d i f f e r e n c e between conventional e l a s t i c i t y theory and the tensor formulation, the basis of the theory w i l l f i r s t be o u t l i n e d . * C a r t e s i a n axes are used throughout the d i s c u s s i o n . When a force F- i s a p p l i e d t o a s o l i d , the stresses i n t  the s o l i d are defined by T  ij  - AAj^O  £A"j  A  1  where AF. i s the force a c t i n g over elemental area AAjof the s o l i d . The T^ are compressional (-ve) or t e n s i l e (+ve) stresses and the Tjj  are shear s t r e s s e s . I f a p o i n t P o f p o s i t i o n vector x i n the s o l i d i s d i s p l a c e d  through a distance u" under the a c t i o n o f a s t r e s s , t o the p o s i t i o n x' =• x b  y  u" , then the deformation of the s o l i d may be described  straps  f  M  =  (  %  _S«)  +  A  Hooke's law i s that f o r small deformations  2  o f a s o l i d , the  s t r e s s i s d i r e c t l y p r o p o r t i o n a l t o the s t r a i n T  ij  ~  'jkl  ?kt  c  The C^^ are e l a s t i c constants and C may be t r e a t e d as a tensor of the f o u r t h order. Under a general r o t a t i o n of axes Tjj , Em and C,^  will  change. I f the r o t a t i o n operator i s c* the the s t r a i n tensor transforms as  £ - i = ot ><* > £ h  L  kk  tt  kl  or  £  kt  =  <*' g' > / t(  k  t  A3  and the e l a s t i c constants tensor transforms as  CiJW -  CukL  *U'«j;'o( * Mk  o< ' u  A4  * See, f o r example, H.B.Huntingdon: S o l i d State Physics vol.7 p.213 McGraw H i l l p u b l i c a t i o n , e d i t e d by S e i t z and T u r n b u l l .  58 Since T i s symmetrical w.r.t. interchange o f i and j and £ i s symmetrical w.r.t. interchange o f k and 1, there are only s i x independent components o f each o f £ and T, C i s symmetric w.r.t. interchange of i and j , k and 1, i j and k l , so that there are are only 21 independent components o f C. Thus T and £ may be w r i t t e n as s i x - v e c t o r s and C as a 6x6 m a t r i x i n accordance w i t h conventional theory, and r e l a b l e the double s u b s c r i p t w i t h a single subscript: 12  6. The  11 -"1,  22  2,  33  3,  23  obtained by t h i s r e l a b l i n g procedure e  same as the conventional s t r a i n s e*.. I n f a c t  Y  4,  31  5,  are not the f o r v = l,2,3,  and £ =. %e^ f o r v = 4,5,6. v  Conventional e l a s t i c i t y theory has the disadvantage that the e and c are not tensors and transformation o f these q u a n t i t i e s Y  MV  requires an i n v o l v e d treatment. CALCULATION OF STRAINS ON SPECIMENS.  (i) G  Consider a specimen of thickness d , area A = a* as shown i n F i g A ( i ) , and  C  choose c a r t e s i a n axes so that the plane a  ABGH i s the z = 0 plane. Suppose a s t r e s s T  y  Ei  i s a p p l i e d t o the surface ABGH so that the displacement o f a p o i n t i n the plane i s  ^sA  F  u = (u° ,ut ,0 ) . A p o i n t on the opposite 0  face of the specimen CDEF, w i l l i n general be d i s p l a c e d through  u  d  = (u, , u , u,) t  where the u*( may be w r i t t e n i n terms of the u? by a T a y l o r expansion J u  I ~  0 u  i  +  ( ^  u  i \d  +  ••••  •• -  The specimen w i l l be deformed i n c r o s s s e c t i o n as shown i n F i g A ( i i ) . For very  F Fig. A  59 t h i n specimens the terms i n d  and higher may be neglected. Thus  the s t r a i n at z = d i n terms of the s t r a i n at z - 0 i s , from equations A l and A5 St  =  ,A6  £° + < * 6 j l d  This i m p l i e s that B'C' and A'D' are s t r a i g h t l i n e s . In t h i s s e c t i o n i t w i l l be shown that the s t r a i n i n the t h i n specimens i s uniform  to a good approximation, i . e . (V, £ j ) d  I n approximation A6 t h i s i m p l i e s For convenience  L  £^ « £°j  £^  0  ... .A7  the plane of the specimen w i l l be taken as  a [100] c r y s t a l plane. This choice w i l l not a f f e c t the f i n a l r e s u l t . I n t h i s coordinate frame  C =•  'C c  Ci C,x  II  I  X  Cn C, J C|t C||  n  a  -Cii.  ic 0 0 !0 c 0 10 0 c< 4 4  0  4 4  In t h i s work the s t r a i n i s uniform i n the z = 0 plane The c o n d i t i o n that T^ = 0 i s £„ = - 2£'* so we have  T = T (l  Now  V f.,< 2V C,z =  Thus  Su = fn/ 1 1  x  2  22  = [c„ + c _l7  On  4;  %  fn-0  ^  ] £„  ( n e g l e c t i n g higher orders)  tt£<3  a % £„ = 7, ?„ = o and  (%  £cj  d ^f' / 1 3  o  -2q/c„| 0  Now since d ~ lO'^cm £„ )•> 4x10" c°ii  .  f,  3  and a ~ 1cm the c o n d i t i o n A7 i s s a t i s f i e d i f  w i l l not be greater than £„ so t h i s i n e q u a l i t y  i s s a t i s f i e d and the s t r a i n i s uniform through the specimen. The shear s t r e s s T Since Thus  where F i s the force a p p l i e d along OB'. 2F i s uniform through the specimen T,, = 2cf^ £„ _ ad.K i„ ~ 2x10 £„ f o r _ ad 23  =  '44  44  germanium. For convenience K has been s u b s t i t u t e d f o r  c„ + c,»- 2c.*" OH  60 So we have  (V £.j) d ~ 1 0 " ^ j  and  S  2  0  1  £ij=^f  \  -2c /c 0 u  0  (l  /  Now suppose that the specimen i s r i g i d l y attached to a t h i c k i s o t r o p i c substrate, and the specimen i s s t r a i n e d by the d i f f e r e n t i a l c o n t r a c t i o n on c o o l i n g . The angular deformation of the T  substrate at the i n t e r f a c e i s since T i s continuous across the i n t e r f a c e , c' i s an e l a s t i c modulus of the substrate m a t e r i a l . T ad K Now -4 = —T- —, E„ which i s a very small deformation. 23  s  C  44  A  C  4 4  Thus the substrate i s not appreciably a f f e c t e d by the deformation of the germanium specimen. In p r a c t i c e the problem i s a three l a y e r problem: a t h i c k substrate, a t h i n l a y e r of epoxy and the germanium specimen. By reasoning  s i m i l a r to that above, the s t r a i n £„ of the specimen i s  unaffected by the epoxy and i s uniform through the specimen, i f the epoxy obeys the r e l a t i o n i / K_ \ T' 4? T' a ( c J epoxy 2 3 ^ n where T' i s now given by the r e l a t i o n 1  1  23  T» = 23 23  £ [ (_T Kd) . + (£,Kd) A " ' specimen epoxy L v  M  J  j  For specimens and epoxy l a y e r s of 10/t thickness, and since £„ w i l l be of the same order f o r both m a t e r i a l s , then t h i s c o n d i t i o n implies  /K \ \c j 4t  I* epoxy L  ±  +  K  1 K poxy g e  5 l Q  J  e  The epoxy used was hard - e s p e c i a l l y at reduced temperatures so i t i s probable that t h i s c o n d i t i o n was obeyed. The f o l l o w i n g e x p e r i ments were performed to determine whether the i n e q u a l i t y holds, a) The germanium specimen was glued to the substrate w i t h d i f f e r e n t glue thicknesses ( < 25/* ) .  61 b) Specimens  o f d i f f e r e n t t h i c k n e s s e s were p r e p a r e d  ( < 18^) .  c) Specimens were mounted w i t h both s u r f a c e s glued to the same substrate material. I n none o f these experiments was a specimen d i f f e r e n t  the a b s o r p t i o n spectrum o f  from the o t h e r s . The e x c i t o n peak p o s i t i o n s  were r e p r o d u c i b l e . The sharpness o f the e x c i t o n peaks ( f o r specimens under tension) was uniform  an i n d i c a t i o n t h a t the s t r a i n  was  through the specimens. The experiments a,b,c, thus  i n d i c a t e t h a t the above i n e q u a l i t y h o l d s . Since  the s u b s t r a t e was  n e g l i g i b l y deformed by the specimen,  the s t r a i n i n the germanium was where AT and the  was  -  -  ( ^ s u b s t r a t e " °^ge )A  the temperature change on c o o l i n g the specimen,  °^ s are expansion c o e f f i c i e n t s  (linear).  T  62 APPENDIX B : TRANSFORMATION OF ELASTIC CONSTANTS UNDER ROTATION OF AXES For a l l the germanium specimens used i n the experiment, n e c e s s a r y to know the s t r a i n s c o - o r d i n a t e system We  (see  a s s o c i a t e d w i t h the p r i n c i p a l  z-direction  crystal  axes.  i s then normal to the a p p l i e d  These axes are the p r i n c i p a l  Appendix A ) . The  and i n the z d i r e c t i o n  There  axes o f the s t r a i n t e n s o r  s t r a i n i n the x-y p l a n e i s g i v e n by £ --XT  is  where X i s found from  3 3  c o n d i t i o n t h a t normal s t r e s s  T  £ , f =T a,  (  1  vanishes.  3 3  p l a n e i s a [110]  a [ l l l ] p l a n e . Since a l l ( l l l j  any convenient p l a n e may  il  the  three cases to c o n s i d e r : When the x-y p l a n e i s a  p l a n e , the x'-y is  a p p l i e d to the c r y s t a l i n the  can take the plane o f the specimen to be the x-y p l a n e  i n a l l c a s e s . The stress.  i t is  plane,and when the x"-y"  [001]  plane  p l a n e s are e l a s t i c a l l y e q u i v a l e n t  be chosen  f o r the c a l c u l a t i o n .  Similarly  f o r the f l l O ^ p l a n e s . axes are the < 1 0 0 ) ,  a) x,y,z,  (010>, < 0 0 l ) ,  C has c u b i c symmetry i n these axes:  /c  crystallographic M  Co  C =  c„  c  C  C  n  C u _Cji  a  c |c«, 0  Then the  £  kL  are d i r e c t l y  U s i n g Hookes Law we  find  £„- £» = T , t  ^ A  b ) x ' , y ' , z , axes are < 1 0 1 ) f  =  £,,= -AT  x',y',z',  , <010> , < T 0 1 ) ,  about  axes to the  axes i s a r o t a t i o n  the y a x i s .  Thus  0  4tl  0  0  c j  , £„ = £ „ *  = 0  !  0  c  4 4  ^£ia c„  The p l a n e c o n s i d e r e d i s [ 1 0 T J . Transformation from the x,y,z,  0  0  h  0  V  0  ( l  through  - —  crystal  axes.  axes.  63  ~  ./1 0 1 0 J2 0 01  *  We have f ' =  .11 0-1 - ~\ 0 42 0  and  = s  •Jl  1 0 1 £ = i-n 2 T 1-A 2 0 UX 0 /  so from equation A3,  / T\ T -AT  0  \ o/  In order t o f i n d X  C'must be known. For T = 0 we r e q u i r e 33  ( from Hooke s Law) :  • Bl  1  ^  C  33ll  +  C  '  22 ~  33  )T=  0  So the complete tensor C' need not be evaluated. From equation A4 i t i s found that C 33/. = % ( 2C„ + 2C, - 4C ) ,  = k( C„ + C,» + 2C )  1  2  C 3322 ' Thus  ~  C  ,  3333  44  44  i  X =  C., + C. + 2C, 4  c r y s t a l axes. Transformation from the x,y,z, axes t o the x",y",z", axes i s a r o t a t i o n o f - ~" about the y-axis followed by a r o t a t i o n of -tan~'^= about the x' a x i s . Here we have ©c = j / 75  0 J3  and  ot~  = . IJ3  -1 -Jl  0 2 -ji * 1/3 1 72/ i  As before, from equation A3 i t i s found that £ — X I 2-X 2-A 2-A  UX 14A  \ 1 + *' And t o f i n d X we use equations B l and A4 A  2( C H  +•  2Cii-  2Cda)  64  Reference no,  BIBLIOGRAPHY  1.  G.Wannier, Phys. Rev. ,52, 191, (1937).  2.  G.Dresselhaus, J.Phys. Chem. S o l i d s , I, 14, (19 56).  3.  L.H.Hall, J.Bardeen,F.J.Blatt, Phys. Rev. 95, 559 , ~~(19 56). G.G.Macfarlane, T.P.McLean, J.E.Quarrington and V.Roberts, Phys. Rev. 108, 1377, (1957). G.G.Macfarlane, T.P.McLean, J.E.Quarrington and V.Roberts, Proc Phys. Soc. (London), 7 1 863, (1958).  4. 5.  r  6.  S.Zwerdling, B.Lax, L.M.Roth and K.J.Button, Phys. Rev. 114 , 80, (19 59) .  7.  G.G.Macfarlane, T.P.MacLean, J.E,Quarrington, and V.Roberts, Phys. Rev. L e t t e r s , 2, 252, (1959).  8.  W.H.Kleiner and L.M.Roth, Phys. Rev. L e t t e r s , 2, 334,  9.  G.E.Picus and G.L.Bir, Sov. Phys. S o l i d State, 1,  U 9 59) .  1502,  (19 597.  10.  J.Bardeen and W.Shockley, Phys.Rev. 80, 72, (1950),  11.  R . J . E l l i o t t , Phys.Rev, 108, 1384, (1957).  12.  W.G.Pfann and F.L.Vogel ( J r ) , Acta Met. 5,377, (1957).  13.  S . G . E l l i s , T r a n s i s t o r s I , p u b l i c a t i o n o f the R.C.A. l a b o r a t o r i e s , P r i n c e t o n N.J.  14.  A . U h l i r ( J r ) , T r a n s i s t o r Technology, B e l l Laboratory s e r i e s , 3 133,(19 5?), e d i t e d by F.J. B i o n d i .  15.  J.M.Buck and F.S.McKim, J.Electrochem. Soc. 103, 593,  16.  S.I.Novikova, Sov. Phys. S o l i d State, 2,37,(1960).  17.  E . K . P l y l e r , N.M.Gailar and T.A.Wiggins, J.Res.Nat.Bur. Stand. 48, 221, (19 52) .  18.  R . J . E l l i o t t , Phys. Rev. 96, 280, (1954).  19.  E.O.Kane, J.Phys.Chem. S o l i d s , 6,236 , (1958).  20.  W.Kohn and J.M.Luttinger, Phys. Rev. 98,915, (1957).  21.  P . J . P r i c e , Phys. Rev. 124, 713, (1961).  22.  M.E.Fine, J.Appl. Phys. 24, 338, (19 53).  ~ T 1 9 56) .  65 ference no. 23.  24.  M . Cardona and W P a u l , J.Phys. Chem. S o l i d s 17, 138, ~~ (1960). H.Y.Fan, M.L.Sheppard and W.Spitzer, Proceedings of the A t l a n t i c C i t y Photoconductivity Conference 19 54.* For instance, W.Paul and D.M.Warschauer, J.Phys. Chem. S o l i d s , 8 196, (19 59) .  25.  J . J . H a l l , Phys.Rev. 128, 68, (1962)..  26.  E.Antoncik, Czech. J.Phys. 5 , 449, (19 5 5).  27.  For instance, E.Burstein, G.Picus and N.Sclar, Proceedings of the A t l a n t i c C i t y Photoconductivity Conference 19 54.*  28.  D.L.Dexter, Proceedings of the A t l a n t i c C i t y Photoc o n d u c t i v i t y Conference 19 54,*  29.  W.Read, D i s l o c a t i o n s i n c r y s t a l s , McGraw H i l l publication.  30.  W.H.Brattain and H. B. Briggs, Phys. Rev. _75, 1705,(1949).  31.  B.W.Levinger and D.R.Frankl, J.Phys. Chem. S o l i d s , 20, 281, (1961) .  * Proceedings of the A t l a n t i c C i t y P h o t o c o n d u c t i v i t y c o n f e r e n c e ^ 5 4 , e d i t e d by R.G. Breckenridge et a l published by J.Wiley and sons Inc N.Y. 1955.  

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