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Infrared absorption lines in boron-doped silicon. Colbow, Konrad 1963

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INFRARED  ABSORPTION  LINES  IN  BORON-DOPED  SILICON  by  KONRAD  COLBOW  B . S C o , McMaster U n i v e r s i t y , M . S c , McMaster U n i v e r s i t y ,  A  THESIS THE  SUBMITTED  IN  REQUIREMENTS DOCTOR  Hamilton, Ont., .1959 Hamilton, Ont., 1960  PARTIAL  FOR OF  THE  FULFILMENT DEGREE  OF  OF  PHILOSOPHY  i n t h e Department of PHYSICS  accept  this  THE  t h e s i s as conforming t o t h e r e q u i r e d  UNIVERSITY  OF  May,  BRITISH 1963  COLUMBIA  standard  In p r e s e n t i n g  t h i s thesis i n p a r t i a l fulfilment of  the requirements f o r an advanced degree a t t h e U n i v e r s i t y British  Columbia, I agree t h a t the  a v a i l a b l e f o r reference  and  study.  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  It i s understood t h a t  s h a l l not  Department o f  Physics  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada.  31 19fi3  representatives.  copying or p u b l i c a t i o n of t h i s t h e s i s f o r  f i n a n c i a l gain  May  Department o r by h i s  be  be a l l o w e d w i t h o u t my  Columbia,  written  permission.  The U n i v e r s i t y o f 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  KONRAD  COLBOW  B . S c , McMaster U n i v e r s i t y , M. Sc., McMaster U n i v e r s i t y ,  1959 1960  WEDNESDAY, MAY 29, 1963, AT 2:00 P.M. IN  ROOM 452, BUCHANAN BUILDING  COMMITTEE IN CHARGE Chairman: F.H. Soward  R. B a r r i e A.M. C r o o k e r F.W. Dalby  J.C. G i l e s K.B. Harvey G.M. V o l k o f f G.B. Walker  E x t e r n a l Examiner: H.J. H r o s t o w s k i Department o f P h y s i c s , U n i v e r s i t y o f Oregon  INFRARED ABSORPTION LINES IN BORON-DOPED SILICON  ABSTRACT  In boron-doped s i l i c o n ,  GRADUATE STUDIES  o p t i c a l absorption  p l a c e through the e x c i t a t i o n o f bound h o l e s ground ixate. spectrum.  to e x c i t e d s t a t e s .  This  allowance  by the f i n i t e s p e c t r o m e t e r t h o r s gave a m i s l e a d i n g  slit  from the  for line  width,  low  and the onset  Physics  P h y s i c s o f the s o l i d s t a t e D i e l e c t r i c s and Magnetism  r e s o l u t i o n and  Noise  distortion  i n P h y s i c a l Systems  E l e c t r o n Dynamics  R. B a r r i e M. Bloom R.E. Burgess R.E. Burgess  p r e v i o u s au-  p i c t u r e o f the low temperature  h a l f - w i d t h , the temperature dependence o f t h i s width,  F i e l d o f Study:  leads t o a l i n e  Due t o a l a c k o f s u f f i c i e n t  a f a i l u r e t o make proper  takes  Related  Studies:  Applied Electromagnetic  half-  Theory  D i g i t a l Computer Programming  o f c o n c e n t r a t i o n broadening a t  G.B. Walker J.R.H. Dempster  temperatures.  New e x p e r i m e n t a l  data are presented  by i n t r o d u c i n g the mechanism o f s t a t i s t i c a l broadening  Stark  1.  due t o i o n i z e d i m p u r i t i e s , and by m o d i f y i n g  B a l t e n s p e r g e r ' s (1953) t h e o r y ening.  PUBLICATIONS  and e x p l a i n e d  a t t r i b u t e d t o phonon broadening  s  f o r c o n c e n t r a t i o n broad-  At low i m p u r i t y c o n c e n t r a t i o n the w i d t h i s ( B a r r i e and N i s h i k a w a  Temperature Dependence o f A b s o r p t i o n L i n e Width i n Boron-Doped S i l i c o n . Konrad Colbow, J„W. B i c h a r d , and J.C. G i l e s . Can. J . Phys. 40 1436 (1962).  2.  A b s o r p t i o n L i n e Width i n Boron-Doped S i l i c o n . Konrad Colb ow. B u l l . Am. Phys. Soc. S e r i e s II I , 485 (1962). S  1962)  and i n t e r n a l s t r a i n s  (Kohn 1957).  ABSTRACT  I n boron-doped s i l i c o n , o p t i c a l a b s o r p t i o n t a k e s p l a c e t h r o u g h - t h e e x c i t a t i o n o f bound.holes from t h e ground s t a t e t o e x c i t e d s t a t e s . leads t o a l i n e spectrum,,  This  Due t o a l a c k o f s u f f i c i e n t r e s o l u t i o n and a  f a i l u r e t o make proper a l l o w a n c e f o r l i n e d i s t o r t i o n by t h e f i n i t e s p e c t r o m e t e r s l i . t w i d t h , , p r e v i o u s a u t h o r s gave a m i s l e a d i n g p i c t u r e o f t h e low t e m p e r a t u r e h a l f - w i d t h , t h e temperature dependence o f t h i s h a l f - w i d t h , and t h e o n s e t o f c o n c e n t r a t i o n b r o a d e n i n g a t low t e m p e r a t u r e s . NewJ#% e x p e r i m e n t a l d a t a a r e e x p l a i n e d by i n t r o d u c i n g t h e r^ew" mechanism o f s t a t i s t i c a l S t a r k b r o a d e n i n g due t o i o n i s e d i m p u r i t i e s , and by m o d i f y i n g B a l t e n s p e r g e r ' s (1953) t h e o r y f o r c o n c e n t r a t i o n b r o a d e n i n g .  At  low i m p u r i t y c o n c e n t r a t i o n t h e w i d t h i s a t t r i b u t e d t o phonon b r o a d e n i n g ( B a r r i e and N i s h i k a w a 1962) and i n t e r n a l s t r a i n s (Kohn 1957).  ACKNOWLEDGEMENTS  My s i n c e r e thanks a r e due t o Dr. R. B a r r i e f o r h e l p f u l a d v i c e and c o n s t r u c t i v e c r i t i c i s m . d u r i n g t h e p r e p a r a t i o n o f t h i s thesis. I a l s o w i s h t o thank D r s . J . W. B i c h a r d , J . C. G i l e s , and A. M. C r o o k e r f o r v a l u a b l e The r e s e a r c h  discussions.  f o r t h i s t h e s i s was s u p p o r t e d by t h e Defence  Research Board o f Canada, Grant No. 9512-26.  I a l s o l i k e to  express my g r a t i t u d e t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r the award o f a s t u d e n t s h i p .  iii  TABLE OF CONTENTS Page Abstract  i i  T a b l e of C o n t e n t s  . i i i  L i s t of F i g u r e s  v  Acknowledgements  , . .  v i i  I Chapter I  -  Chapter I I  Introduction  -  Experimental  1.  Apparatus and E x p e r i m e n t a l P r o c e d u r e  2.  S p e c t r o m e t e r Broadening  3.  R e s u l t s and D i s c u s s i o n of E r r o r s  Chapter I I I -  1  .  4 . .  7 12  Theory and I n t e r p r e t a t i o n of Data  1.  I m p u r i t y S t a t e s and the Hydrogenic A p p r o x i m a t i o n  .  2.  G e n e r a l I n t e r p r e t a t i o n of L i n e Broadening  3.  S t a t i s t i c a l Theory of S t a r k Broadening  21  4.  The E f f e c t of S c r e e n i n g  26  5.  Temperature Dependence of t h e H a l f - w i d t h due t o t h e  ....  L i n e a r and t h e Q u a d r a t i c S t a r k E f f e c t 6.  15 20  27  E v a l u a t i o n of t h e H a l f - w i d t h due to the Q u a d r a t i c Stark Effect  „ . .  29  7.  S t a t i s t i c a l Broadening due t o van der Waals F o r c e s .  31  8.  B r o a d e n i n g due t o O v e r l a p F o r c e s  32  9.  B r o a d e n i n g due to I n t e r n a l S t r a i n s  35  Phonon Broadening  36  10.  -  iv -  Page Chapter IV  -  Conclusions.  40  Table 1:  Standard Voigt profiles  43  Table 2;  Integrated absorption, A  Table 3:  Integrated absorption cross-section,  Appendix I;  . . . . . . .  44 44  Distribution of holes between the ground state, excited states, and the valence band as a function temperature  45  Appendix II: Second-order Stark shift for hydrogenic Is and 2p states  Bibliography  48  49  V  LIST OF FIGURES To f o l l o w page: FIG.  1.  A b s o r p t i o n c o n s t a n t v s . wave number f o r boron-doped s i l i c o n a t v a r i o u s t e m p e r a t u r e s and  t h r e e boron  concentrations  6 8  FIG.  2.  Water v a p o r l i n e - w i d t h v s . wave number  FIG.  3.  T y p i c a l V o i g t a n a l y s i s (water v a p o r l i n e at 298.6 cm"'')' 10  FIG.  4.  E x t r a p o l a t i o n of the t r u e h a l f - w i d t h to  zero  c o n c e n t r a t i o n of i m p u r i t i e s FIG.  5.  12  True h a l f - w i d t h v s . temperature f o r t h r e e boron concentrations  FIG.  6.  12  H a l f - w i d t h minus z e r o - c o n c e n t r a t i o n h a l f - w i d t h v s . temperature f o r l i n e 4  FIG.  7.  12  A b s o r p t i o n c o n s t a n t v s . wave number f o r 11 ohm  cm  boron-doped s i l i c o n a t f o u r temperatures  12  FIG.  8.  Energy l e v e l diagram f o r boron-doped s i l i c o n  FIG.  9.  D i s t r i b u t i o n , of h o l e s between the ground e x c i t e d s t a t e s , and  FIG.  10.  11.  state,  the v a l e n c e band v s . t e m p e r a t u r e .  as a f u n c t i o n . o f s e p a r a t i o n  12.  14  Screening  26  parameter v s . temperature f o r t h r e e  c o n c e n t r a t i o n s of boron FIG.  14  Screened e l e c t r i c f i e l d , and Coulomb f i e l d w i t h c u t off,  FIG.  ....  F i e l d d i s t r i b u t i o n f u n c t i o n f o r v a r i o u s v a l u e s of s c r e e n i n g parameter  27 the 27  vi  To f o l l o w page: FIG. 13.  H a l f - w i d t h o f t h e f i e l d d i s t r i b u t i o n as a f u n c t i o n of t h e s c r e e n i n g parameter  FIG. 14.  27  H a l f - w i d t h i n u n i t s of t h e s t r e n g t h parameter v s . temperature f o r the l i n e a r S t a r k e f f e c t  FIG. 15.  28  H a l f - w i d t h i n u n i t s of the s t r e n g t h parameter v s . temperature f o r the q u a d r a t i c S t a r k e f f e c t  FIG. 16.  28  Broadening of h y d r o g e n i c l e v e l s v s . d i s t a n c e between i m p u r i t i e s i n u n i t s of the e f f e c t i v e Bohr r a d i u s  . .  32  1  CHAPTER  I  INTRODUCTION  The perfect  i n t r o d u c t i o n of a group I I I impurity,, l i k e boron, i n t o a  silicon  l a t t i c e produces a h o l e ( e l e c t r o n d e f i c i e n c y ) l o o s e l y  bound to the i m p u r i t y takes  ion.  In boron-doped s i l i c o n ,  p l a c e through the e x c i t a t i o n of bound h o l e s  excited states.  This  leads to a l i n e  were u n a f f e c t e d by, each o t h e r  and  by  spectrum.  infrared  absorption  from the ground s t a t e to I f the bound h o l e  states  l a t t i c e v i b r a t i o n s , these s p e c t r a l  l i n e s would have no w i d t h o t h e r  than t h e i r n a t u r a l w i d t h , which i s about  10"^  Herzberg 1944).  electron volts  ( e v ) , (e.g  been observed a r e c o n s i d e r a b l y b r o a d e r .  t h i s t h e s i s apply The  interesting  to i n t e r a c t i o n w i t h t h e  i n a s i m i l a r way reason  l i k e boron i s p u r e l y Neglecting  each o t h e r .  Most of the d i s c u s s i o n s i n  f o r choosing  a more c o m p l i c a t e d  acceptor  impurity  historical. i n s t r u m e n t a l broadening, B u r s t e i n et a l (1953) found i n (10"  ev) at 4 2 ° K and  c a l c u l a t i o n s of Lax at  4.2  °K and  l a t e r concluded and  the  B u r s t e i n (1955) which gave a h a l f - w i d t h of 3.6  mev  Lax  and  due  to the simultaneous emission  i n reasonable  percent  agreement w i t h  predicted roughly  to be  an  0  i n c r e a s e of the h a l f - w i d t h ( f u l l width at half-maximum) of about 4 d T h i s was  to  to group V donor i m p u r i t i e s (bound  boron-doped s i l i c o n h a l f - w i d t h s of about 1 mev  at ,77 °K.  lattice  s o - c a l l e d c o n c e n t r a t i o n broadening, that i s broadening due  the i n t e r a c t i o n of bound h o l e s w i t h  electrons).  l i n e s t h a t have  T h i s g i v e s r i s e to two  f i e l d s of study,- namely l i n e broadening due v i b r a t i o n s , and  However, the  the observed  i n c r e a s e i n h a l f - w i d t h at 77  B u r s t e i n (1955) suggested t h a t the w i d t h of the i m p u r i t y or a b s o r p t i o n  of one  accompanying the change of s t a t e of the bound h o l e .  levels is  or more phonons Sampson and  Margenau  (1956) f u r t h e r improved on the agreement between the c a l c u l a t e d and  the  °K.  2 observed w i d t h s .  U s i n g t h e s i m p l e L o r e n t z broadening approach  they  c a l c u l a t e d a low temperature h a l f - w i d t h o f 1.6 mev. Kane (1960) p o i n t e d out t h a t t h e observed b r o a d e n i n g a t 4.2 °K may be e n t i r e l y i n s t r u m e n t a l .  Kane suggested t h a t due t o t h e weakness o f t h e  e l e c t r o n - p h o n o n i n t e r a c t i o n i n s i l i c o n and germanium t h e dominant o p t i c a l l y induced t r a n s i t i o n i s p u r e l y e l e c t r o n i c w i t h o u t any change o f phonon o c c u p a t i o n number.  He suggested t h a t t h e w i d t h a r o s e from a f i n i t e  time o f t h e e x c i t e d s t a t e due t o t h e e l e c t r o n - p h o n o n i n t e r a c t i o n , .  life-  Kane  -3 e s t i m a t e d a w i d t h o f about 3 x 10  mev.  I n t h i s l i f e - t i m e broadening, the  w i d t h s o f t h e a b s o r p t i o n l i n e s a r e m o s t l y d e t e r m i n e d by those o f t h e e x c i t e d levels.  Thus t h e w i d t h of t h e l i n e s w i l l c r i t i c a l l y depend on t h e energy  l e v e l s t r u c t u r e , and one e x p e c t s d i f f e r e n t l i n e s t o have d i f f e r e n t w i d t h s . T h i s was i n disagreement w i t h t h e e x p e r i m e n t a l r e s u l t s o f B u r s t e i n e t a l (1953), and t h e t h e o r y by Lax and B u r s t e i n (1955), which p r e d i c t e d t h a t t h e w i d t h s were m o s t l y determined by t h e ground s t a t e and thus s h o u l d be t h e same f o r a l l l i n e s . Colbow et a l (1962) showed t h a t t h e observed b r o a d e n i n g a t 4.2 °K i s n o t e n t i r e l y i n s t r u m e n t a l as suggested by Kane (I960)  a  However,  taking  proper c a r e o f i n s t r u m e n t a l broadening they found a h a l f - w i d t h o f 0.2 mev a t 4.2 °K and an i n c r e a s e o f t h e h a l f - w i d t h between 4.2 °K and 90 °K by a f a c t o r of  6, r a t h e r than t h e 40 p e r c e n t p r e d i c t e d by L a x a n d - B u r s t e i n (1955).  Their  d a t a a l s o showed t h a t d i f f e r e n t l i n e s have d i f f e r e n t h a l f - w i d t h s , and t h a t t h e l i n e shape a t 4.2 °K i s p r e d o m i n a n t l y  lorentzian.  The t h e o r y o f phonon broadening has been c l a r i f i e d by B a r r i e and N i s h i k a w a (1962). Barrie  T h e i r r e s u l t s a r e i n f u l l agreement w i t h Kane's (1960).  and N i s h i k a w a o b t a i n e d e x p l i c i t l y t h e l i n e shape f u n c t i o n o f t h e z e r o -  phonon p r o c e s s , w h i c h t u r n s out t o be a p p r o x i m a t e l y l o r e n t z i a n near t h e peak.  3 They pointed out that the multiphonon processes which Lax and Burstein (1955) considered contribute to the continuous background and become important only at higher temperatures (about 90°K).  However, the theory of Barrie and  Nishikawa did not account for the rather steep temperature dependence observed by Colbow et al (1962). Up to this point i t had always been assumed that for the rather low impurity concentrations used by Colbow et al (1962) and Burstein et al (1953), concentration broadening was negligible.  This appeared to follow  from the experimental data of Newman (1956), which showed that for borondoped silicon concentration broadening starts above 10^ boron impurities 3  per cm .  In addition Baltensperger•s (1953) calculations suggested that  concentration broadening in boron-doped silicon should start at approximately 6 x lO-^ boron impurities per cm"*.  In contrast to this, the data presented 15  here show that at an impurity concentration of 1.2 x 10 broadening is already important.  -3  cm  concentration  It is believed that the main shortcoming  of Baltensperger s (1953) theory lies in the assumption of a regular 1  lattice qf impurities.  The implications of an alternative, and more likely  assumption of a random distribution of impurities are discussed in this thesis. A new mechanism is suggested to account for the observed rapid increase of the half-width with temperature, namely statistical Stark broadening due to ionized impurities.  4  CHAPTER  lo  II  Apparatus  and E x p e r i m e n t a l Procedure  R a d i a t i o n from a G l o b a r source was d i s p e r s e d i n a Model 83 P e r k i n Elmer spectrometer m o d i f i e d t o house a Bausch  and Lomb g r a t i n g w i t h 30  grooves per m i l l i m e t e r b l a z e d a t 333 cnf * i n t h e f i r s t r e s o l v i n g power was X/AX = 315, f o r a s l i t s h o r t e r wavelength  order.  w i d t h o f 0.8 mm.  plates.  The i n t e n s i t y of  r a d i a t i o n was reduced by s o o t i n g the m i r r o r i n the  e n t r a n c e o p t i c s and by u s i n g two r e f l e c t i o n s from sodium ray  The t h e o r e t i c a l  The remaining s h o r t wavelength  fluoride  residual  r a d i a t i o n was measured and  c o r r e c t e d f o r by p a s s i n g the r a d i a t i o n through a R o c k s a l t window, which  will  t r a n s m i t o n l y r a d i a t i o n s h o r t e r than about 500 c m ° ^ , The e f f e c t of the s h o r t wavelength  photons  negligible.  on t h e o c c u p a t i o n number o f the i m p u r i t y s t a t e s i s  The s h o r t wavelength  c o n t i n u o u s background,  and was w i t h i n e x p e r i m e n t a l e r r o r t h e same through a  pure and the doped sample. radiation  r a d i a t i o n merely c o n t r i b u t e s to the  Any p o s s i b l e h e a t i n g of t h e specimen  i s n e g l i g i b l e as w e l l -  Near t h e e n t r a n c e s l i t  due to t h i s  the radiation i s  chopped by a s e m i - c i r c u l a r d i s c a t a frequency o f 13 c y c l e s sec°^,  After  d e t e c t i o n by a thermocouple w i t h a cesium i o d i d e window, the a m p l i f i e d s i g n a l was d i s p l a y e d on a Brown s t r i p - c h a r t r e c o r d e r . light  t h e number o f photons  boron  impurities  is  equilibrium.  i s v e r y s m a l l compared t o t h e t o t a l number o f  i n the sample.  stopped by t h e chopper  In each b u r s t o f  D u r i n g the time i n t e r v a l when t h e r a d i a t i o n  b l a d e , the sample i s b e l i e v e d to r e t u r n to  Thus no s a t u r a t i o n e f f e c t s a r e expected t o o c c u r .  The spectrometer was c a l i b r a t e d by means o f the atmospheric  water  vapor a b s o r p t i o n spectrum, u s i n g the data o f P l y l e r and A c q u i s t a (1956), and R a n d a l l et a l (1937).  In t h e c o u r s e o f a measurement the spectrometer was  f l u s h e d continuously, w i t h dry n i t r o g e n gas t o reduce t h e a b s o r p t i o n due to .. a tmo s ph er i'cf wa't'e'r^Vapo r s; . 1  5  The experiment c o n s i s t s o f measuring t h e t r a n s m i s s i o n r a t i o s o f boron-doped s i l i c o n and i n t r i n s i c  s i l i c o n as a f u n c t i o n o f t h e wave number  of t h e r a d i a t i o n f o r v a r i o u s temperatures and r e s i s t i v i t i e s o f boron-doped silicon.  The measurements were made a t f o u r temperatures c o r r e s p o n d i n g t o  the b o i l i n g p o i n t s o f h e l i u m (4.2°K), n i t r o g e n (77.4°K), oxygen (90.1°K), and pumped n i t r o g e n (60  +  3°K).  The r e f r i g e r a n t s were c o n t a i n e d i n a m e t a l dewar v e s s e l , t h e base of which was l o c a t e d near a f o c u s o f t h e r a d i a t i o n l e a v i n g t h e monochromator. The r a d i a t i o n p o r t s were c o v e r e d w i t h cesium i o d i d e d ' windows.  Parallel-  s i d e d specimens were a t t a c h e d t o a f l a t copper b l o c k a t t h e base o f t h e c o o l a n t c o n t a i n e r , w i t h f a c i l i t i e s f o r r o t a t i n g a l t e r n a t i v e l y t h e doped o r an i n t r i n s i c  sample i n t o t h e path o f t h e r a d i a t i o n .  Thermal c o n t a c t  between t h e copper b l o c k and t h e samples was a c h i e v e d by means of vacuum grease c o n t a i n i n g a suspension p l a c e by a f l a t copper s t r i p the r a d i a t i o n ) ,  o f s i l v e r powder.  The samples were h e l d i n  ( w i t h a h o l e i n i t s c e n t e r f o r t h e passage o f  Carewas p a i d t o a v o i d s t r a i n i n g the samples.  The samples were c u t from i n g o t s grown by M e r c k and Company by t h e f l o a t i n g - z o n e technique.  They had t h i c k n e s s e s between 0.0361 cm (most  h i g h l y doped m a t e r i a l ) and 0.5085 cm ( l i g h t l y doped m a t e r i a l ) , t h e l a t t e r b e i n g t h e l a r g e s t t h i c k n e s s t h a t c o u l d be r e a d i l y accommodated i n t h e m e t a l dewar a v a i l a b l e .  These t h i c k n e s s e s were chosen t o g i v e a compromise  between t h e s i g n a l s t r e n g t h ( h i g h t r a n s m i s s i o n ) and t h e o b s e r v a b l e absorption (low transmission).  The specimen s u r f a c e s were ground w i t h  p r o g r e s s i v e l y f i n e r grades o f carborundum and f i n a l l y p o l i s h e d w i t h l e v i g a t e d alumina..  B e f o r e mounting i n the d,ewar v e s s e l , the,.surfaces ,were degreased i n  an u l t r a s o n i c c l e a n i n g b a t h .  I m p u r i t y c o n c e n t r a t i o n s were d e t e r m i n e d from  the room-temperature r e s i s t i v i t y o f each sample.  The r e s i s t i v i t i e s were 1.3  ohm cm, 11 ohm cm, 130 ohm cm, and 3600 ohm cm ( i n t r i n s i c ) .  A l l samples were  6  e s s e n t i a l l y uncompensated. Maita 1.2 cm"  (1954),  x 10 3  (1962) and Morin  and  the c o r r e s p o n d i n g c o n c e n t r a t i o n s of boron a r e r e s p e c t i v e l y  cm" ,  1 6  U s i n g the d a t a of I r v i n  1.2  3  x 10  cm" ,  1 5  1.0  3  x 10  1 4  cm" ,  and  3  l e s s than 5 x  10  1 2  (intrinsic). With one  e x c e p t i o n , the specimens used  i n the p r e s e n t  i n v e s t i g a t i o n were t h i c k enough t h a t i n t e r f e r e n c e f r i n g e s c o u l d not observed  w i t h the r e s o l u t i o n a v a i l a b l e . cm)  F o r the one  t h i c k , , 1.3  ohm  narrow and  s h a l l o w enough to be r e a d i l y averaged  be  specimen (0.0361  cm  f o r which i n t e r f e r e n c e f r i n g e s d i d appear, they were  the r e c o r d e r c h a r t .  out in. r e a d i n g the data of  Under these c o n d i t i o n s , the monochromatic a b s o r p t i o n  c o e f f i c i e n t c< i s o b t a i n e d from the t r a n s m i s s i o n at a g i v e n wave number by  (Moss 1959) _  T  (1 - R ) 1 - R  The  2  exp.  2  exp.  (-  d)  .  ( - 2 c< d)  t r a n s m i s s i o n of the i n t r i n s i c m a t e r i a l i s then g i v e n by T  =  Q  (1 - R )  /  2  (1 - R )  Hence the t r a n s m i s s i o n r a t i o of doped and (1 - R T/T  °  Y-  1 - R  T h i s i s the r a t i o t h a t was  the same f o r the i n t r i n s i c +  0.03  by B i c h a r d and G i l e s  was  and  intrinsic  ( - *  s i l i c o n becomes  d)  2—*  ~ ( - 2 o(d)  exp.  The  .  (1-3)  experiment.  s u r f a c e r e f l e c t i v i t y R was  the boron-eloped  determined  (1962).  remained c o n s t a n t over  exp.  (1-2)  a c t u a l l y measured i n . a t y p i c a l  d i s the specimen t h i c k n e s s .  a va l u e of 0.31  )  2  = ~  0  2  .  2  silicon.  assumed to be  U s i n g Eq.  (1-2)  f o r R, which i s the same as t h a t used  W i t h i n experimental  the frequency  Here  e r r o r the  r e g i o n under study  reflectivity  (360 cm°^ to  cm"*-), and t h e temperature  range (4.2°K to 90°K) i n v e s t i g a t e d .  v a l u e f o r the r e f l e c t i v i t y  E q . ( l - 3 ) leads to the graphs o f a b s o r p t i o n  240  With t h i s  FIGURE  Absorption  1.  c o n s t a n t v s . wave number f o r boron=doped  s i l i c o n at v a r i o u s temperatures  (T) and t h r e e  boron  concentrations ( N ) . A  To f o l l o w page 6.  7  constant  as a f u n c t i o n of wave number shown in, F i g u r e 1.  Instead  of  measuring t h e t r a n s m i s s i o n f o r a doped sample r e l a t i v e to t h a t f o r an i n t r i n s i c sample and u s i n g E q . ( l - 3 ) , one c o u l d have determined  the  t r a n s m i s s i o n f o r a doped sample r e l a t i v e t o a i r , u s i n g Eq„(l~l)o t h e r e a r e s e v e r a l advantages t o t h e f i r s t method. r e f l e c t i v i t y i s c o n s i d e r a b l y l e s s important  The  exact v a l u e f o r the  f o r c a l c u l a t i n g the  c o e f f i c i e n t i f E q . ( l - 3 ) , r a t h e r than Eq,(l-=2), i s used.  However,  absorption  In a d d i t i o n ,  s c a t t e r e d s h o r t w a v e l e n g t h r a d i a t i o n can be expected to be more n e a r l y  the  same and t h u s c a n c e l i n the r a t i o , when the doped sample i s compared w i t h an i n t r i n s i c one,  r a t h e r t h a n w i t h no sample i n t h e l i g h t p a t h .  The  same  a p p l i e s f o r t h e c a n c e l l a t i o n o f atmospheric w a t e r vapor a b s o r p t i o n . In t h e p r e c e e d i n g c o e f f i c i e n t i t was  o u t l i n e f o r the c a l c u l a t i o n o f the  i m p l i c i t l y assumed t h a t a p a r a l l e l beam of l i g h t i s at  normal i n c i d e n c e on a p a r a l l e l - s i d e d specimen. specimen s u r f a c e s were l e s s than 0.01°. parallel. beam was 2.  absorption  Any  a n g l e s between the  A c t u a l l y , t h e l i g h t beam was  However, s i n c e the maximum a n g l e of i n c i d e n c e i n the  not  converging  o n l y 8°, no c o r r e c t i o n f o r convergence needed to be a p p l i e d .  Spectrometer B r o a d e n i n g U s i n g the c u r v e s of F i g u r e 1 and " p e e l i n g o f f " the c o n t r i b u t i o n s  of n e i g h b o r i n g  l i n e s , one can. d e t e r m i n e an."observed l i n e shape" and  "observed h a l f - w i d t h " . line profile.  The  The  an  p e e l i n g o f f i s a c h i e v e d by assuming a symmetric  r e s u l t i n g a b s o r p t i o n l i n e s ( d o t t e d l i n e s ) may  F i g u r e 6, w h i c h g i v e s a m a g n i f i e d  be seen i n  p i c t u r e of the l i n e s 3 and 4 of F i g u r e  T h i s " p e e l i n g o f f " becomes d i f f i c u l t a t h i g h e r t e m p e r a t u r e s .  1(a).  However, i f one  assumes t h a t t h e areas under the weaker l i n e s i n F i g u r e 6 s t a y n e a r l y constant  and t h a t t h e i r h e i g h t s d e c r e a s e w i t h t e m p e r a t u r e i n t h e same way  the s t r o n g a b s o r p t i o n l i n e , one can. decompose the e x p e r i m e n t a l  curves  into  as  8 the dotted lines, as shown in Figure 60 Errors in these assumptions give rise to considerably smaller errors' in the observed half-width of the strong line. These observed absorption lines are not only broadened by processes in the specimen but also by the f i n i t e resolution of the spectrometer.  It is the  purpose of the present section to outline the method used for obtaining from the observed line the "true line shape" and the "true half-width", that is the line profile obtained by using a spectrometer with infinite resolution. The intensity distribution in a spectral  line broadened by two  independent effects is expressed by a convolution integral of the form (Unsold 1955) -t-00  f (x) = / f ( x - y) f " (y) d y  .  (2-1)  -00  Here f '(x)  or f (x) is the profile the line would assume i f only one  broadening effect were present; x is the distance from the centre of the line in terms of either wavelength or frequency units.  If f (x) is the observed  line shape, Eq.(2-1), determines the true line,profile f "(y), know the s l i t function f'(y).  provided we  This s l i t function is the profile a  monochromatic signal assumes after passing through the spectrometer.  It may  be obtained experimentally from the observed profiles of single atmospheric water vapor absorption lines, which have a true width considerably smaller than the observed width and thus approximate well enough to monochromatic lines.  The small true width of single water vapor lines was inferred from  a plot,of their observed half-width as a function of spectrometer s l i t width. The observed half-widths of suitable wafer vapor absorption  lines have been  plotted in Figure 2. The corresponding spectrometer s l i t width is 0.8 mm in the region from 370 cm"*- to 260 cm"*-, and 1.2 mm in the region from 260 cm"*" to 240 cm"*-. The same s l i t widths were used to obtain the spectra in Figure 1.  FIGURE  2.  Water vapor l i n e - w i d t h v s . wave number.  To f o l l o w page 8.  FIG.  2  Numerous methods f o r s o l v i n g t h e i n t e g r a l e q u a t i o n (2-1) have been a p p l i e d ( U n s b l d 1955, van de H u l s t 1946). are v e r y l a b o r i o u s . and  However, a l l t h e g e n e r a l  The p r o b l e m becomes q u i t e s i m p l e i f t h e s l i t  methods function  t h e observed l i n e p r o f i l e may be f i t t e d e i t h e r both by l o r e n t z i a n o r both  by g a u s s i a n c u r v e s .  I n t h i s c a s e t h e t r u e l i n e p r o f i l e comes out t o be  r e s p e c t i v e l y l o r e n t z i a n o r g a u s s i a n as w e l l , and one has t h e s i m p l e r e l a t i o n between t h e h a l f - w i d t h s l o r e n t z i a n % h" = h - h' ir 2 2 '2 : h = h -h  gaussian  (2-2)  Here h" i s t h e t r u e h a l f - w i d t h , h t h e observed h a l f - w i d t h , and h' t h e h a l f width of the s l i t f u n c t i o n .  However, no s i n g l e f u n c t i o n o f e i t h e r type g i v e s  a good f i t t o both t h e o b s e r v e d l i n e shape and the s l i t f u n c t i o n . I t has been found t h a t t h e p r e s e n t e x p e r i m e n t a l c u r v e s f o r the observed l i n e and  t h e s l i t f u n c t i o n can be q u i t e a d e q u a t e l y f i t t e d by V o i g t  ( V o i g t 1912).  profile  functions  These f u n c t i o n s a r e d e f i n e d as t h e c o n v o l u t i o n i n t e g r a l  between a g a u s s i a n and a l o r e n t z i a n f u n c t i o n .  They may be used t o f i t any  p r o f i l e w h i c h l i e s between a g a u s s i a n and a l o r e n t z i a n c u r v e .  I f f ( x ) and  f ' ( x ) a r e b o t h taken to be V o i g t f u n c t i o n s , then t h e t r u e l i n e shape i s a l s o a V o i g t f u n c t i o n by Eq. (2-1). observed  line shape, +oo  "f fr,( x )^ = w MJ  e x  P ' [b. " (y / P 2 ) ]±~  ,  2  1 + t ( x - y) / P ]  -oo slit  One may thus w r i t e  L  dy ;  , ,v (2-3) 0  2  function,  -  +oo £  ,  w  _  .  j  M  »ft\ 2 1 1  **P.[- ( y / P >  ]  2  1  + C ( x - J* / P [ ]  D  Y  .  ( J  .  4 )  2  t r u e l i n e shape, f••(.).-  M  " -007 ° -[~ xp  i  j  '  r  i  >  2  i  *  >  «-»  10 II  I  where M, M , and M and Eq. ( 2 - 1 )  are constants.  I t i s a p r o p e r t y o f these  functions  that  - ?i  -  p;  (2-6)  and  Figure 3 i l l u s t r a t e s a t y p i c a l Voigt analysis.  a t o n e - t e n t h t h e h e i g h t , b2 a t  w i d t h h a t h a l f maximum h e i g h t , t h e w i d t h two-tenth  In a d d i t i o n to the  t h e maximum h e i g h t , e t c . , a r e measured.  A f t e r c a l c u l a t i n g the  r a t i o s b^/h, where i goes from one t o e i g h t , T a b l e 1 ( v a n de H u l s t and Reesinck and  V o i g t parameters p]_/h  1947) was used t o o b t a i n t h e c o r r e s p o n d i n g  ^2 ^ / ^ > w h i c h were a p p r o p r i a t e l y averaged o v e r d i f f e r e n t b f f o r a g i v n  absorption  line. A l o r e n t z i a n l i n e shape g i v e s P^/h = 0.500, ?2 P2  g a u s s i a n . p r o f i l e l e a d s t o B^/h =0,  /"  =  0-36  .  2  =  0> w h i l e a  The a t m o s p h e r i c  w a t e r v a p o r l i n e s i n v e s t i g a t e d t o d e t e r m i n e t h e s l i t f u n c t i o n ( F i g u r e 2) were a l l found t o l e a d t o  P[/h'  =  0.28 t 0.08 ; Pj  2  / ' h  2  =  °'  1 6  - °'  •  0 6  ("> 2  8  U s i n g t h e w a t e r v a p o r a b s o r p t i o n l i n e h a l f - w i d t h s from F i g u r e 2, Eq. (2-8) enables us t o f i n d t h e s l i t f u n c t i o n V o i g t parameters Performing  B[  and  P2  2  •  t h e a n a l y s i s i n t h e same f a s h i o n f o r t h e o b s e r v e d  a b s o r p t i o n l i n e s i n boron-doped s i l i c o n , one f i n d s t h e V o i g t parameters J3^ and  p2«  0  n e  m  a  then u s e Eq. (2-6) and (2-7) t o o b t a i n t h e V o i g t  v  D"  D"  parameters IT  error  P2  , P2  2 f° * r t  ie  t  r  u  e  absorption l i n e s .  Within  experimental  O w  a  s  found t o be equal t o z e r o , w h i c h means t h e V o i g t f u n c t i o n s  for  the true absorption l i n e s s i m p l i f y to l o r e n t z i a n p r o f i l e s .  Eq.  (2-5) becomes  That i s ,  FIGURE 3.  Typical Voigt analysis (water vapor line at 298.6 cm" ). The numbers b^/h (where i goes from 1 to 8) give the line width at i/10 of the maximum height, relative to the line width at half maximum height, (h). The values Bi/h where obtained from Table 1.  To follow page  —I  b / h = 0.53 8  299  298 Wave number  FIG.  3  (cm'h  0.300  f"(x) = c" (1 + x / f ^ ) ~ , 2  2  (2-9)  l  where c" i s the maximum height of the curve, occurring at x = 0; and one has I!  |,  = 0.50o  £ Vi/h  It was found that for the observed absorption lines in boron-doped silicon P^/h varied between 0.45 for narrow lines to about 0.5 for broad lines.  Under these conditions, with h h":= gh - g'h  1  = 2p  , Eq. (2-6) leads to (2-10)  ,  where g = 2 P^/h varies between 0.9 and 1.0 for different absorption lines, and by, Eq. (2-8) one has g' = 2 pj/h' = 0.56 + 0.16. The values listed in Table 1 under p may be used to evaluate the integrated absorption A. Both A and p are defined by +oo  A  =  (  -oo  (2-11)  f(x) dx = phc  for the observed line, or similarly for the true line by A" =  -oo  f"(x) dx = p" h" c" ,  '  (2-12)  where c and c" are respectively the observed and the true central ordinate (maximum height).  Since the integrated absorption is not changed by  spectrometer broadening provided one scans over a reasonably wide wave number range, one has p h c= p h c Knowing the observed integrated absorption, h", and p^'/h", Eq, (2-13) and Table 1 enables one to find the true central ordinate c", i f desired. With the present experimental data i t is a good enough approximation to take in Eqs.  (2-11), (2-12), and (2.13) from Table 1 p ^ p" =  1.54 ±  for a l l absorption lines investigated.  0.03,  (2-14)  12  3o  R e s u l t s and  D i s c u s s i o n of E r r o r s  C o r r e c t i n g the data  1 f o r spectrometer d i s t o r t i o n by the  in Figure  methods o u t l i n e d i n the p r e v i o u s  s e c t i o n , the t r u e h a l f - w i d t h s h" were  These have been p l o t t e d i n F i g u r e 5 as a f u n c t i o n of  obtained.  for three concentrations  o f boron impurities..  e x t r a p o l a t e d to zero i m p u r i t y c o n c e n t r a t i o n . shown i n F i g u r e 5 as d o t t e d  lines.  For  temperature  In F i g u r e 4 these data The  r e s u l t i n g curves  are  are  the purpose of e x t r a p o l a t i o n a  W3  II  p l o t of  In h  vs.  straight  lines.  was found convenient  and  results  temperatures and  reasonably  h" ( F i g u r e 5)  The main c o n t r i b u t i o n to the u n c e r t a i n t y i n varying origins for different  in  l i n e s under study.  has  This  will  p a r t l y be i l l u s t r a t e d by r e f e r e n c e to F i g u r e 7, which shows i n m a g n i f i e d of l i n e s 3 and  form the a b s o r p t i o n c o n s t a n t silicon  at f o u r d i f f e r e n t  temperature and  temperatures.  g' i n Eq.  s u b t r a c t s two s i m i l a r numbers. 130  ohm cm at 4o2°K.  Eq.  (2-10) one  wide l i n e s  For r a t h e r sharp l i n e s  impurity concentration)  the u n c e r t a i n t y o f g and  the main source  o b t a i n s h " = 0.08 f 0.04 mev.  Two cases.  The  to n e i g h b o r i n g other  first  sources  extreme, f o r  the u n c e r t a i n t y i n  lines.  of u n c e r t a i n t y a r e important  i s the r a t h e r l a r g e a b s o r p t i o n  4 at 4 , 2 ° K and  using  i n p e e l i n g - o f f the  60°K,  The  p r e s e n c e of some background a b s o r p t i o n  second source  l i n e s due  only f o r special  i n the 1,3 ohm cm m a t e r i a l ,  which g i v e s r i s e to an u n c e r t a i n t y of about f i v e percent of l i n e s 2 and  and  On the o p p o s i t e  impurity concentration)  one  is line 1 in  h ' = 0.095 mev,  the h a l f - w i d t h r e s u l t s p r i m a r i l y from the d i f f i c u l t y a b s o r p t i o n due  the f a c t t h a t  extreme c a s e f o r t h i s  Here h = 0.153 mev,  (low  of e r r o r comes from  (2-10), t o g e t h e r w i t h  The  ( h i g h temperature and  4 i n 11 ohm cm boron~doped  i n the peak-heights  of e r r o r i s the  to atmospheric water vapor.  FIGURE  4.  Extrapolation of the true half-width (logarithmic scale) to zero concentration of boron impurities (N ). A  To follow page 12.  0  2  4  6  8  10 12 14 N / (10 cm" ) X  A  FIG. 4(a)  3  4  1  16  18  20  FIGURE  True h a l f - w i d t h  5.  ( c o r r e c t e d f o r spectrometer broadening) v s .  temperature f o r boron c o n c e n t r a t i o n s ii)  1.2  x 10  1 5  cm" ,  low c o n c e n t r a t i o n s  3  iii)  1.2  x 10  of 1 6  i ) 1.0 x 1 0 ^ cm" , 3  and  cm" , 3  i v ) very  (extrapolated).  i  To f o l l o w page  T (°K) FIG.  5(a)  FIGURE  6.  Half-width minus zero-concentration half-width (extrapolated) vs. temperature for line 4.  To follow page 12.  0  10  FIG.  6  20  .30  40  50  T (°K)  60  70  80  90  FIGURE  7.  A b s o r p t i o n c o n s t a n t v s . wave number f o r 11 ohm cm 15 (1.2 x 10  ^ cm  temperatures.  ) boron-doped This  f i g u r e i l l u s t r a t e s some o f t h e  d i f f i c u l t i e s and u n c e r t a i n t i e s off  s i l i c o n at four  involved  the influence of neighboring  i n "peeling"  lines.  To f o l l o w page 12.  13 Incomplete c a n c e l l a t i o n o f water v a p o r l i n e s may be r e s p o n s i b l e f o r t h e s m a l l bump on t h e h i g h energy s i d e o f l i n e 4 ( s e e F i g . 7 ) . dip  S i m i l a r l y , the  i n l i n e 2 r e p o r t e d e a r l i e r (Colbow e t a l . 1962) may be due t o water v a p o r .  The f a c t t h a t a r a t h e r s t r o n g water v a p o r a b s o r p t i o n l i n e o c c u r s r i g h t a t t h e p o s i t i o n o f l i n e 2 g i v e s t h e d a t a f o r t h i s l i n e ( F i g u r e 5b) l e s s r e l i a b i l i t y than those f o r l i n e 4 and 1 ( F i g u r e s 5a, c ) . S i n c e a more complete s e t of h a l f - w i d t h measurements was a c c e s s i b l e f o r l i n e 4, F i g u r e 5a w i l l be used e x c l u s i v e l y i n t h e q u a n t i t a t i v e i n t e r p r e t a t i o n s i n Chapter I I I , i  Other p i e c e s o f i n f o r m a t i o n e x t r a c t e d from t h e d a t a i n F i g u r e 1 a r e t h e p o s i t i o n s o f t h e a b s o r p t i o n l i n e s ( F i g u r e 8) and t h e i n t e g r a t e d absorption (Table 2 ) .  The energy l e v e l s i n F i g u r e 8 have been p l o t t e d by  assuming an i o n i z a t i o n energy o f 46 mev ( B u r s t e i n e t a l . 1956, Kohn 1957), The p r e s e n t  author b e l i e v e s t h a t an i o n i z a t i o n energy o f p o s s i b l y as low as  44 mev cannot be r u l e d o u t by t h e p r e s e n t  experimental  would g i v e b e t t e r agreement between t h e e x p e r i m e n t a l (Schechter  data.  This  value  and t h e c a l c u l a t e d  1962) p o s i t i o n s o f e x c i t e d s t a t e s ( F i g u r e 8 ) . The e x p e r i m e n t a l l y determined p o s i t i o n s o f t h e a b s o r p t i o n  lines  a r e e s s e n t i a l l y i n agreement w i t h t h o s e o b t a i n e d e a r l i e r by H r o s t o w s k i and K a i s e r (1958),  A t 4.2°K an a d d i t i o n a l l i n e was r e s o l v e d near t h e  i o n i z a t i o n edge i n t h e 130 ohm cm m a t e r i a l . The u n c e r t a i n t y o f t h e i n t e g r a t e d a b s o r p t i o n i n T a b l e 2 i s about 10 p e r c e n t i,e.  Eq. ( 2 - 1 1 ) ,  f o r a l l l i n e s from t h e u n c e r t a i n t y i n t h e product I n a d d i t i o n an u n c e r t a i n t y o f a p p r o x i m a t e l y  p h c,  5 percent  e x i s t s due t o t h e f a c t t h a t t h e wings o f t h e a b s o r p t i o n l i n e s may n o t be well fitted  by V o i g t p r o f i l e s .  to an o v e r l a p o f n e i g h b o r i n g  However, t h i s i s d i f f i c u l t  lines.  t o e s t i m a t e due  14  The obtained  integrated absorption  cross-sections  by d i v i d i n g the i n t e g r a t e d a b s o r p t i o n  boron i m p u r i t i e s i n the ground s t a t e ( F i g u r e 9). i s estimated  to be a p p r o x i m a t e l y 25  percent.  (]E)  i n T a b l e 3 were  ( T a b l e 2) by the number of The  uncertainty  of  FIGURE  8.  Energy level diagram for holes in boron-doped silicon. The "experimental" energy level diagram (left side) was obtained by taking the ground state at -46 mev.  Along  the vertical arrows the positions of the observed absorption lines are given in mev and cm"*-. The numbers in the center of the diagram (1 to 9) serve to identify the levels for discussion in this thesis.  The relative  intensities quoted in the diagram refer to a boronconcentration of 1.0 x 1 0  14  cm" at 4.2 °K. 3  For comparison, the right side of Figure 8 shows the calculated positions (Schechter 1962) for the ground state and the lowest four excited states.  The numbers in bracket  give the theoretical degeneracies.  To follow page  W/MIIIIIIIIIIIIIIIIIIIIIIIIIIIIII,, VALENCE BAND W//M.  Theoretical ( S c h e c h t e r ,1962) B<0 7 •4  ~  ; -—-  B>0 (2) (2)  •3 (4) 12 (4) -16  -20  >  ——y relative intensities 4.7 .73 4.7 0.3 0.4  ..0  0.6  0.3  0.1  'a  'a  -24  B -28  -32  'B  'a  st  m  o  CJ  •  •  00  r~  st  'flo •  ON  o  CO  'a o  'e  00  to  ON  St  0>  I—1  CO  CO CO  •  CO  o  •  CO  'a  , o  •  'a o  o  O  o  CO  •  •  •  St  St  bo  CO  CO  CO  CO  "0  LO  • 36  (4) > B  > a  LO  -40  st  o• o +1  o• o  +  CNI St  -44  o  iO  •  •  st  co  CO  FIG.  > a  <D  8  > 6  St  CM  o• o +1  o« o +1  O  <t  CO  NO  00  ON  co  •  CO  > B  > a  <D  o • o +1  O• o +1  o• o +1  ON  NO  CD  .  <t  St  ©  •  > a  CM  CM  St  St  > a to  > a LO  o« o +1  o• o +1  LO  ON  i—ie  CO  St  CO  St  FIGURE  9. i  D i s t r i b u t i o n o f h o l e s between t h e ground s t a t e ( a ) , e x c i t e d s t a t e s ( b ) , and t h e v a l e n c e band ( c ) v s . temperature. boron c o n c e n t r a t i o n s a r e : 2) 1.2 x 1 0  1 5  cm" , 3  1) 1.0 x 10  and 3) 1.2 x 1 0  1 6  The  cm" , cm" . 3  The o r d i n a t e  g i v e s t h e number o f h o l e s i n u n i t s o f 1 0 , 1 0 , and 1 0 1 4  1 5  1 6  -3 cm  f o r cases 1 ) , 2 ) , and 3) r e s p e c t i v e l y .  To f o l l o w page  15  CHAPTER  lo  III  Impurity S t a t e s and the Hydrogenic Approximation Silicon  i s a group IV semiconductor, and i t s e l e c t r o n i c  i s such t h a t at zero temperature the e l e c t r o n s f i l l l e a v i n g the c o n d u c t i o n band empty. a group V element  i s added,  i m p u r i t y atom i s formed.  structure  the v a l e n c e band,  When a s m a l l q u a n t i t y o f a group I I I o r  a s e t of energy l e v e l s ,  l o c a l i z e d a t the  For group V i m p u r i t i e s t h e s e a r e r e f e r r e d to as  donor  levels.  They correspond to an e l e c t r o n l o o s e l y bound to the i m p u r i t y  ion.  T h i s e l e c t r o n can be "donated" to the c o n d u c t i o n band w i t h a s m a l l  e x p e n d i t u r e of energy.  In the c a s e o f group I I I elements the l e v e l s are  r e f e r r e d to as a c c e p t o r l e v e l s , s i n c e they can accept an e l e c t r o n from the v a l e n c e band. holes  A l t e r n a t i v e l y one may  l o o s e l y bound to the i m p u r i t y  p i c t u r e a c c e p t o r l e v e l s as c o n t a i n i n g ion.  Boron i s a group I I I element  and  g i v e s r i s e to a c c e p t o r l e v e l s i n s i l i c o n . At room temperature the e l e c t r o n s and h o l e s are removed from the i m p u r i t y s i t e s by thermal e x c i t a t i o n s , and e x i s t as f r e e charge c a r r i e r s i n the c r y s t a l , w h i l e at l i q u i d h e l i u m temperature n e a r l y a l l these c a r r i e r s are bound to s p e c i f i c  i m p u r i t y s i t e s and occupy the lowest energy  These bound s t a t e s have been e x t e n s i v e l y s t u d i e d (Kohn  state.  1957), and i t has II  been shown t h a t they can be approximately, d e s c r i b e d by the same S c h r o d i n g e r e q u a t i o n as the s t a t e s of an e l e c t r o n i n a hydrogen atom, but w i t h the Coulomb p o t e n t i a l m o d i f i e d by 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 H , and the e l e c t r o n mass r e p l a c e d by an e f f e c t i v e mass m*, L u t t i n g e r and Kohn (  ( K i t t e l and M i t c h e l l  1954,  1955): 2ni* V  2  . e /Kr) F ( ? ) 2  = EF  ( r ).  (1-1)  In analogy w i t h the s o l u t i o n s of the hydrogen atom, one expects an l e v e l scheme w i t h the e n e r g i e s reduced by  m*/mX  2  and the Bohr  energy  orbits  16  enlarged  by  Xm/m  .  Taking  one  obtains roughly  The  r a d i u s of the f i r s t o  f o r boron-doped s i l i c o n  the observed v a l u e Bohr o r b i t  e x p l a n a t i o n of why The  f o r the i o n i z a t i o n  t u r n s out  r a t h e r than the 0o53 A i n hydrogen.  such a simple d e s c r i p t i o n of i m p u r i t y  be concerned w i t h  f u n c t i o n r i g h t at the i m p u r i t y  site.  an  states i s possible.  the i m p u r i t y i s  l i k e the d i e l e c t r i c  A l s o , the o r b i t s of the bound c a r r i e r s are  l a r g e t h a t we need not  0,5  energy of 46 mev„  These l a r g e Bohr o r b i t s g i v e  l a r g e f o r a macroscopic q u a n t i t y  to be u s e f u l .  and m /m =  o 13 A f o r a c c e p t o r s ,  to be  mean d i s t a n c e between the bound c a r r i e r and  sufficiently  X =12  sufficiently  the d e t a i l e d n a t u r e  T h i s i n any  constant  of the wave  c a s e i s t r u e of e x c i t e d  s t a t e s , such as 2 p s t a t e s , whose w a v e f u n c t i o n s v a n i s h at the i m p u r i t y 'The  s - s t a t e s are  l i k e l y to be much more c o m p l i c a t e d  s i n c e the e i g e n f u n c t i o n s potential  i s no  are l a r g e r i g h t  longer m o d i f i e d  depressed i n energy,  at the i m p u r i t y  by the d i e l e c t r i c  T h i s d e s c r i p t i o n of i m p u r i t y  and  s i t e where the  constant.  s t a t e s would be q u i t e good, i f the  energy v s , wave number v e c t o r diagram were of p a r a b o l i c shape and simple  extremum at k = 0.  site,  The wave f u n c t i o n s of the a c c e p t o r  had  a  s t a t e s are  l i n e a r combinations of B l o c h waves chosen from near the top of the  valence  I!  band, and  the e f f e c t i v e mass S c h r o d i n g e r  equation  (1-1)  corresponds  simply  to an expansion of the energy around k = 0, keeping terms to o r d e r k . fact  t h a t we  principle. extension  are concerned o n l y w i t h The  large spatial  s m a l l k f o l l o w s from the u n c e r t a i n t y  extension  of the Bohr o r b i t  implies a small  f o r the c r y s t a l momentum k. Unfortunately,  Information  the v a l e n c e band i n s i l i c o n  i s not  simple.  about the s t r u c t u r e of t h i s band *has been o b t a i n e d  c y c l o t r o n resonance e x p e r i m e n t s by Lax, Dresselhaus,  K i p , and  c o u p l i n g the v a l e n c e  Kittel  (1955).  The  Z e i g e r , and  I f i t were not  from  Dexter (1954) and for spin-orbit  band maximum at k = 0 would be s i x f o l d  degenerate,  17 i n c l u d i n g the double degeneracy a r i s i n g understand  from s p i n .  The  t h i s degeneracy i s to c o n s i d e r the t i g h t b i n d i n g l i m i t ,  the wave f u n c t i o n s c o r r e s p o n d i n g to the maximum go over wave f u n c t i o n s f o r s i l i c o n . partially.  The  s p l i t t i n g away from  corresponding  k — 0 i n the (100)  the v a l e n c e band c o r r e s p o n d i n g to gives  X  = 35  one  expects  J =  to atomic  J. = 3/2  l i e s an amount X  1/2.  states,  A theoretical  degenerate  below the top of estimate  mev. the h i g h e s t a c c e p t o r s t a t e s (lowest  t h a t a l l s i x v a l e n c e bands w i l l  i n the a c c e p t o r s t a t e s .  e q u a t i o n , Eq. equations.  £  I n s t e a d of a s i n g l e p a r t i a l  0  ^,  W  (which a r e determined The  (1954) and  •$  ( 1  are numbers r e l a t e d  s p l i t t i n g X).  differential  They are  i s an a c c e p t o r s t a t e energy  by K i t t e l  enter w i t h a p p r e c i a b l e  These have been d e r i v e d by K i t t e l and M i t c h e l l  err ^ [  mev,  (1-1), one f i n d s a s e t of s i x c o u p l e d p a r t i a l d i f f e r e n t i a l  L u t t i n g e r and Kohn (1955).  The  atomic  s t a t e s f o r bound h o l e s ) l i e above the v a l e n c e band by about 40  amplitude  Here  i n which  the degeneracy  d i r e c t i o n into twofold  state  S i n c e i n boron-doped s i l i c o n energy  to  (1954) the top of the v a l e n c e band  Another t w o f o l d degenerate  (Kohn 1957)  i n t o 3p  spin-orbit coupling l i f t s  A c c o r d i n g to E l l i o t t  remains f o u r f o l d degenerate,  bands.  simples way  and M i t c h e l l  to t h r e e e f f e c t i v e mass c o n s t a n t s A, B, C  form of the  data£, and £b»wthe s p i n - o r b i t  j^, may  be found  i n the  articles  (1954) and L u t t i n g e r and Kohn (1955).  In terms of the e f f e c t i v e mass c o n s t a n t s , the e n e r g i e s of t h r e e v a l e n c e bands are g i v e n by  2 )  r e l a t i v e to the v a l e n c e band, maximum.  from c y c l o t r o n resonance  specific  _  the  18 E  (k) = Ak  1 > 2  + [B k  2  2  4  + C  E (I) = 3  (k  2  -X  2 x  +  k  + k  2 y  2 y  k/ + k  2 z  k )] 2  1 / 2  x  ,  (1=3)  Ak , 2  each eigenvalue being doubly, degenerate. The total wave functions of these acceptor states have the form V(?)  =  IZ  An  n,k  (k) V n,l?  6  ,  ~1  F. (r) j  ->  0. (r). j  (1-4)  Here the (j^j (?) are the Bloch functions at the top of the valence bands in the unperturbed crystal.  The Fj (r) are slowly varying functions which  modulate the Bloch functions. For this reason they are often referred to as envelope functions. In the limit where the mass constants B and C become zero (corresponding to spherical constant energy surfaces), the envelope functions reduce to simple hydrogenic functions. Schechter (1962) attempted to solve Eq. (1-2).  In order to simplify the calculations he  f i r s t calculated the energy levels in the extreme limits of X = 0 and X =0*. First-order perturbation calculations were then made, in \ and X~^ in order to obtain estimates for the energy levels corresponding to the actual, finite value of X.  Expanding the envelope functions F.. ("r) in  spherical harmonics, Schechter»s acceptor state t r i a l functions had the form V ^ Z I r 1  1  exp. <-r/r ) 1  XI j,m  C lm j  Y < 9 , <J> ) 0 <r). lm ' j  (1-5)  The complete acceptor Hamiltonian is invariant under the operations of the f u l l tetrahedral group T^ (Margenau and Murphy 1943) i f spin is ignored, or, the tetrahedral double group (Dresselhaus 1955) i f spin is included from the beginning.  The acceptor state wave functions  19 for  each- l e v e l form bases f o r i r r e d u c i b l e r e p r e s e n t a t i o n s of  a p p r o p r i a t e symmetry group.  the  S i n c e the s p h e r i c a l harmonics f o r each 1 go  i n t o s p h e r i c a l harmonics of the same 1 under the symmetry o p e r a t i o n s , each term i n the sum  over  1 i n E q . ( l - 5 ) must t r a n s f o r m i n the same way  complete wave f u n c t i o n .  A f t e r u s i n g group t h e o r e t i c a l arguments to  the number of s p h e r i c a l harmonics a p p e a r i n g  i n the t r i a l  wave f u n c t i o n and  the energy were o b t a i n e d by maximizing  v a l u e of the energy  f o r an e l e c t r o n s t a t e .  m i n i m i z i n g the energy The determined,  f o r a hole  the e x p e c t a t i o n  ( T h i s corresponds  s i n c e the v a l e n c e band e n e r g i e s depend o n l y on B linearly  i n the  D  j j> and  o  f  E  fl*  (l- )» 2  these c a l c u l a t e d  The  8  agreement w i t h experiment  group.  and  In  The  T% of '  The ground s t a t e i s predominantly  2p s t a t e s r e s p e c t i v e l y ,  v a l e n c e bands c o l l a p s e i n t o  (1962)  positions  ( i n brackets)  c o r r e s p o n d i n g to the  the lowest f o u r e x c i t e d s t a t e s 2p - l i k e i n the sense Is  (1-3).  i s seen to be o n l y  , o r t w o f o l d c o r r e s p o n d i n g to V, and o  t e t r a h e d r a l double  by Eq.  those of the h i g h e s t f o u r e x c i t e d  l e v e l s a r e e i t h e r f o u r f o l d degenerate  representation P  2  l e v e l s , together with t h e i r degeneracies  a r e shown i n F i g . 8.  hydrogenic  to  Schechter  e l e c t r o n a c c e p t o r s t a t e s f o r both p o s i t i v e and n e g a t i v e B.  The  to the  state).  c a l c u l a t e d the ground s t a t e energy  fair.  reduce  s i g n of the e f f e c t i v e mass c o n s t a n t B i s e x p e r i m e n t a l l y not  However, B o c c u r s  of  the  f u n c t i o n s , and  1 to v a l u e s l e s s than some IQ , the b e s t approximations  restricting  as  the Is-like  and  t h a t they reduce  i n the l i m i t when a l l t h r e e  one.  the f o l l o w i n g s e c t i o n s t h e s e r a t h e r c o m p l i c a t e d wave  f u n c t i o n s w i l l not be used. w i l l be a r b i t r a r i l y  I n s t e a d simple h y d r o g e n i c wave f u n c t i o n s  employed i n which the f i r s t  by an e f f e c t i v e Bohr o r b i t  a*.  It w i l l  to  Bohr o r b i t  i s replaced  be r e q u i r e d t h a t t h i s  effective  20 Bohr orbit satisfy the hydrogenic energy equation E where E  2.  n  =  - e/  a* n  2X  2  2  ,  (1-6)  is the experimental binding energy.  n  General Interpretation of Line Broadening Consider a semiconductor with a random distribution.of  and  -  neutral acceptor or  donor impurities.  ionized  At an absorbing neutral  impurity there w i l l be a certain electric field, the magnitude of which depends on the specific spatial configuration of the surrounding impurities. This electric field may give rise to a f i r s t order Stark splitting, or second order Stark shift, of different magnitudes for different impurity sites.  Thus one obtains a broadening of the resultant absorption line. Let us suppose the distribution of intensity of absorption for  one neutral impurity which is in an electric field F, is given by I (F,W); then in order to obtain the total intensity distribution in the Stark broadened line one must integrate over a l l such distributions weighted by the probability of the distribution occurrences I  (fiW),diO =  oo  dOO j"  I (F,(0) W (F)  dF.  (2-1)  o Here W (F) is the field strength probability function, which has been studied by Holtsmark (1919, 1924)  and others in connection with  gravitational problems and pressure broadening in gases.  This probability  function w i l l be discussed in the following section. Each component of the statistically- broadened absorption line w i l l be treated as a sharp line, that is to say the I (F,C0) in Eq.(2-1) will be replaced by a delta function.  Actually the true width of I(F,W) may be  obtained by extrapolating the experimental true width in Figure;; 5, to zero concentration of impurities (dotted lines).  These widths are seen to be  21 already quite considerable, and are believed to result primarily from phonon broadening, and broadening by internal strains due to dislocations. The latter two broadening mechanisms w i l l be discussed in sections 10 and 9 respectively. After the width resulting from statistical Stark broadening of two sharp energy levels has been obtained i t has to be combined with the width at zero concentration by a convolution integral of the form fc (x) = where  (x - y) I(y) dy  -oo  ,  (2-2)  f^ (x) is the true line shape at concentration c, f  Q  (x - y) is  the true line shape extrapolated to zero concentration of impurities, and I(y) is the line shape obtained from, statistical Stark broadening under the assumption that the I(F,Cu>) simplifying assumption that "  then the half-width of f I(y).  c  in Eq. (2-1) are sharp lines. If one makes the "  f  c  it  , f  0  , and I(y) have lorentzian profiles, tt  is simply the sum of the half-widths of f  This assumption is at least approximately  0  and  justified (Colbow et a l .  1962).  3.  Statistical Theory of Stark Broadening The basic question i s , "What is the probability that the frequency  of a sharp absorption line is displaced by an amount between A CO and ACO + dACO"?  Let us assume that this frequency displacement  the displacement  results from  of only one of the two levels involved in the absorption,  by means of either the linear or the quadratic Stark effect.  This w i l l be  justified later. The probability of a certain frequency displacement due the surrounding impurities is then simply related to the probability of a certain electric field at the absorbing impurity.  This in turn depends on  22  the probability of finding another impurity at a distance r from. i t . the  Since  I(F,to) in Eq. (2-1) is taken to be a delta function, the probability  function for a frequency displacement between ACO and  ACJ+ dACO w i l l give  directly the desired line shape of the absorption line The problem may  I(AtO).  be divided into several parts according to the  dependence of the f i e l d on the distance from the f i e l d producer.  In the  present section i t w i l l be assumed that the field at the absorbing neutral impurity results mainly from the surrounding ionized impurities, and has the form of a Coulomb f i e l d :  F = (e/4H"€) r  .  This case becomes important  above 40°K, when an increasing number of impurities w i l l be ionized (Figure 9).  (In the following section the effect of screening by free holes will be  discussed in an approximate fashion, and in sections 7 and 8 line broadening resulting from the interaction with neutral impurities will be considered). For the purpose of illustration effect of only the nearest neighbor.  i t is useful to consider f i r s t the  This approximation may  be referred to  as the binary form of the Holtsmark theory. Consider a neutral impurity atom and ask what is the probability W(r) dr  that there is no ionized impurity closer than r and one ionized  impurity in the spherical shell volume element be  N.j_ ionized impurities per cm .  4TCr  dr  at r.  Let there  The probability of having one ionized  impurity in the shell between r and  r + dr is assumed to be given by  p(r, dr) = (4TTr  2  dr)  N. t  The probability of no ionized impurity closer than r, P(r) is obtained from the relation P(u + du) = P(u) (1 - 4 TTu du r  2  Thus dP  / P  =  - 4TTu  2  N  £  du  ,  N ). t  23  which gives after integration P(r)  =  exp.[ - (4TT/3) r  Ni] .  3  Thus one has W(r) dr =  P(r) p(r, dr)  =  After introducing the mhan s^ae*«g r (4TT/3) r  =  3 0  1/ %  4 l N  r dr exp.[- (4TC/3) N r ] . 2  3  t  t  , defined by  Q  ,  (3-L)  one may finally write W(r) dr =  exp.(-r/r )  d(r/r )  3  0  0  3  .  (3-2)  Since a given r defines an electric field F at a neural impurity site, and an electric field defines a frequency displacement via the Stark effect, Eq. (3-2) also represents the distribution of these last two quantities. Thus i f the field is given, by F = ( e / 4 T 6 ) r  -  2  , substitution into Eq. (3-2)  gives W(F) dF = C exp. [- ( F / F )  3 / 2  0  =  (3/2F) ( F / F ) / 3  2  0  ]  d (F /F)  3 / 2  0  exp. [- ( F / F )  3/2  0  ' ] dF , (3-3)  2  where F — (e/4TT£) r . C was taken as minus one, in order that the field probability be normalized to D  Q  T o  W(F) dF =  1 .  (3-4)  For the linear Stark, effect AW = s F and A C0 = s F , where s Q  is a constant.  Using a delta function for  l(F,W) , one obtains from  Eq. (2-1) oo  KAOO) = =  =  S <f(AlO - s F) W(F) dF o oo /  I !" s  I s|"  Substituting for W(AW/s)  1  1  Q  £(F - i W / s ) W(F) dF W(kW/s) .  from Eq. (3-3), this becomes  24  ( s / | s | ) (3/2 LU)  I(AU>) ,=  (AW /AU3)  exp.[-(L» /*0)  3 / 2  0  3 / 2  0  ]  .  (3-5) One  o b t a i n s two  peaks c o r r e s p o n d i n g  to p o s i t i v e and n e g a t i v e s, each having  the i n t e n s i t y d i s t r i b u t i o n f u n c t i o n  I(AU)  For the q u a d r a t i c S t a r k e f f e c t where t i s another I(ACO)  constant. =  oo / o  =  oo /  =  (1/2  Au?= t F  Again u s i n g Eqs.  ^(AW  Itl"  as g i v e n by Eq.  - t F )  W(F)  2  1  «f(F  2  1 / 2  W(F)  J  since F i s positive. I(AU»  =  (l/2|t|)(AU)/t)Or f i n a l l y ,  1 / 2  A bJ  and  (3-3)  6~[  F  -  t F  2 Q  ,  t h i s gives  (AW/t)  + (AcO/t)  I / 2  ] W(F)  dF  exp.[ - ( A t 0 o / A u ; )  ].  w[(AW/t) / 3 1  u s i n g Eq.  -  Q  dF  [^[F +  =  (2-1)  and  dF  - A(0/t)  |t|)(A(0/t)"  9  (3-5).  2  1 / 2  ]j  ,  (3-3)  (t/|t|)(3/4A(0)(ALJ /AW)  3/4  o  3/4  (3-6) For any  g i v e n energy l e v e l t can be e i t h e r p o s i t i v e or n e g a t i v e ,  on the s p e c i f i c  The approximations  depending  energy, l e v e l s t r u c t u r e .  f o r e g o i n g elementary c o n s i d e r a t i o n s , which y i e l d to the i n t e n s i t y d i s t r i b u t i o n Eq.  superseded by the work of Holtsmark (1919, 1924) the c o o p e r a t i v e e f f e c t of many i o n s .  (3-5)  and  the b i n a r y  (3-6), have been  and o t h e r s , who  included  Review a r t i c l e s on t h i s f i e l d where  w r i t t e n by Chandrasekhar (1943), Breene J r . (1957), and Margenau and  Lewis  (1959). The  f i e l d F, which appears i n Eq.  (3-3), must be w r i t t e n as a  v e c t o r sum when i t i s compounded from the f i e l d s of many i m p u r i t i e s i n different  places;  25 F  =|2__F |  .  n  (3-7)  The analysis leading to Eq. (3-3) must be carried out in a configuration space of 3N dimensions.  In place of Eq. (3-3) i t yields the function W(B) dp  plotted in Figure 12 (curve labelled Holtsmark), where  F/F . For small Q  and large values of B one obtains respectively the two alternate expressions: W(B ) =  oo  (2/TtB) / o  f(4/3TT) B  v sin v  (1 - 0.463 B  2  L 1.496 496 B "  exp. [ -(v/ B )  (1 + 5.107  5J // 2i  2  ]  3 /  + 0. 1227  B  dv + ... ), p small  A  BB"~ ' + 14.93 p " + ..),B large. J3// 2  3  (3-8) In the binary theory  F was defined by  F  =  Q  Q  =  (e/4TC6) r "  =  2.60 (e/4TU6) N i  2  0  (e/4TC6) 2 / 3  [(4T/3)  2  /  3  .  The many-ion calculation gives nearly the same parameter: F  D  =  2.61 (e/4TTe) N  .  2 / 3 t  (3-9)  QJne then passes to the frequency distribution in the same way as before: In the linear Stark effect ku?/Au5  0  =  F/F  I(kiJ)  d(&u3) = W ( £ ) dB,  KAO)  =  c  = B , and since in general (3-10)  one finds i/LW  0  w(Au;/t^u; ) .  (3-11)  0  1 /o  For the quadratic Stark effect  (&U?/tW ) ' 0  = F/F  Q  = |3 , and Eq. (3-3)  gives KA00) ,=  1/2 (£\0J &vOr o  1/2  W t(W/fcu) ) Q  1 / 2  ]  .  (3-12)  26 4.  The  E f f e c t of  Screening  When.the c r y s t a l c o n t a i n s w i l l a l s o be an equal number  p =  i m p u r i t i e s per cm  of f r e e charge c a r r i e r s ,  of which has been n e g l e c t e d so f a r . c a r r i e r s may  ionized  The  , there  the i n f l u e n c e  e f f e c t of these f r e e charge  be approximated by u s i n g a screened Coulomb p o t e n t i a l of  the  form  U ,=  (e/4Tre)  e"  r / X  ,  (4-1)  IT  .  Where A  1/r  i s the Debye-Huckel (1923) s c r e e n i n g l e n g t h .  In c l a s s i c a l  statistics A  =  [ kT/4TT p e ] 2  1 / 2  ,  (4-2)  k b e i n g the Boltzmann c o n s t a n t . D i f f e r e n t i a t i n g the e x p r e s s i o n f o r U i n Eq. screened Coulomb F>  =  (4-1)  one o b t a i n s  field (e r / 4 T l * r ) [ ( 1 / r ) + ( 1 /  )]  2  A  which i s to r e p l a c e the simple Coulomb f i e l d  e"  r / X  i n Eq.  ,  (4-3)  (3-3). However,  a n a l y s i s of the l i n e - b r o a d e n i n g problem even w i t h t h i s s i m p l i f i e d remains f o r m i d a b l e . approximated  the  U s i n g the same approach as Ecker  (1957), F  field has  been  by  '4TC6 r  if  3  r <  X  (4-4)  if r U  0  ,  F i g u r e 10 shows a comparison between the screened Coulomb f i e l d w i t h  i t s c u t - o f f at r = A  field  at 90°K and  the  f o r t h r e e c o n c e n t r a t i o n s of  i m p u r i t i e s , as a f u n c t i o n of r i n u n i t s of r . s  Where the raeaw apae-iog  r  s  i s d e f i n e d by  (4TU/3) U s i n g Eq.  r  3 s  =  1/N . A  (4-5)  (4-4), the method of s e c t i o n 3 leads to a dependence of the  shape on the s c r e e n i n g parameter  line  FIGURE  10.  Screened electric field (dotted lines), and Coulomb field with cut-off ( f u l l lines), as a function of separation. The separation between field point and field producer is expressed in units of the ntonni uporing "hr.twr.r.n . impnr "I ti r.n 14 (r ). s  The boron concentrations are:  i i ) 1.2 x 10  15  i) 1.0 x 10  cm" , and i i i ) 1.2 x 10 3  16  cm . -3  -3 cm  ,  The screened  electric fields were calculated from the equation F = (e/4Tie) r ( r " f + X " ) e ° - 1  1  r / X  , using the calculated  screening length X at 90°K.  To follow page 26.  27 £ =  (4Tt\  /3) %  3  =  (1/6)TU°  1/2  (kT/e ) 2  3 / 2  p~  1 / 2  .  (4-6)  The physical meaning of the screening parameter is the number of ionized impurities within the Debye radius A  . Clearly as X tends to infinity the  results must agree with those of Holtsmark.  Figure 12 shows the results of  Ecker's (1957) computations for several values of £ one can obtain the half-widths  . From these curves  h ( B ) , which have been plotted in Figure 13  as functions of the screening parameter.  The dependence of the screening  parameter on temperature has been shown in Figure 11 for three different concentrations of impurities. Figure 11 was obtained by using Eq. (4-6) and the values for p from- Figure 9.  5.  Temperature Dependence of the Half-width due to the Linear and the Quadratic Stark Effect For the linear Stark effect  &(0 =  s  F  =  s F B 0  ,  and hence in units of frequency, the half-width of h^AtO) =  s F  D  I(A.(0)  is given, by  h(B ) .  (5-1)  For the quadratic Stark effect =  t F  2  =  t F  2  p  Q  2  ,  and hence in units of frquency, the half-width of h (Lo!>) 2  -  t F  h (B  2  2  0  I(&CO)  is given by  ) .  (5-2)  Taking for silicon the dielectric constant 6/ 6  Q  = 12 , one finds, using  Eq. (3-8) F  Q  =  2.61 (e/4Tue) N i  2 / 3  =3.13  x 10'  8  Ni  2 / 3  volt cm" , 1  (5-3)  -3 where Ni is in units of cm  .  Using Eqs. (5-1), (5-2), (5-3), and Figures 11 and 13, the curves for  h^(AcJ) / s  and  h2(AoO) / t  as a function of temperature for three  FIGURE  11.  S c r e e n i n g parameter, £,,vs. t e m p e r a t u r e f o r boron concentrations of and  iii)  1.2 x 1 0  i ) 1.0 x 1 0 1 6  cm"  3  1 4  cm" , 3  i i ) 1.2 x 1 0  1 5  cm" , 3  .  To f o l l o w page 27.  FIGURE  Field of  distribution  12.  function, W(B),  t h e s c r e e n i n g parameter, <T.  f o r various values  (Ecker  1957).  To f o l l o w page 27.  0 FIG. 12  0.5  1  1.5  2  2.5  3  3.5  4  FIGURE  13.  H a l f - w i d t h of the f i e l d d i s t r i b u t i o n , f u n c t i o n of the (logarithmic  h(B),  as  a  s c r e e n i n g parameter, 6*;  scale).  To  f o l l o w page  27.  28 impurity concentrations have been calculated. These graphs are shown in Figures 14 and 15 respectively, and their temperature dependence must be compared with Figure 6, which was obtained by subtracting the half-width extrapolated to zero concentration of impurities (Figure 4) from the true half-width in Figure 5. Let us assume that the electric field at the impurity center has its direction along the z-axis. «fV  =  The perturbation Hamiltonian is then  -Fez.  (5-4)  The wave functions of the bound carrier have the form yr (!)(•?) = ZIoCj F .  ( i )  (r)  0.(r)  (5-5)  in the effective mass theory of donor or acceptor states.  Here the  are numerical coefficients, the 0 j ( r ) Bloch waves, and the  F^^(r)  hydrogen-like envelope functions. The matrix element of <fV  are  between two  such states is given by (  y(i),  y(i') )  .  F j  (O  f  £  v  Fj  (i»> ) .  ( 5  .  6 )  Since the effective mass Hamiltonians for both types of states are invariant under inversion and there are no accidental degeneracies, such as the 2s, 2p degeneracy in hydrogen, the first-order Stark effect vanishes.  However, the  f u l l Hamiltonian of the impurity, problem has only tetrahedral symmetry and is not invariant under inversion. As a result, i f the effective mass theory is seriously in error, states belonging to the representation T^, can have an appreciable first-order Stark effect. significance for the acceptor ground state  (Yq)  an<  *  Vg  This might possibly be of in silicon (Kohn 1957),  but the resulting effect on the half-width is d i f f i c u l t to estimate.  By  Eq. (3-5) one would presumably obtain two peaks, corresponding to positive and negative strength parameter, whose unresolved resultant may give rise to  FIGURE  14.  Half~width in units of the strength parameter s vs. temperature for the linear Stark effect. concentrations are:  i) 1.0 x 10  i i ) 1.2 x 10  and i i i ) 1.2 x 10  15  cm" , 3  14  The boron  cm" , 3  16  cm"  3  .  To follow page 28.  0  10  20  30  40  50 T (°K)  FIG. 14  60  70  80  90  FIGURE  15.  Half-width in units of the strenght parameter t vs. temperature for the quadratic Stark effect. 14 boron concentrations are: i i ) 1.2 x 10  15  cm" , 3  i) 1.0 x 10  and i i i ) 1.2 x 10  The 3  cm 16  ,  cm"  3  .  To follow page 28.  29 an appreciable broadening of the absorption line.  Fortunately, however,  Figures 6, 14, and 15 seem to suggest by their temperature dependence that the second-order Stark effect might be the dominant effect and, as w i l l be seen in the next section, sufficient to explain the experimental data.  If  the broadening in Figure 6 were caused predominantly by the first-order Stark effect a steeper rise of the half-width at low temperature would have to be expected, as will be seen from Figure 15, especially for the 1.3 ohm cm material.  6.  Evaluation of the Half-width due to the Quadratic Stark Effect The second-order Stark shift for a nondegenerate state y ^  is  given by * E<D 2  =  ' | y(i), vV (  Replacing the denominators  i , )  X  ) | / ( E < ^ - E^')) .  E ^ ^ - E^*'^  assuming that the linear Stark effect  (6-1)  2  by an average value A E  (y  ( i )  &V  ,  and  vanishes, Eq. (6-1)  becomes * E  ( 1 )  2  =  1/AE  £!' |  =  1 / A E 5Z* i  (>  £v y ( i )  ,  «sv y  ( i , )  ( i , )  )l  )(>  2  ( i , )  ,  iv  y  ( i )  )  1  =  1 / A E (>  =  1/AE  ( F  ( i )  ( i )  ,  (cfV)  2  ,  (cJV)  2  ) F  ( 1 )  ) .  (6-2)  The shifts w i l l be of a similar order of magnitude for degenerate states. Using Eq. (5-4) for £ V  and simple hydrogenic wave functions for  F^^,  Eq. (6-2) has been evaluated in Appendix II. The result is £E 2  Here  a*  =  t F  2  =  c (e F a*) / AE . 2  (6-3)  is the effective Bohr radius, and c is a constant which turns out  to be 1 for the Is state and 18.for the 2p  0  state (Appendix II).  Thus, the  30 q u a d r a t i c S t a r k e f f e c t may be n e g l e c t e d f o r t h e ground s t a t e , when compared with that f o r the e x c i t e d s t a t e . What i s t h e v a l u e o f t r e q u i r e d , such t h a t  h2(A^)/t  g i v e s t h e temperature dependence o f t h e h a l f - w i d t h in> F i g u r e 6?  i n F i g u r e 15 The c u r v e  f o r t h e 1.3 ohm cm boron-doped s i l i c o n seems best s u i t e d f o r n u m e r i c a l c o m p a r i s o n , s i n c e i t shows t h e l a r g e s t change w i t h  temperature.  From F i g u r e 6 t h e change i n h a l f - w i d t h between 0°K and 80°K i s MhJ!  - h^) .=  1.06 mev.  (6-4)  Over t h e same temperature range one o b t a i n s from F i g u r e 15: A[h (AO) / t ]  =  2  Equating  k^t^AvO)] t  =  0.95 x 1 0 v o l t 6  and A (h£ - h^)  1.1 x 1 0 " mev v o l t " 6  2  2  cm"  2  .  (6-5)  one o b t a i n s cm  .  2  (6-6)  L e t us assume t h a t energy l e v e l 4 ( F i g u r e 8) may be approximated by a hydrogenic  2p s t a t e . E  2  =  The e x p e r i m e n t a l v a l u e f o r i t s b i n d i n g energy i s  (46 -  39.7) mev  =  U s i n g t h i s v a l u e and t h e h y d r o g e n i c E with  n  =  - m* e  4  / 2 Ag Y 2  2  6.3 mev .  energy n  2  =  (6-7)  equation - e  2  / 2 X a* n  2  ,  (6-8)  n = 2 , one o b t a i n s f o r t h e e f f e c t i v e Bohr r a d i u s o f s t a t e 4: a*  =  24 1 .  (6-9)  U s i n g Eqs. ( 6 - 6 ) , ( 6 - 9 ) , and t a k i n g  c = 18  for a 2p  0  s t a t e , Eq. (6-3)  leads to A E  =  c e  2  a*  2  / t  =  1 mev ,  w h i c h has t h e o r d e r of magnitude one would expect  (6-10) from t h e g e n e r a l n a t u r e of  the energy, l e v e l diagram ( F i g u r e 4 ) . In equating  A[h2(AW)J  and A ( h c - h©) i t was assumed i m p l i c i t l y ,  t h a t t h e i n t e r a c t i o n between n e u t r a l i m p u r i t i e s , w h i c h i s r e s p o n s i b l e f o r t h e low temperature b r o a d e n i n g ,  changes l i t t l e w i t h temperature below 80°K.  31 Screening and a decrease of N  A  -  w i l l tend to reduce this contribution  to the half-width with increasing temperature.  However, since these forces  w i l l be appreciable only for nearest neighbors, for which the screening is small below 80°K, the temperature dependence of their contribution to the half-width should be relatively small below 80°K.  7.  Statistical Broadening due to van der Waals Forces In this and the following section the interaction between neutral  impurities w i l l be discussed.  The present sectionideals with van der Waals  forces, which arise when the overlap between the wave functions is negligible.  A l l other forces between neutral atoms may be classed as  "overlap forces" and arise only when the wave function of one atom overlaps that of the other.  These forces w i l l be considered in the following section.  Margenau (1933) developed the statistical theory for broadening by van der Waals forces in order to explain pressure broadening in gases.  The  line shape of this distribution is I(AV>)  =  2TUb  N Atf"  1/2  3 / 2  exp.^Ti:  3  N  2  / 9 A*) .  (7-1)  The half-width of this distribution is h (AV) w  =  0.82TC b N 3  2  .  (7-2)  Where b is the strength constant of the van der Waals forces. Vi  =  - b r f  That is (7-3)  6  is the van der Waals interaction between the absorbing impurity and the ith neutral impurity, and r i is the distance between them. When applied to the present problem neutral impurities per cm . 3  N =  - Ni  is the number of  Thus at very low temperatures one has  N = N^.  Figure 6 shows that at low temperatures when the number of ionized impurities is negligible, the half-widths are 0.03,  0.12,  and 0.60 mev, for  i m p u r i t y c o n c e n t r a t i o n s of  ( 1 , 12, and  120) x 10  cm  respectively.  If  t h e van der Waals f o r c e s were t o e x p l a i n t h i s c o n c e n t r a t i o n b r o a d e n i n g at t e m p e r a t u r e s , then a c c o r d i n g to Eq. t h e square of the c o n c e n t r a t i o n . 1 : 144  : 14400.  low  (7-2) t h e h a l f - w i d t h s h o u l d i n c r e a s e w i t h  That i s , they s h o u l d be i n the  However, they a r e o b s e r v e d t o go l i k e  ratios  1.: 4 : 20 .  I f t h e e x p r e s s i o n by London and E i s e n s c h i t z (1930) f o r the s t r e n g t h parameter b i s m o d i f i e d t o s u i t t h e p r e s e n t problem, one b  =  6.48  (e  2  / Ka*)  S u b s t i t u t i n g t h i s i n t o Eq. h (A\^)  =  w  where  a*  2.0  i s i n cm and  3  1 6  cm"  =  w  0.60 to  5  (7-4) taking X  N  2  = 12 f o r s i l i c o n ,  leads to  mev,  Taking  (7-5) f o r the e f f e c t i v e Bohr r a d i u s  a* = 13 X, Eq.  (7-5) g i v e s f o r  N = 1.2  x  3  h (Av>) T h i s has  a*  i n cm".  of the ground s t a t e a v a l u e of 10  .  ( 7 - 2 ) , and x 10"  N  a*°  obtains  1.1 x 10"  5  mev.  t o be compared w i t h an e x p e r i m e n t a l  mev.  (7-6) half-width  (he - h^)  of  Thus i t would seem t h a t the van der Waals i n t e r a c t i o n i s too weak  e x p l a i n t h e v a r i a t i o n o f t h e h a l f - w i d t h w i t h c o n c e n t r a t i o n of i m p u r i t i e s  a t low t e m p e r a t u r e s ( F i g u r e 6 ) .  8.  B r o a d e n i n g due  to Overlap  Forces  I n t h i s s e c t i o n i t w i l l be shown i n a q u a l i t a t i v e way f o r c e s may  how  overlap  account f o r the o b s e r v e d e a r l y onset of c o n c e n t r a t i o n b r o a d e n i n g  a t low t e m p e r a t u r e s ( F i g u r e 6 ) . Baltensperger s 1  (1953) work, the r e l e v a n t p a r t of w h i c h i s shown  i n F i g u r e 16, s u g g e s t s t h a t c o n c e n t r a t i o n b r o a d e n i n g should s t a r t at r  s  =  12 a* .  (8-1)  Here a* i s the e f f e c t i v e Bohr r a d i u s of a bound c a r r i e r of e f f e c t i v e mass  m*  FIGURE  16.  Broadening of hydrogenic levels vs. distance between impurities in units of the effective Bohr radius, a*. (Baltensperger 1953).  To follow page 32.  33 in a crystal with dielectric constant X, and r (4 T/3)r If one takes (Conwell  3 s  is defined by  g  = 1/N .  (8-2)  A  1956) for boron-doped silicon "K ~ 12, m /m = 0.5,  and hence a* = X a m/m*  = 13A ,  (8-3)  then Eqs. (8-1) and (8-2) would suggest that concentration broadening should set in at about N  = 6 x 10  16  cm"  .  3  (8-4)  This, as was mentioned earlier, seemed to agree with Newman's (1956) data. In contrast to this, however, the present data (Figure 6) shows that concentration broadening starts below N  A  = 1.2 x 10  15  cm"  .  3  (8-5)  How can one bring Baltensperger's calculated values in agreement with this earlier onset of broadening? involves replacing r  s  Two ways suggest themselves.  in Eq. (8-1) by Xr , where X<1. s  The f i r s t  This takes care of  the fact that the impurities do not form a regular lattice as assumed by Baltensperger (1953).  The second modifiction, that of a*, takes into  consideration that the simple hydrogenic model used to obtain Figure 16 gives rather inadequate values for the binding energy. Instead of a regular spacing of impurities, let us assume a random distribution.  Since for overlap forces we are only concerned with  small separations, we may use the binary approximation of Holtsmark's theory. r = Xr  s  By Eq. (3-2) the probability of a nearest neighbor at a distance is given by W (X) dX = exp. (-X )dX 3  Now  3  .  (8-6)  let us ask the question: "What is the value of X = Xjy2> such that half  the impurities find themselves closer than X]y  r 2  t o s  t n e  neighboring  one,  34  and half of them are further away?" Eq. (8-6) gives ^1/2 1/2 = f o  if3 J  exp.(-X ) dX 3  l/2  or,  i  _  (_  v  = 1 - exp. (- X a  v  n  J  3  ) ,  0.885.  =  X  _  The mean spacing of the impurities closer than  r  s  t o  t  n  e  i  r  nearest  neighbor is 0.648 r , which suggests that the effective mean spacing in s  Eq. (8-1) should be about r * = 0.7 r g  s  .  (8-7)  For computing the edges of the Is and the 2p.bands (Figure 16) Baltensperger (1953) used a simple hydrogenic model in conjunction with the cellular method. Schrodinger  This involves assuming the validity of the effective mass  equation  * /2m* V V 2  + (e /Xr + E )V  2  2  n  = 0  (8-8)  within a sphere of radius r ,,defined by Eq. (8-2). s  The general solution  of Eq. (8-8) has the form Yn,l,m = n , l R  ( r )  Y  l,m <e,4>) .  (8-9)  and is well known from the hydrogen problem. The energy is given by E  = - m* e/2«fi K n 4  n  2  2  2  = - e /2Xa*n . 2  2  (8-11)  where, in the cellular method, n is to be determined by appropriate boundary * conditions.  A value of a  o = 13A for boron doped silicon gives one the  observed ionization energy E^ — 46 mev for the bound hole. In estimating the concentration where the overlap forces become appreciable, one is concerned with the 2p states, since these broaden long before the Is state (Figure 16).  It is now realized that acceptor states  are much more complicated than simple hydrogenic wave functions. However, i f one wants to approximate the particular line under study by a simple  35 hydrogenic  ls-*2p  transition, the most reasonable approach might be to  require that a*, in the 2p wave function V2IO with E  2  n = 2.  '  Eq  *  (8-1-0), satisfy Eq. (8-11)  Thus, using the experimental value for the binding energy of  = 6.3 mev for energy level 4 (Figure 8), one obtains from.Eq. (8-11) a* = 24 %' .  (8-12)  With the modifications given by Eqs. (8-7) and (8-12), one obtains from Eqs. (8-1) and (8-2), for the onset of concentration broadening at low temperatures, a value of N  =  A  3 x 10  15  cm"  ,  3  (8-13)  which compares rather well with Eq. (8-5), considering the crudeness of the assumptions involved.  Not only is the hydrogenic model a rather crude  approximation, but the usefulness of the band scheme itself becomes doubtful, when one is dealing with a random distribution of impurities.  9.  Broadening due  to I n t e r n a l S t r a i n s  In t h i s and f o r the h a l f - w i d t h , It  the f o l l o w i n g s e c t i o n an attempt  extrapolated  to zero  i s b e l i e v e d t h a t an a p p r e c i a b l e  s t r a i n s due  to d i s l o c a t i o n s .  expected to be  impurity  i s made to account  concentration  p a r t of t h i s w i d t h r e s u l t s from  The  internal  T h i s c o n t r i b u t i o n to the h a l f - w i d t h i s  e s s e n t i a l l y temperature independent over the  range i n v e s t i g a t e d .  (Figure 5).  temperature  remaining c o n t r i b u t i o n to the h a l f - w i d t h w i l l  be  a t t r i b u t e d to phonon broadening i n the f o l l o w i n g s e c t i o n . The 1962)  samples used i n t h i s  d i s l o c a t i o n d e n s i t y of about  experiment have a quoted (Merck and n = 5 x 10  4  Co.  d i s l o c a t i o n l i n e s per cm  2  .  These d i s l o c a t i o n l i n e s are expected to g i v e r i s e to i n t e r n a l s t r a i n s whose magnitude i s g i v e n s  .=  a p p r o x i m a t e l y by  ( n / ) x 10" 1  2  8  cm  ^  (Kohn 2.2  1957) x 10"  6  .  (9-1)  36 As d i s c u s s e d i n s e c t i o n  III-l,  i n the absence o f s t r a i n s a l l  a c c e p t o r l e v e l s a r e a c c o r d i n g to t h e o r y e i t h e r Under a shear s t r a i n s^ t h e f o u r f o l d t w o f o l d degenerate l e v e l s , arde»=±ja=s.  LIVI  degenerate l e v e l s w i l l  s£ ^  =  (10  _o  € ^  n £  15 ev (Lax and B u r s t e i n 1955). from t h e ground s t a t e  broadened by about 0.03 ground s t a t e  0.06 mev.  For  Thus f o r kT > s £  T<Tl/2 °K  (9-2)  deformation p o t e n t i a l  ( i . e . T > 1/2 ° K ) ,  to a t w o f o l d degenerate l e v e l w i l l be strains; while transitions  from  degenerate l e v e l w i l l be broadened by about  only t h e t w o f o l d degenerate lower energy  level  Hence no broadening o f a b s o r p t i o n l i n e s due to the ground  s p l i t t i n g would be expected, and o n l y a t r a n s i t i o n  e x c i t e d s t a t e would be broadened by about 0.03  10.  first  o f t h e s t r a i n and i s o f the o r d e r  mev due to i n t e r n a l  to a f o u r f o l d  w i l l be o c c u p i e d . state  I.<-.-J-Q  0.03 mev .  i s an e f f e c t i v e  c o n s t a n t , which depends on the geometry  the  i n t o two  1/2  cm)  t h i s and t h e f o l l o w i n g s e c t i o n  transitions  degenerate.  The magnitude o f the s p l i t t i n g i s  AE  of  split  pron n__t"Vi <r_twr> f rt 1 rl nnrr. r m i i n - i w ^ p l  ft  In  twofold or f o u r f o l d  to a f o u r f o l d  mev .  Phonon Broadening Let  us c o n s i d e r a b s o r p t i o n l i n e 4,  from t h e ground s t a t e  to the energy l e v e l  electron-phonon i n t e r a c t i o n to have a f i n i t e  c o r r e s p o n d i n g to the t r a n s i t i o n  Tp ( l e v e l 4 i n F i g u r e 8 ) .  w i l l cause both t h i s  life-time, resulting  The  l e v e l , and t h e ground  state  i n a broadening o f these l e v e l s , and  thus a l s o  i n a broadening o f the c o r r e s p o n d i n g a b s o r p t i o n l i n e .  magnitude  o f t h i s broadening w i l l depend on the exact p r o b a b i l i t i e s  d i f f e r e n t modes o f decay and e x c i t a t i o n  available  The  t o these s t a t e s .  f o r the The  l i f e - t i m e o f t h e ground s t a t e w i l l be much longer than t h a t o f the e x c i t e d state,  and hence we s h a l l n e g l e c t  i t s contribution  to t h e h a l f - w i d t h of t h e  37 Let us denote by X the state that has the most influence  absorption line.  on the life-time of the state jj . According to Nishikawa (1962), the contribution of this state X AU)  K  to the half-width of the zero-phonon line is  y^DeJtP]  (6 /v a* 2TT) 2  =  2  3  ?  ^  ,  q=y/a*  J  x  P  ^  v  T  X  <  T  (10-1) where •y  and  T  ,=  p x  I T  p  v>  =  c  = -K v  - T|  ( c/ T  e  a* / *f v ,  x  T  - 1 y  l  (10-2)  ,  (10-3)  / a* k  (10-4)  is the characteristic temperature, above which the half-width starts increasing. e  p x  (t)  =|dr  FpCft*  F (r) x  e t ' 1  ?  ,  (10-5)  F(r) being the eigenfunctions of the unperturbed electronic Hamiltonian, which will be approximated- by simple hydrogenic functions. The other parameters have the following meanings "q* =  wave number vector of the phonon,  v  =  sound velocity in s i l i c o n = 8.3 x 10^ cm sec."*",  €  =  deformation potential constant fa 15 ev ,  ^  =  density of silicon = 2.33 g cm  k  =  Boltzmann's constant = 1.38 x 10"-16 erg deg~-1  ,  Unfortunately, the nature of the states ^, and X> the value of  T^, and  the proper choice for the effective Bohr radius a*, are rather uncertain. Let us start with the experimental value  T  c  47 °K (Figure 5) and the  assumption that £ has 2p character (Kohn 1957). (Nishikawa 1962), Eq. (10-4) gives obtains | T^ - T^| = 4.0 mev.  y  If one takes  a* = 13.5 A*  = 1 , and thus by Eq. (10-2) one  This would suggest identifying the state X  38 with the state 2 in Figure 8. Assuming this state is also 2p-like, one obtains by using simple hydrogenic wave functions  With this value Eq. (10-1) gives (0.106 mev AU) = 1 (0.170 mev  for T = T  (10-6)  A  for T = 2 T  If on the other hand one takes from Eq. (8-11) for the energy T  D  .  c  a* = 24 A* (the value one obtains  , with  n = 2), Eq. (10-4) gives  P  y  = 1.8 . By Eq. (10-2) this implies |T^ - T^| =4.1 mev, and thus  suggests again identifying the state X with the state 2 in Figure 8. However, for  — L.8 one obtains  y x [le^l ]^ 2  P  * - o- * 70  v / a  1 0 - 3  >  which after substitution into Eq. (10-1) leads to ( 0.0029 mev for T = T Au) = 1 (.0.0046 mev for T = 2 T c  x  .  c  (10.7)  Another possibility is that the state X , responsible for the shortened life-time of B, is not the 2p-like state 2 in Figure 8, but rather a 2s-like state, which one may expect in this general region, and which one would presumably not see in optical absorption from the Is-like ground state. Making this assumption one obtains for a* = 13.5 8, corresponding to y ^ = 1 , that At*^= 0. For a* = 24 8, corresponding to  y ^ = 1.8 one obtains  V0^1)|^  q  =  y  p  x  /  a  ,  - 2,75 x 10- .  Substituting this value into Eq. (10-1) one obtains atJ1=  '0.0074 mev for T = T < ° ' 0.0117 mev for T = 2 T  c  .  (10-8)  39  If one considers the crudeness of the wave functions used, and the nature of the approximations involved,, the agreement between the above estimates, especially, Eq. (10-6), and the width extrapolated to zero concentration in Figure 5, seems to be amazingly good.  One may observe,  that the increase in width of the extrapolated line in Figure 5 is somewhat steeper than that predicted by Eq. (10-6) between T = 2 T. c  T = T  c  and  However, this would be expected, i f one remembers that while  the multi-phonon processes are of l i t t l e importance to the half-width below  T = T , their relative importance when compared with the zero-phonon c  process increases with temperature.  They may be expected to make an  appreciable contribution to the experimentally observed half-width above  40 CHAPTER IV Conclusions The data presented in this thesis concerning a study of absorption line width in boron-doped silicon, differs considerably from that obtained by previous authors.  The reason for this disagreement is  a previous lack of sufficient resolution and a failure to make proper allowance for line distortion by the finite spectrometer s l i t width. The low tempjerature half-width is considerably smaller, and its temperature dependence above 50°K much steeper than found previously.  The onset of  concentration broadening is seen to occur at a considerably smaller impurity concentration than seemed to follow from previous experiments. The positions of the absorption lines are essentially in agreement with those obtained by Hrostowski and Kaiser (1958). It is believed that four effects make significant contributions to the true half-widths.  These are a) statistical Stark  broadening, b) phonon broadening, c) broadening due to the overlap of impurity wave functions, and d) broadening by internal strains.  To the  approximation that the absorption lines (corrected for spectrometer distortion) have lorentzian profiles, their half-widths are just the sums of the widths for the four broadening mechanisms when each is considered by i t s e l f .  This follows, since the absorption lines may be thought of as  resulting from a combination of the separate broadening effects by three consecutive convolution integrals. In a semiconductor containing a random distribution of neutral and ionized impurities, different absorbing impurities w i l l be in different electric fields due to the surrounding ionized impurities. . These fields (F) should be expected to give rise to appreciable second-order Stark shifts  41 ((f E = tF ) of the excited states, resulting in a broadening of the total 2  absorption lines.  The contribution to the half-width from this effect was  obtained from a knowledge of the field strength probability function W(F), which had been extensively studied previously in connection with gravitational problems, gases, and plasmas.  Ecker (1957) computed W(F),  including the effect of screening by mobile charge carriers.  Using these  results, and the calculated dependence of the ionized impurity concentration on the temperature, this statistical Stark broadening was found to account satisfactorily for the rapid increase of the half-width above 50°K. Agreement with the data was achieved by using a strength parameter of -9  2  -2  t.= 1.1 x 10" ev volts  cm' , which has the magnitude one would expect  from an approximate calculation. At very low impurity concentrations an essential contribution to the half-width is expected to result from the finite life-time of the excited state due to the electron-phonon interactions. Theoretical calculations show that the half-width for this process is given by h.= h / (1 - e - V ) T  c  The calculated value of h temperature T . c  Q  .  depends on the choice of the characteristic  The best agreement between this theory and the data was  obtained for T = 47°K  and h = 1.0 x 10"^ ev. For these values the  c  theory suggests that the life-time of the state responsible for the absorption line under study is mostly influenced by a state about 4.0 x 10-3  e v  below i t .  The multi-phonon processes are of l i t t l e importance to the half-width below T = T , but their relative importance when compared to c  the zero-phonon processes increases with temperature.  They may be  expected to make an appreciable contribution to the experimentally  42 observed half-width above T = T . c  In addition, a nearly temperature independent contribution to the half-width of about 3 x 10"^ ev can be expected from internal strains due to dislocations (corresponding to a quoted dislocation density of 4 9 about 5 x 10 dislocation lines per cnr). A cellular calculation, when modified to f i t the assumption of a random distribution of impurities, gives an order of magnitude estimate for the onset of concentration broadening at low temperatures. Broadening of a 2 p state is predicted to start at about 3 x 10  iJ  impurities per cm , i f  one replaces the mean spacing ( r ) between impurities by an effective mean s  spacing of 0.7 r , and takes g  o -r/2a* a* equal to 24A in the expression e '  occurring in the "2p" wave function under consideration.  (This value of  -3 a*  gives the experimental binding energy of 6.3 x 10 ev for this state). Considering the crudeness of some of the approximations  involved, in particular the use of hydrogenic wave functions, the agreement between theory and experiments seems to be rather good. An important question regarding the theoretical approach chosen in this problem, concerns the validity of the assumption of a random distribution of impurities.  This assumption is believed to be more  realistic than that of a regular array of impurities, and i t is the only possible one in the absence of any concrete evidence for a clustering of boron impurities in silicon.  T a b l e 1 : Standard V o i g t  Parameters  O r d i n a t e s i n Terms of C e n t r a l O r d i n a t e 0.8  p /h  ^ / ^  P /h  p- /h  0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500  0.00 0.04 0.09 0.14 !0.19 0.24 0.30 0.36 0.43 0.51 0.59 0.69 0.79 0.92 1.07 1.26 1.50 1.83 2.38 3.54 oo  0.60 0.59 0.57 0.55 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.35 0.33 0.30 0.27 0.23 0.19 0.13 0.00  0.36 0.34 0.32 0.31 0.29 0.27 0,25 0.23 0.21 0.20 0.18 0.16 0.14 0.12 0.11 0.09 0.07 0.05 0.04 0.02 0.00  x  2  Profiles  2  2  2  0.7  p  1.06 1.08 1.11 1.13 1.16 1.18 1.20 1.23 1.25 1.28 1.30 1.33 1.35 1.38 1.40 1.43 1.45 1.48 1.51 1.54 1.57  0.6  0.5  0.4  0.3  0.2  Widths i n Terms o f H a l f - w i d t h  0.57 0.72 0.56 0.72 0.56 0.71 0.56 0.71 0.56 0.71 0.56 • 0.71 0.55 0.71 0.55 0.70 0.55 0.70 0.54 0.70 0.54 0.70 0.53 0.69 0.53 0.69 0.53 0.68 0.52 0.68 0.52 0.68 0.52 0.67 0.51 0.67 0.51 0.66 0.51 0.66 0.50 0.66  0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.83 0.83 0.83 0.82 0.82 0.82  1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  1.15 1.15 1.15 1.16 1.16 1.17 1.17 1.17 1.18 1.18 1.18 1.19 1.19 1.19 1.20 1.20 1.21 1.21 1.22 1.22 1.22  1.32 1.33 1.33 1.33 1.34 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.44 1.45 1.47 1.48 1.50 1.52 1.53  1.52 1.53 1.54 1.56 1.57 1.59 1.60 1.62 1.64 1.66 1.68 1.71 1.74 1.77 1.81 1.85 1.88 1.92 1.96 1.98 2.00  0.1  0.05  0.02  0.01  2.38 2.49 2.63 2.79 3.00 3.24 3.52 3.80 4.14 4.44 4.73 5.03 5.32 5.57 5.83 6.07 6.30 6.55 6.76 6.92 7.00  2.58 2.82 3.13 3.56 4.08 4.58 5.05 5.50 5.96 6.40 6.78 7.15 7.52 7.86 8.21 8.55 8.86 9.18 9.50 9.77 9.95  (b^h)  1.82 1.84 1.87 1.90 1.94 1.98 2.02 2.06 2.10 2.15 2.19 2.24 2.29 2.34 2.40 2.46 2.54 2.64 2.74 2.87 3.00  2.08 2.12 2.19 2.25 2.34 2.42 2.54 2.64 2.75 2.87 2.98 3.12 3.26 3.39 3.54 3.70 3.85 4.00 4.13 4.25 4.36  44  Table 2.  2 Integrated absorption, A (cm" )  Resistivity Line No. (ohm cm) frig. 1)  Temperature (°K) 4.2 60 77 90  130  1  1.4  1.8  130  2  6.7  4.4  -  130  4  6.6  5.3  1.8  11  1  18  20  23  -  11  2  54  109  50  53  11  3  6.6  -  -  -  11  4  44  73  60  45  1.3  1  160 - 220  250  470  1.3  2  630  920  600  650  1.3  4  620  800  810  950  Table 3. Resistivity (ohm cm)  Integrated absorption cross-section, Line No. (Fig.l)  Temperature (°K) 4.2 60 77 90  130  1  1.4  4.1  -  130  2  6.7  10  -  -  130  4  6.6  12  17  -  11  1  1.5  2.2  4.3  -  11  2  4.5  12  9.4  ,19  11  3  0.55  -  -  -  11  4  3.7  7.9  11  16  1.3  1  1.3  2.0  2.9  7.2  1.3  2  5.2  8.4  6.9  10  1.3  4  5.2  7.3  9.3  15  cm).  45 Appendix I:  Distribution of holes between the ground state, excited states, and the valence band as a function of temperature.  The present appendix outlines the formalism used to calculate the curves shown in Figure 9.  Let us begin by assuming that the number of  donor impurities and free electrons is negligible. impurities with energy levels holes.  E ,= - £  There are N  acceptor  A  above the valence band, and p free  a  The probability P(E) of an acceptor level having four electrons  with paired spins when an extra electron is attached (ionized condition) is given by P(E) where  =  { l + 2 exp.[(E - E )/kT]}  ,  - 1  F  Ep is the Fermi energy.  (l-l)  The number of ionized acceptors is thus  given by p  =  N  A  p  =  N  v  ( l + 2 exp.[( e  + E )/kT]|  .  _ 1  a  p  (1-2)  Writing exp. (Ep/kT) ,  (1-3)  one obtains a quadratic equation for p, whose appropriate solution is p Here  N  y  =  -N /2 v  +  (1/2)(N  2 V  + 4 <  N )  .  1 / 2  A  (1-4)  = (N /2) exp.(- € /kT) , and the effective density of states v  a  N  v  is given by N  v  =  2 (2TTkT/h ) / 2  3  (m  2  3 hl  / + m 2  3 h2  /) .  (1-5)  2  Eq. (1-5) has been derived under the assumption that the actual energy surfaces in silicon may be approximated by two spherical constant-energy surfaces with effective masses (E - E)/kT»l. F  m i and n  m  n2  .  It was assumed that  This approximation is good as long as  p<0.4 N  1958), which applies for a l l cases of interest in this thesis.  y  (Smith  The valence  band may be regarded to this approximation as a single level with  46  degeneracy  N ,, placed at the top of the band. V  In deriving Eq. (1-2) from the expression for the probability of occupation P(E), i t was assumed that one is only concerned with a single level.  However, the impurity levels have excited states, and these should  be included in Eq. (1-2) with the appropriate probabilities. One then has for the number of un-ionized acceptors A P  N  where  =  N  A  + S i " exp, [ ( E  1  1  F  (1-6)  - E.)/kT] ,  E^ is the energy of the i th excited state, and g^ is a number to  take account of degeneracy and spin ( E = - 6 , g = 2). Q  a  0  Since below 90°K  one has (E^ - E ) / k T » l , the f i r s t term in the sum predominates, and 0  hence Eq. (1-2) is a good approximation. Taking for the effective masses the values (Bube 1960) 0.5 m  =  e  ;  m  =  h2  0.16 m  e  ,  (1-7)  Eq. (I-t5) leads to  N  .= 2.02 x 1 0 T 7 15  V  3  2  cm  -3  (1-8)  .  Substitution into Eq. (1-4), taking the ionization energy of  = 46 mev  for boron-doped silicon, leads to the curves of Figure 9 for the number of ionized impurities as a function of the absolute temperature T. The number of bound carriers in excited states N  E X  is then  computed from N  ex  =  (% " P) ^  Gi exp.(- fi/kT) ,  where the assumption was made that  N  6 X  <^(NA  - p).  (1-9) The energy differences  €^ between the i th excited state and the ground state are obtained from Figure 8.  The degeneracies of the excited states relative to the ground  state, G^, were taken as 1, 1, 1/2, 1/2, for the lowest four excited states  47 (Kohn 1957), w h i c h make the main c o n t r i b u t i o n t o N . e x  A v a l u e of  e q u a l t o one was t a k e n f o r the r e m a i n i n g f i v e e x p e r i m e n t a l l y observed energy l e v e l s ( F i g u r e 8 ) . The number o f i m p u r i t i e s i n t h e ground s t a t e , N , was  then  o b t a i n e d from N  =  N  A  -  p  -  N  e x  .  (I-  48  Appendix I I ;  In  Second-order S t a r k s h i f t  t h e p r e s e n t appendix t h e m a t r i x elements o c c u r r i n g i n  Chapter I I I , Eq. ( 6 - 2 ) , w i l l <F< >,  a v )  I  be e v a l u a t e d :  F<*>>  2  ,  £ v = - F e z , and f o r F ^ ) hydrogenic wave f u n c t i o n s  where will  f o r h y d r o g e n i c Is and 2p s t a t e s .  ^ >  1  be taken.  m  In p a r t i c u l a r one has f o r the Is and t h e 2p s t a t e s :  = T T  Yi  0 0  (r,©,c£)  >2  1 0  ( r , ©,tf» = ( 1 / 4 ) ( 2 T T ) -  1  /  2  a*"  " ^ 2 1 ± l ' » 0> " (1/8) T r " (r  9  1 / 2  exp.(-r/a*)  3 / 2  a*  1 / 2  a*"  _ 5 / 2  r cos9  exp.(-r/2a*)  r s i n 9 e x p . [ ( - r / 2 a * ) ± i £J .  5 / 2  Hence one o b t a i n s 2TC  di00>  (^ > V  /  >100> -  2  it  0  d(J) | d e  0  0  / 0  0  dr r  2 2 -3 9 = 2 F " e* a* J w^ dw -1 J  a*  2  sin6  T  1  0  0  r c o s ^ Q Q  ^ 4 r exp.(-2r/a*) d r  J  .  2  Similarly,  <V$10»  <* > V  >210>  "  .2 2 (  1  =  /  1  6  )  F  6  a  -5 j w  '• dw  L  J  <>21+1' (<f V ) >2H-1> -  2  =  (  = Here w = c o s G .  exp.(-r/a*) d r  18- a * . 1  2  °° r  1  /  3  2  )  F  2 e 2 a  *"  oo  / w ( l - w ) dw J r e x p . ( - r / a * ) dr 1 0  5  2  2  6  6a* . 2  Thus one o b t a i n e d Eq. (6-3) w i t h c g i v e n by 1, 18, and  6 f o r t h e Is s t a t e , 2 p  0  s t a t e , and 2 p s t a t e s +  respectively.  49  BIBLIOGRAPHY Baltensperger, W.  1953. Phil. Mag. 44, 1355.  Bichard, J.W. and Giles, J.C. Breene, Jr., R.G. Bube, R.H.  1962. Can. J. Phys. 40, 1480.  1957. Revs. Modern Phys. 29, 94.  1960. Photoconductivity of Solids (John Wiley & Sons, Inc., New York, London), p.209.  Burstein, E., Bell, E.E.,  Davisson, J.W.,  and Lax, M. 1953. J. Phys. Chem. 57, 849.  Burstein, E., Picus, G.S., Henvis, B., and Wallis, R. 1956. J. Phys. Chem. Solids I, 65. Chandrasekhar, S. 1943. Revs. Modern Phys. _15, 1. Colbow, K.,  Bichard, J.W.,  and Giles, J.C. 1962. Can. J. Phys. 40, 1436.  Colbow, K.,  1962. Bull. Am. Phys. S o c , Series II, ]_, 485.  Conwell, E.M. 1956. Phys. Rev. 103, 51. Debye, P. and Huckel, E. 1923. Physik. Z. 24, 185. Dexter, D.L., Zeiger, H.J., and Lax, B. 1956. Phys. Rev. 104, 637. Dresselhaus, G., Dresselhaus, G. Ecker, G.  Kip, A.F.,  and K i t t e l , C.  1955. Phys. Rev. 98, 368.  1955. Phys. Rev. 100, 580.  1957. Z. Physik JL48, 593; _149, 254.  E l l i o t t , R.J. Herzberg, G.  1954. Phys. Rev. 96, 266. 1944. Atomic Spectra and Atomic Structure (Dover Publications, New York), 2nd ed., p.51.  Holtsmark, J. 1919. Am. Physik, 58, 577; Physik. Z. 20, 162. Holtsmark, J. 1924. Physik. Z. 25, 73. Hrostowski, H.J., and Kaiser, R.H. 1958. J. Phys. Chem. Solids, 4, 148. Irvin, I. C. 1962. Bell System Tech. Journ. Kane, E.O.  41, 387.  1960. Phys. Rev. _119, 40.  Kittel, C. and Mitchell, A.H. 1954. Phys. Rev. 96, 1488.  50 Kohn, W.  and  L u t t i n g e r , J.M.  Kohn, W.  1957.  1955.  Phys. Rev. 97,  869.  S o l i d S t a t e P h y s i c s , e d i t e d by F. S e i t z and D. T u r n b u l l , V o l . 5 (Acadmeic P r e s s , Inc., New  Lax, M.  and  London, F.  B u r s t e i n , E. and  1955.  Phys.Rev.  E i s e n s c h i t z , R.  Margenau,  H.  1933.  Margenau,  H.  and  1930.  Phys. Rev. 43, Murphy, G.M.  100,  York),p.257.  592.  Z. Physik  60,  491.  129.  1943.  The Mathematics of P h y s i c s and  Chemistry (Van Nostrand, New Y o r k ) , p.559. Margenau,  H.  and  Merck and Co., Morin, F . J .  1962.  and  Moss, T.S.  Lewis, M.  1959.  569.  P r i v a t e communication.  M a i t a , J.P.  1959.  Revs. Modern Phys. _3_1,  1954.  Phys. Rev. 96,  28.  O p t i c a l p r o p e r t i e s of semiconductors ( B u t t e r w o r t h s , London),  Newman, R.  1956.  Phys. Rev.  Nishikawa, K.  1962.  Nishikawa, K.  and  Nishikawa, K.  1962.  P l y l e r , E.K.  and  R a n d a l l , H.M.,  103,  103.  Phys. L e t t e r s , B a r r i e , R.  1962.  I,  140. B u l l . Am.  Phys. S o c , S e r i e s I I , 7_,U85.  Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia.  A c q u i s t a , N.  Dennison, D.M.,  1956.  J.Res. N.B.S., ^6,  G i n s b u r g , N.  and  149.  Weber, L.R.  1937.  Phys. Rev. 52, Sampson, D.  and  Schechter, D.  Margenau,  1962.  H.  1956.  Phys. Rev.  J . Phys. Chem. S o l i d s , 23,  103,  160.  879.  237.  Smith, R.A.  1958.  Semiconductors (Cambridge U n i v e r s i t y P r e s s ) ,  U n s o l d , A.  1955.  P h y s i k der Sternatmospharen ( S p r i n g e r - V e r l a g ,  p.80. Berlin,  Gottingen,. H e i d e l b e r g ) , Chap. IX. Van de Hulst,, H.C.  1946.  Van de H u l s t , H.C.  and  V o i g t , W.  1912.  p.14.  B u l l . Astron. R e e s i n c k , I.I.M.  Munch. Ber.  603.  I n s t . Neth. _10, 1947.  75.  A s t r o p h y s . J . 106,  121.  

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