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Low temperature behavior of krypton monolayers on graphite Shrimpton, Neil Douglas 1987

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LOW TEMPERATURE BEHAVIOR OF KRYPTON MONOLAYERS ON GRAPHITE by N E I L D.  B. S c . , P h y s i c s ,  SHRIMPTON  U n i v e r s i t y of V i c t o r i a  M. S c . , The U n i v e r s i t y  of B r i t i s h Columbia  1978 1981  A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF P H Y S I C S ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g  to the required  THE UNIVERSITY OF B R I T I S H COLUMBIA April © Neil  Douglas  1987 Shrimpton,  1987  standard  In  presenting  degree  this  at the  thesis  in  partial  fulfilment  of  University of  British  Columbia,  I agree  freely available for reference and study. copying  of  department publication  this or of  thesis by  for scholarly  his  this thesis  or  her  Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6G/81)  H~  2<T-  ?7  requirements  for  may  representatives.  It  be is  advanced  that the Library shall make it  I further agree that permission  purposes  an  granted  for extensive  by the head  understood  that  of  my  copying  or  for financial gain shall not be allowed without my written  permission.  Date  the  11  ABSTRACT  The  low t e m p e r a t u r e b e h a v i o u r of a system of incommensurate  krypton monolayers was c a l c u l a t e d  on g r a p h i t e was a n a l y s e d .  f o r a v a r i e t y of monolayer  t h e m i s f i t and o r i e n t a t i o n  The f r e e  energy  c o n f i g u r a t i o n s and  o f t h e minimum e n e r g y  configuration  d e t e r m i n e d as a f u n c t i o n of t h e t e m p e r a t u r e and c h e m i c a l potential  of the system.  significantly  The f r e e e n e r g y d i d n o t v a r y  o v e r t h e t e m p e r a t u r e r a n g e f r o m 0 K t o 4 K.  zero point energy c o n t r i b u t e s s i g n i f i c a n t l y  to the free  The  energy  and c o u l d n o t be n e g l e c t e d . The  l o w e s t e n e r g y v i b r a t i o n a l modes were d e t e r m i n e d ; t h e s e  modes c o r r e s p o n d t o m o t i o n o f t h e d o m a i n w a l l s .  For  c o n f i g u r a t i o n s w i t h c l e a r l y s e p a r a t e d domain w a l l s t h e v i b r a t i o n a l modes s e p a r a t e d i n t o g r o u p s o f t h r e e the lowest energy t r i a d , c o m p r e s s i o n a l mode. s h e a r i n g modes.  (triads).  In  t h e l o w e s t e n e r g y mode was a  The s e c o n d a n d t h i r d  l o w e s t modes were  These t h r e e modes d e s c r i b e t h e f u n d a m e n t a l  forms o f domain w a l l m o t i o n . t r i a d s a r e more e n e r g e t i c  The modes o f t h e h i g h e r  forms of t h e l o w e s t e n e r g y  energy triad.  iii  TABLE OF CONTENTS  T I T L E PAGE  i  ABSTRACT  i i  TABLE OF CONTENTS L I S T OF TABLES  i i i '.  vi  L I S T OF FIGURES  v i i  ACKNOWLEDGEMENT  X  1.  1  2.  INTRODUCTION (I-)  MOTIVATION  (II)  FORMALISM  10  POTENTIAL ENERGY  14  (I)  MICROSCOPIC CONSIDERATIONS  14  (II)  SIMPLIFICATIONS  23  (III)  METHOD OF SOLUTION  30  (IV)  NUMERICAL APPROXIMATIONS  34  (V)  CALCULATIONAL PROCEEDURE  43  (VI)  RESULTS AND DISCUSSION  47  iv  3.  4.  5.  DYNAMICAL ASPECTS  54  (I)  BACKGROUND  54  (II)  HARMONIC APPROXIMATION  59  (III)  ZERO POINT ENERGY  65  (IV)  CALCULATION  69  (V)  RESULTS AND DISCUSSION  73  LOW ENERGY PHONON MODES  77  (I)  HELMHOLTZ FREE ENERGY  (II)  NORMAL MODE CALCULATION  ..79  (III)  REVIEW OF PREVIOUS WORK  84  (IV)  QUANTIZED NATURE OF WALL MOTION  86  (V)  DOMAIN WALL I N S T A B I L I T I E S  88  (VI)  FREE ENERGY CALCULATION  91  (VII)  RESULTS AND DISCUSSION  94  CONCLUSION  77  101  V  APPENDIX A: GEOMETRICAL  PROPERTIES  106  APPENDIX B: SOLITON EQUATIONS  109  APPENDIX C: SUBSTRATE ENERGY UNCERTAINTY  112  APPENDIX D: NUMERICAL  116  INTEGRATION METHOD  FIGURES  119  REFERENCES  149  vi  L I S T OF  TABLES  T a b l e 1.  E  T  values for a m i s f i t  o f 3.7%  95  T a b l e 2.  E  T  values for a m i s f i t  o f 3.0%  95  T a b l e 3.  E  T  values for a m i s f i t  o f 2.8%  96  vii  L I S T OF FIGURES  F i g u r e 1.  Diagram of the g r a p h i t e s u b s t r a t e  119  Figure  2.  Types  120  Figure  3.  T y p i c a l adatom p o s i t i o n s an  of domain  walls incommensurate  monolayer w i t h h e x a g o n a l symmetry  121  Figure  4.  D i s c r e t e v a l u e s of m i s f i t and o r i e n t a t i o n . . . 1 2 2  Figure  5a.  E n e r g y p e r a d a t o m when Vg = 3. OK  123  F i g u r e 5b.  E n e r g y p e r a d a t o m when V  124  Figure  5c.  E n e r g y p e r a d a t o m when Vg = 7. OK  125  Figure  5d.  E n e r g y p e r a d a t o m when Vg = 8.OK  126  Figure  5e.  E n e r g y p e r a d a t o m when V  127  Figure  6a.  E n e r g y d i f f e r e n c e w i t h r o t a t i o n Vg = 5.OK  ..128  F i g u r e 6b.  Energy d i f f e r e n c e with r o t a t i o n V  ..129  F i g u r e 6c.  E n e r g y d i f f e r e n c e w i t h r o t a t i o n Vg = 10.OK  Figure  A comparison of t h e adatom p a i r p o t e n t i a l <Mr) a n d t h e p o l y n o m i a l f i t a r + b r + c r + d ...131  7a.  g  g  = 5.OK  = 9. OK  3  = 8.OK  g  2  .130  viii  Figure  7b.  A c o m p a r i s o n o f 0 ' ( r ) a n d 3 a r + 2br+c  132  Figure  7c.  A c o m p a r i s o n o f # ' ( r ) a n d 6 a r + 2b  133  F i g u r e 7d.  A c o m p a r i s o n o f # ' ' ' ( r ) a n d 6a  134  Figure  E n e r g y p e r a d a t o m when V = 7.OK when t h e p a i r p o t e n t i a l # ( r ) i s r e p l a c e d by t h e polynomial ar +br +cr+d  135  8.  2  1  q  3  F i g u r e 9.  2  Dispersion curves V  g  = 6.OK, M% = 2.22%  ....136  F i g u r e 1 0 a . Domain w a l l m o t i o n w i t h e n e r g y O.OllmeV  ....137  F i g u r e 1 0 b . Domain w a l l m o t i o n w i t h e n e r g y 0.021meV  ....138  F i g u r e 1 0 c . Domain w a l l m o t i o n w i t h e n e r g y 0.027meV ....139 F i g u r e 11.  Dispersion curves V  g  = 6.OK, M% = 1.75% ....140  F i g u r e 1 2 a . Domain w a l l m o t i o n w i t h e n e r g y 0.037meV ....141 Figure  12b. Domain w a l l m o t i o n w i t h e n e r g y 0.315meV  ....142  Figure  1 3 a . Domain w a l l m o t i o n w i t h e n e r g y 0.201meV  ....143  F i g u r e 1 3 b . Domain w a l l m o t i o n w i t h e n e r g y 0.201meV  ....144  F i g u r e 14a. D i s p e r s i o n curves V  g  = 8.OK, M% = 3.33% ....145  F i g u r e 1 4 b . D i s p e r s i o n c u r v e s V = 8.OK, M% = 3.29% the monolayer i s r o t a t e d s l i g h t l y  146  IX  F i g u r e 15.  Domain w a l l m o t i o n o f u n s t a b l e mode  F i g u r e 16.  Dispersion curves V  g  = 8.OK  , M% = 3.7%  147 ....148  X  ACKNOWLEDGEMENTS  I would  like  t o thank  Birger  B e r g e r s e n and B e l a Joos f o r  p r o v i d i n g s u p p o r t and encouragement over t h e p e r i o d of t h i s thesis.  The p r o g r a m s f o r t h e t h e s i s  computer systems and  were d e v e l o p e d  a t b o t h U.B.C. a n d t h e U n i v e r s i t y  I am g r a t e f u l  on t h e of Ottawa,  f o r the f i n a n c i a l support provided  through  NSERC a t b o t h i n s t i t u t i o n s . T h i s m a n u s c r i p t was p r o d u c e d developed  by B i l l  program a v a i l a b l e I would  also  u s i n g a word p r o c e s s i n g p r o g r a m  U n r u h a n d I am g r a t e f u l t o h i m f o r m a k i n g t h e t o me.  like  t o thank  my w i f e M i r i a m , f o r s u p p o r t i n g me  c l o s e l y over t h e y e a r s t a k e n t o produde t h i s  thesis.  1  1.  INTRODUCTION  (I)  MOTIVATION R a r e g a s p h y s i s o r b e d s y s t e m s have been e x t e n s i v e l y s t u d i e d  b e c a u s e o f t h e two d i m e n s i o n a l film.  Experimental  possible  (2D) n a t u r e  of the surface  r e s u l t s have i n d i c a t e d a l a r g e number o f  2D p h a s e s a n d t h i s h a s p r o m p t e d c o n s i d e r a b l e  interest.  One s u c h  system  At low t e m p e r a t u r e s ,  i s k r y p t o n on g r a p h i t e .  graphite w i l l  p h y s i s o r b k r y p t o n , and  under s u i t a b l e c o n d i t i o n s , a monolayer f i l m of k r y p t o n i s formed.  I n a d d i t i o n t o t h e gas and l i q u i d  phases of t h e  m o n o l a y e r , a commensurate and an incommensurate s o l i d occurs.  phase  The p u r p o s e o f t h i s work i s t o s t u d y t h e  i n c o m m e n s u r a t e p h a s e o f k r y p t o n on g r a p h i t e . Krypton with  i s a r a r e g a s and t h e r e f o r e i t i n t e r a c t s v e r y  i t s surrounding environment.  fluctuations forces).  of the atomic  S i n c e no c h a r g e  The i n t e r a c t i o n s r e s u l t  exchange o c c u r s , t h r e e body  s u b s t r a t e or t o the i n d i v i d u a l  neighbouring  from a  t h e number o f p a r a m e t e r s of the system  i s p r o v i d e d by  Bruch . 1  required to  i s reduced.  e x c e l l e n t d i s c u s s i o n of the i n t e r a c t i o n s present physisorbed system  t o the  k r y p t o n atoms.  M o r e o v e r , b e c a u s e r a r e g a s atoms a r e s p h e r i c a l ,  describe the microscopic nature  effects  The f o r c e s p r e s e n t a t t h e  ( m i c r o s c o p i c ) l e v e l c a n t h u s be a s c r i b e d e i t h e r  t h e o r e t i c a l p o i n t of view,  from  moments ( L o n d o n - v a n d e r Waals  c a n be t r e a t e d a s p e r t u r b a t i o n s . atomic  weakly  ina  An  2  Experimentally, large,  defect  exfoliated  free  surfaces  a r e a s m i n i m i z e the  g r a p h i t e s u b s t r a t e s w i t h many  can  exfoliated  the  films.  s y s t e m and  the vapour p r e s s u r e  3  I n d i c a t e d the  diffraction studies  This spacing  i s not  of the  lattice  smaller spacing. this thesis  (From t h e  the  o f 4.26  spacing  in  favoured  the  is calculated  they  (Figure 1).  centers  site.  2D  the the  liquid  s o l i d phases'*.  s o l i d p h a s e where  atoms as  t o be  o f 4.26  lattice  the  A.  their lattice  with  4.02  A.)  The  A  variation  s i t e s on t h e  a  used  s u b s t r a t e does  i t s surface.  The  i s 1.42  not in  graphite  bond l e n g t h o f t h e c a r b o n atoms  A.  Since  the a d s o r p t i o n  with spacing  /3xl.42  of the monolayer r e f l e c t s the  Instead,  of  phases.  p h a s e , a 2D  f a c t t h a t the  ^3x1.42 A i s t o o c l o s e a p a c k i n g adsorption  films  sites  o f t h e h e x a g o n s f o r m e d by t h e c a r b o n  form a t r i a n g u l a r  observed spacing  surfaces  form of the adatom i n t e r a c t i o n  a t t r a c t i o n over  g r a p h i t e plane  are a t the  above the  with spacing  the a t t r a c t i o n causes c e r t a i n a d s o r p t i o n t o be  of  temperature of  for a triangular  A r e f l e c t s the  exert a uniform  gas  by k r y p t o n  i n t e r a c t i o n energy i s minimized  spacing  gas  f o u n d o n l y one  favoured  study  the measurable p r o p e r t i e s  p o s s i b i l i t y o f two 5  surface  g r e a t number o f s u c h  o b s e r v e d a 2D  adatoms formed a t r i a n g u l a r  in  s i z e on t h e  be d r i v e n b e t w e e n i t s v a r i o u s studies  Large  2  F u r t h e r m o r e , by c o n t r o l l i n g t h e  Volumetric  Early  The  graphite magnifies  monolayer can  p h a s e , and  prepared .  impact of f i n i t e  monolayer phase t r a n s i t i o n s . in  be  fact  bonds,  A. that  f o r t h e Kr atoms t o f i l l  t h e atoms c h o o s e t o f i l l  The  every  every third  3  adsorption site  so t h a t t h e i r  sublattice with spacing commensurate  positions  4.26 A.  T h i s phase, which i s  with the substrate,  or t h e / 3 X / 3 r e g i s t e r e d  form a t r i a n g u l a r  i s known a s t h e  phase.  Further experimental work , 6  however, r e v e a l e d a second  phase w h i c h i s c h a r a c t e r i z e d by a s h i f t peak away f r o m i t s commensurate  o f t h e main  location.  a v e r a g e d adatom s p a c i n g i n t h i s  misfit  phase.  The p r e c i s e d e f i n i t i o n  i s more d e n s e t h a n t h e commensurate  between t h e  of the  The i n c o m m e n s u r a t e phase.  the degree of r o t a t i o n  phase  The i n c o m m e n s u r a t e .  m o n o l a y e r c a n a l s o be r o t a t e d w i t h r e s p e c t t o t h e 7  results  i n c o m m e n s u r a t e p h a s e and t h a t  i s p r o v i d e d b y e q u a t i o n 2.8.  configuration ;  solid  diffraction  The s h i f t  f r o m a d i s c r e p a n c y , o t h e r w i s e known a s t h e m i s f i t ,  of t h e commensurate  commensurate  commensurate  i s termed the  Q orientational epitaxy . 0  positions  Satellite  i n d i c a t e t h a t t h e adatoms  hexagonal s u p e r l a t t i c e f r o m t h e commensurate  structure  .  peaks a r e o b s e r v e d and t h e i r w i t h i n t h e m o n o l a y e r have a The m o n o l a y e r c a n be d r i v e n  phase t o t h e incommensurate phase by  i n c r e a s i n g t h e vapour p r e s s u r e of t h e gas above t h e monolayer. The m i s f i t ,  M%,  i s observed t o vary with the chemical  potential  M as  M% oc ( M - P ) *  (1.1)  c  where £ - 1/3 a n d M  c  i s the chemical p o t e n t i a l  of the  q commensurate  phase  .  This r e l a t i o n s h i p  i s f o u n d t o be  4  independent of the temperature of the system m i s f i t s t h e monolayer substrate; is  i s not rotated with  however, f o r m i s f i t s g r e a t e r  rotated  1 0  .  For small  respect  t o the  t h a n 3.5% t h e m o n o l a y e r  .  Many t h e o r i e s were d e v e l o p e d t o e x p l a i n t h e b e h a v i o r o f t h e incommensurate monolayer. N o v a c o a n d McTague® interaction  A c a l c u l a t i o n was p e r f o r m e d b y  f o r a s y s t e m where t h e s u b s t r a t e  i s nearly uniform across  system, the substrate  i t s surface.  c a n be c o n s i d e r e d  t o be a  For t h i s perturbing  i n f l u e n c e on t h e p o s i t i o n s o f t h e a d a t o m s a n d t h e i n t e r a c t i o n can  be l i n e a r i z e d .  The r e s u l t s o f t h i s c a l c u l a t i o n  t h a t the incommensurate monolayer w i l l to the s u b s t r a t e . substrate  For krypton  interaction  be r o t a t e d  on g r a p h i t e ,  indicate  with  respect  however, t h e  i s s i g n i f i c a n t l y m o d u l a t e d a n d a s shown  11 by  Shiba  x  , i t s i n f l u e n c e c a n n o t be r e g a r d e d a s b e i n g  Shiba's c a l c u l a t i o n s  predicted  u n i a x i a l phase as t h e d e n s i t y value. will  With a f u r t h e r  linear.  t h a t t h e monolayer would form a increases  increase  f r o m i t s commensurate  i n the density,  t h e monolayer  f o r m a h e x a g o n a l p h a s e t h a t be a l i g n e d a n d t h e n w i l l  to r o t a t e with  respect  exceeds a t h r e s h o l d predicted  to the substrate  value.  orientation  when t h e d e n s i t y  Although the r o t a t i o n a l  by S h i b a matched t h e o b s e r v e d  start  behaviour  misfit-dependent  o f t h e i n c o m m e n s u r a t e monolayer- - , t h e u n i a x i a l 1  phase has never been o b s e r v e d Shiba's theory  considered  0  experimentally. t h e a d a t o m i n t e r a c t i o n t o be  harmonic, an a p p r o x i m a t i o n t h a t w i l l  compromise t h e r e s u l t s  1 2  .  5  Calculations not  require  f o r the  e n e r g y of the  a p p r o x i m a t i o n s t o the  static adatom  i n v o l v e a r e l a x a t i o n p r o c e s s whereby the adatoms a r e achieved. for  the  shifted until a static Gooding et a l .  the  i s d e p e n d e n t on substrate  will  3  be  configuration  the  registered with  configuration  will  configuration  has  the  t h a t are  substrate  walls.  with  be  where t h e  These c o n f i g u r a t i o n s  can  l a r g e enough f o r the  m o n o l a y e r t o be  v a l u e s of the  The  s t r i p e d phase i s not  therefore  the  higher  e n t r o p y of the  h e x a g o n a l c o n f i g u r a t i o n ^ ^ has enough t o  influence  the  monolayer  the  incommensurate registered  density  substrate  with  domain  corrugation  registered at  m o n o l a y e r upon c o m p r e s s i o n w i l l  .  x  The  the  If  have a h e x a g o n a l or u n i a x i a l  For  phase  K,  higher  ( s t r i p e d ) symmetry.  temperature, the  adatom  than 11.0  adatoms a r e  s e p a r a t e d by  and  corrugation.  otherwise  is  studies  substrate  incommensurate.  regions  configuration  l o w e s t e n e r g y per  substrate,  do  the  of k r y p t o n m o n o l a y e r s  i s greater  the  p o s i t i o n s of  force-free  magnitude of the  corrugation  that  pair interaction  have p e r f o r m e d r e l a x a t i o n  non-rotated configurations  conclude that  the  1  configuration  observed  zero  form a s t r i p e d  experimentally,  domain w a l l s  in  been e x p e c t e d t o be  equilibrium configuration  the  significant of  the  monolayer. M o l e c u l a r d y n a m i c c a l c u l a t i o n s by Abraham e t describe  the  temperatures. that  the  classical The  b e h a v i o u r of the  system at  p e r i o d i c boundary c o n d i t i o n s  monolayer c o u l d  al.^-  5 - 1  - ' 0  finite  were c h o s e n  f o r m e i t h e r a u n i a x i a l or a  so  hexagonal  6  superlattice structure.  No s t r i p e d p h a s e s  were o b s e r v e d ,  a l t h o u g h t h e i r e x i s t e n c e c o u l d n o t be r u l e d configurations monolayer  was  close  out f o r  t o t h e commensurate t r a n s i t i o n  f o u n d t o f o r m a honeycomb d o m a i n w a l l  t h a t approached  1 7  .  The  network  hexagonal p e r i o d i c i t y a t low t e m p e r a t u r e s  and  coverages . 1 7  The  r e l a x a t i o n and m o l e c u l a r d y n a m i c s  limited the  c a l c u l a t i o n s are  by c o m p u t a t i o n a l c o n s t r a i n t s and h e n c e c a n n o t e v a l u a t e  b e h a v i o r of the monolayer  commensurate phase.  The  through the t r a n s i t i o n t o a  r e s u l t s of these s t u d i e s  can, however,  be e x t e n d e d t o t h e c o m m e n s u r a t e limit ® by u s i n g a 1  phenomenological d e s c r i p t i o n of the system based  on t h e  r e n o r m a l i z e d s y s t e m of domain w a l l s ^ . 1  The  concept of d e s c r i b i n g  the p r o p e r t i e s  terms of domain w a l l s t r u c t u r e s s o l i t o n systems  followed  of the monolayer  f r o m e a r l y work  b y F r e n k e l and K o n t o r o v a .  on  Interactions  2 0  p r e s e n t w i t h i n the monolayer  in  can p r o v i d e a system of e q u a t i o n s  w h i c h d e t e r m i n e t h e p o s i t i o n s of t h e adatoms  (Appendix B ) .  o1  F r a n k and v a n d e r M e r w e ^ the  x  recognized that,  f o r systems  a d a t o m s e e k s t o have a s p a c i n g d i f f e r e n t f r o m t h a t  where of the  a d s o r p t i o n p o i n t s p a c i n g , t h e s e e q u a t i o n s c a n have s o l i t o n solutions.  These s o l i t o n s p r o d u c e  domain w a l l s which s e p a r a t e  r e g i o n s where t h e a d a t o m s a r e r e g i s t e r e d . configuration the  The  energy of the  c a n be d e t e r m i n e d e n t i r e l y f r o m t h e s e p a r a t i o n  d o m a i n w a l l s , and a r e n o r m a l i z e d d e s c r i p t i o n o f t h e  based  on t h e p o s i t i o n s o f t h e d o m a i n w a l l s  i s possible.  of  system When  7  this to  r e n o r m a l i z e d d e s c r i p t i o n was  include wall  Villain The  1  f o r m a honeycomb n e t w o r k  phases  could also  An e x c e l l e n t r e v i e w o f t h i s t h e o r y i s p r o v i d e d  and  by  2 2  monolayer  unfavourable. transition  be  Gordon . i s expected  to form a s t r i p e d  phase i f the i n t e r s e c t i o n of the w a l l s The  monolayer  will  which  t h e n have a c o n t i n u o u s striped  i s f o l l o w e d by a h e x a g o n a l  the d e n s i t y i s i n c r e a s e d .  If wall  intersections  e n e r g e t i c a l l y f a v o u r a b l e , the monolayer p h a s e and  the t r a n s i t i o n  hexagonal  incommensurate phase w i l l  above, the s t r i p e d  incommensurate  is energetically  from the commensurate phase t o a  incommensurate phase,  will  not  phase  p h a s e has  order, although this  are form a  striped  never  be  first  order.  been o b s e r v e d .  As  noted  Experimental  suggests t h a t the t r a n s i t i o n i s  J  has  not been s u b s t a n t i a t e d .  The  l a t e s t e x p e r i m e n t a l e v i d e n c e ^ s u g g e s t s t h a t the monolayer 2  a continuous t r a n s i t i o n  as  f r o m the commensurate phase t o the  e v i d e n c e by N i e l s e n e t a l . first  phenomenologically  i n t e r s e c t i o n s ^ , h e x a g o n a l l y symmetric  where t h e d o m a i n w a l l s analyzed.  extended  from a commensurate t o a  incommensurate c o n f i g u r a t i o n , a l t h o u g h near domain w a l l s t r u c t u r e s appear  has  hexagonal  the t r a n s i t i o n  t o become d i s o r d e r e d  2 5  .  the  This  d i s o r d e r i n g o f t h e d o m a i n w a l l s t r u c t u r e s has b e e n d i s c u s s e d by Coppersmith  et a l . ^ .  t h e s t r i p e d and to  2  Their calculations  indicate that  honeycomb d o m a i n w a l l n e t w o r k s  d i s l o c a t i o n s which d e s t r o y t h e i r T h e o r i e s have b e e n d e v e l o p e d  2 7  are  both  succeptible  structure.  which d e s c r i b e c o n f i g u r a t i o n s  8  with  domain w a l l s a r r a n g e d  honeycomb w a l l n e t w o r k s . the  ln patterns  sublattice the  be  identified  of the  as  b e i n g an  substrate.  The  site  t y p e of w a l l  substrate,  0,  only  two  possible  (Figure  2  as  and  a,  b,  the  patches.  This  are  +/-120  0  ).  A s y s t e m so  s i t u a t i o n can patches are  be  f o r e x a m i n i n g the  b e t w e e n them, t h a t  particular,  near the  8  The  predict that  s y s t e m o f an  graphite  and  d y n a m i c s on  constrained the  light walls) can  be  described depend  the  to a 3 s t a t e  p a r t i c l e s and  the  Potts domain  model c a l c u l a t i o n s  p h a s e s , and  the  monolayer can  domain w a l l s w i l l thesis  have.  become  In  fluid.  i s to c a l c u l a t e the  free energy  incommensurate k r y p t o n monolayer adsorbed  t o d e t e r m i n e the the  are  commensurate phase t r a n s i t i o n , C a f l i s c h e t  the  i n t e n t of t h i s  the  the  adsorption  i t s energy w i l l  Potts  various  transitions  2  (and  renormalized  represent t h e i r interactions.  useful  one  w a l l s are  constrained  t r e a t e d as  to  s u b l a t t i c e s with  symmetry d i r e c t i o n s of  t y p e s of heavy w a l l s  x -'3  correspond  of o n l y  the  has  the  S3  or c,  t y p e s of domain w a l l s w h i c h s e p a r a t e  model where t h e walls  Assuming t h a t  c p a t c h e s o f a d a t o m s , and  s o l e l y on  The  b,  site,  domain w a l l s  involving a shift  between domains.  t o l i e a l o n g the  the  a,  b o u n d a r y between domains of d i f f e r i n g  simplest  or  Because each r e g i s t e r e d domain  adatoms s i t u a t e d a t e v e r y t h i r d a d s o r p t i o n  domains can  al.  other than s t r i p e d  impact of the  b e h a v i o u r of the  c a l c u l a t i o n i s b a s e d on  adatoms w i t h i n the  the  m o n o l a y e r and  system at microscopic  of  on  monolayer's low  temperatures.  i n t e r a c t i o n s of  takes i n t o account  the  the  9  quantized differs  n a t u r e o£ t h e a d a t o m m o t i o n .  classical.  f r e e energy of t h e monolayer  static  with  Chapter  configuration.  the motion of  and t h e i r c o n t r i b u t i o n t o t h e i s determined.  thermodynamics of the system i s d i s c u s s e d  following section.  molecular  t h e m o t i o n o f t h e adatoms i s  The v i b r a t i o n a l modes a s s o c i a t e d  domain w a l l s a r e s t u d i e d ,  The  calculation  from m o l e c u l a r dynamic c a l c u l a t i o n s i n t h a t  d y n a m i c c a l c u l a t i o n s assume t h a t  the  This  2 considers  The m i c r o s c o p i c  w i t h i n the monolayer a r e d i s c u s s e d .  i n the  the energy of the i n t e r a c t i o n s present  The a v e r a g e p o t e n t i a l  e n e r g y p e r a d a t o m i s c a l c u l a t e d and i t s d e p e n d e n c e on t h e configuration  of the monolayer  Chapter 3  i s shown.  considers  the  dynamic a s p e c t s of t h e system.  the  m o t i o n o f t h e adatoms i s p r e s e n t e d w i t h a d i s c u s s i o n  approximations required  The d y n a m i c a l m a t r i x f o r  in i t s derivation.  The z e r o  point  e n e r g y i s c a l c u l a t e d and i t s i m p a c t i o n t h e c o n f i g u r a t i o n monolayer i s a n a l y s e d .  Chapter  4 considers  v i b r a t i o n a l modes o f t h e d o m a i n w a l l s . modes i s d i s c u s s e d on  and t h e i r i n f l u e n c e  previous sections  discussion  The b e h a v i o u r o f t h e s e at finite  that t i e s  t o g e t h e r and s t a t e s  of t h e a u t h o r o f t h i s  thesis.  of the  the low energy  temperatures  t h e f r e e energy of t h e monolayer i s examined.  presents a concluding  of the  Chapter 5  the r e s u l t s of the  the o r i g i n a l  contribution  10  (II)  FORMALISM  The  behaviour of t h e monolayer  must be c o n s i d e r e d i n t h e  context of the t o t a l system of krypton vapour, s u b s t r a t e and condensed k r y p t o n f i l m . configuration  graphite  The e q u i l i b r i u m  i s the c o n f i g u r a t i o n that minimizes the t o t a l  free energy of t h i s  system.  An e x a c t c a l c u l a t i o n  f o r the t o t a l  free energy which c o n s i d e r s a l l aspects of the system simultaneously  i s not p r a c t i c a l .  However, a p p r o x i m a t i o n s c a n  be made w h i c h s e p a r a t e t h e f r e e e n e r g y  into several  independent  components t h u s r e d u c i n g t h e c o m p l e x i t y o f t h e p r o b l e m . s e p a r a t i o n has been c o n s i d e r e d by V i l l a i n their  and G o r d o n  2 2  This and  arguments a r e r e p e a t e d here so t h a t a c o n s i s t e n t  t e r m i n o l o g y may be e s t a b l i s h e d . The  g e n e r a l system  V containing N  T  i s t a k e n t o be a s e a l e d chamber o f v o l u m e  k r y p t o n atoms a n d a g r a p h i t e s u b s t r a t e .  t e m p e r a t u r e o f t h e s y s t e m i s m a i n t a i n e d a t T. appropriate conditions the krypton w i l l s u b s t r a t e t o form a s o l i d monolayer  will  monolayer.  The  Under  condense onto t h e  The c o n f i g u r a t i o n o f t h e  d e p e n d on t h e t e m p e r a t u r e a n d p r e s s u r e , o f t h e  gas a b o v e t h e s u b s t r a t e . C o n s i d e r t h e s y s t e m , a t t e m p e r a t u r e T, when N k r y p t o n atoms are In  adsorbed  onto t h e g r a p h i t e  leaving  Nrp-N  atoms a s v a p o u r .  order t o determine t h e p r o p e r t i e s of t h e system i t i s  necessary t o construct the p a r t i t i o n  function  11  Z = £ e"^ s  (1.2)  E s  where s r a n g e s o v e r a l l p o s s i b l e s t a t e s o f t h e s y s t e m the  c o n s t r a i n t s V,N )  , E  T  £=l/k T.  When t h e v a p o u r  B  (given  i s t h e e n e r g y o f t h e s t a t e , and  s  is dilute,  atoms a r e w e a k l y i n t e r a c t i n g s o t h a t  t h e v a p o u r and  condensed  t h e e n e r g y c a n be  split  i n t o two c o m p o n e n t s  E (N) = E S  E  where  E  the  (1.3)  c  i s t h e e n e r g y o f t h e v a p o u r and  y  i s the energy of the condensed  c  graphite This  + E  v  function  v a p o u r and t o t h e c o n d e n s e d  |  T e  -0P(N -N) T  N=0  £ s  I n p r a c t i c e , t h e chamber  e  By u s i n g  as  -0E (N) c  c  i s t a k e n t o be s u f f i c i e n t l y  t h a t t h e c h e m i c a l p o t e n t i a l o f t h e gas i s c o n s t a n t range  i n N relevant  phases.  This  allows  the chemical  the c o n t r i b u t i o n of the  v a p o u r , t h e p a r t i t i o n c a n be g i v e n  =  i n t o c o n t r i b u t i o n s due t o  system.  o f t h e gas M t o d e s c r i b e  z  and t h e  substrate.  separates the p a r t i t i o n  potential  krypton  to the formation the p a r t i t i o n  of the s o l i d  large  over the monolayer  f u n c t i o n t o f a c t o r i z e as  12  Z  Z-y  —  Z„ = 'v  where  ZQ  e"^ T  (1.5)  N  Nn  and  Z  c  = e " ^  =  £  ,T  N=0  Because  the monolayer  equilibrium state w i l l  A  where F The  c  c  e ^  (  E  ^  M  N  )  c  i s large,  o n l y one  configuration will  t h e s u m m a t i o n o v e r N c a n be n e g l e c t e d .  s i g n i f i c a n t and  free energy  £ s  The  t h e n be t h e s t a t e t h a t m i n i m i z e s t h e  A , C  =  F  c  -  (1.6)  MN  i s t h e H e l m h o l t z f r e e e n e r g y of the condensed  study i s therefore r e s t r i c t e d  condensed  be  system.  The  effect  phase.  t o an e x a m i n a t i o n o f t h e  of the vapour  i s p r o v i d e d by t h e c h e m i c a l p o t e n t i a l  on t h e t o t a l  M which  system  i s t r e a t e d as a  parameter. The  energy of the condensed  p o t e n t i a l and k i n e t i c  energies.  system w i l l Because  system a r e q u i t e m a s s i v e , the dynamic phase  involve  both  the components of the  a s p e c t s of the  condensed  a r e , a t low t e m p e r a t u r e s , expected t o p r o v i d e o n l y  minor  c o r r e c t i o n s t o the i n f o r m a t i o n o b t a i n e d from c a l c u l a t i o n s the  static  properties.  The  p o t e n t i a l energy of the  s y s t e m i s c o n s i d e r e d f i r s t and  of  condensed  the dynamical c o n t r i b u t i o n s  are  discussed  later.  14  2.  POTENTIAL ENERGY  (I)  MICROSCOPIC CONSIDERATIONS  The p o t e n t i a l e n e r g y o f t h e c o n d e n s e d i n t o two p a r t s . substrate;  The f i r s t  the second  consider  monolayer  i s the p o t e n t i a l energy of the bare  i s t h e e n e r g y change when k r y p t o n i s  adsorbed onto the s u r f a c e . to  s y s t e m c a n be d i v i d e d  The c a l c u l a t i o n c a n be r e s t r i c t e d  o n l y t h e p o t e n t i a l e n e r g y change due t o t h e  because  equilibrium  c o n s t a n t t e r m s have no i m p a c t on t h e  configuration.  A g i v e n adatom w i l l  be i n f l u e n c e d  by p a i r  interactions  with  n e i g h b o u r i n g k r y p t o n atoms a n d t h e i n t e r a c t i o n w i t h t h e substrate.  The i n t e r a c t i o n b e t w e e n k r y p t o n atoms  fluctuations an a t t r a c t i v e  i s due t o  i n t h e m u l t i p o l e moments w h i c h , on a v e r a g e , potential.  create  A p a r a m e t e r i z e d form f o r t h i s  p o t e n t i a l c a n be d e t e r m i n e d f r o m t h e b u l k p r o p e r t i e s o f t h e r a r e gas* because  .  S i n a n o g l u and P i t z e r  noted that the s u b s t r a t e ,  of i t s m e t a l l i c n a t u r e , a c t s as a m i r r o r t o t h e d i p o l e  moments o f t h e k r y p t o n a t o m s ^ .  T h i s c r e a t e s an a d d i t i o n a l  q u a d r u p o l e moment w h i c h m o d i f i e s t h e i n t e r a c t i o n k r y p t o n atoms  i n the monolayer.  between  A more e x a c t f o r m f o r t h i s  m o d i f y i n g e f f e c t h a s been d e t e r m i n e d b y M c L a c h l a n ^ . 1  Additional  i n f l u e n c e s have b e e n d i s c u s s e d  in detail  by B r u c h  w i t h t h e c o n c l u s i o n t h a t t h e y do have a n i m p a c t on t h e  x  15  c o n f i g u r a t i o n of the m o n o l a y e r .  Thus, the  o f an a d a t o m w i t h  i t s n e i g h b o u r s can  summation of p a i r  potentials.  i s taken  t o be  the  of Rauber e t a l . the  3  bulk  be d e t e r m i n e d by  In t h i s study,  potential  of A z i z ^  have been u s e d  2  Interaction  9  the and  a  pair the  energy  potential  parameters  i n the M c L a c h l a n ^ l  form f o r  substrate modification. The  substrate  i n t e r a c t i o n , which i n v o l v e s the  attraction  b e t w e e n s u b s t r a t e c a r b o n atoms and  the adsorbed k r y p t o n ,  h o w e v e r , n o t as  The  within  the  easily described.  s u b s t r a t e , due  atoms, c o m p l i c a t e s not  figure  substrate  1, t h e  bonding between carbon  i n t e r a c t i o n t o the  form of the  symmetry of the  charge d i s t r i b u t i o n  to chemical  y e t been m o d e l l e d r e l i a b l y .  parameterized  in  the  extent  Because of t h i s ,  substrate potential i s used  s u b s t r a t e has  is  i n the  a Bravais  t h a t i t has a  that r e f l e c t s  calculation. lattice  with  As  shown  basis  vectors  1.42  x  ^  i . 42  y (2.1)  Bj  where t h e I f we lattice,  /3  units  1.42  are  y  Angstroms.  d e f i n e g t o be a r e c i p r o c a l  lattice  the  the  substrate potential  has  form  vector  of  the  the  16  V(r,z)  (2.2)  = V (z) a  where z i s t h e h e i g h t  of t h e k r y p t o n atom above t h e s u b s t r a t e  and  r i s a vector  confined  its  o r i g i n a t an a d s o r p t i o n  all  reciprocal lattice  t o the plane of the s u b s t r a t e site.  vectors  The s u m m a t i o n p r o c e e d s  having over  of the s u b s t r a t e .  The d o m i n a n t t e r m i n ( 2 . 2 ) , V ( z ) , i s t h e s u r f a c e a v e r a g e o f a  the  p o t e n t i a l energy V ( r , z ) .  provide  the l a t e r a l  shown by S t e e l e potential  3 3  The r e m a i n i n g t e r m s i n ( 2 . 2 )  v a r i a t i o n of the s u b s t r a t e p o t e n t i a l .  , the l a t e r a l v a r i a t i o n  i n the substrate  i n f l u e n c e s t h e a d s o r b e d atoms i n two ways:  the s u b s t r a t e can e x e r t a l a t e r a l secondly, the height  Firstly,  f o r c e on t h e a d a t o m s , and  z a t w h i c h e a c h atoms s i t s  f u n c t i o n of i t s l a t e r a l  As  position r.  v a r i e s as a  F o r t h e commensurate  m o n o l a y e r , each atom s i t s above an e q u i v a l e n t s i t e s u b s t r a t e and t h e m o n o l a y e r i s p l a n a r .  on t h e  For the incommensurate  m o n o l a y e r s , however, t h e adatoms a r e not a l l s i t u a t e d a t equivalent sites  o f t h e s u b s t r a t e and t h e m o n o l a y e r w i l l  t h e r e f o r e n o t be  planar.  Gooding e t a l . *  3  have c o n s i d e r e d  of t h e adatoms i n t h e m o n o l a y e r . p o s i t i o n s o f t h e adatoms difference  i n heights  be n e g l i g i b l e .  the v a r i a t i o n  They f i n d  i n the height  t h a t because the  ( r , z ) are smoothly modulated, the  b e t w e e n a n a d a t o m and i t s n e i g h b o u r s  will  The i n t e r a c t i o n s b e t w e e n t h e a d a t o m s c a n  t h e r e f o r e be c a l c u l a t e d a s t h o u g h t h e m o n o l a y e r was p l a n a r ; t h e  17  heights  o f t h e adatoms a r e d e p e n d e n t o n l y on t h e s u b s t r a t e  interaction. Because the s u b s t r a t e the  potential  determines the heights  adatoms a s a f u n c t i o n z ( r ) o f t h e t h e i r p o s i t i o n  z of  r , the  monolayer's c o n f i g u r a t i o n i s determined e n t i r e l y from the lateral  p o s i t i o n s r of the adatoms.  the  m o n o l a y e r c a n be assumed  lie  on t h e s u r f a c e  surface  defined  the p o t e n t i a l  T h i s does not i m p l y  planar,  r a t h e r t h a t t h e adatoms  by t h e f u n c t i o n z ( r ) .  of the s u b s t r a t e  as a f u n c t i o n of r , w i t h t h e r e s u l t i n g  V(r)  a form s i m i l a r  substrate  potential  to (2.2).  lattice  after  This  V(r)  shell-^.  = V  +  Q  this  potential  The v a r i a t i o n  i n the  i s s o s m o o t h on t h i s s u r f a c e , h o w e v e r ,  summation over r e c i p r o c a l the f i r s t  On  c a n be e x p r e s s e d <  exclusively having  that  V  vectors  leads  truncated  to a s i m p l i f i e d  V ( i - e ?' 1  g  c a n be  1  that  form  )  (2.3)  g  where t h e s u m m a t i o n lattice  vectors.  involves only the f i r s t  Vg i s a p a r a m e t e r , known a s t h e  c o r r u g a t i o n , which provides substrate  interaction.  interaction,  shell  the l a t e r a l v a r i a t i o n  reciprocal substrate i n the  Because of t h e symmetry of t h e  Vg i s a c o n s t a n t  t h a t c a n be f a c t o r e d  from the  summat i o n . Given the pair interaction  potential  between k r y p t o n  <Mr),  which d e s c r i b e s  atoms  i n t h e m o n o l a y e r , and t h e  the  18  substrate krypton  p o t e n t i a l V ( r ) , which gives  a t o m and t h e s u b s t r a t e ,  the i n t e r a c t i o n between a  the p o t e n t i a l energy of the  m o n o l a y e r c a n be c a l c u l a t e d .  E = T  Y *(r'-r)  +  r, r  V V(r)  (2.4)  r  1  where r and r ' a r e p o s i t i o n s o f t h e adatoms w i t h i n t h e m o n o l a y e r , and t h e f i r s t s u m m a t i o n e x c l u d e s t h e that both p o s i t i o n s  coincide.  The p o t e n t i a l e n e r g y i s m i n i m i z e d monolayer  possibility  for configurations  of the  i n w h i c h t h e p o s i t i o n s of t h e adatoms s a t i s f y t h e  condition  0 =  r> >  -»  -»  -»  W(r'-r)  -»  -»  + W(r)  (2.5)  r'  As b e f o r e ,  t h e summation e x c l u d e s the p o s s i b i l i t y of  coincidence. forces  felt  Equation  (2.5) r e f l e c t s  by t h e adatoms a r e z e r o  the f a c t that the s t a t i c  when t h e p o s i t i o n s o f t h e  adatoms m i n i m i z e t h e p o t e n t i a l e n e r g y . r i g h t hand s i d e  of t h i s equation  More g e n e r a l l y , t h e  describes  the force  felt  by a n  a d a t o m a t r due t o i t s n e i g h b o u r i n g adatoms and t h e s u b s t r a t e . This  force,  i f non-zero, w i l l a c c e l e r a t e  t h e adatoms and t h e  r e s u l t i n g b e h a v i o u r o f t h e m o n o l a y e r c a n be a n a l y s e d molecular dynamics c a l c u l a t i o n s .  by  19  The d e n s i t y m o d u l a t i o n s due t o t h e d o m a i n w a l l s a r e n o t a s s h a r p a s shown i n f i g u r e 2; i n s t e a d , t h e r e g i s t e r e d r e g i o n s smoothly blend  into the higher  the monolayer always m a i n t a i n configuration.  density w a l l s a hexagonal  = R  +  The adatoms i n  packing  lattice  (2.6)  v e c t o r s , cif^ and d  2  o f v e c t o r s R*, have t h e i r  by t h e a v e r a g e d s p a c i n g  u  o f v e c t o r s R*  u(R)  The p r i m i t i v e l a t t i c e triangular  .  T h e i r p o s i t i o n s r have s l i g h t d i s p l a c e m e n t s  from the averaged t r i a n g u l a r l a t t i c e  r  1 3  , f o r the length  determined  o f t h e adatoms i n t h e m o n o l a y e r .  For  t h e c o m m e n s u r a t e p h a s e u(R*) = 0, and t h e b a s i s v e c t o r s a r e t h e same a s t h e s u b s t r a t e v e c t o r s  and D*, 2  defined  from (2.1) t o  be  Di = 2 B i + B  2  (2.7) D  2  = -Bi + B  2  The e x p e r i m e n t a l l y m e a s u r e d q u a n t i t i e s o f m i s f i t M% and orientation  9 are r e l a t e d to these  M% = [ l -  ) * 100  basis vectors  by  (2.8)  20  with  6 b e i n g t h e a n g l e b e t w e e n t h e two v e c t o r s  T h e s e q u a n t i t i e s a r e a l s o r e l a t e d t o t h e s i z e and of t h e domain  D\ .  cfj, and  arrangement  network.  M o l e c u l a r dynamics  c a l c u l a t i o n s show t h a t  f o r m s a honeycomb n e t w o r k  of domain w a l l s a t  the  monolayer  low  17 temperatures substrate this  .  The  domain w a l l s a r e not p i n n e d t o the  and move e a s i l y .  3 4  For systems  b e h a v i o u r c a n be d e s c r i b e d  v i b r a t i o n s about a p e r f e c t deviation  honeycomb.  of the domain w a l l s  configuration  i n terms  i s a breathing  with  f i x e d coverage,  of domain  The  wall  lowest energy  from the p e r f e c t  honeycomb  d i s t o r t i o n which v a r i e s the s i z e  of t h e r e g i s t e r e d domains w i t h o u t c h a n g i n g t h e t o t a l the  domain w a l l s  i n the m o n o l a y e r .  length  I f energy i s not  1 4  required  to b r e a t h e the domain w a l l s , the p o t e n t i a l energy w i l l  be  the  same f o r a l l b r e a t h e d c o n f i g u r a t i o n s  and c a n be c a l c u l a t e d  the  I f the breathing  configuration  requires  that  is periodic.  energy, the p e r i o d i c c o n f i g u r a t i o n w i l l  configuration  o f minimum e n e r g y .  I f the p e r i o d i c  has v i b r a t i o n a l modes t h a t a r e u n s t a b l e , configuration monolayer with  be l o w e r i n e n e r g y .  a t a g i v e n m i s f i t and  i s tested  from  motion  the  configuration  then another Thus, the energy of the  o r i e n t a t i o n c a n be  the assumption that the c o n f i g u r a t i o n  assumption are  will  be  of  calculated  is periodic.  when t h e v i b r a t i o n a l modes o f t h e  This system  calculated. Assuming  superlattice  the s t a t i c  incommensurate  periodicity,  monolayer  has  f i g u r e 3 shows t h a t an a d a t o m i n one  21  registered  region  counterpart  -4  i  2  be s e p a r a t e d  i n another r e g i s t e r e d  Ri  nD* +mB  will  -4  =  -4  nDi+mD  -  4  -  region  Identical  by a  distance  4  + £Bi+kB  2  from i t s  (2.9a)  2  i s t h e p o s i t i o n o f t h e c o u n t e r p a r t adatom i n t h e  absence of t h e i n t e r v e n i n g  w a l l ; S.f +kB i  + 2  i s the s h i f t  in its  p o s i t i o n due t o t h e d o m a i n w a l l . Given the p e r i o d i c possible  nature of t h esystem, there w i l l  s e p a r a t i o n s between i d e n t i c a l c o u n t e r p a r t adatoms, and  R^ c a n be t a k e n t o be one o f t h e l a t t i c e this  be many  group.  v e c t o r s which  By symmetry, t h e o t h e r p r i m i t i v e  generate  l a t t i c e v e c t o r R*  2  w i l l be  R  = -mDi + (n-m)D  2  2  -  k i \ + (S,-k)B  These s u p e r l a t t i c e v e c t o r s a r e a l s o lattice  vectors  k  -4 2  -4  (2.10)  -4  -md( + (n-m)d;  integers indicate  t o the primitive  nd i + md  2  The  (2.9b)  f o r t h e averaged adatom p o s i t i o n s by  -4  Ri -t R  related  £  n and m s p e c i f y  t h etype of w a l l s  This thesis  t h e s i z e o f t h e domains and % and that  separate t h e domains.  has been r e s t r i c t e d t o s u p e r l a t t i c e  22  configurations substrate  with walls that correspond to s h i f t s  adsorption  site.  The  d i s c r e t e s e t of  m i s f i t s and  o r i e n t a t i o n s of the  restriction  i s given  reduce the This  ability  considering  value  o f m i s f i t and  irrational  r o t a t i o n angles, for  s a m p l i n g t h a t the  k which correspond to s h i f t s  of  The be  However, a t  extension obtained  the  low  not  t o an a r b i t r a r y  by a  m i s f i t s and  c a l c u l a t i o n to  k = 0 ( f i g u r e 4) p r o v i d e s general  be  procedure  o f a s e q u e n c e o f r a t i o n a l numbers  restricting  w h i c h 8, = 1 and  does  by  r o t a t i o n can  limit.  k=0  augmented  point.  extension  and  this  set.  o r i e n t a t i o n s can  o f % and  adsorption  analagous t o the an  to generate t h i s  values  more t h a n one  S e t t i n g &=1  one  possible  monolayers t h a t conform to  i n f i g u r e 4.  s e t o f m i s f i t s and  of o n l y  b e h a v i o u r of the  to  small configurations  a d e n s e enough monolayer can  be  determined. B e c a u s e f u r t h e r work w i l l quantities  i n v o l v e summations of  ( a l l of w h i c h r e q u i r e  i n t e g e r v a r i a b l e s Sl, k, contexts.  To  m and  summation i n d i c e s ) , the  n will  remove c o n f u s i o n ,  be  used i n d i f f e r e n t  a l l f u r t h e r mention of  monolayer c o n f i g u r a t i o n w i l l  be  i n terms of m i s f i t  orientation;  k,  m and  will  not  be  the  integers  associated  %,  with  The  r e l a t i o n s h i p between the  the  m o n o l a y e r and  Appendix  A.  these  the  various  n  (aside  and  from appendix  c o n f i g u r a t i o n of the  various  the  system.  geometric properties  integer variables  is described  in  of  A)  23  (II)  SIMPLIFICATIONS  From z(r).  ( 2 . 2 ) t h e adatoms w i l l  be f o r c e d t o l i e on a  surface  The p o s i t i o n s o f t h e adatoms on t h i s s u r f a c e w i l l  be  d e t e r m i n e d by t h e s u b s t r a t e ' s c o r r u g a t i o n , w h i c h i m p o s e s a lateral  f o r c e on t h e a d a t o m s , and t h e i n t e r a c t i o n s b e t w e e n t h e  adatoms. slight, if  The v a r i a t i o n  i n height  z o f t h e adatoms i s so  t h a t t h e a d a t o m p a i r i n t e r a c t i o n s c a n be c a l c u l a t e d a s  t h e m o n o l a y e r was p l a n a r  substrate  interaction  1 3  .  A l s o , w i t h the form of the  (2.3) t h e l a t e r a l  p o s i t i o n s r of the  a d a t o m s c a n be c a l c u l a t e d a s t h o u g h t h e m o n o l a y e r was The h e i g h t s (2.2).  planar.  o f t h e adatoms c a n be s u b s e q u e n t l y c a l c u l a t e d f r o m  This s i m p l i f i e s  the c a l c u l a t i o n .  I n a d d i t i o n , o n l y p e r i o d i c c o n f i g u r a t i o n s , where t h e r e g i s t e r e d d o m a i n s a r e s e p a r a t e d by d o m a i n w a l l s w h i c h o n l y one a d s o r p t i o n s i t e ,  need be c o n s i d e r e d  i n order  shift to  d e t e r m i n e t h e v a r i a t i o n of t h e m o n o l a y e r ' s p o t e n t i a l energy as a f u n c t i o n o f i t s m i s f i t and o r i e n t a t i o n . calculation,  because the p o t e n t i a l energy of the monolayer can  be d e t e r m i n e d by e x a m i n i n g o n l y t h e adatoms p e r i o d i c domains. are  This reduces the  Further  simplifications  i n one o f t h e to the c a l c u l a t i o n  p o s s i b l e , because t h e d i s p l a c e m e n t s of t h e adatoms u  s m o o t h l y m o d u l a t e d and t h e d i s t a n c e s between adatoms c l o s e t o t h e a v e r a g e d s p a c i n g s R* d e t e r m i n e d  be e x p a n d e d a s a T a y l o r  remain  from the m i s f i t .  T h i s a l l o w s t h e adatom p a i r p o t e n t i a l s used i n ( 2 . 4 ) to  are  s e r i e s about the averaged  and ( 2 . 5 ) spacing.  24  Given  ( 2 . 6 ) , t h i s c o n t r i b u t i o n t o t h e p o t e n t i a l of an adatom a t  -4 -»  r ( R ) c a n be e x p r e s s e d  -4  E (R)  =  a  as  1  r>  £  ) [1 + (u-u')-V  f  - 4 - 4 - +  1  +  -4  (2.6)  gradient The  -»  ,  <  *(R-R«)  t h e p o s s i b i l i t y t h a t R=R'.  -4  u and u  -»  , ••• J  +  where t h e s u m m a t i o n e x c l u d e s  -4  i . ( (u-u') • v " ) +  —)  2  -  1  1  }  From  -4  a r e f u n c t i o n s o f R a n d R' r e s p e c t i v e l y , a n d t h e  1  operator  w o r k s on t h e a r g u m e n t R* o f t h e f u n c t i o n  c o n t r i b u t i o n of the p a i r i n t e r a c t i o n s t o the force  felt  by a g i v e n a d a t o m a t r i s l i k e w i s e  F  a  (  R  )  "X l .  =  R  1+  (u-u')-V  +  )-V)  2  +  ..-)W(R-R') (2.12)  the d i s t a n c e s  after  the f i r s t  few terms because  between t h e adatoms r e m a i n hear t h e s p a c i n g s  triangular lattice.  continuum approximation  of  The most f r e q u e n t l y u s e d t r u n c a t i o n  for a n a l y t i c studies excludes  the anharmonic terms.  Within the  (appendix B ) , the displacements  a d a t o m s must s o l v e a s e c o n d o r d e r equations.  1  '  These s e r i e s c a n be t r u n c a t e d  the  j{ ( u - u  These a p p r o x i m a t i o n s  of the  n o n l i n e a r system of have been d i s c u s s e d  in a  io previous  work J  i  w i t h the c o n c l u s i o n t h a t the expansion of  (2.11) must i n c l u d e a t l e a s t t h e c u b i c t e r m s t o p r o d u c e agreement w i t h r e l a x a t i o n s t u d i e s .  Other s t u d i e s  3 5  on s i m i l a r  25  s y s t e m s have c o n s i d e r e d that truncating  t h e q u a r t i c t e r m and have  (2.11) a t t h e c u b i c  concluded  i s adequate.  Therefore,  to  a l l o w t h e i n f l u e n c e o f p a i r i n t e r a c t i o n s t o be c a l c u l a t e d r a p i d l y , a l l terms past t h e c u b i c a r e  neglected.  A n o t h e r b e n e f i t of t h e smooth m o d u l a t i o n i n t h e adatom positions describe  i s t h a t t h e amount o f i n f o r m a t i o n  required to  t h e c o n f i g u r a t i o n o f t h e m o n o l a y e r c a n be  reduced . 1 1  From t h e p e r i o d i c i t y o f t h e m o n o l a y e r , t h e d i s p l a c e m e n t s o f a d a t o m s u (R*) c a n be e x p r e s s e d a s a F o u r i e r  sum  1 1  '  J D  (2.13)  where  q  a  = lq  m  superlattice the  first  L  +  mq  (A.S).  is a reciprocal lattice  2  The F o u r i e r  after  few l o n g w a v e l e n g t h s h e l l s b e c a u s e t h e d i s p l a c e m e n t s  truncation  i s established  The r e s u l t i n g r e d u c t i o n to describe  The e x a c t c r i t e r i a f o r  i n s e c t i o n IV of t h i s  chapter.  i n t h e amount o f i n f o r m a t i o n  required  t h e s y s t e m makes t h e c o m p u t a t i o n s f o r t h e  d i s p l a c e m e n t s u more When t h e e x p a n s i o n due  of the  s e r i e s c a n be t r u n c a t e d  have s u c h a s m o o t h s p a t i a l m o d u l a t i o n . the  vector  efficient. (2.13) i s i n s e r t e d i n t o ( 2 . 1 2 ) ,  t o t h e a d a t o m i n t e r a c t i o n s becomes  the force  26  'a< > = R  a  F  L,M  £ L M e L, M ' f  i j  LM'  UL,M'QL,M  "  "  i q  n  7  +  where  R  "L-fc,  ^  &,m  and  Q  with  R  = £ 2 sin (q ' n M  -  L M  '  (  R  L - * . , -  -r L M  7W(h) ->->->  -h)  4  b a s i s v e c t o r s df a n d d? x  Similarly,  "T  WW(h)  the h summations range over t h e t r i a n g u l a r the  L,M  R  M-m  -h)  V  = / -sin(q h '  L / M  :  M-m  2  L  (2.14)  lattice  g e n e r a t e d by  (A.2).  2  when ( 2 . 1 3 ) i s i n s e r t e d  into  e n e r g y p e r a d a t o m due t o t h e c o n t r i b u t i o n  (2.11), the average of the p a i r  interactions i s  r - i f - i - *  Ua = Uo +  l\ \ u  L,M  tt L,M 6 U  where  U  0  =  L  L  /  U  M  Z-L,  -» u . '  ' %,m m-M'  ^ #(h)  u  =  :Q  (2.15)  +  M  - i l , - m :l. L - S l , ~ M-m R  R  L,M '  +  R  a,,mJ '  i s a summation of t h e p a i r p o t e n t i a l s  the averaged t r i a n g u l a r  >  over  lattice.  From ( 2 . 3 ) , t h e f o r c e e x e r t e d substrate i s  L  on a g i v e n a d a t o m due t o t h e  27  -i  F (R)  = -vV<2)  S  = V  g  ^ -i9jk e 3 k  i g  :k'  R  -i  -i  e^k'"**'  (2.16)  f  ~*  ~*  where S j ] ^ j g restricted  A  + kg  (A.4) and t h e summation  2  t o the f i r s t  substrate. of  ~>  shell reciprocal  Because o f t h e s u p e r l a t t i c e  over j and k i s  lattice  vectors of the  periodicity,  functions  u(R*) c a n be e x p r e s s e d a s  e  ig  j  k  -u(R)  I  =  .3k  L,M  Equations  F (K> S  e  i5  (2.17) and (A.7) a l l o w  = - I  (  iVg l 9 j,k  L,M  where t h e s u m m a t i o n  L M  -R  ( .i7) 2  7  j  ( 2 . 1 6 ) t o have t h e f o r m  k  M  _  R  ) e^LM'S  -  (  2  .  1  8  over j and k m a i n t a i n s t h e r e s t r i c t i o n s of  (2.16) . Similarly, substrate  If  the average p o t e n t i a l  p e r a d a t o m due t o t h e  becomes  "s  =  V  o  )  +  6  V  g " V  g  1  A^ D,k  t h e d i s p l a c e m e n t s o f t h e adatoms  (2.19) and (2.15) w i l l  (2.19)  a r e known, t h e sum o f  provide thet o t a l potential  energy per  28  adatom.  The  solutions the  displacements  of  (2.5).  are determined  From the r e s u l t s  from the f o r c e - f r e e  of  ( 2 . 1 4 ) and  (2.18),  force-free solution requires that  . -*  F(R)  = - £ F  L,M  e  L M  i q  -*  LM'  *LM = "L,M*5LM TI +  4,m  V \  will  t h a t values of u ^  Brillouin extend  m  consequence, the  of F  (2.20b)  L  and  / M  A^  -k are d e f i n e d over  k L M  Sim and  r a t h e r t h e s u m m a t i o n o f F*^  similar  of a l l B r i l l o u i n  p o i n t s LM  requirement  will  i n t h e o r y , be s o l v e d .  o f F*  LM  of u ^  and  m  which extend  ignored.  Thus F*  LM  A^  L M  m  and  the system of  first  t o , be  Brillouin  zero  p r o v i d e , t o a l l i n t e n t s and  force-free configuration.  the  equations the  the  shell  coefficients be  This  the s o l u t i o n of  purposes,  to  zone c a n  i n (2.20b).  of  From  the higher  a r e n e g l i g i b l e and  p r o v i d e s a s i m p l e r s y s t e m of e q u a t i o n s , will  imply  the s e t  However, c o n s i d e r i n g t h a t  beyond the i s taken  over  zones i s z e r o .  m o n o l a y e r s t r u c t u r e i s so s m o o t h s p a t i a l l y , coeffients  a  p r o v i d e enough e q u a t i o n s  m a t c h t h e number o f v a r i a b l e s u ^ can,  As  f o r c e - f r e e c o n d i t i o n (2.20a) does not  =  this  the  j k i n (2.20b)  p a s t t h e zone e d g e .  L M  F*LM fj f o r a l l LM,  (2.20b),  h,m)  k  zone, the summations over  the values  +  M-m  j A _j  J / K  first  L,M  M-m -» 9jk  (2.20a)  R  :  v  Given  where  *L-l, [*L-l,-  %,m +  o  =  R  general  which  29  The  following three sections relate  a n a l y s i s and d e s c r i b e i n d e t a i l values  of the c o e f f i c i e n t s  u<j . m  of the c o n f i g u r a t i o n i s o b t a i n e d  p u r e l y t o the  t h e method Given  f o r c a l c u l a t i n g the  these  from (2.15  numerical  values the energy and 2 . 1 9 ) .  The  d i s c u s s i o n o f t h e p h y s i c a l p r o p e r t i e s o f t h e s y s t e m resumes i n s e c t i o n VI of t h i s  chapter.  30  (III)  METHOD OF SOLUTION  A Newton s t e p method of e q u a t i o n s  (2.20b).  where t h e v a l u e s  the  forces F  L  M  T h i s method i s a n i t e r a t i v e  ofu^  An i n i t i a l  steps.  i s chosen t o s o l v e the n o n l i n e a r  m  approach the s o l u t i o n  estimate  (2.20b).  from  t h e F^Lj^j t o a s m a l l p e r t u r b a t i o n ^ u ^  i n a series of  r e t a i n i n g only the l i n e a r  F  L M  (u  + Au) =  F  L M  (u) +  of u ^ , and m  The r e s p o n s e o f  i s then c o n s i d e r e d .  m  expanding the force c o e f f i c i e n t s a l g e b r a i c a l l y and  process  i s made f o r t h e v a l u e s  are calculated  system  By  i n terms of A u £  terms  X  1,  5r  M  • AUfcm  ,  (2.21)  m Sim  where  °LM, Sim  = ~§LM  S(L-Sl)S(M-m) - \ u _ : (R _< M-m M-m L  + V 3  and  advantage i s taken  L,M  t  > '  L  g  k  g  j  k  - R  A  ^  L  /  .  M-m-k  k  of the f a c t  51,m  j  1 /  that  M-m  M  ^  +  5l,m.)  R  (2.22)  m  31  The  perturbations  which  F* (u+Au)=0  s e t of  equations  F  (u)  =  B  - I £,m  (2.20b),  t h e s e new  are  t a k e n t o be  are  ( 2 . 2 2 ) and  therefore  added t o the  next set  is repeated u n t i l  the  change i n the  E  L,M  falls  <3fc )  F  • Au  m  below a s p e c i f i e d  linear  u^  m  values  and  the  process  of p e r t u r b a t i o n s . e n e r g y , AU  This  from  process  determined  from  (2.25)  value.  The  minimum e n e r g y  t h u s be  (or  d e t e r m i n e d as  accurately  required. The  above d i s c u s s i o n p r e s e n t s  the  Newton s t e p method i n i t s  most s t r a i g h t f o r w a r d  form.  consideration  q u i t e so e a s i l y h a n d l e d .  discussed  i s not  domain w a l l s .  dynamical matrix center,  However, t h e  p r e v i o u s l y , the adatoms w i l l  phonon modes, i n f a c t two the  the  L M  f o r c e - f r e e ) c o n f i g u r a t i o n can as  must s o l v e  for  (2.24) i s t h e n used t o d e t e r m i n e ,  the  =  values  (2.24)  values,  A U  the  • Au^  LM/  Sim  These p e r t u r b a t i o n s of  ? m  ( 2 . 2 1 ) and  LM  L M  Au  the  Since of the  matrix  unique s o l u t i o n .  modes a r e the  matrix  As  has  been  have e x t r e m e l y  soft  a c o u s t i c , due  causes the  to motion  D i n (2.24) i s s i m p l y  system c a l c u l a t e d a t the  i s s i n g u l a r and This  s y s t e m under  Newton s t e p  the  Brillouin  (2.24) does not  of  zone  have a  method t o  fail.  32  To  r e s o l v e t h i s p r o b l e m , a s p e c i f i c c o n f i g u r a t i o n must  s e l e c t e d and only this  t h e Newton s t e p method r e s t r i c t e d  solution.  The  be  so t h a t i t s e e k s  c o n f i g u r a t i o n c h o s e n has  an  adatom  s i t u a t e d a t t h e c e n t e r o f t h e h e x a g o n a l d o m a i n s f o r m e d by domain w a l l s .  S i n c e the c e n t e r s of the domains a r e  the c e n t r a l adatoms a r e a t a d s o r p t i o n s i t e s Given  the  registered,  of the g r a p h i t e .  t h e h e x a g o n a l s y m m e t r y o f t h e s u p e r l a t t i c e and  the  s y m m e t r y o f t h e s u b s t r a t e , t h e minimum e n e r g y c o n f i g u r a t i o n should  be  i n v a r i a n t u n d e r r o t a t i o n s of 60°  adatoms.  I t i s therefore reasonable  r o t a t i o n s o f 180° (2.24). be  (2.24) can  their be  central  t o impose i n v a r i a n c e u n d e r  ( o r i n v e r s i o n s y m m e t r y ) on t h e s o l u t i o n s t o  This r e s t r i c t i o n  z e r o and  about these  f o r c e s the v a l u e s  of u  Q O  and  F*  to  QO  c o n t r i b u t i o n s t o the s y s t e m of e q u a t i o n s  neglected.  The  r e s u l t i n g reduced  in  matrix i s  nons i n g u l a r . An a d d i t i o n a l b e n e f i t o f i m p o s i n g the u ^ 0  values are t o t a l l y  m  similar  requirement  i m a g i n a r y and  a p p l i e s t o t h e F*  o f t h e m a t r i x D need be c o n s i d e r e d solution.  Therefore,  i n v e r s i o n symmetry i s t h a t  LM  u^ n\ *-!l,-m •  , and  =-u  o n l y one  quarter  i n (2.24) t o o b t a i n a  obtained,  of c o m p u t a t i o n The  i t also significantly  reduces  the  of  the  amount  necessary.  Newton s t e p method was  accurate  full  i n v e r s i o n symmetry i s imposed t o ease  c a l c u l a t i o n s i n c e i t not o n l y a l l o w s a unique s o l u t i o n ( 2 . 2 4 ) t o be  *  s e l e c t e d because, w i t h  s t a r t i n g p o i n t , i t converges r a p i d l y .  s e a r c h methods r e q u i r e a c a l c u l a t i o n  o f t h e F*  an  A l l nonlinear LM  values; a  33  numerically tedious process. s h o u l d be m i n i m i z e d .  Therefore,  This c r i t e r i a  repeated  computations  e l i m i n a t e s the other  commonly u s e d s e c a n t and g r a d i e n t methods b e c a u s e t h e s y s t e m has  l o w e n e r g y phonon modes a s s o c i a t e d w i t h d o m a i n w a l l  motion.  Even a f t e r  e l i m i n a t i n g the degeneracy i n the  c o n f i g u r a t i o n due t o t h e a c o u s t i c modes, t h e l o w e n e r g y modes s t i l l  optical  a l l o w l a r g e f l u c t u a t i o n s t o be made i n t h e  displacements the c a l c u l a t e d  o f t h e adatoms w i t h o u t  significant  energy of the c o n f i g u r a t i o n .  impact  upon  T h i s can s e v e r e l y  s l o w g r a d i e n t methods w h i c h t a k e s t e p s a c c o r d i n g t o t h e d i r e c t i o n and m a g n i t u d e o f t h e c u r r e n t f o r c e , a n d c a n c a u s e s t a b i l i t y problems i n secant  methods.  Modified secant  a r e s t a b l e , b u t e a c h s t e p r e q u i r e s more c o m p u t a t i o n o f t h e Newton s t e p m e t h o d .  methods  than  those  34 (IV)  NUMERICAL  The to  following  facilitate  efficient  displacements. the to  first  quantities.  included  adatom i n t e r a c t i o n truncated  to  shells.  adatom introduce  However, t h e s e  Once t h e  errors  m i s f i t , M%,  calculated  error  and  Another  the  first  to  R* M  for  the  errors  will  of  be  into shown  summation can  few  nearest  (2.14)  IN  and  configuration  h used  for U  is specified, U  the  c a l c u l a t i o n for U  the  summations  over  in  of  summation. values  displacements are  wavelength terms are  the  (2.11)  From t h e of  U £  M  has  many terms a r e  can  reflects  Q  of  (2.14). e q u a t i o n (2.13) in  described  preceeding  by  section  been e s t a b l i s h e d .  s m o o t h l y modulated, the sufficient  Q  and  h required  adatoms a r e  the  examined.  t h i s p r o v i d e s a means  summation  in  (2.15).  in  Q  in truncating  displacements  how  be  truncation  (2.11),  the  neighbour  s e t t i n g bounds on  L/  of  Since  the  the  the  number  (2.11).  numerical approximation concerns  series  the  e f f e c t of  bounds f o r  of d e t e r m i n i n g  Exactly  the  and  in truncating  correspondingly,  Fourier  the  error  establishing  where t h e  only  concerns  summation  range,  is equivalent  for  Because the  i n the  is short  involve  This  calculation  the  the  will  numerical approximation  adatom s h e l l s  the  c a l c u l a t i o n of  controllable.  The  be  i s c o n c e r n e d w i t h a p p r o x i m a t i o n s made  These a p p r o x i m a t i o n s  calculated be  section  APPROXIMATIONS  to d e s c r i b e  needed depends on  first  the the  few  a a  method Because long  system. accuracy  35  required 2.19).  f o r the c a l c u l a t e d values A t r i a l and  of the energy (2.15  and  e r r o r method of determining the number of  terms i s i n e f f i c i e n t , and  t h e r e f o r e , an a n a l y t i c method i s  sought. Truncation describe  of the F o u r i e r s e r i e s r e s t r i c t s the a b i l i t y  sharp s t r u c t u r e s  v e r t i c e s , the sharpest domain w a l l s .  The  i n the monolayer.  structures  Apart from the  i n the monolayer w i l l be  the  e f f e c t of t r u n c a t i n g the F o u r i e r s e r i e s  can  be examined by c o n s i d e r i n g  how  the c a l c u l a t e d energy of  domain w a l l changes as the t r u n c a t i o n a n a l y s i s the simpler suitable.  to  i s changed.  For  the this  system of u n i a x i a l domain w a l l s i s  Corrections  f o r the v e r t i c e s w i l l be  discussed  later. The  t h e o r e t i c a l work by Frank and  van der Merwe  21  provides  an a n a l y t i c d e s c r i p t i o n of a u n i a x i a l system of domain w a l l s . This  i s reviewed i n appendix B to e s t a b l i s h the  required  i n the context  monolayer has  of t h i s t h e s i s .  For  equations  the case where the  a s i n g l e domain w a l l , displacements of  adatoms w i l l be given as  (B.3 and  the  B.6)  -4  u(R) ->  = 2?f 4  where S = Bj. »R  4 tan ( _ 1  e*  s  )  (2.26)  ->  and  B i i s a b a s i s vector  of the  substrate.  For the s o l u t i o n (2.26) the energy of a given adatom can be reduced to  (B.4)  36  E(R)  where for  = E  + a(9'(S))  0  9(S) i s defined  Given the  the Fourier  9'(s)  f(q)  error  determined the  total  portion  provides  ( 2 . 1 3 ) c a n be  of 9(S) i n (B.6),  JJ f ( q ) -Q  e  ^  1  an a n a l y t i c  form  and t h e impact of tested.  i n the limit  Q -+ oo ,  as  dq  (2.28)  is  f(q)  The  =  -  (2.27)  a domain w a l l ,  c a n be e x p r e s s e d  9» ( S )  where  This  with  series  the d e f i n i t i o n  function  + b6'(S)  by (B.6).  the energy associated  truncating  2  lot ^7  =  e  —  due t o t r u n c a t i n g by c o n s i d e r i n g energy  "  q T t / 2  * -T77J  (2.29)  the Fourier  the impact  series  of f i n i t e  of the system determined  of the t o t a l  energy which  from  i s dependent  i s then values  o f Q on  (2.27).  The  on t h e v a l u e o f  Q i s  oo  E(Q)  where  9'(S) v a r i e s  =  aj  with  ,  (6'(S))  2  dS  Q as defined  (2.30)  by  (2.28).  37  From  (2.29)  E(Q)  =  The r e l a t i v e  ( i -  a  2e  -Qlt/ot  (2.31)  1 + e-Qit/ot  e r r o r %S(Q) i n t r o d u c e d  by h a v i n g  a f i n i t e value of  Q i s then  %S(Q)  2 1 + eQrf/ot  =  (2.32)  The Q s e n s i t i v e t e r m s p r e s e n t energy w i l l  i n the c a l c u l a t i o n  be t h e h a r m o n i c a n d c u b i c t e r m s o f ( 2 . 1 5 ) and t h e  s u b s t r a t e c o r r u g a t i o n terms of (2.19). (2.32) has n e g l e c t e d  To t e s t t h e e s t i m a t e  calculations  Since  the v e r t i c e s present  s t r u c t u r e , the r e s u l t i n g estimate accurate.  the  provided  by ( 2 . 3 2 ) ,  lattice  v e c t o r s g^ a n d q  v e c t o r s q^  m  to include only  2  vectors are spaced c l o s e r together  because  a r e l a r g e o n l y a few  the r e c i p r o c a l l a t t i c e  a n d more a r e r e q u i r e d .  v a r i e t y o f c o n f i g u r a t i o n s were c o n s i d e r e d ;  values  terms  are required, f o r configurations  w i t h s m a l l e r m i s f i t v a l u e s , however,  repeated  sample  For c o n f i g u r a t i o n s with large m i s f i t s ,  reciprocal lattice  reciprocal  i n the monolayer's  were p e r f o r m e d t o o b t a i n t h e e n e r g y p e r a d a t o m  I q ^ I = Q. m  the d e r i v a t i o n of  o f t h e e r r o r may n o t be  when t h e s u m m a t i o n ( 2 . 1 3 ) was t r u n c a t e d with  of the  the c a l c u l a t i o n  f o r each c o n f i g u r a t i o n w i t h a v a r i e t y of c u t o f f  f o r Q.  In a l l cases,  expression  (2.32) a c c u r a t e l y  A was  38  estimated  the e r r o r a s s o c i a t e d w i t h the c u t o f f v a l u e of  Thus, the e s t i m a t e  of  ( 2 . 3 2 ) a l l o w s bounds t o be s e t , i n  a d v a n c e , f o r t h e number o f c o e f f i c i e n t s calculate The  numerical approximation  e v a l u a t i o n of A '  k  coefficients  be  3  Bessel  can  functions  -*  u^  m  required to  the energy of the monolayer to a s p e c i f i e d  final  i n (2.17).  L M  3 6  obtained  .  Q.  From  -» -»  accuracy.  r e q u i r e d i s f o r the  Analytically,  these  by t a k i n g sums and  products  of  (2.13)  • ~*  -4  ~*  -4  iqo -R m  (2.33)  The  coefficients  f u n c t i o n s can terms.  u^  m  a r e s m a l l , and  t h e r e f o r e be  the summation over  truncated after  the  Moreover, f o r the higher s h e l l terms,  summation can  be  truncated after  magnitude of the c o e f f i c i e n t s increasing  I q^  m  I.  This  U£  the m  first  decreases  first the  reflected zero  term because rapidly  i n the v a l u e s of A ^ p ^  number o f c o e f f i c i e n t s  the  for  i s e q u i v a l e n t to r e p l a c i n g the u ^  T h i s l a c k of s h a r p  f o r the higher order  few  Bessel  ( 2 . 1 3 ) by t h e t r u n c a t e d s e t w h i c h i n v o l v e s o n l y t h e low o r d e r s h e l l s .  Bessel  shells.  A^^  and  m  significant  structure is also  which can With  be assumed t o  be  t h i s r e s t r i c t i o n the  the computation  in  required to  39  d e t e r m i n e them c a n be r e d u c e d . However, e v e n w i t h t h e s e  reductions, the numerical  e v a l u a t i o n of (2.33) f o r A g ,  becomes  3 k  the  number  operations  of c o e f f i c i e n t s required  w i t h t h e number  u^  m  i n c r e a s i n g l y t e d i o u s as  i s increased.  i n the c a l c u l a t i o n  of c o e f f i c i e n t s  c o e f f i c i e n t s may be r e q u i r e d not  m  u^ .  increases Since  m  The number o f exponentially  a l a r g e number o f  i n the c a l c u l a t i o n ,  t h i s method i s  suitable. A b e t t e r method o f e v a l u a t i n g  f a c t t h a t a domain w a l l induces adatom w i t h r e s p e c t displacements,  a shift  to the s u b s t r a t e  u  2 1  of  from the  i n t h e phase of an  . - Because t h e  o f t h e adatoms a r e measured r e l a t i v e t o  the averaged s p a c i n g g j k ' u ( R ) i s !4rt. the  ( 2 . 1 7 ) c a n be d e r i v e d  o f t h e m o n o l a y e r , t h e maximum v a l u e o f  T h i s bound on t h e v a l u e s  o f g j " u (R*) k  allows  expansion of  e  z  =  £ n  i-j  z  n  where  z = igj -u(R)  (2.34)  k  >  t o be t r u n c a t e d expression evaluated  after  the f i r s t  by a s h o r t s u m m a t i o n i n v o l v i n g r e p e a t e d m  operations  Since  this  i n v o l v e s p o w e r s o f z, t h e c o e f f i c i e n t s A ^ £  of t h e c o e f f i e n t s U £ . utilizing  few t e r m s .  self  m  convolutions  T h i s c a n be f u r t h e r i m p r o v e d b y  c o n v o l u t i o n s , w h i c h r e q u i r e h a l f a s many  as the general  case.  From  c a n be  40  e  A  n  (  =  z  A  )  n  i s obtained  2.34)  n  e ^  where  2  from  n =  ^  +  n  a truncated Taylor  with the r e s u l t i n g  much s m a l l e r t h a n  Z//  error  S .  fewer  obtained  by p e r f o r m i n g  result.  The  error  terms  than  n repeated  in e  z  due  (2.35)  n  series  Since  n  !4Tt f o r l a r g e v a l u e s  be t r u n c a t e d a f t e r  g  (as i n  the argument w i l l  o f n, t h e e x p a n s i o n  (2.34).  squares  to this  expansion  e  be  can  i s then  z  of the summation's  method  c a n be  calculated  from  (A  n  ±  S )  n  An  n  Sn A  )  (2.36)  n  (  If  the series  will  for A  n  1 ±  2  n  S  n  )  i s truncated after  since  A  m terms,  n  -* 1  the error  n  be  m  with  S  the r e l a t i v e  method  (2.37)  error  of c a l c u l a t i o n  %S  being  i n the value  for e  z  due  to  this  41  The c o m p u t a t i o n a l  effort  the  a b o v e scheme w i l l  U£  and n s e l f  m  restricted  so that  non-zero.  Given  the  involve  convolutions.  smoothly modulated,  required  f o r obtaining m general Because  the s e t of U £ only  these  the long  conditions,  reasonable  of c o e f f i c i e n t s . than  the exponential  using  m  of the s e t  the displacements are  and t h e s e t of A ^  varies  k  m  rate  of  0 m  operations  as the square of  of increase  increase  are  k  coefficientsare  t h e number  A^ £  This  £  convolutions  wavelength  t o c a l c u l a t e the values  number  m  the s e t A  i s more  of the Bessel  series  approach. f o r %S  From t h e e x p r e s s i o n the  i n (2.38),  l e a s t amount o f c o m p u t a t i o n  self  convolutions  convolutions rounding, than  (i.e.  m=2  the error  due t o t h i s  representative  w h e n m=4  bound  %S.  error  b o u n d %S,  that  With  values  t h e number ).  significant o f m.  configurations that  and n i s v a r i e d the condition the value  of  general  f o r large  Because  values  of t h i s ,  of n  the actual  for several the  general  a c c u r a c y and t h e l e a s t work  to suit  that  that  However, due t o machine  of the monolayer;  the best  appear  r e s u l t i f t h e number o f  method were c a l c u l a t e d  was e s t a b l i s h e d  occurs  than  and n l a r g e  i s more  i t i s f o r large  errors  rule  i s greater  would  i twould  _z£Yirt,  the specified  error  and f o r t h e s p e c i f i e d  o f n m u s t be t h e s m a l l e s t  integer  satisifies  log <*/ > ^- 41 %S log(8) 2  n  h  } (  2.39)  42  T h i s value  of n can be l a r g e r than necessary, s i n c e some  smoothly modulated c o n f i g u r a t i o n s have maximum values are w e l l below !4rt.  Thus a c r i t e r i o n  that a l s o takes i n t o  account the c o n f i g u r a t i o n of the monolayer provides efficient  scheme.  This  i s discussed  of z t h a t  i n appendix C.  the most  43  (V)  CALCULATIONAL PROCEDURE  The  p o t e n t i a l energy of t h e monolayer c o n f i g u r a t i o n s a r e  calculated  f o rvarious  at the points  values V  g  of the substrate  o f m i s f i t a n d o r i e n t a t i o n shown i n f i g u r e 4.  S i n c e t h e Newton s t e p  method r e q u i r e s  an a c c u r a t e  configuration  t o s t a r t , a means o f e s t i m a t i n g  configuration  must be f o u n d .  requires  Furthermore, the c a l c u l a t i o n  a p p r o x i m a t i o n s used i n t h e c a l c u l a t i o n .  numerical  The e r r o r bounds u s e d  t h e summation (2.13) and t o c a l c u l a t e t h e A ^ M i n k  L  ( 2 . 1 7 ) must be e x p r e s s e d a s r e l a t i v e e r r o r s  (2.32 and 2.38).  S e t t i n g t h e s e r e l a t i v e e r r o r bounds a l s o r e q u i r e s of t h e f i n a l the  initial  this  e r r o r bounds t o be s e t f o r t h e v a r i o u s  to truncate  corrugation  result.  an e s t i m a t e  A p r o c e d u r e whereby a c c u r a t e e s t i m a t e s o f  c o n f i g u r a t i o n are obtained  p r i o r t o c a l c u l a t i o n i s thus  required. From r e l a x a t i o n s t u d i e s , t h e s t r u c t u r e changes s m o o t h l y w i t h configuration with  misfit.  configurations  As a r e s u l t , t h e u*£ f o r one m  c l o s e l y approximates the u ^  a s l i g h t l y smaller  misfit.  t h e domain w a l l s  triangular and  the i n i t i a l  lattice  the error  m  of the c o n f i g u r a t i o n  For large m i s f i t blend t o g e t h e r and t h e  d i s p l a c e m e n t s o f t h e adatoms a r e s l i g h t . configuration  of t h e monolayer  For such a  s t a t e c a n be t a k e n t o be t h e  of the averaged p o s i t i o n s  bounds  (i.e. u  0 j i n  =0  ),  ( 2 . 3 2 ) a n d ( 2 . 3 8 ) c a n be made more  s t r i n g e n t without s i g n i f i c a n t computational consequences.  This  44  e s t a b l i s h e s a procedure whereby a l a r g e m i s f i t c o n f i g u r a t i o n c a l c u l a t e d , so that the smaller s e q u e n t i a l l y using the  m i s f i t can  the r e s u l t s of one  next c o n f i g u r a t i o n  of smaller  obtained  c a l c u l a t i o n to estimate  misfit.  type of procedure as the bootstrap  be  This  i s the same  procedure used by  A p o s s i b l e danger of t h i s procedure i s that the may  converge to metastable s t a t e s .  f o r c e - f r e e s o l u t i o n s to  minimum energy c o n f i g u r a t i o n .  For  they w i l l  1 3  always r e l a x e d  Regardless of the  initial  the minimum energy c o n f i g u r a t i o n .  solutions  be  will  be  the  configurations,  s t a t e , the  monolayer  concluded to  Because the  c a l c u l a t i o n a l procedure matches the  For  35  relaxation  to the c o n f i g u r a t i o n which was  s t u d i e s , the method i s considered  not  non-rotated  such metastable s t a t e s were sought by calculations .  Novaco .  Such c o n f i g u r a t i o n s  (2.20b), but  is  be  present  r e s u l t s of these r e l a x a t i o n  to be  accurate.  r o t a t e d s t a t e s , however, r e l a x a t i o n r e s u l t s were not  a v a i l a b l e to t e s t the c a l c u l a t i o n .  Instead, t e s t s f o r  metastable s t a t e s were made by s e q u e n t i a l l y s o l v i n g f o r s o l u t i o n s along a v a r i e t y of paths through the f i g u r e 4. and  More than one  i t is therefore  configuration  f o r c e - f r e e s o l u t i o n was  believed  However, with  the  increased  appears to be a sudden f l i p between domain w a l l  systems with h i g h l y sheared domain w a l l s and  very smoothly  modulated systems where commensurate domains are absent. behaviour may  in  never found  that the method f i n d s  of minimum energy.  r o t a t i o n , there  set of p o i n t s  This  be a r e s u l t of s t r e t c h i n g the cub^ic approximation  45  of  (2.11) beyond v a l i d i t y .  which are quite  outside  high  the  Fortunately  these  configurations,  r a n g e o f v a l u e s shown i n f i g u r e 4,  i n e n e r g y and  do  not  influence  the  choice  They c a n  The  t o t h i s s t u d y have m i s f i t s  relevant  o r i e n t a t i o n s w i t h i n the the  range of  f i g u r e 4.  p r o c e d u r e f o r c a l c u l a t i n g the For  the  calculated  in this  t o an  0.005 K.  s o u r c e s o f e r r o r t o be total  of the  0.005 K.  By  four  the  e r r o r , the  truncating 0.003 K, substrate  The  the  the  bounds a r e  Fourier  error  expansion  involved  corrugation  (C.8)  error  (2.26) a s s o c i a t e d  steps  i s l e s s t h a n 0.001  It  i s recognized  that separating final  that  them o u t  uncertainty  i n the  sample c a l c u l a t i o n s w i t h that portioning  out  s e t so  For  that  the  four  be  the  t o be  accurate  e r r o r due  i n (2.13) i s l e s s  limiting  were  summation over  to  than  influence  i s l e s s t h a n 0.0005 K,  with  are  set to  i n c a l c u l a t i n g the  number o f  and  of the  Newton  K. these e r r o r s are  i n t e r r e l a t e d , and  as  underestimate  i n d i c a t e d can  calculation.  e r r o r s as  the  However, p e r f o r m i n g  a v a r i e t y o f e r r o r bounds  indicates  shown p r o v i d e s a v a l u e o f  e n e r g y a c c u r a t e t o 0.005 K w i t h effort.  energies  c a l c u l a t i o n , the  n e a r e s t n e i g h b o u r adatoms i n (2.11) i s r e q u i r e d t o 0.0005 K.  region  accurate.  Because t h e r e  e s t i m a t e d u n c e r t a i n t i e s was out  and  this  is  c h a p t e r , the  c o n t r o l l e d w i t h i n the  portioning  disregarded.  Within  configurations  r e s u l t s presented a c c u r a c y of  be  of  equilibrium configuration. configurations  therefore  are  a minimum o f  the  computational  a more a c c u r a t e c a l c u l a t i o n o f t h e  energy,  the  the  46  i  e r r o r bounds can be u n i f o r m l y  scaled  smaller.  47  (VI)  RESULTS AND  The monolayer's misfit.  DISCUSSION  energy i s found to vary p r i m a r i l y with the  In comparison, the v a r i a t i o n  o r i e n t a t i o n s of the monolayer  i n energy f o r d i f f e r e n t  i s q u i t e minor.  The lowest  energy c o n f i g u r a t i o n i s found to 'be non-rotated f o r s m a l l m i s f i t s , and r o t a t e d when the m i s f i t exceeds a value which v a r i e s with the s u b s t r a t e c o r r u g a t i o n Vg.  However, when the  energy of the r o t a t e d c o n f i g u r a t i o n i s compared to the energy of the non-rotated c o n f i g u r a t i o n of same m i s f i t , the d i f f e r e n c e i s extremely s l i g h t .  Thus, the change with m i s f i t  energy of the monolayer  i n the  can be determined by c o n s i d e r i n g  only  the non-rotated c o n f i g u r a t i o n s . P l o t s of energy versus m i s f i t  f o r the non-rotated  c o n f i g u r a t i o n s are shown i n f i g u r e s 5a-f .  The s o l i d  lines  i n d i c a t e the p o t e n t i a l energy per adatom i n the monolayer. c a l c u l a t i o n s have been performed ranging up to 11.0 been shown.  for substrate  The  corrugations  K and the most r e p r e s e n t a t i v e p l o t s have  Since r o t a t i n g the monolayer  does not cause  any  s i g n i f i c a n t decrease i n the p o t e n t i a l energy, the g e n e r a l r e s u l t remains that the monolayer  forms  incommensurate  i f the s u b s t r a t e c o r r u g a t i o n i s l e s s than 9.8 differs  from the value of 11.0  because a d i f f e r e n t the  K . x2  islands  This value  K quoted by Gooding et  al.  1 3  form of the s u b s t r a t e s c r e e n i n g i s used i n  calculation . 1 2  As noted i n the i n t r o d u c t i o n  (1.1), the e q u i l i b r i u m  48  «  c o n f i g u r a t i o n of the monolayer depends on the chemical potential  l-i of the vapour above i t s s u r f a c e .  exponent & of (1.1) i s found 0.3 .  The chemical p o t e n t i a l  9  the chemical  The c r i t i c a l  e x p e r i m e n t a l l y to be  approximately  P of the vapour, which i s a l s o  p o t e n t i a l of the monolayer, can be determined  by  t a k i n g d e r i v a t i v e s of the t o t a l energy of the monolayer with r e s p e c t to the number of adatoms i n the monolayer. e q u i v a l e n t to seeking the minimum of (1.5).  This i s  To f i t these  c a l c u l a t e d values of H t o (1.1), because there i s no data near the commensurate phase, M  c  and  P  c  i s not known a c c u r a t e l y and both B  are t r e a t e d as parameters.  U n f o r t u n a t e l y , given the l a t i t u d e of values f o r M  c  and f o r  d e c i d i n g which data p o i n t s should be c o n s i d e r e d , l i n e a r l o g f i t s c o u l d be made f o r a wide range of B v a l u e s .  B was  found  to  l i e anywhere between 0.28 and 0.33.  of  B i s i n s e n s i t i v e to the value chosen f o r the s u b s t r a t e  corrugation.  T h i s remains true even when p o i n t s are i n c l u d e d  that correspond Vg  Furthermore, the value  to n o n p h y s i c a l c o n f i g u r a t i o n s c a l c u l a t e d f o r  < 9.8 K which would r e q u i r e negative spreading pressures t o  exist. The  o r i e n t a t i o n a l e p i t a x y has been analysed  f o r three  d i f f e r e n t s u b s t r a t e c o r r u g a t i o n s which r e p r e s e n t the general behaviour for  of the monolayer.  S e l e c t i v e data has been c a l c u l a t e d  other Vg t o c o n f i r m the trends  illustrated.  above, the energy of the monolayer i s r e l a t i v e l y its  orientation.  In order to separate  As noted i n s e n s i t i v e to  out the o r i e n t a t i o n a l  49  behaviour from the misfit-dependent  behaviour,  the r e s u l t s  are  d i s p l a y e d as energy d i f f e r e n c e s between the r o t a t e d c o n f i g u r a t i o n and  the non-rotated  configuration.  For  the  s p e c i f i e d s u b s t r a t e c o r r u g a t i o n s , the energies are c a l c u l a t e d for a l l the p o i n t s of m i s f i t and f i g u r e 4.  The  o r i e n t a t i o n shown i n  energy d i f f e r e n c e s are mapped out as contours  a s u r f a c e of p o s s i b l e m i s f i t s and  o r i e n t a t i o n s ( f i g u r e s 6a-c).  When the s u b s t r a t e c o r r u g a t i o n has a value Vg = 5.0 minimum energy path matches q u a n t i t i v e l y the d e s c r i b e d by S h i b a . 1 1  are  K the  behaviour  T h i s i s expected s i n c e harmonic t h e o r i e s  found to be adequate f o r such a c o r r u g a t i o n .  However,  1 2  for l a r g e r c o r r u g a t i o n s  on  the behaviour changes and  the monolayer  r o t a t e s at m i s f i t s much s m a l l e r than Shiba's theory would predict.  Furthermore, the l i n e shape of the minimum energy  c o n f i g u r a t i o n r o t a t i o n verses m i s f i t curve  w i l l not match the  Shiba r e s u l t even i f a r b i t r a r y s c a l i n g of Shiba's parameters i s used.  To understand t h i s behaviour, the s t r u c t u r e of  monolayer must be  the  considered.  For c o n f i g u r a t i o n s with s u b s t r a t e c o r r u g a t i o n s g r e a t e r 10.0  K the r o t a t e d c o n f i g u r a t i o n s at small m i s f i t s t h a t would  correspond  to the higher energy contours  converge.  S t r u c t u r e s t h a t would correspond  walls  than  (figure  4) could not be produced.  a l s o found by Gooding e t . a l .  1  3  of f i g u r e 6c d i d not to heavy domain  T h i s behaviour  was  f o r s t r i p e d heavy w a l l s at high  s u b s t r a t e c o r r u g a t i o n s s i n c e the r e l a x a t i o n c a l c u l a t i o n would not s t a b i l i z e .  The  energies  of these  s t r u c t u r e s are expected  50  to be s i g n i f i c a n t l y higher configurations  i n energy than the s t a b l e  that are l e s s r o t a t e d .  They w i l l t h e r e f o r e not  have an i n f l u e n c e on the e q u i l i b r i u m c o n f i g u r a t i o n of the monolayer. A general  a n a l y s i s of the s t r u c t u r e of non-rotated  monolayers having small m i s f i t s r e v e a l s that the d e n s i t y of the adatoms i s constant along This  a t the center  of the domains, i t i n c r e a s e s  the domain w a l l s , and reaches a maximum a t the v e r t i c e s . i s i n agreement with the concept of s o l i t o n  modulations arranged i n a honeycomb network. d e n s i t y a t the vertex  i s due to the focused  density  The peak i n compression of the  domain w a l l s which i n t e r s e c t a t the v e r t e x .  The compression of  the adatoms a t the v e r t i c e s becomes stronger  f o r higher  substrate  corrugations  at the v e r t e x  being  r e s u l t i n g i n the spacings of the adatoms  l e s s than 4.02 A which i s the spacing  favoured by the f l o a t i n g monolayer. Vg  = 8.0 K, the spacing  For example, when  a t the vertex  the monolayer i s overcompressed. to r o t a t e , the domain w a l l s  i s found to be 3.99 A and  I f the monolayer i s allowed  form a staggered i n t e r s e c t i o n  the compression due t o the domain w a l l s  is diffused.  and  The  d e n s i t y of the adatoms a t the v e r t i c e s f o r r o t a t e d configurations  i s not as high as that f o r the non-rotated  c o n f i g u r a t i o n and spacings c l o s e r to 4.02 A are found. decreases the i n t e r a c t i o n energies  of the adatoms a t the  v e r t i c e s , and s i n c e the change i n the s u b s t r a t e d i f f u s i n g the vertex  This  i s n e g l i b l e , the r o t a t e d  energy due t o  configuration  51  w i l l have v e r t i c e s of lower energy.  When the domains are small  the monolayer w i l l have many v e r t i c e s and the r o t a t e d c o n f i g u r a t i o n s w i l l be e n e r g e t i c a l l y more For  favourable.  l a r g e r domains the p r e f e r r e d c o n f i g u r a t i o n i s  non-rotated d e s p i t e the overcompression of the v e r t i c e s . l  This  i s due to the f a c t that the number of adatoms a t the v e r t i c e s becomes s i g n i f i c a n t l y l e s s than the number of adatoms on the domain w a l l s .  Since  the domain w a l l s  for rotated  c o n f i g u r a t i o n s must c o n t a i n a shear as w e l l as a compression, the domain w a l l adatoms would favour configuration.  At s m a l l e r  a non-rotated  m i s f i t s , the adatoms on the domain  walls dominate the adatoms a t the v e r t i c e s and the monolayer becomes  non-rotated.  This describes  the o r i e n t a t i o n a l behaviour found i n  f i g u r e s 6b,c. However, and  not an a r t i f a c t  t h i s behaviour must be v e r i f i e d as r e a l  of the c o n s t r a i n t s of the c a l c u l a t i o n . The  c o n f i g u r a t i o n s examined have i n v e r s i o n symmetry about the domain c e n t e r s .  By breaking  v e r t i c e s can be d i f f u s e d . by Duesbery and J o o s  3 8  this  i n v e r s i o n symmetry, the  T h i s p o s s i b i l i t y has been examined  who concluded that breaking the  i n v e r s i o n symmetry does not change the energy s i g n i f i c a n t l y . The o r i e n t a t i o n a l behaviour observed i n f i g u r e 6 considered  i s thus  t o be c o r r e c t even with symmetry breaking.  The  consequences to the o r i e n t a t i o n a l behaviour of f o r c i n g the monolayer t o be p e r i o d i c are d i s c u s s e d modes.  i n the s e c t i o n on phonon  52  As  noted  previously,  the o r i e n t a t i o n a l behaviour  c o m p l e t e l y agree w i t h the c a l c u l a t i o n s discrepancy present  harmonic t h e o r i e s the sharp  Taylor series  impact  expansion  will  add  their contribution  Thus, f o r a  system, the o r i e n t a t i o n a l behaviour  p r e d i c t e d by  t h e t h e o r y due  t o be  to i n c l u d i n g  adatom the  cubic  but as  noted  be s l i g h t c o m p a r e d t o  the c u b i c term.  i s expected  real; additional  more T a y l o r s e r i e s  cause s l i g h t changes i n the m i s f i t v a l u e  In  terms i n the  t o the a c c u r a c y ,  will  terms  (2.11).  however, i n c l u d i n g Additional  not  This  p o t e n t i a l minimum o f t h e  f i t considerably.  of i n c l u d i n g  calculation  .  t o the p a i r p o t e n t i a l  i s not w e l l modelled,  terms improves the  previously,  1 1  i s d i r e c t l y a t t r i b u t a b l e to the anharmonic  i n the approximation  interaction  of S h i b a  does  the  periodic this  improvements i n terms w i l l  f o r the  onset  only  of  rotat ion. G o r d o n and vertex  Lancon  in relaxation  spreading  have c o n s i d e r e d calculations.  the d i f f u s i o n of  They c o n c l u d e  t h e v e r t e x d o e s n o t c a u s e any  the energy. difference calculation  However, a s s e e n i n f i g u r e s i s v e r y s l i g h t and t h e y may  not  within  change i n  energy  the a c c u r a c y  of  match the r e s u l t s of S h i b a  1 1  .  i s observed  high temperatures  p h a s e s t o be  to  S i n c e t h e . m e a s u r e m e n t s were ( > 30 K  ), t h e  thermal  m o t i o n o f t h e a d a t o m s s h o u l d d i f f u s e t h e v e r t i c e s and non-rotated  their  have d e t e c t e d t h i s e f f e c t .  E x p e r i m e n t a l l y the o r i e n t a t i o n a l behaviour  taken at f a i r l y  that  observable 6 the  the  favoured  where t h e  zero  a l l o w the  temperature  53  r e s u l t s shown i n f i g u r e 6 would i n d i c a t e otherwise. not  This i s  unreasonable s i n c e the domain walls are p r e d i c t e d to  broaden with t e m p e r a t u r e ' . 2 2  3 9  Furthermore, the thermal  broadening of the v e r t i c e s i s suggested by the molecular dynamics study of Schobinger and Abraham . 18  They f i n d that a t  f i n i t e temperatures, hexagonal  phases are e n e r g e t i c a l l y more  favourable  for configurations  than s t r i p e d phases  temperature are p r e d i c t e d  to be s t r i p e d  1 3  .  that a t zero  54  3.  DYNAMICAL ASPECTS  (I)  BACKGROUND  The  p r e v i o u s chapter  has t r e a t e d the system c l a s s i c a l l y and  the c o n f i g u r a t i o n s of minimum energy assumed t h a t the krypton atoms are s t a t i o n a r y .  T h i s i s an incomplete  d e s c r i p t i o n of the  system s i n c e the krypton and carbon atoms v i b r a t e about t h e i r e q u i l i b r i u m p o s i t i o n s (even a t zero temperature) and a c a l c u l a t i o n of the free energy must a l s o c o n s i d e r t h i s motion. For the commensurate c o n f i g u r a t i o n of the monolayer de Wette et al. *-* have performed e x t e n s i v e c a l c u l a t i o n s , with the 4  harmonic approximation,  f o r the system's dynamics.  They  that the in-plane motion of the adatoms i s decoupled  found  from the  out-of-plane motion and can be c a l c u l a t e d u s i n g the r i g i d s u b s t r a t e approximation  (2.2).  The out-of-plane  long  wavelength modes of the monolayer are h e a v i l y coupled  to the  motion of the s u b s t r a t e , and, f o r the out-of-plane motion, the rigid The  s u b s t r a t e approximation  appears to be i n a p p r o p r i a t e .  c o u p l i n g between the out-of-plane motion of the  monolayer and s u b s t r a t e , however, has not been observed r a r e gas physisorbed s y s t e m . 4 1  r e s u l t s can best be e x p l a i n e d  i n any  Instead, the experimental (except f o r Xe on the h i g h l y  corrugated KCl s u b s t r a t e ) by t r e a t i n g the adatoms as 4 2  independent E i n s t e i n o s c i l l a t o r s a c t i n g under the h o l d i n g p o t e n t i a l V ( z ) ( c f 2.2) of a r i g i d a  substrate.  For g r a p h i t e  55  this result  i s reasonable  s i n c e the s u b s t r a t e  comparison to the m o n o l a y e r ' . 4 0  to  be t o t a l l y r i g i d ,  4 3  i s quite r i g i d  I f the s u b s t r a t e  the h o l d i n g p o t e n t i a l V ( z ) a  s u b s t r a t e so dominates the out-of-plane s u b s t r a t e c o r r u g a t i o n and adatoms are of l i t t l e  In  i s assumed  of  the  behaviour t h a t the  i n t e r a c t i o n s of the  surrounding  influence.  For A g ( l l l ) H a l l et a l .  4 4  have performed c a l c u l a t i o n s f o r an  e l a s t i c s u b s t r a t e , which i n d i c a t e t h a t the monolayer's out-of-plane  motion can become h y b r i d i z e d with the  R a y l e i g h modes.  substrate  Furthermore, h y b r i d i z a t i o n with the  continuum  of bulk modes of the s u b s t r a t e broadens the f r e q u e n c i e s long wavelength modes.  ( H a l l et a l .  r a d i a t i v e damping by the s u b s t r a t e . ) dynamics of the s u b s t r a t e and  r e f e r to t h i s  4 4  of  the  as  However, when the  monolayer i s compared with  that  of the bare s u b s t r a t e , the d i f f e r e n c e i s a motion of the monolayer, whose major frequency  out of the plane  is  d i s p e r s i o n l e s s except near the c r o s s i n g p o i n t of the mode.  Therefore  the out-of-plane  Rayleigh  motion of the adatoms,  r e l a t i v e to the s u b s t r a t e , would appear to be E i n s t e i n l i k e . In any  case,  experimental  evidence  i n d i c a t e s t h a t the motion  s p e c i f i c to the monolayer appears to be  l e s s s e n s i t i v e to the-  s u b s t r a t e ' s dynamics than de Wette et a l . discrepancy  i n de Wette's c a l c u l a t i o n may  that only 13 sheets s u b s t r a t e , and  4 0  suggest.  be due  The  to the  fact  of g r a p h i t e were used to model the  the monolayers may  on dynamics of the t o t a l system.  thus have an undue Apart  Influence  from the f a c t t h a t a  56  finite  s l a b c a l c u l a t i o n can not d e s c r i b e r a d i a t i v e damping, i t  i s expected t h a t a c a l c u l a t i o n  i n v o l v i n g more sheets of  g r a p h i t e w i l l produce f r e q u e n c i e s which c l u s t e r c l o s e r to the dispersionless Einstein  frequency.  In summary, i t appears reasonable motion from the out-of-plane  to separate  the in-plane  motion of the monolayer.  For  in-plane motion the s u b s t r a t e can be t r e a t e d as r i g i d . out-of-plane  motion, most of the dynamics w i l l  with the s u b s t r a t e .  For  be a s s o c i a t e d  Out-of-plane motion s p e c i f i c to the  monolayer i s expected to be " E i n s t e i n l i k e " ,  although the  R a y l e i g h mode and the long wavelength bulk modes of the s u b s t r a t e w i l l a l t e r the behaviour s l i g h t l y .  Because the  i n t e n t of t h i s work i s to determine the e q u i l i b r i u m m i s f i t and o r i e n t a t i o n of the monolayer, any q u a n t i t y that does not vary with the monolayer's c o n f i g u r a t i o n w i l l be i r r e l e v a n t . The plane  s h o r t wavelength motion of the monolayer out of the i s d i s p e r s i o n l e s s , even i f the long wavelength motion may  not b e  4 u  '  4 i / 4 4  .  This i m p l i e s t h a t the adatom i n t e r a c t i o n s are  i n s i g n i f i c a n t and can be neglected incommensurate monolayers, having  f o r these  modes.  For the  d e n s i t y f l u c t u a t i o n s that do  not o v e r l y compress the adatoms, t h i s c o n c l u s i o n w i l l be valid.  Furthermore, S t e e l e ' s  corrugation substrate  3 3  estimates  of the s u b s t r a t e  i n d i c a t e t h a t the i n f l u e n c e of c o r r u g a t i o n on the  interaction  i s t o t a l l y overshadowed by the magnitude  of the h o l d i n g p o t e n t i a l V ( z ) and, f o r the out-of-plane a  behaviour,  the s u b s t r a t e  i n t e r a c t i o n can be regarded  as  57  uniform.  Thus, f o r motion p e r p e n d i c u l a r  adatoms move independently their  of each other and  l o c a t i o n on the s u b s t r a t e .  d i s t i n c t i o n between out-of-plane commensurate and  without  This removes any  the  regard  to  possible  motion of adatoms i n the  incommensurate monolayers, and  c o u p l i n g between the monolayer and be,  to the s u b s t r a t e ,  the mode  the s u b s t r a t e  i s expected to  on average, the same r e g a r d l e s s of the c o n f i g u r a t i o n of the  monolayer.  For very dense monolayers, the mass of  krypton  atoms on the s u r f a c e w i l l s t a r t to i n f l u e n c e the motion even i f the adatom i n t e r a c t i o n s remain n e g l i g i b l e .  However, the i n t e n t  of t h i s work i s to examine incommensurate phases near commensurate t r a n s i t i o n and  f o r such systems, the  the  out-of-plane  dynamics are not expected to i n f l u e n c e the e q u i l i b r i u m configuration. The  average h o l d i n g p o t e n t i a l of the s u b s t r a t e , V ( z )  (2.2), which dominates the out-of-plane in-plane behaviour.  The  motion does not  between the adatoms.  Since the p o s i t i o n s of the adatoms change  f r e e energy c o n t r i b u t i o n w i l l  i n order  plane  the i n t e r a c t i o n s  with d i f f e r e n t c o n f i g u r a t i o n s of the monolayer, the  Thus, the  affect  behaviour of adatoms w i t h i n the  i s determined by the s u b s t r a t e c o r r u g a t i o n and  The  in  a  in-plane  be c o n f i g u r a t i o n dependant.  in-plane dynamics of the monolayer must be determined to f i n d the e q u i l i b r i u m c o n f i g u r a t i o n .  in-plane modes are of c o n s i d e r a b l e  dynamics are s i g n i f i c a n t l y d i f f e r e n t incommensurate monolayers.  i n t e r e s t , because  f o r the commensurate  the  and  For the commensurate monolayer,  58  each adatom  i s at an a d s o r p t i o n s i t e .  T h i s l o c k s the monolayer  onto the s u b s t r a t e and does not a l l o w v i b r a t i o n a l  energies  below a given band gap v a l u e .  phase,  The incommensurate  however, c o n t a i n s domain w a l l s which have a great d e a l of m o b i l i t y and the monolayer w i l l  have low energy modes.  as mentioned i n the i n t r o d u c t i o n , V i l l a i n  1 4  has p r e d i c t e d t h a t  a domain w a l l b r e a t h i n g mode could become completely consequences f o r the commensurate  Indeed,  - incommensurate  soft  with  phase  transition . 2 6  Therefore, assumed  f o r the c a l c u l a t i o n of the dynamics, i t i s  t h a t the motion of the monolayer can be s p l i t  in-plane and out-of-plane  components.  into  The out-of-plane  motion  i s not expected to have any i n f l u e n c e on the e q u i l i b r i u m m i s f i t and o r i e n t a t i o n of the monolayer and w i l l not be However, the in-plane motion w i l l  considered.  i n f l u e n c e the e q u i l i b r i u m  c o n f i g u r a t i o n and must be c o n s i d e r e d .  In the f o l l o w i n g  s e c t i o n s , any f u r t h e r mention of dynamics or motion w i l l to  t h a t w i t h i n the plane  of the monolayer.  refer  59  (II)  HARMONIC APPROXIMATION  Krypton atoms a r e q u i t e massive and a t low temperatures t h e i r v i b r a t i o n a l amplitudes r e l a t i v e to t h e i r neighbours w i l l  be s m a l l .  nearest  C a l c u l a t i o n s of the dynamics of  f l o a t i n g monolayers i n d i c a t e that anharmonic e f f e c t s  provide  only s m a l l p e r t u r b a t i o n s to the general r e s u l t s obtained harmonic t r e a t m e n t s . 4 5  from  Thus, a t low temperatures, the general  behaviour of the adatoms i n response to each other can be obtained For  i f the motion i s assumed to be harmonic.  incommensurate monolayers, the adatoms are a l s o  i n f l u e n c e d by the s u b s t r a t e .  The domain w a l l motion causes  s i g n i f i c a n t movement of the adatoms r e l a t i v e to the s u b s t r a t e and  the i n f l u e n c e of the s u b s t r a t e  only f o r small displacements move f r e e l y  3 4  ,  i n t e r a c t i o n w i l l be harmonic  of the w a l l s .  Since the w a l l s  the energy a s s o c i a t e d with t h e i r motion w i l l be  small and the amplitude of t h e i r v i b r a t i o n s w i l l be l a r g e even at  zero temperature.  Thus, the motion of adatoms cannot be  c a l c u l a t e d w i t h i n the harmonic approximation.  However, t h i s  does not imply t h a t the mode f r e q u e n c i e s cannot be c a l c u l a t e d with the harmonic  approximation.  From a renormalized  d e s c r i p t i o n of the incommensurate  monolayer, the p r o p e r t i e s of the monolayer can be determined e n t i r e l y from the shape of i t s domain w a l l s t r u c t u r e s . l o c a t i o n of these  s t r u c t u r e s on the s u b s t r a t e  s i n c e the p i n n i n g  i s so s m a l l .  The  i s not r e l e v a n t  In the renormalized  picture,  60  the domain w a l l s w i l l have v i b r a t i o n a l modes a s s o c i a t e d t h e i r motion.  Since the behaviour of the domain w a l l s i s  i n s e n s i t i v e to t h e i r effects will  l o c a t i o n on the s u b s t r a t e , anharmonic  r e s u l t only from the w a l l s c o l l i d i n g .  f o l l o w i n g c a l c u l a t i o n assumes t h a t these occur.  with  The  c o l l i s i o n s do not  The v i b r a t i o n a l modes can then be determined by  c o n s i d e r i n g the behaviour of the system under p e r t u r b a t i o n s of the domain w a l l s .  infinitesimal  The consequences of t h i s  assumption w i l l be d i s c u s s e d with the r e s u l t s . For harmonic motion of the adatoms, the normal modes of the incommensurate monolayer can be c a l c u l a t e d from the dynamical matrix.  The f r e q u e n c i e s  of the dynamical matrix. domain w a l l s w i l l eigenvectors. be  of the modes w i l l  be the eigenvalues  The low frequency  motion of the  not, however, be given d i r e c t l y by the  Instead,  the form of the domain w a l l motion must  i n t e r p r e t e d from the movement of the adatoms on the domain  wa 11 s . The  p o s i t i o n s of the v i b r a t i n g adatoms are d e f i n e d to be  r ( R , t ) = r(R) + v(R,t)  where r(R*) i s the s t a t i c v(R*,t) provides  f o r c e f r e e p o s i t i o n of (2.6) and  the time dependent behaviour.  assumption of harmonic behaviour, adatom w i l l  (3.1)  With the  the f o r c e f e l t by a given  be l i n e a r l y dependent on the motion of the adatoms  i n the monolayer.  This force w i l l  cause the adatom t o i  61  a c c e l e r a t e ; the r e s u l t i n g motion w i l l obey  M-  3  3t  -4  v(R,t)  2 2  where M  a  -4  D(r(R). r (R» ) ) •v(R,t)  (3.2)  i s the mass of the adatom, and the summation over R'  i n c l u d e s the p o i n t R*.  The p e r i o d i c nature of r(R:) allows the  matrix D i n (3.2) to be expressed as  _ -4  B(r(R) r(R»))  £5jt (h) e *nt  =  /  -4  i q  (3.3)  , R  m  %m  where  h* = R*-R*' and q%  i s the r e c i p r o c a l l a t t i c e v e c t o r as  m  d e f i n e d f o r (2.13). With the d e f i n i t i o n s of (2.13) and (2.17), and l i m i t i n g the expansion of the adatom p a i r in  i n t e r a c t i o n to o n l y c u b i c terms as  (2.11), D£ (h*) can be c a l c u l a t e d  from  m  ^2,m ) = (n>  SU)S(m) [ Ws&(h) - 5 ( h ) ^ W^(h') ] h'  s(h)  -*  5(h)  [v Z g 3 q  r  V  jk  jk  -4 -iqi a o  I I (1-e ^m  A^.  / m  _  -4  n  k  ]  +  +  -h' -* -+ ) u -WV^(h') 5tm  62  -T  (l-  -7  3Stm  lc e  )  n  U s i m  -WW(h)  where S(2,) and S(m) are i n t e g e r d e l t a f u n c t i o n s , d e l t a f u n c t i o n , and £(Fi)  vector  (3.4)  S(h*) i s a  i s the adatom p a i r i n t e r a c t i o n .  For harmonic systems, the e q u i l i b r i u m p o s i t i o n s of the adatoms w i l l be the force configuration.  free p o s i t i o n s r(R*) of the s t a t i c  From the p e r i o d i c i t y of the s t a t i c  c o n f i g u r a t i o n , v(R*,t) must have the form  v(R,t)  I  =  k  I %  v^ (k) m  e^+W-K  where it can be r e s t r i c t e d to the f i r s t s u p e r l a t t i c e and q ^ For  equation  in  (3.5) must  M  o'(ic) v  a  This  L M  "  i"<iot  (3  .  5)  m  m  B r i l l o u i n zone of the  i s d e f i n e d above.  (3.2) to be s a t i s f i e d , the c o e f f i c i e n t s v ^ (?) m  solve  (k) =  I  vj^k)  •  I  B _j, L  M  _ (I?) m  eI(Wta»' ' h  (3.6)  i d e n t i f i e s the dynamical matrix of the incommensurate  monolayer  g  LM,5lm( ) k  =  ^ . ^ ^ ( h ) h  e  1  (3.7)  63  The  summation over h* i n v o l v e s nearest neighbour  surrounding adatoms and  s h e l l s of the  i n c l u d e s the case when F?=0.  adatom i n t e r a c t i o n s are s h o r t range, the terms i n decrease  r a p i d l y away from the o r i g i n and  t r u n c a t e d a f t e r the f i r s t  few s h e l l s .  Since  the  (3.4)  the summation can  be  This i s e q u i v a l e n t to  assuming that D _<^ _ ( h*) i s zero f o r values of I h* I past a L  M  m  given cut o f f v a l u e . The and  p o s s i b l e s o l u t i o n s to equation  (3.6) can be  calculated  provide the in-plane normal modes (CJ (J<), v ^ ( s k ) ) of the 4  s  system.  The  m  /  index s i s used to i d e n t i f y the mode.  system i s two  dimensional,  Since the  the number of modes w i l l be  the number of adatoms per domain.  twice  With these normal mode  f r e q u e n c i e s , the in-plane component of the average Helmholtz f r e e energy per adatom i s  *c =  where  U  c  H  +  F  c  ^' s(k> s, k  +  J  = Nf  c  NJ  I_m(l s, k  - -^s(i0/2)  f o r e q u a t i o n - ( 1 . 6 ) , and  e  U  c  i s the energy per  adatom of the s t a t i c c o n f i g u r a t i o n (equations 2.15 Equation  (3.8)  has s p l i t  ( 3 > 8 )  plus 2.19).  the f r e e energy i n t o a s t a t i c  energy c o n t r i b u t i o n , a zero p o i n t energy, and a temperature dependent c o n t r i b u t i o n . calculated  The  s t a t i c energy, U  i n the p r e v i o u s s e c t i o n .  zero p o i n t energy, E  Q  The  c  , has  been  c a l c u l a t i o n f o r the  , i s given i n the f o l l o w i n g s e c t i o n .  c a l c u l a t i o n f o r the temperature dependent c o n t r i b u t i o n , E  T  The ,  64  are presented phonon modes.  later  In chapter  4 which c o n s i d e r s the low  energy  65  ZERO POINT ENERGY  (III)  From (3.8) the zero p o i n t energy for the eigenvalues of the dynamical  can be obtained by matrix and  solving  then summing  for a l l values of wavevector i< i n the B r i l l o u i n zone.  For  incommensurate monolayers, the number of adatoms per domain i n c r e a s e s d r a m a t i c a l l y as the d e n s i t y of the c o n f i g u r a t i o n approaches the commensurate l i m i t .  As an example, the  non-rotated c o n f i g u r a t i o n with a m i s f i t of 2.22% adatoms per domain and thus 1350  has  675  i n - p l a n e normal modes.  S o l v i n g f o r the eigenvalues of such a l a r g e system i s c o m p u t a t i o n a l l y unwieldy.  Furthermore,  calculating  the  eigenvalues f o r s e v e r a l vt p o i n t s compounds the problem. T h e r e f o r e , a more p r a c t i c a l method of c a l c u l a t i n g E  is  Q  required. I f the eigenvalues of (3.6) are d e f i n e d to be x (k*), s  frequency  "  S  ( J 4 )  -*  Xg(k)  The  of the mode can be determined  =  2  from  ~*  (3.9)  Og(IC)  zero p o i n t energy can then be expressed  as  (3.10)  d\  where x  m  and x  Q  the  are the upper and  lower  bounds of >^ (k) s  66  r e s p e c t i v e l y , and g ( x ) i s t h e d e n s i t y o f s t a t e s  9  ( x )  = N I S(x-x (k)) s, k  (3.11)  s  The d e n s i t y o f s t a t e s , b e i n g a f u n c t i o n o f x , c a n be expanded  a s a sum o f o r t h o g o n a l  two d i m e n s i o n a l maximum x . Legendre  polynomials  where  Since  (  M  x  =  a  XJKJ  a  ^ +  5  XQ  n  (  J  , and  (  x<3 = x  m  (3.11), a  - x  n  where  o  =  1 (2n+l) y 2( x ( k ) - x N X ( j ZJ_J n* X s, k s  JTMa  J  a  n  c (x ,x ) n  m  0  '  1  2  )  must have t h e f o r m  t h e f o r m (3.12) f o r g ( x ) i n (3.10)  E  3  0  a  )  }  D  Using  using  a  ?  ( 3 . 1 2 ) must s a t i s f y  n  form a r e w e l l d e s c r i b e d  2(x-x )  A  =  of t h i s  and  Q  4 6  £ 9  I n the case of  s y s t e m s , g ( x ) i s n o n - z e r o a t i t s minimum x  Functions  m  polynomials.  (3.13)  produces  (3.14)  67  r c (x ,x ) n  m  0  = J x  ,, x  2(X"X ) a  P (  X i  ) dx  a  n  x  (3.15)  0  Thus E  Q  can be c a l c u l a t e d  moments a  once the orthogonal  of the dynamical  n  polynomial  matrix are determined.  While  many  moments may be r e q u i r e d t o a c c u r a t e l y d e s c r i b e the d e n s i t y of s t a t e s g ( x ) ( 3 . 1 2 ) , c a l c u l a t i o n s of smooth averages require r e l a t i v e l y The  few moments . 46  method of Wheeler and B l u m s t e i n  over s i n ( 3 . 1 3 ) t o be expressed polynomial m a t r i x . in  (3.14)  4 7  a l l o w s the summation  as the t r a c e of an orthogonal  Because of the form of the dynamical  matrix  ( 3 . 7 ) , the orthogonal polynomial matrix must have the form  LM, 5Cm  The  p(n) (h) e i ( k L-8.,M- m  recurrence r e l a t i o n  •4  <3!lm>  h  (3.16)  t o p r o v i d e these orthogonal  polynomial  matrices i s  ( n  +  l >  ^ ( S )  (  2  n  +  1  \ [x ( L-*,M-m<h > 5tm,h'  )  5  d  ,  - x  a  ! _i, _ (h'>) L  P< >(h-h') n  St, m  -  n P ^ L  M  (h)  M  m  e i ^ m ^ ' l  -i  (3.17)  68  where  = (0) -» = PrV (h) = I M  L / M  -» (h)  recurrence p r o c e s s . i d e n t i t y matrix n u l l matrix;  a  n  d  P  ( I I M(h*)  =(-1) -* = -» L,M < > = ° L , M ( ) n  i s  h  s t a r t the  d e f i n e d to be the 2x2  i f L, M and h are zero, otherwise  i t i s the 2x2  0 ^ (h*) i s d e f i n e d to be the n u l l matrix L  M  for a l l  L, M and h*. ) S ince  I Y i ( k + qfc )-h e  m  =  S(n  »)  ( 3  .  1 8 )  k, Sim  performing in  the summations  (3.16) w i l l r e s u l t  a  n  in a  of (3.13) on the t r a c e of the matrix n  being given by  = ^ T r [ P^ (h=0) ] o  (3.19)  69  (IV)  CALCULATION  The method f o r c a l c u l a t i n g the zero p o i n t energy, above, allows s e v e r a l approximations the e x e c u t i o n speed  to be made which improve  of the c a l c u l a t i o n without degrading the  accuracy of the r e s u l t s .  Firstly,  be t r u n c a t e d a f t e r the f i r s t  the summation i n (3.14) can  few terms because i t converges  r a p i d l y g i v e n an a p p r o p i a t e c h o i c e of X as n i n c r e a s e s , the weights an i n c r e a s i n g significant of e r r o r  error  m  and x .  0  i n the moments a  m  Furthermore,  i n magnitude and  can be permitted  n  This increasing  a l l o w s approximations  recurrence process The  n  Q  n  Q  c ( x , x ) decrease  l o s s of accuracy i n E .  in a  presented  without  tolerance  to be made to the  (3.17).  r e c u r r e n c e process  i n (3.17) i n v o l v e s a summation over hi  and a summation over £ and m.  Advantage can be taken of the  -4  form of D^ (h) i n (3.4) which allows the summations to be m  separated.  I f each of these summations can be l i m i t e d , the  amount of computation significantly  r e q u i r e d by the r e c u r r e n c e process i s  reduced.  From (3.4), the summation over h*' i n (3.17) w i l l depend on the magnitude of the adatom p a i r interaction the f i r s t  interaction.  Because the p a i r  i s s h o r t range the summation can be t r u n c a t e d a f t e r  few s h e l l s .  Furthermore,  the r e s u l t i n g values of the  P ^ ( F i ) w i l l d i m i n i s h r a p i d l y with i Fx I and, as i n (3.7), the L  M  summation of (3.16) can be t r u n c a t e d . the assumption  t h a t the values of P  T h i s i s e q u i v a l e n t to ^(h ) are zero f o r values 4  L  70  of  I r i | l a r g e r than a g i v e n c u t o f f .  With the c u t o f f value s e t ,  the accuracy of (3.16) (and the subsequent a c c u r a c y of decreases  with i n c r e a s i n g n because  (3.19))  (3.17) i n v o l v e s a  c o n v o l u t i o n over h* and the s u c c e s s i v e l y generated  polynomials  have v a l u e s t h a t are s i g n i f i c a n t beyond the c u t o f f . The  r e c u r r e n c e process  & and m.  (3.17) a l s o  i n v o l v e s a summation  Because the monolayer has a smoothly  r a p i d l y with  I q ^ I.  i s n u m e r i c a l l y analogous  to the r a p i d  decrease  with I h* I i n the terms of (3.4), and s i m i l a r result. to  m  consequences  The summation over Sim i n (3.17) can thus be r e s t r i c t e d  the long wavelength s h e l l s , and the values of the r e s u l t i n g  orthogonal polynomials P Iq  modulated  decrease  s t r u c t u r e , terms i n (3.4) w i l l T h i s behaviour  over  L M  L  ^(n*) can be taken t o be zero f o r  I l a r g e r than a g i v e n c u t o f f .  it, the t r u n c a t i o n f o r Sim w i l l  cause  As with the t r u n c a t i o n f o r increasing  inaccuracy i n  the higher moments. The  moments a  n  decline  i n accuracy a t a r a t e which i s  dependent on the values of bounds s e t f o r I f? I and I q £ I i n the m  recurrence process. moments w i l l  When the bounds are s e t low, o n l y a few  have s i g n i f i c a n t a c c u r a c y .  I f many a c c u r a t e  moments are r e q u i r e d , the bounds must be s e t h i g h e r .  Since  high bound v a l u e s i n c r e a s e the amount of computation  required  by the r e c u r r e n c e p r o c e s s , i t i s d e s i r a b l e to c a l c u l a t e E  Q  with  as few moments as p o s s i b l e . Given t h a t E  Q  must be c a l c u l a t e d to w i t h i n a s p e c i f i e d  u n c e r t a i n t y , the number of moments r e q u i r e d i s determined  by  71  the convergence o£ the summation i n (3.14) and the r a t e of convergence of (3.14) depends on the values of x most r a p i d convergence occurs when x  Q  and x  m  and x .  Q  The  m  are, r e s p e c t i v e l y ,  the minimum and maximum eigenvalues of the system. U n f o r t u n a t e l y , the approximations to  the t r u n c a t i o n s f o r h* and q ^  unstable i f x eigenvalues.  Q  and x  m  m  i n the r e c u r r e n c e process due cause the method t o become  are e x a c t l y the mimimum and maximum  This i n s t a b i l i t y  the d e n s i t y of s t a t e s g(x)  a r i s e s from the d e s c r i p t i o n of  i n (3.12).  beyond the i n t e r v a l d e f i n e d by x  Q  and x  Should m  r e c u r r e n c e process has been approximated, s t a t e s w i l l extend  Because the  the r e s u l t i n g d e n s i t y  beyond the bounds s e t by the maximum and  minimum e i g e n v a l u e s , and the c a l c u l a t e d moments a diverge.  be non-zero  , the polynomial  moments c a l c u l a t e d w i l l d i v e r g e t o i n f i n i t y .  of  g(x)  the method, values f o r x  To s t a b i l i z e  Q  n  will and x  m  must be  chosen which are o u t s i d e the range of p o s s i b l e e i g e n v a l u e s . This w i l l have the consequence t h a t the summation of (3.14) w i l l not converge as q u i c k l y as expected, reasonable c h o i c e of x significantly  slower.  Q  and x  m  , the convergence w i l l not be  The system does not have eigenvalues  l e s s than zero, and t h e r e f o r e , x x  negative.  m  but given a  should be s l i g h t l y  Q  i s chosen t o be s l i g h t l y l a r g e r than the maximum  e i g e n v a l u e , but determining t h i s value i s n u m e r i c a l l y tedious and not necessary because o n l y an estimate Instead, x  m  i s required.  i s s e t t o be twice the maximum frequency of a  f l o a t i n g monolayer which has a d e n s i t y equal t o the h i g h e s t  72  d e n s i t y present  i n the monolayer; the p o i n t o£ h i g h e s t d e n s i t y  i s a t the center of the domain w a l l v e r t i c e s . x  m  T h i s value f o r  can be e a s i l y c a l c u l a t e d and p r o v i d e s a reasonably r a p i d  convergence f o r (3.14). Given these values f o r x  Q  and x  m  , E  w i t h i n 0.0005 K with o n l y 11 moments.  Q  can be c a l c u l a t e d to  For these moments t o  have s u f f i c i e n t accuracy, the bounds on h must be s e t so t h a t I Fi I 4. 25 A.  The bounds on q ^  m  are adequately s e t i f they are  taken to be the same as the bounds used  to c a l c u l a t e U  to the  c  accuracy of 0.0005 K. To check the r e s u l t s , the c a l c u l a t i o n was performed  with  more than 11 moments and with l a r g e r bounds on fi and q ^ . m  values f o r E  Q  were confirmed  p r e s c i b e d e r r o r of 0.0005 K.  The  to be a c c u r a t e to w i t h i n the A more g e n e r a l t e s t  convergence of the method was performed  f o r the  f o r the e a s i l y  handled  commensurate c o n f i g u r a t i o n and the bounds i n d i c a t e d above produced  the r e q u i r e d accuracy when compared to the r e s u l t s of  d i r e c t summation of the e i g e n v a l u e s .  73  (V)  RESULTS AND  DISCUSSION  For s e v e r a l d i f f e r e n t values the zero p o i n t energies m i s f i t s and  of the s u b s t r a t e  corrugation,  have been c a l c u l a t e d f o r a v a r i e t y of  orientations.  The  zero p o i n t energy i s r e l a t i v e l y  i n s e n s i t i v e to the o r i e n t a t i o n of the monolayer. accuracy  of the c a l c u l a t i o n , E  Q  appears i n v a r i a n t over  range of o r i e n t a t i o n s i n f i g u r e 4. of the monolayer w i l l not of the monolayer.  The  Thus the  zero p o i n t energy  zero p o i n t e n e r g y d o e s have s i g n i f i c a n t of the monolayer.  As shown i n  T h i s i s not unexpected because as  adatoms become more c l o s e l y packed, the thus the  the  the zero p o i n t energy i n c r e a s e s with the  of the monolayer.  frequencies,  the  i n f l u e n c e the o r i e n t a t i o n a l behaviour  v a r i a t i o n with the m i s f i t f i g u r e s 5a-f,  Given  increase.  density  the  force constants,  and  Since the zero p o i n t energy  does not change as the monolayer r o t a t e s , i t would appear t h a t a c a l c u l a t i o n based on a f l o a t i n g monolayer with an averaged substrate  i n f l u e n c e should  c a l c u l a t i o n was  d u p l i c a t e these r e s u l t s .  performed and  agreed, the values  of E  Q  although the g e n e r a l  zero p o i n t energy but the a c t u a l values  The  behaviour  could not be made to match.  compression of the monolayer produces the general  rigorous  Such a  Thus,  trend  i n the  are s u b j e c t to more  considerations.  most s i g n i f i c a n t consequence of these r e s u l t s i s t h a t  the zero p o i n t energy changes the value  of the  substrate  c o r r u g a t i o n at which the monolayer switches from an  74  Incommensurate ground s t a t e to a commensurate ground s t a t e . From c o n s i d e r a t i o n of the s t a t i c energy of the monolayer the c r i t i c a l value  value  i s higher  of s u b s t r a t e c o r r u g a t i o n i s 9.8  K.  alone,  This  than that p r e d i c t e d by t h e o r i e s which c a l c u l a t e  the s u b s t r a t e c o r r u g a t i o n from the m i c r o s c o p i c i n t e r a c t i o n s between the adatom and provided  the  the s u b s t r a t e .  l a t e s t estimate  p o i n t energy i s c o n s i d e r e d ,  of Vg = 7.4 the c r i t i c a l  c o r r u g a t i o n i s reduced to 7.0 c o r r u g a t i o n of 7.4 unless  K,  incommensurate.  i n c r e a s i n g the chemical  to an  Cole  When the  value  K ( f i g u r e 5c).  evidence  from a commensurate s o l i d  The  K.  the monolayer should  i t i s f o r c e d to be  with the experimental  V i d a l i and  1 8  be  of  have  4 8  zero  substrate  For a s u b s t r a t e commensurate  T h i s i s i n agreement  t h a t the monolayer can be d r i v e n incommensurate s o l i d  by  potential . 6  p o t e n t i a l between the adatoms i s approximated by a  Taylor s e r i e s that i s truncated  past the c u b i c term.  The  zero  p o i n t energy depends on the second d e r i v a t i v e of the p a i r p o t e n t i a l and  the c u b i c approximation may  t e s t the e f f e c t of t h i s approximation, i n t e r a c t i o n was  not be v a l i d .  the adatom p a i r  r e p l a c e d by c u b i c polynomials  segments over which the nearest  To  f i t to the  neighbours range.  T h i s i s an  a l t e r n a t i v e d e s c r i p t i o n to the truncated T a y l o r s e r i e s expansion and  the d i f f e r e n c e s are d i s p l a y e d i n f i g u r e 7.  s u b s t r a t e c o r r u g a t i o n of 7.0  K the two  same behaviour of the system ( f i g u r e  For a  c a l c u l a t i o n s produce  8).  Though the behaviour of the adatoms can g e n e r a l l y be  the  75  described  as harmonic, anharmonic i n f l u e n c e s w i l l a l t e r the  results slightly.  From s e l f - c o n s i s t e n t c a l c u l a t i o n s f o r the  f l o a t i n g m o n o l a y e r , the zero temperature motion of the 45  adatoms causes the zero p o i n t energy to be 6% higher value  obtained  under the harmonic approximation.  than the  The increase  i n energy can be a t t r i b u t e d to the i n t e r a c t i o n s of the adatoms s i n c e the f o r c e constants  increase  f a s t e r upon compression than  they decrease on d i l a t i o n . The  incommensurate monolayer i s more complex s i n c e i t s  s t r u c t u r e allows  many d i f f e r e n t phonon modes.  Some of these  modes correspond to domain w a l l motion and move blocks of adatoms as a u n i t .  These modes do not c o n t r i b u t e  significantly  to the zero p o i n t energy s i n c e they are low i n energy and, as will  be seen i n the f o l l o w i n g s e c t i o n , few i n number.  modes which do c o n t r i b u t e energy a r e the higher  s i g n i f i c a n t l y to the zero  The  point  energy modes which i n v o l v e motion of the  adatoms r e l a t i v e t o t h e i r neighbours. anharmonic e f f e c t s w i l l  For these modes the  be the same as those f o r the  commensurate c o n f i g u r a t i o n .  Therefore,  the zero p o i n t  i n f i g u r e s 5a-e of the incommensurate c o n f i g u r a t i o n s commensurate t r a n s i t i o n w i l l a c t u a l l y be higher  energies  near the  by 6%.  Away  from the commensurate t r a n s i t i o n , the d e n s i t y of the monolayer is greater  and, from the nature of the adatom i n t e r a c t i o n , the  anharmonic c o n t r i b u t i o n to the zero p o i n t energy decrease. significant  should  I t i s a n t i c i p a t e d that t h i s decrease w i l l impact on the general  trend of the zero  not have a point  76  energy.  As a r e s u l t , where the zero p o i n t energy c a l c u l a t e d  with the harmonic approximation p r e d i c t s t h a t the s u b s t r a t e c o r r u g a t i o n value  i s 7.0  critical  K ( f i g u r e 5c), the  anharmonic c o n t r i b u t i o n to the zero p o i n t energy w i l l s h i f t value  to be s l i g h t l y  g r e a t e r than 7.0  K.  the  77  4.  (I)  LOW ENERGY PHONON MODES  HELMHOLTZ FREE ENERGY  The  first  two terms  i n (3.8) f o r the Helmholtz f r e e  energy  were determined i n the p r e v i o u s c h a p t e r s , and r e p r e s e n t the energy of the monolayer i d e n t i f i e d as E free  energy.  Eip  T  The t h i r d  term,  , p r o v i d e s the temperature dependence of the o  ^  =  a t zero temperature.  I^lnd  - e-^ s(k)/2 u  }  (  4  >  1  )  s, k  where N i s the number of adatoms i n the monolayer At  and £=l/k T. B  low temperatures only the lowest energy modes of the  monolayer  will  be s i g n i f i c a n t  commensurate monolayer adsorption s i t e s w i l l band gap v a l u e .  i n t h i s e x p r e s s i o n . The  with a l l the adatoms locked i n not have any modes with e n e r g i e s below a  The incommensurate monolayer,  on the other  hand, has h i g h l y mobile domain w a l l s and low energy modes a r e possible.  From (4.1) c o n f i g u r a t i o n s with extremely s o f t modes  could have a large free energy; the m i s f i t or o r i e n t a t i o n of the  monolayer  may thus change s i g n i f i c a n t l y with temperature  even when the chemical p o t e n t i a l It  M i s kept constant i n (1.6).  i s t h e r e f o r e i n t e r e s t i n g t o determine the magnitude of E a t  d i f f e r e n t temperatures f o r v a r i o u s c o n f i g u r a t i o n s of the  T  78  monolayer. E>p c a n be c a l u l a t e d was  used  rapidly  to calculate  E . Q  However,  v a r y i n g f u n c t i o n a s u-*0.  calculated  for this  corresponding moments a is  by t h e p o l y n o m i a l  (4.1) i s s e e n  The c o e f f i c i e n t s  e x p r e s s i o n would c a u s e  t o (3.14),  method  which  t o be a c (x n  m /  \ ) Q  t h e summation,  t o converge very s l o w l y .  must be c a l c u l a t e d  n  expansion  and t h e v i a b i l i t y  Thus, many  o f t h e method  reduced. Fortunately,  t h e low energy  allow approximations compromise  difficulties  dynamical  matrix  evaluated  of the r e s u l t i n g  These a p p r o x i m a t i o n s  discussed  eigenvalues  nature,  a s s o c i a t e d with  and a l l o w a p r a c t i c a l  a direct  e i g e n v a l u e s and  overcome t h e p r e v i o u s l y  and e i g e n v e c t o r s o f ( 3 . 6 ) .  through  of t h e i r  t o be made t o (3.6) w h i c h do n o t  the accuracy  eigenvectors.  modes, b e c a u s e  the large s i z e solution Thus  of the  f o r the  (4.1) c a n be  summation o f e i g e n v a l u e s .  79  (II)  NORMAL  The known this  static to  be  fact  smoothly  of  configuration  allowed the and  the  motion  In  corresponding  approximation can  The  .  adatom p o s i t i o n s  comparable  adatoms  1 3  approximations  efficiently.  the  structure  modulated  calculated  of  CALCULATION  incommensurate  has  description  MODE  It  is  of  the  the to  monolayer  previous  be  made  (2.13)  so  energy  per  therefore  be  made  for  is  described  to  that  is  chapters the the  static  adatom can  assumed  that  the  energy  low  be  a motion  (3.5). of  the  adatoms  by the  truncated  expression  (4.2)  The  cutoff  parameter  approximation, is  correct,  will The  be  of  The  then  only  set  only  of  a  S  where  determines be  portion in  of  (3.6)  the  of  can  accuracy  below.  determining  relevant portion  the  discussed  equations  these  significant  submatrix  will  significance  linear  consider  and  n,  of  If  this  dynamical the then  low be  the assumption  matrix  energy reduced  (3.7)  modes. to  terms. the  dynamical  matrix  will  be  the  80  ST M  8,  ( >  =  k  m  0  F  O  I St I , I ml , IX. I  R  or  IM|  >  n  '  (4.3)  SLM,X,m( >  = *>LM,Slm( >  k  This  produces the reduced  Ma x  WL (k) = I  2  otherwise  k  M  S  L M  s e t of equations  ^ ( ^ . ^ ( k )  (4.4)  with w  L M  (k)  The be  =0  i f ILI>n o r IMI>n  eigenvalues  t r e a t e d as t r i a l  trial  solutions  (3.6).  M  In  fact  Iu -x 2  a  2  (4.5)  x (j<) and e i g e n v e c t o r s w ^ ( i c ) 2  m  solutions  to (3.6).  The v a l u e s x  are c l o s e t o being the actual 4 9  o f (4.4) c a n 2  of these  eigenvalues  o  2  of  ,  | i s  (4.6)  where  e  Using  =  \X  ^LM,Hm( )-Wjt (k) k  m  (4.4), equation  calculated  from  - mX ^ 2  L M  (k)]  ( 4 . 7 ) c a n be r e d u c e d  |  so t h a t  (4.7)  e c a n be  81  1  or  (4.2),  From  i n c r e a s e as the value will  I I  ILI>n IMI>n  IHUn Iml^n  DLM^mfkVwj^k) I"  the a c c u r a c y  of the a p p r o x i m a t i o n  n i s i n c r e a s e d , thus  c h o s e n f o r n.  a l s o depend on  the v a l u e  known t o be  from  (3.7)  the energy  that  e i n (4.8)  of  will  IL-SU  LM,Slm(^  rapidly  IM-ml.  and  depend on  how  I SI I and  I ml.  variations  i n determining  free  energy  e i g e n v e c t o r s can  values  (4.1).  and  the  for  monolayer  s i n c e the  domain  decrease  4  m  trial low  energy  these  be d e t e r m i n e d  not  decrease  o f t h e phonon mode  values x  temperature  For  structural  w^ (ic) w i l l  the energy  of the  F o r t u n a t e l y , o n l y the  significant  and  the a c c u r a c y  consequence  w^fk ) w i l l  a s s o c i a t e d with  Thus s i n c r e a s e s a s  i n c r e a s e s and  Helmoltz  be  monolayer  i n c r e a s i n g phonon e n e r g i e s ,  t h a t a r e much s h a r p e r  rapidly.  decrease.  With  the  modes o f t h e  smoothly modulated s t r u c t u r e s ,  however, t h e modes w i l l  as  low e n e r g y  s  the  r a p i d l y Wo^fic) d e c r e a s e s  t o m o t i o n o f t h e domain w a l l s , and  r a p i d l y with  of  (calculated  T h i s has  correspond  The  From  on  i n magnitude f o r  l&l  walls are  I ml.  D  increasing  and  depend  c o n f i g u r a t i o n of the  ( 3 . 4 ) ) must d e c r e a s e  increasing values  s will  o f t h e phonon mode.  s m o o t h l y m o d u l a t e d and  and  of  should  For a g i v e n c h o i c e of n the value  p r e c e d i n g work, t h e g r o u n d s t a t e is  (4.8)  will  modes a r e dependence of  modes t h e  eigenvalues  a c c u r a t e l y from  o f n s m a l l enough t o make t h e c a l c u l a t i o n  the  (4.4)  for  practical.  From t h e p r e c e d i n g d i s c u s s i o n , t h e a p p r o x i m a t i n g  system of  82  equations  (4.4)  determinable Imply, be  error  however,  determined  from  will  (4.2)  to  of  that  the a l l  by t h i s  do  solutions  produce  indeed (3.6)  solutions  solutions the  provide is  to  To show the  are  (3.6).  low energy  method.  it  that  This  modes that  complete  necessary  to  and  (3.6)  of  the  set  show  within  a  does  the  not  system  trial  of  a  low  can  solutions energy  one-to-one  correspondance. The agree  approximate when  the  one-to-one  are  strength for  a l l  modes of  modes.  of  is  exact not  exists  Since  the  Vg, this  the  Vg.  from  together  combination  Thus  monolayer  by t h i s  calculated  one-to-one  between  of  method. (4.2);  the  dispersion as  a  exact  and  a  and the  function  of  corrugation  o  must for  to  The e i g e n v e c t o r s when  Vg = 0,  of  determined  however,  equations  curves  frequencies  be  of  i.e.  correspondence  a l l  can  system  corrugated,  continuously deformable  values  (4.6)  close  substrate  correspondence  approximate modes  (4.2)  the  be  maintained  the  low  energy  within  the  accuracy  can a l s o  mode  calculated  normal  be  frequencies  the  eigenvectors  may b e  a  the  corresponding nearly degenerate  are  linear modes  of  (3.6). Dispersion  curves  configurations. determined error little would  calculated  The a c c u r a c y  by v a r y i n g  bound  e  is  with  n  for  suggest.  are  the  This  the  parameter  generous, values  of  of  overly  since  the  for  a  range  values n.  It  superlattice  calculated is  apparent  eigenvalues  n much l e s s  than  large  bound  error  of  the is  change value a  is that very of  result  e of  the  83  the  sharp  linear this to  cutoff  a t n present  i n the eigenvectors  mix o f t h e e i g e n v e c t o r s  cutoff,  resulting  i t s corresponding  v (ic) L M  i n w (ic) L M  eigenvector  being  less  v (k*).  than  This  L M  L M  in equation  ( 4 . 6 ) i t becomes s i g n i f i c a n t .  however, c a n be e x p e c t e d value  of s suggests,  direction obtained  o f v^Cl?).  t o be a c l o s e r  since w (ic) L M  A  branches a t the r p o i n t .  g i v e an i n d i c a t i o n  eigenvalues For  x,  match t o o t h a n t h e  o f t h e two l o w e s t  n, t h e i r  of the absolute  i n the  of the e r r o r i n x i s  These v a l u e s a r e found  the d i s p e r s i o n curves  phonon  t o approach computed  e r r o r i n the  c a l c u l a t e d , the parameter  e i s s m a l l enough t h a t a l l b u t t h e t h r e e  modes a r e a c c u r a t e  t o b e t t e r than  modes a n a n a l y s i s a t t h e r p o i n t bounds.  the  calculated  the  eigenvalues  eigenvalues  frequencies w i l l zero.  0.1% .  i s roughly  For the three  i s necessary  The f r e q u e n c i e s a r e g i v e n  modes c o n s i d e r e d ,  away f r o m  be n o t i c e d , b u t  o f t h e low e n e r g y phonon modes.  chosen so t h a t  error  match  discrepancy  i s predominantly  z e r o as n i s i n c r e a s e d and, f o r a g i v e n values  a pure  The e i g e n v a l u e  better indication  by e x a m i n i n g t h e v a l u e s  A  L M  i s required to describe  between w ( i < ) and v ( i c ) w o u l d n o t o r d i n a r i l y LM  w (ic).  the percent  softest softest  t o o b t a i n the  by t h e square  and because the a b s o l u t e constant  n was  root of  error i n  f o r t h e low e n e r g y  phonon  e r r o r i n the c a l c u l a t e d  be p r o p o r t i o n a l t o 1/u f o r f r e q u e n c i e s o f u  84  (III)  REVIEW OF PREVIOUS WORK  In a p r e v i o u s p a p e r , the behaviour 3 4  of the three  lowest  phonon modes was d i s c u s s e d f o r monolayer c o n f i g u r a t i o n s with m i s f i t values between 1.75% and 2.22%.  For reasonable  values  of the s u b s t r a t e c o r r u g a t i o n , the domain w a l l s w i l l be c l e a r l y separated a t these m i s f i t s .  These m i s f i t v a l u e s a r e l a r g e  enough t h a t the c a l c u l a t i o n f o r the low energy normal modes i s p r a c t i c a l , and given the renormalized nature of the domain w a l l s , the i n f o r m a t i o n obtained about these domain w a l l modes w i l l a p p l y t o incommensurate monolayers i n the l i m i t of the commensurate  transition.  A t y p i c a l d i s p e r s i o n curve i s shown i n f i g u r e 9.  f o r the incommensurate monolayer  Of the normal modes p o s s i b l e f o r the  monolayer, three modes a r e s i g n i f i c a n t l y s o f t e r than the r e s t . For the wavevector k a t M, f i g u r e s lOa-c  i l l u s t r a t e each of the  three lowest modes of f i g u r e 9; the dashed l i n e s show the deformation (solid  of the w a l l s about t h e i r e q u i l i b r i u m p o s i t i o n s  lines).  The s c a l e i s provided so t h a t the s i z e of the  domains can be determined, deformations visible.  although the magnitudes of the  a r e s c a l e d a r b i t r a r i l y t o make the modes c l e a r l y  The value used f o r the s c a l i n g of the w a l l  deformations  i s , however, c o n s i s t e n t i n a l l of f i g u r e s 10.  From f i g u r e 10a the s o f t e s t mode i s the b r e a t h i n g mode p r e d i c t e d by V i l l a i n  1 4  ,  with the other a c c o u s t i c mode and the  o p t i c a l mode producing a s h e a r i n g of the domain w a l l s .  85  When the c o n f i g u r a t i o n of the monolayer was shear modes were found to i n c r e a s e i n energy.  r o t a t e d the The  two  dispersion  of the b r e a t h i n g mode, however, d i d not s i g n i f i c a n t l y change. When the m i s f i t  of the monolayer was decreased, the b r e a t h i n g  mode became s i g n i f i c a n t l y s o f t e r , while the shear modes showed little  change.  86  (IV)  QUANTIZED  The modes  previous as  these  influence energy.  on  a  The  misfit  grouping  of  modes  the  1.75%  modes. triads  lowest  energy  mode.  of  is  shown  in  for  the  sixth  mode  is  Where with  a  the two  higher  is  found  walls  of  most  of  the  the  the  mode.  It  the  modes  is  version  vertices)  the  lowest  the  lower  ninth  energy  with  four  for  with  13a. the  the The  lowest  motion fourth  lowest  at in  the  for  the  the  r  figure  higher  energy  motion  vibrate  except  12b  energy  vibrate  with  at  the  three  (not  the  point  mode.  to  mode  the  shown)  domain  nodes.  example,  the  lowest of  triad  for  c o r r e s p o n d i n g mode  the  type  curves  the  Furthermore,  same  triad  domain w a l l s  to  the  calculated  the  domain w a l l s  have  not  form a  and  that  the  to  were  calculated  clear of  causes  of  energy  significant  curves  curves  shape  lowest  Helmholtz Free  dispersion  third  for  of  dispersion  12a  mode  the  causes  M point  figure  repeats  the  an a l t e r n a t e  compared  form  (at  be  The d i s p e r s i o n  energetic  nodes  the  motion  lowest  three  energy  domain w a l l  low e n e r g y  vibrate  As the  figure  energy  nodes.  the  more  higher  shows  .  to  on t h e  dependance  The shape  energy  form  expected  the  11  higher  The  are  of  Figure of  concentrated  3 4  temperature  forms  considered. at  paper  N A T U R E OF WALL MOTION  the  energy  of  the  lowest  energy  motion  mode.  of  the  domain walls  mode  figure  10a  fourth  lowest  energy  mode  is  clearly a  Because  this  can  more  mode  at  be mode energetic  causes  the  87  domain w a l l s t o v i b r a t e with a wavelength t h a t i s twice l e n g t h , the frequency  their  of the mode provides the v e l o c i t y a t  which v i b r a t i o n s w i l l propagate along the domain w a l l s . form of t h i s mode i s constant through  t o the r p o i n t  13b), and should t h e r e f o r e be d i s p e r s i o n l e s s . c o n f i r m a t i o n f o r the value of the domain w a l l ' s  The  (figure  Further wavevelocity  can be obtained by examining the seventh mode ( f i g u r e 11). T h i s mode Is twice as e n e r g e t i c as the f o u r t h mode and causes the domain w a l l s t o v i b r a t e with a wavevector equal t o t h e i r length. Since no c o n t r a d i c t i o n s t o t h i s quantized behaviour found,  were  the g e n e r a l c o n c l u s i o n can be made t h a t the domain w a l l  modes a r e quantized a c c o r d i n g t o the motion of a fundamental t r i a d of modes. determined  a c c o r d i n g to which of the three t r i a d modes i t  corresponds behaviour  The motion of a domain w a l l mode can be  t o , and the number of the t r i a d ' s group.  i s expected  This  to break down when the energy of the  modes approaches the band gap energy of the commensurate monolayer.  At t h i s energy the adatoms i n the r e g i s t e r e d  r e g i o n s a l s o v i b r a t e and the monolayer w i l l have v i b r a t i o n a l modes other than domain w a l l modes.  88  (V)  DOMAIN WALL INSTABILITIES  From the s t a t i c r e s u l t s , at m i s f i t s g r e a t e r than 2.5%  the  monolayer w i l l be r o t a t e d when the s u b s t r a t e c o r r u g a t i o n i s g r e a t e r than 8.OK.  The  non-rotated  c o n f i g u r a t i o n , because of  the overcompression of the v e r t i c e s , w i l l be e n e r g e t i c a l l y unfavourable.  T h i s observed r o t a t i o n , however, may  be  an  a r t i f a c t of the c a l c u l a t i o n s i n c e i t Is p o s s i b l e to d i f f u s e v e r t i c e s by breaking monolayer.  the  i n v e r s i o n symmetry imposed on  the  the  T h i s allows the domain w a l l s to s h i f t r e l a t i v e  to  the s u b s t r a t e so t h a t the domain w a l l s no longer meet at a point.  Should the monolayer seek to form a l e s s  c o n f i g u r a t i o n by s h i f t i n g the domain w a l l s , the modes c a l c u l a t e d f o r the monolayer with should  be u n s t a b l e .  unstable  Within  energetic translational  i n v e r s i o n symmetry  the accuracy  of the  calculation,  t r a n s l a t i o n a l modes have never been observed.  The  monolayer does, however, have other modes which can become unstable. F i g u r e 14a non-rotated  shows the d i s p e r s i o n curves  monolayer with m i s f i t  c o r r u g a t i o n 8.0 becomes unstable f i g u r e 15 and  K.  3.33%  c a l c u l a t e d f o r the  and  I t i s apparent t h a t one  at the M p o i n t .  substrate of the modes  T h i s mode i s shown i n  corresponds to a s h e a r i n g of the domain w a l l s .  Since the energy of the monolayer w i l l decrease with r o t a t i o n ( f i g u r e 6b), t h i s suggests t h a t the monolayer i s t r y i n g to rotate.  A l t e r n a t e l y , the  i n s t a b i l i t y may  indicate that  the  89  domain w a l l p e r i o d i c i t y imposed on the monolayer w i l l be broken so t h a t the monolayer can s t a y  non-rotated.  However, t h i s i s not l i k e l y s i n c e the unstable  mode  ( f i g u r e 15) i n d i c a t e s t h a t the domain w a l l s wish t o be sheared.  Because the r o t a t e d c o n f i g u r a t i o n imposes a shear on  the domain w a l l s the non-rotated c o n f i g u r a t i o n can not r e s i s t being  rotated.  This  i s c o n t r a r y to the behaviour of the  monolayer a t m i s f i t s below 2.5%. mode i s not unstable The  and the domain w a l l s r e s i s t  shearing.  non-rotated c o n f i g u r a t i o n s of the monolayer are unstable  when the s u b s t r a t e misfit  At these m i s f i t s the shear  corrugation  i s greater  i s between 2.5% and 3.5% .  than 7.0 K and the  This i n s t a b i l i t y i s also  found t o a l e s s e r degree i n some c o n f i g u r a t i o n s s l i g h t l y rotated.  that a r e  For a g i v e n m i s f i t the r o t a t e d  c o n f i g u r a t i o n s of lowest energy from f i g u r e 6 w i l l be s t a b l e as w i l l c o n f i g u r a t i o n s that a r e r o t a t e d f u r t h e r  ( f i g u r e 14b).  m i s f i t s l a r g e r than 3.5% , the monolayer no longer  For  has d i s t i n c t  domain w a l l s and i t becomes smoothly modulated; t h i s eases the l o c a l i z e d overcompression a t the v e r t i c e s and the motion of the monolayer i s s t a b i l i z e d . configuration w i l l s t i l l  Despite  t h i s f a c t , the lowest energy  be r o t a t e d .  The d i s p e r s i o n curves f o r  the non-rotated c o n f i g u r a t i o n with m i s f i t  3.7% i s shown i n  f i g u r e 16. In summary, overcompression of the v e r t i c e s i s expected to be a s i g n i f i c a n t  i n f l u e n c e , not o n l y on the s t a t i c energy of  the monolayer, but a l s o on the s t a b i l i t y of the c o n f i g u r a t i o n .  90  The monolayer  w i l l seek t o ease the overcompression by  or b r e a k i n g the p e r i o d i c i t y imposed  rotating  on the c o n f i g u r a t i o n .  At  higher temperatures, thermal motion of the adatoms i s expected to d i f f u s e the v e r t i c e s so that configurations self-consistent prediction  become s t a b l e .  the p e r i o d i c non-rotated Such an e f f e c t r e q u i r e s  a  phonon c a l c u l a t i o n , and c o n f i r m a t i o n of t h i s  awaits f u r t h e r  work.  91  (VI)  FREE ENERGY CALCULATION i  The  expression  (4.1)  which determines  E  involves a  T  summation over a l l modes of the monolayer. calculate E  E  T  i t i s necessary to separate  T  =  jj  £ d  k  p  s  In order to  (4.1)  as  £(JO  (4.9)  where  f<JO  =  The  j ln( 1 - -0Ku (k)/2 e  N  s  )  (  t o t a l number of adatoms i n the monolayer N has  factored  i n t o the number of adatoms per domain N  number of domains N^.  >  1  0  )  For temperatures  near  p  been and  the  zero o n l y the  lowest energy modes of the monolayer need be c o n s i d e r e d . it  4  e  Since  i s p o s s i b l e to c a l c u l a t e the f r e q u e n c i e s of these modes,  f(k)  can be determined.  f u n c t i o n with r e c i p r o c a l superlattlce Equation  From ( 3 . 6 ) ,  f(k) i s a  l a t t i c e vectors q ^  m  periodic  b e i n g those  of the  (A.5). (4.9)  can be d e s c r i b e d as an average of f ( k )  e v a l u a t e d a t a l l p o i n t s it i n the f i r s t superlattice.  B r i l l o u i n zone of the  For a monolayer of i n f i n i t e  summation i n ( 4 . 9 )  Is a c t u a l l y an  integral.  extent,  the  Chadi and  Cohen  5 0  92  have d e v i s e d  a method f o r n u m e r i c a l l y  functions that areanalytic.  integrating periodic  W i t h t h e i r method, e q u a t i o n ( 4 . 9 )  becomes a w e i g h t e d sum o f t h e f u n c t i o n f ( i ? ) e v a l u a t e d d i s c r e t e s e t o f s p e c i a l p o i n t s ic. appropriate  ata  The s e t o f s p e c i a l p o i n t s  t o ( 4 . 9 ) have b e e n d e t e r m i n e d b y C u n n i n g h a m . 5 1  number o f p o i n t s to the accuracy  i n the s e t and t h e i r required  method i s r e v i e w e d Unfortunately,  positions vary  f o r the numerical  The  according  integration.  This  i n a p p e n d i x D. f(kf) d e t e r m i n e d f r o m ( 4 . 1 0 ) i s n o t a n a l y t i c .  The a c o u s t i c modes o f t h e m o n o l a y e r c a u s e f ( k ) t o d i v e r g e l o g a r i t h m i c a l l y whenever i? a p p r o a c h e s a r e c i p r o c a l l a t t i c e vector qj, . m  discussed,  I n order  t o use t h e numerical  f(£) must be m o d i f i e d  The a s y m p t o t i c w i t h ic n e a r q  i n t e g r a t i o n scheme  so that i t i s a n a l y t i c .  behaviour of the acoustic frequencies  is  x > m  o(k) -* c lk-q^ l tf  tf  The mode i n d e x transverse  f(k)  "^(J?)  (4.11)  m  cr c o r r e s p o n d s t o e i t h e r t h e l o n g i t u d i n a l o r t h e  a c c o u s t i c mode.  =  j j ^ l n  s  I f f(ic) i sexpressed as  ( 1 - ~0 e  fi(J  s(io/2 j  JtS 9 < 1 - e-^^'k-qjiml/ ) 2  (4.12)  £ I I \ ln( 1 - -** tflk-qV l/2 , N  p  Sim 5r P  e  c  m  93  the  l o g a r i t h m i c a l l y divergent  out and  integrated  of f (ic) i s now  over ic a n a l y t i c a l l y .  a n a l y t i c and  Cunningham s p e c i a l  p o r t i o n of f ( k ) can  points . 5 1  can  be  The  integrated  be  separated  remaining using  the  portion  94  (VII)  RESULTS AND  DISCUSSION  At low temperatures, the magnitude of E  T  i s extremely  s e n s i t i v e to e r r o r s i n the f r e q u e n c i e s c a l c u l a t e d f o r the energy modes. misfit  With the present  low  computer r e s o u r c e s , when the  i s s m a l l or the s u b s t r a t e c o r r u g a t i o n i s l a r g e , the  frequency  v a l u e s obtained  from (4.4)  w i l l not be  low  accurate . 3 4  Thus the c a l c u l a t i o n must be c o n s t r a i n e d to c o n f i g u r a t i o n s with l a r g e r m i s f i t values is unfortunate,  or s m a l l e r s u b s t r a t e c o r r u g a t i o n s .  because the c o n f i g u r a t i o n s with s m a l l m i s f i t  large substrate corrugation w i l l and  p o t e n t i a l l y the g r e a t e s t E The  T  .  f r e e energy c a l c u l a t i o n i s a l s o not p o s s i b l e i f the  s u b s t r a t e c o r r u g a t i o n i s g r e a t e r than 7.0 c o n f i g u r a t i o n s become unstable 3.5%.  K,  When the the  when the m i s f i t  Thus the c a l c u l a t i o n f o r E  c o n f i g u r a t i o n s t h a t permit  T  non-rotated  i s between  2.5%  i s c o n f i n e d to s t a b l e  the f r e q u e n c i e s to be  determinined  accurately. The 2.8%,  c o n f i g u r a t i o n s chosen f o r t h i s study have m i s f i t s 3.0%,  and  3.7%.  When the m i s f i t  r o t a t e d c o n f i g u r a t i o n s are c o n s i d e r e d . corrugation  i s taken to be 8.OK  c l e a r l y separated 3.7%  or  have the s o f t e s t phonon modes,  underlying s t a t i c c o n f i g u r a t i o n i s unstable.  and  This  at 2.8%  i s 2.8% The  or 3.0%  of  , only  substrate  so t h a t the domain w a l l s are  m i s f i t , while the c o n f i g u r a t i o n at  i s a modulated s t r u c t u r e .  T h i s allows the f r e e energy  a s s o c i a t e d with s o f t domain w a l l modes to be compared to the  95  more r i g i d system that does not have d i s t i n c t domain w a l l s . The values  E  T  f o r these c o n f i g u r a t i o n s are as f o l l o w s :  Table r . Values of E  T  c a l c u l a t e d f o r various  temperatures and o r i e n t a t i o n s when M%=3.7%  0.00° 0. 52° 0. 79° 0.95°  T=0.5K  T=1.0K  T=2.0K  T=4.OK  -0.0019K -0.0011K -0.0011K -0.0009K  -0.0086K -0.0058K -0.0055K -0.0051K  -0.0325K -0.0252K -0.0240K -0.0231K  -0.1329K -0.1171K -0.1145K -0.1130K  Table 2. Values of Eip c a l c u l a t e d f o r v a r i o u s temperatures and o r i e n t a t i o n s when M%=3.0%  0. 09° 0. 71°  T=0.5K  T= l.OK  T=2.0K  T=4.OK  -0.0034K -0.0020K  -0. 0116K -0. 0073K  -0.0366K -0.0260K  -0.1419K -0.1192K  96  Table Values  of  temperatures  and  for  orientations  various  when  M%=2.8%  T=1.0K  T=2.0K  T=4.0K  0.21°  -0.0022K  -0.0081K  -0.0276K  -0.0941K  0.68°  -0.0024K  -0.0080K  -0.0265K  -0.0916K  immediate  temperatures neglible. to  the  invalidate  of  greater  than  impact  of  E  T  4.OK. with  in  energies  the  that  the  been  energies value  lowest  monolayer  deteriorates  domain w a l l  additional  of  make  the E  T  low energy modes  will can  not  be  have  monolayer.  significant  may  F u r t h e r m o r e , the  does  been  greater of  cause  the  modes  rapid  any  twice  temperatures since  no  that  Because  increasing band gap  the  the  modes  the  The commensurate  energy  may b e  for  imposed  than  11.OK.  have.  with  than  free  determined  has  possible can  at  values.  limit  much g r e a t e r  Furthermore,  the  This  the  frequencies  to  T  modes  not  gap  commensurate  E  r e q u i r e d to  have  band gap  is  that  domain w a l l  commensurate value  of  made  equilibrium orientation  increase  calculated  be  harmonic approximation.  the  values  can  contribution  temperatures  the  significant The  the  the  softening  observation  The o n l y  change  However,  are  calculated  T  T=0.5K  The  is  E  3.  band  modes  of  the  accuracy  of  the  energy, will  be  present  in  values  with  suspect. the  monolayer  97  t h a t are missed by the c a l c u l a t i o n .  Thus the temperatures must  be kept s m a l l so the i n a c c u r a c i e s caused by the approximate equations The  (4.4)  are minimized.  f r e q u e n c i e s have been c a l c u l a t e d under the assumption of  harmonic motion.  T h i s approximation w i l l  domain w a l l s c o l l i d e with each other. domain w a l l s w i l l  not be v a l i d  Since c o l l i s i o n s of the  i n c r e a s e the energy of w a l l modes, anharmonic  e f f e c t s a r e expected to d i m i n i s h the values Confirmation  i f the  of Eip.  of t h i s p r e d i c t i o n awaits a s e l f - c o n s i s t e n t phonon  c a l c u l a t i o n of the incommensurate monolayer. The  o n l y mechanism that c o u l d s o f t e n the domain w a l l motion  i s the formation  of domain w a l l d i s l o c a t i o n s  based on the e l a s t i c constants system should  be performed.  2 6  . Further work  of the renormalized  domain w a l l  In the absence of domain w a l l  d i s l o c a t i o n s , however, a t low temperatures domain w a l l motion should not i n f l u e n c e the m i s f i t and o r i e n t a t i o n f o r the e q u i l i b r i u m c o n f i g u r a t i o n of the monolayer. To extend the r e s u l t s t o the commensurate l i m i t , necessary  to estimate  decreases.  the change l n E  i t is  as the m i s f i t  T  From previous w o r k , when the domain s i z e i s l a r g e 34  the s o f t e s t mode i s the b r e a t h i n g mode. t h i s mode w i l l dominate the value of E  T  At low temperatures and the v a r i a t i o n of  i t s c o n t r i b u t i o n due t o m i s f i t should be examined. The  b r e a t h i n g mode i s i n f l u e n c e d by the domain w a l l  interaction.  The s t r e n g t h of t h i s  i n t e r a c t i o n w i l l decrease as  98  U (L)  = c L  W  for  tf  e  (4.13)  _ ) C L  I n c r e a s i n g s e p a r a t i o n L of the domain w a l l s  constants c, c and £ a r e determined e n e r g e t i c s of the system.  2 2  .  The  from the c a l c u l a t e d  Correspondingly,  the frequency of  the b r e a t h i n g motion should decrease as  u(k)  =  f(ic,L)  (4.14)  _ , < : L / 2 e  where f(ic,L) i s a f u n c t i o n which i s p e r i o d i c i n k and a l g e b r a i c i n L.  To l e a d i n g order, the c o n t r i b u t i o n of t h i s mode t o (4.1)  w i l l decrease as  £  ln(  )  _ , C L / 2  e  with i n c r e a s i n g L.  (4.15)  Since Np v a r i e s as L  w i l l decrease  as 1/L.  approximation  f o r motion i s v a l i d .  vibrational and  2  , the c o n t r i b u t i o n  T h i s a n a l y s i s assumes t h a t the harmonic  amplitudes  will  For low energy modes the  be l a r g e even a t zero  the domain w a l l s may c o l l i d e with each o t h e r .  c o l l i s i o n s occur, the motion w i l l  temperature, If  not be harmonic and (4.15)  w i l l not be c o r r e c t . Coppersmith e t a l .  2 6  have presented a model which d e s c r i b e s  the domain w a l l s as being f r e e t o move provided t h a t t h e i r displacements  do not exceed L/4.  T h i s ensures  t h a t the domain  99  w a l l s move independently of each other, and simulates  extent,  the anharmonic r e p u l s i o n between the doman w a l l s .  They p r e d i c t that the wall breathing restricting valid  to some  f r e e energy c o n t r i b u t i o n of the domain  w i l l decrease as l n ( L ) / L  l i m i t than 1/L  predicted  2  by  .  This  i s a more  (4.15) and  f o r c o n f i g u r a t i o n s approaching the  should  be  commensurate  transition. At higher  temperatures, other  i n f l u e n c e the value  of E  T  appear as the domain s i z e could  More low energy domain w a l l modes  i n c r e a s e s , and 3 4  be expected to i n c r e a s e as w e l l .  these modes does not modes N  p  and,  of  E  T  However, the number of  i s expected to remain s m a l l  T  as  increases.  r e s u l t s presented are  substrate  the value  increase as q u i c k l y as the t o t a l number of  from (4.10), E  the domain s i z e The  .  domain w a l l modes begin to  corrugation  f o r c o n f i g u r a t i o n s where the  i s 8.OK.  Because the domain w a l l modes  decrease i n energy as the value i n c r e a s e with Vg.  of Vg  increases, E  For reasonable values  can be estimated from the renormalized  T  will  of Vg the change i n  d e s c r i p t i o n based  E  T  on  domain w a l l s . Increasing  the s u b s t r a t e  w a l l s ; from a renormalized  corrugation  w i l l sharpen the domain  d e s c r i p t i o n based on the  continuum  model, the c o n f i g u r a t i o n w i l l resemble a c o n f i g u r a t i o n that a lower value  of Vg and  v a r i e s as  , scaling considerations  'Vg  smaller  number of adatoms per domain N  D  misfit.  has  Because < i n (4.13) p r e d i c t the r a t i o of  f o r these two  the  configurations is  100  (4.16)  For the c o n f i g u r a t i o n with the  l a r g e r Vg  , the domain w a l l s  v i b r a t e f a s t e r because they c o n t a i n fewer adatoms and massive.  The  e f f e c t of t h i s on  be minor compared to the  (4.10),  small.  the values  of E  T  impact of r e s c a l i n g Np  f o r these  Correspondingly,  less  however, i s expected to .  Because the  s u b s t r a t e c o r r u g a t i o n cannot be s i g n i f i c a n t l y g r e a t e r 8.OK,  are  c o n f i g u r a t i o n s should  i t i s obvious t h a t d e c r e a s i n g  s u b s t r a t e c o r r u g a t i o n w i l l decrease the value  of E  T  .  than remain the  101  5.  CONCLUSION  The  purpose  temperature graphite. variation  of t h i s  Of  specific  system.  To  in  energy free  and  understand with  and  the  substrate. The  interaction  potentials.  of the  that  temperature  m u s t be  i s dependent  on  of the  To free  of the monolayer  was  On  interaction,  only  as of  a the  of  the  the  interactions  krypton  simplify energy  the  that  varies  considered.  This  interactions  between k r y p t o n atoms  in  the  interactions  between  and  form  of the p a i r  substrate.  modelled  potential  of k r y p t o n and  t h e s e adatoms  modified to  Because  the substrate  the can  account one  of the  parameter,  by  was  obtained include  not  be  was  n a t u r e of the substrate  from  the  between  determined  f o r the substrate  the  pair  interaction  the symmetry of the s u b s t r a t e ,  calculation.  observed  calculated.  the adsorbed  substrate.  a parameterized form  reflects  on  the  k r y p t o n atoms and  accurately,  adsorbed  orientation  orientation  b e t w e e n a d a t o m s was  bulk properties  screening the  The  low  behaviour, the v a r i a t i o n  the component  and  and and  krypton vapour,  involves  monolayer,  and  the graphite  only  the  misfit  of the system  the c o n f i g u r a t i o n  component  the  misfit  between the  calculation,  this  the  i s the e x p e r i m e n t a l l y  chemical potential  energy  monolayers,  with  interest  of the monolayer's to the  The  i s to determine  behaviour of k r y p t o n monolayers  response  free  thesis  potential,  used  i n the  substrate corrugation,  was  102  necessary was  to d e f i n e the s u b s t r a t e p o t e n t i a l .  v a r i e d so that the  This parameter  impact of the s u b s t r a t e c o r r u g a t i o n  on  the p r o p e r t i e s of the monolayer could be determined. The  s u b s t r a t e c o r r u g a t i o n i n f l u e n c e s the way  the m i s f i t  o r i e n t a t i o n of the monolayer v a r i e s with the temperature chemical  p o t e n t i a l of the system.  At constant  K,  in this  ( 0 - 4 K), the m i s f i t does not change with  temperature. 7.0  and  chemical  p o t e n t i a l , over the range of temperatures c o n s i d e r e d thesis  and  When the s u b s t r a t e c o r r u g a t i o n i s g r e a t e r than  the monolayer forms a commensurate s o l i d t h a t becomes  incommensurate when the chemical  p o t e n t i a l of the system i s  increased.  However, when the s u b s t r a t e c o r r u g a t i o n i s l e s s  than 7.0  the monolayer does not  The  K,  s i g n i f i c a n c e of t h i s r e s u l t  c o r r u g a t i o n value  of 7.4  form a commensurate  i s that the  solid.  substrate  K c a l c u l a t e d by V i d a l d i and  produces a commensurate s o l i d at zero temperature.  Cole  4 8  This  produces agreement between a t h e o r e t i c a l l y d e r i v e d value f o r the s u b s t r a t e c o r r u g a t i o n and the monolayer can  form a commensurate s o l i d .  most e n e r g e t i c a l l y favourable chemical  the experimental  evidence The  that  m i s f i t of the  c o n f i g u r a t i o n v a r i e s with  the  p o t e n t i a l as shown i n (1.1), r e g a r d l e s s of the  substrate corrugation. observations  although  T h i s r e s u l t agrees with the  i n f o r m a t i o n does not  experimental provide  c o r r o b o r a t i o n t h a t the s u b s t r a t e c o r r u g a t i o n should  be 7.4  K.  Monolayers with s m a l l m i s f i t values are not r o t a t e d , however when the m i s f i t exceeds a given t h r e s h o l d v a l u e , the monolayer  103  is rotated.  The  r e l a t i o n s h i p between the o r i e n t a t i o n of  monolayer and  its misfit  i s dependent on the  corrugation.  For s u b s t r a t e  corrugations  substrate  l e s s than 7.0  o r i e n t a t i o n a l behaviour matches t h a t p r e d i c t e d by However, when the s u b s t r a t e  corrugation  the  K,  the  Shiba . 1 1  i s greater  than 7.0  K  the anharmonic nature of the adatom p a i r i n t e r a c t i o n causes Shiba's r e s u l t s to be m i s f i t values  inaccurate.  The  monolayer i s r o t a t e d at  l e s s than those p r e d i c t e d by Shiba because  non-rotated c o n f i g u r a t i o n s have adatoms that are at the v e r t i c e s of the domain w a l l s . m i s f i t values  In f a c t ,  where the r o t a t e d c o n f i g u r a t i o n s  e n e r g e t i c a l l y favourable,  the  overcompressed  f o r a range of are  the overcompression at the v e r t i c e s  causes the non-rotated c o n f i g u r a t i o n s  to be u n s t a b l e .  The  o r i e n t a t i o n of the monolayer i s s e n s i t i v e to the temperature of the system.  As the temperature i s i n c r e a s e d ,  configurations  that are r o t a t e d w i l l become l e s s r o t a t e d . The  f r e e energy r e q u i r e d s p l i t t i n g  the  f r e e energy i n t o three components - the p o t e n t i a l energy,  the  zero  c a l c u l a t i o n f o r the  p o i n t energy, and  a temperature dependent component.  The  most s i g n i f i c a n t of these components i s the p o t e n t i a l energy which, f o r the range of temperatures considered determines the  in this  o r i e n t a t i o n behaviour of the monolayer.  study The  zero p o i n t energy of the monolayer i s not n e g l i g i b l e , and i n f l u e n c e s the r e l a t i o n s h i p between the m i s f i t of the energy c o n f i g u r a t i o n and  the chemical p o t e n t i a l .  t h i s , r e l a x a t i o n s t u d i e s and  low  lowest  Because of  temperature molecular dynamics  104  c a l c u l a t i o n s do not p r e d i c t the correct ground s t a t e of the monolayer.  The temperature dependent  component of the f r e e  energy, which i s determined by the low energy v i b r a t i o n a l modes of the monolayer,  i s not s i g n i f i c a n t  i n d e t e r m i n i n g the  c o n f i g u r a t i o n of the monolayer. The low energy modes of the incommensurate  monolayer were  c a l c u l a t e d and correspond t o motion of the domain w a l l s .  A  r e n o r m a l i z e d model based on domain w a l l s would p r o v i d e not only the s t a t i c p r o p e r t i e s of the monolayer, but the dynamical p r o p e r t i e s as w e l l .  The lowest energy mode i s the b r e a t h i n g  mode p r e d i c t e d by V i l l a i n  1 4  ,  and though t h i s mode and other  domain w a l l modes are q u i t e s o f t , the entropy a s s o c i a t e d with them does not c o n t r i b u t e s i g n i f i c a n t l y to the f r e e energy. domain w a l l v i b r a t i o n a l modes a r e grouped  The  i n t o t r i a d s with the  motion of a given mode i n the t r i a d c o r r e s p o n d i n g t o one of three fundamental forms of domain w a l l motion.  By s p e c i f y i n g  the t r i a d group and the mode of the t r i a d , the motion a s s o c i a t e d with the mode i s determined. In the course of the work f o r t h i s t h e s i s two papers were published examined  1 2  '  3 4  .  The f i r s t  of these papers, Shrimpton e t . a l .  approximations that are made to c a l c u l a t e the s t a t i c  p r o p e r t i e s of the monolayer.  The r e s u l t s of t h i s  paper  i n d i c a t e d t h a t f o r krypton on g r a p h i t e t h e o r i e s which assumed t h a t the adatom p a i r  i n t e r a c t i o n s were harmonic would not be  quantitatively correct.  The paper a l s o e s t a b l i s h e d the  v a l i d i t y of the techniques used i n t h i s work t o c a l c u l a t e the  1  2  105  s t a t i c configuration  of the monolayer.  The t h e s i s extends the  r e s u l t s presented i n t h i s paper to cases where the monolayer i s rotated  with r e s p e c t  to the s u b s t r a t e .  that harmonic t h e o r i e s  may  The r e s u l t s  not d e s c r i b e  c o r r e c t l y the r o t a t i o n  v e r s e s m i s f i t behaviour of the monolayer The second paper p u b l i s h e d , the dynamics  work there was  either.  Shrimpton e t . a l .  of the incommensurate no q u a n t i t a t i v e  monolayer.  information  by V i l l a i n  1 4  incommensurate  monolayer  was  concerned  P r i o r to t h i s in-plane  monolayer.  With t h i s  about the low energy modes, and with other f o r the zero p o i n t  ,  was v e r i f i e d and  other low energy modes were d e s c r i b e d .  calculated  3 4  about the  motion of the adatoms i n the incommensurate energy mode p r e d i c t e d  indicate  The  low  several information  information  energy, the f r e e energy of the calculated.  The r e s u l t s of the  c a l c u l a t i o n f o r the f r e e energy i n d i c a t e t h a t the low energy modes of the monolayer do not have a c a t a s t r o p h i c e x t e n s i o n of zero temperature c a l c u l a t i o n s t o temperatures.  However, the r e s u l t s a l s o  p o i n t energy of the monolayer the  influences  impact on the  finite  i n d i c a t e that the zero the c o n f i g u r a t i o n  monolayer, and c a l c u l a t i o n s based s o l e l y on the s t a t i c  properties  of the monolayer  may  not be  correct.  of  106  APPENDIX  The  monolayer  c o n f i g u r a t i o n s are  p a r a m e t e r s m and superlattice only and  n.  From e q u a t i o n  domains a r e  information this  (2.9),  vectors  R\  ,  [Ri  This to  R*  the  the  =  Ra]  2x2  From e q u a t i o n averaged  lattice  commensurate  l_CU  /  d  Referring also  be  2  The  top  contains  (2.10),  lattice  with  case  -m  m  n-m  "  row  of  6\  "l o" 0 1  (2.8)  the  the  51=1  integer  of  the i s not  f r o m m and  the n,  quantities. and  k=0  p a r a m e t e r s and  the  basis  periodicity  +  , B ]  (A.1)  vectors are  combined  £  contains  y  the  x  components  components.  the  and  vectors  2j  determined  geometric  n  the  v e c t o r s d^  back t o  other  D ]  size  directly  n o t a t i o n whereby two  matrix.  b o t t o m row  obtained  superlattice  ,  the  by  However, t h i s  between t h e s e  [Di  the  (2.9)  restricted  of the  2  introduces  form a  and  and  be  consider  produces a r e l a t i o n s h i p  determined  determined.  t h a t can  section will  From e q u a t i o n  A  help  d*  2  (2.7),  can  be  determined  D*  to  be  and  2  1  2 -1 1 1  T  misfit  of e q u a t i o n  and  i n t e r m s o f m and  n.  -i  -1  n m  -m n-m  -l  from  the the  v  o r i e n t a t i o n can  (A.2)  then  107  Given the p e r i o d i c i t y o£ the s u p e r l a t t i c e of  r e c i p r o c a l l a t t i c e v e c t o r s with b a s i s cf , q i  RE J  L J  <3i ->  R  -i  -4  -4  -4  -4  -4  -4  -4  =  q  <32  The  from  ,  R  2  i  q  ( A . l ) the s e t  are d e f i n e d by  2  L-i  Ra  oj  n o t a t i o n f o r the matrix formed by the q v e c t o r s has  (A.3)  the  l e f t column c o n t a i n i n g the x components of the v e c t o r s and r i g h t column c o n t a i n i n g the y components. cause  the i n d i c a t e d dot products between the v e c t o r s .  Given the R matrix from L  conventions  the matrix m u l t i p l i c a t i o n of the R* and q v e c t o r s e t s to  produce  q  These  the  and q )  ( A . l ) the q matrix  can be determined  2  i n equation  -4  (A.3).  -4  A  subsequent  by simple m a n i p u l a t i o n of matrices  By analogy to the p r e v i o u s procedure, vectors g  (and  and g  2  [Bj  the r e c i p r o c a l  lattice  f o r the s u b s t r a t e can be d e f i n e d so that  , B ] 2  =  2rt  0 -1  (A.4)  92  from which the r e c i p r o c a l l a t t i c e v e c t o r s of the s u b s t r a t e f o r a g i v e n m and n can be  1 .-2  -4  q  2  —  1 1.  calculated.  _  -m _n-m  -n -m_  +  "l .0  0 1.  -1  9i 92  (A.5)  108  One of the q u a n t i t i e s t h a t i s of i n t e r e s t i s the dot product between averaged l a t t i c e v e c t o r s of the adatoms and the reciprocal product  l a t t i c e v e c t o r s of the s u b s t r a t e .  Any v e c t o r dot  can be determined from the dot products  b a s i s v e c t o r s of the two l a t t i c e s . the help of (2.7) and  [di  , d ] 2  92  , d ] 2  Given equation  (A.2),  with  (A.4),  = 2rt 0 -1  n -m 1-1 m n-m  1 0  From (A.3) and (2.10) i t i s c l e a r  [di gi -» ga  between the  <3i <3  2  [di  2 1  -1 1  (A.6)  that  , d ] 2  2rt  1 -2  (A.7)  109  APPENDIX B  Much e a r l y i n t e r e s t  i n the problem of mismatched spacings  between an o v e r l a y e r of adatoms and the u n d e r l y i n g s u b s t r a t e stemmed from the f a c t t h a t the displacements,  u, of the adatoms  i n the minimum energy c o n f i g u r a t i o n , given s u i t a b l e approximations  t o the m i c r o s c o p i c  i n t e r a c t i o n s , were obtained  by s o l v i n g a second order n o n l i n e a r s e t of d i f f e r e n t i a l equations .  By c o n s i d e r i n g c o n f i g u r a t i o n s where the  displacements  occur o n l y along one d i r e c t i o n , the displacements  2 1  were found  to have s o l i t o n v a r i a t i o n .  T h i s v a r i a t i o n produces  regions where the adatoms are i n r e g i s t r y with the s u b s t r a t e , separated  by s t r i p e s of higher  (or lower) d e n s i t y domain w a l l s .  The minimum energy c o n f i g u r a t i o n i s determined the c l a s s i c a l interactions.  f o r c e f r e e s o l u t i o n t o the m i c r o s c o p i c The adatam i n t e r a c t i o n $ i s assumed t o be  harmonic and the s u b s t r a t e i n t e r a c t i o n sinusoidal  by f i n d i n g  i s assumed to be  ( c f 2.3). A c c o r d i n g t o these c o n d i t i o n s the energy  per adatom i s  E(R)  =  A L 1 + (u-u')-V + Vz (. (u-u').7.) J *(R-R') R' 3 o -+ -* (B.l) + V + 2V £ ( 1 " c o s ( g - R ) ) k= l D  g  3  i s d e f i n e d from g  A  g  and g  2  The continuum approximation  k  ( c f A.4) t o be g  3  = ~q L  g  i s made which t r e a t s the  2  110  displacements as a continuous R*.  f u n c t i o n of the averaged  I f the displacements are s l o w l y v a r y i n g , the  approximation  position  following  can be made  = LI + (R'-R)-v"  u' = u(R')  With the assumption  + 'A. (. (R'-R)-V.) J u(R)  of u n i a x i a l modulations,  (B.2)  the  displacements correspond to  u(R)  where B^ Using  = ~  6(S) Bi  with  S = B\-R  (B.3)  i s a b a s i s v e c t o r f o r the s u b s t r a t e . (B.3), and  (B.2) which i s s u b s t i t u t e d  into  (B.l),  gives  E(R)  =  E(S) = E  0  where  E  and  a  = J^jz  with  b  =  Q  + tta ( § § ) ' + t> | |  = ^ #(h) h  +  V  g  (h-Bi)  (B.4a)  (B.4b)  Q  § (h-Bi) h  £  + 4 V ( l - cos8)  2  (Bi-V)  2  *(h)  (Bi-V) 0(h)  (B.4c)  (B.4d)  Ill  S i m i l a r i l y , when the f o r c e on a given adatom Is c a l c u l a t e d the  f o r c e f r e e c o n d i t i o n r e q u i r e s t h a t the displacements  satisfy  d 8  . „ sin©  2  =  . where  The s o l u t i o n s to (B.5)  oL- =  4V _^ q  (B.5)  are s o l i t o n s with the s p e c i a l case of  a s i n g l e s o l i t o n having the form  9 = 4tan~ (e* ) i  T h i s provides other  (B.6)  s  the shape of the domain w a l l s o l i t o n , with the  p o s s i b l e s o l u t i o n s , where the domain w a l l s are w e l l  separated, c o r r e s p o n d i n g to p e r i o d i c r e p e t i t i o n of the domain walls  over the s u r f a c e  large,  i n d i c a t i n g a s t r o n g l y corrugated  walls w i l l strong be very  of the monolayer.  be q u i t e sharp; c o n v e r s e l y  From (B.6),  substrate,  i f ot i s  the domain  i f oc i s s m a l l , i n d i c a t i n g  i n t e r a c t i o n s between the adatoms, the domain walls broad.  will  112  APPENDIX C  From (2.38), the maximum e r r o r approximations  i n (2.17) r e s u l t i n g  i n c a l c u l a t i n g the A^ jwf k  L  However, the q u a n t i t y of i n t e r e s t  from  can be determined.  i s the energy of the  monolayer and t h i s must be c a l c u l a t e d t o a p r e s e t accuracy. The c o n t r i b u t i o n of the s u b s t r a t e to t h i s energy average  of a l l the i n d i v i d u a l adatom-substrate  energies.  i n v o l v e s an  interaction  Thus an e r r o r bound which r e f l e c t s the u n c e r t a i n t y  in the average value of the adatom-substrate  energy must be  calculated. In  order to estimate the e r r o r , the displacements u a r e  c o n s i d e r e d to be a n a l y t i c f u n c t i o n s of the mean p o s i t i o n s R*. Two extreme cases f o r the behaviour of the displacements are c o n s i d e r e d with the assumption  that the a c t u a l r e s u l t w i l l l i e  somewhere i n between the two. First,  the domain w a l l s are c o n s i d e r e d t o be step  functions.  The displacements u v a r y l i n e a r l y with R*, up to the  edge of the domain, which f o r ease of c a l c u l a t i o n be  i s taken to  circular.  u(R) = cR  i f IR I = a  (C.l)  where c i s to be determined, and a i s the r a d i u s of the domain. The q u a n t i t y of concern to (2.38) i s  113  z = ig-u(R)  (C.2)  Performing the average  of <z > u s i n g ( C . l ) over the area of 2  the c i r c l e r e s u l t s i n  <z > =  j (aclg I)  2  (C.3)  2  The estimate of the e r r o r and  i n the energy  i s <%S>, from  (2.38)  ( C . l ) i t i s c a l c u l a t e d to be  < % 5 >  =  24  8  k  = n  lacLgJl 8  24  8  n  (C.4) with  The  (C.3)  -  < '>" 12 8 z  n  other extreme p o s s i b i l t y i s that the displacements of  adatoms are so smoothly modulated  z = i d g I s i n ( - IRI ) cos(6)  that z can be given as  f o r IRI ^ a  where 8 i s the angle between R* and g.  (C.5)  For such a case <z > i s 2  c a l c u l a t e d to be  <z > 2  =  j (acl g I )  2  (C.6)  114  and  the average e r r o r  <  %  s  >  k  =  16  (  a  c  l  24  i n <%S> f o r such a system i s  ^ '  4 n  8  n  (C.7) with (C.6)  8  8n  Because the c o n f i g u r a t i o n of the monolayer l i e s two  between the  extreme cases i t i s assumed t h a t s e t t i n g the u n c e r t a i n t y t o  be  < % 5 >  will  =  7 ^  <C.8)  overestimate the a c t u a l e r r o r present i n the c a l c u l a t e d  energy c o n t r i b u t i o n of the s u b s t r a t e . Equation can  (C.8) r e q u i r e s a value f o r <z >. 2  From (2.13) t h i s  be c a l c u l a t e d by  <z > = - £ (g-u^m* (g-u-a, Sim ' 2  allowing  )  (C9)  an estimate of <%£> t o be determined from (C.8).  The  parameter value f o r n used to c a l c u l a t e the values A^ M i s k  L  chosen from the e r r o r bound %S t o be the s m a l l e s t satisfies  integer  that  115  n i  - 6 %S log(8)  V  (C.10)  }  where <z > i s given by (C.9). 2  It  i s r e a l i z e d that <z > can be determined d i r e c t l y 4  (2.13).  However, only a quick estimate  r e q u i r e d , and (C.8) with  from  of the u n c e r t a i n t y i s  (C.9) i s adequate to determine  Furthermore, the q u a n t i t y (C.9) can be gleaned c a l c u l a t i o n s r e q u i r e d to evaluate  the A  <%S>.  from the  of (2.34) whereas <z > 4  N  would have to be c a l c u l a t e d s e p a r a t e l y . The  u n c e r t a i n t y i n the c a l c u l a t i o n  (C.8) has been  d e l i b e r a t e l y chosen to be an overestimate (C.7). A-'^LM  T h i s i s necessary will  from.(C.4) and  because the e r r o r s i n c a l c u l a t i n g the  i n f l u e n c e the o v e r a l l c a l c u l a t i o n of the  displacements  u^ . m  Since the values  of U £ a r e used t o m  c a l c u l a t e the s e t of A ^ j ^ , the e r r o r w i l l (C.4) ch oi ce  Thus  and (C.7) underestimate the consequences of a s p e c i f i c of n.  By a n a l y s i n g the observed e r r o r i n the s u b s t r a t e  energy c a l c u l a t e d f o r v a r i o u s n v a l u e s , r e l i a b l e estimate %S,  be compounded.  (C.8) i s found to be a  of the u n c e r t a i n t y , and given an e r r o r bound  (C.10) can be used t o s e t the value  of n.  116  APPENDIX D  Equation  (4.9) r e q u i r e s  i n t e g r a t i n g the p e r i o d i c  f(ic) over the B r i l l o u i n zone of the s u p e r l a t t i c e . Cohen  50  have developed  numerically.  The  function Chadi  a technique to evaluate such an  technique  i s reviewed  and  integral  here to e s t a b l i s h the  terminology. f(k*) i s a p e r i o d i c  f(k) =  X f£ R  e  function  i k  '  and  can be expanded as  (D.l)  R  where R* i s a s u p e r l a t t i c e v e c t o r generated  by  (2.9).  The  symmetry of the s u p e r l a t t i c e a l l o w s the c o e f f i c i e n t s f ^ to be combined so that  I  f(k) =  (D.l) becomes  f  m  m  A (k)  (D.2)  m  where A^k)  =  ^ £ e " l=^m  l k  '  (D.3)  R  R  The v a l u e s of c  m  range over a l l p o s s i b l e  ordered to i n c r e a s e with m. coefficients A weighted  f  m  i n (D.2)  v a l u e s of  If f(k) varies  w i l l decrease  IRI and  are  smoothly with k the  r a p i d l y with  m.  summation of the A (j?) evaluated over a s e t of m  117  points  can be performed so t h a t  2  oC  i  \  i  ( k  i  )  =  0  (D.4)  for a l l m i n the range 0 < m £ N .  The weights oc^ a r e  normalized  so t h a t they t o t a l to u n i t y .  Given  (D.2),  performing  the same weighted sum on the f u n c t i o n f(i?) w i l l  result in  ^c^f'kj) i  Since  f  m  =  f  +  I m>N  f  m  Ioi A [k ) i i  m  Q  as N i s i n c r e a s e d .  f(k*) i n the B r i l l o u i n  2, 04 f ( k i )  =  zone.  |J  to i n c r e a s i n g accuracy For a g i v e n value satisfy  (D.4).  f  Q  i s the average value of  Thus,  f ( k ) dk  (D.6)  as N i s i n c r e a s e d .  of N there are many s e t s of p o i n t s k^ t h a t The p o i n t s i n the minimal s e t a r e c a l l e d  the s p e c i a l p o i n t s of the B r i l l o u i n zone. i n c r e a s e d , the number of s p e c i a l p o i n t s Cohen points  50  (D.5)  i  decreases r a p i d l y with m, the r i g h t hand s i d e of (D.5)  w i l l approach f  will  Q  have d e v i s e d  As the value  increases.  of N i s  Chadi and  a procedure whereby a s e t of s p e c i a l  i s generated by p o i n t group o p e r a t i o n s  on the p o i n t s i n  118  the two  preceding  sets.  Cunningham  01  has  few s e t s of s p e c i a l p o i n t s a p p r o p r i a t e l a t t i c e with hexagonal symmetry. points,  c a l c u l a t e d the  a two  dimensional  By examining these s p e c i a l  i t i s c l e a r that when a given s e t of p o i n t s  with a l l the preceding  i s combined  s e t s of p o i n t s , a t r i a n g u l a r l a t t i c e i s  formed.  Furthermore, none of the  than one  s e t of s p e c i a l p o i n t s .  s p e c i a l points  first  l a t t i c e p o i n t s belong to more When the next s u c c e s i v e  set of  i s i n c l u d e d , a more dense t r i a n g u l a r l a t t i c e  formed of which the previous  is  combined s e t forms a / 3 x / 3  sublattice. For the c a l c u l a t i o n of this fact.  (4.9), advantage has  A triangular lattice  i s generated with  s m a l l enough t h a t the s p e c i a l p o i n t s d e r i v e d provide  the r e q u i r e d accuracy.  extracted set  been taken of  The  spacing  from i t w i l l  s p e c i a l points  are  from the t r i a n g u l a r l a t t i c e by s u b t r a c t i n g from t h i s  a l l p o i n t s t h a t are common with  the  ' 3 x / 3 sublattice.  The  weight oi^ a s s o c i a t e d with t h i s s e t can be determined from the successive for  one  nature of the s e t s of s p e c i a l p o i n t s .  s e t w i l l be 1/3  the weight of the previous  knowing the number of preceding be c a l c u l a t e d .  The  weight  set.  By  s e t s of p o i n t s , the weight  can  I t i s remembered t h a t the weight a s s o c i a t e d  with p o i n t s at the edge of the B r i l l o u i n zone must be  (  modified.  119  F i g u r e 1. Diagram of the g r a p h i t e s u b s t r a t e . The honeycomb network i n d i c a t e s the carbon bonds, the a d s o r p t i o n s i t e s are a t the c e n t e r s of the hexagons formed by the c a r b o n bonds. and B a r e t^e basis^ v e c t o r s w h i c h g e n e r a t e t h e s e t o f a d s o r p t i o n sites. D and D are the basis v e c t o r s for the /3X/3 sublattice. 2  A  £  H  1-42 A I —  120  F i g u r e 2. Types of domain w a l l s between the '3x/3 l a t t i c e s of krypton atoms. Krypton atoms i n an a s u b l a t t i c e are i d e n t i f i e d by •. Atoms i n a b s u b l a t t i c e are i d e n t i f i e d by a •. Atoms i n a c s u b l a t t i c e are i d e n t i f i e d by 1 .  121  F i g u r e 3. T y p i c a l p o s i t i o n s of adatoms f o r an incommensurate monolayer with hexagonal symmetry. For t h i s p a r t i c u l a r c o n f i g u r a t i o n n=4, m=-4, k = 0, and *, = -! i n equation (2.9).  122  F i g u r e 4. P o s s i b l e m i s f i t and o r i e n t a t i o n values f o r hexagonally p e r i o d i c incommensurate monolayers. The angle of r o t a t i o n i s given l n degrees. The monolayers are c o n s t r a i n e d so t h a t , over the length of the p r i m i t i v e l a t t i c e v e c t o r s , the adatoms are s h i f t e d by only one a d s o r p t i o n s i t e r e l a t i v e to the commensurate monolayer. The c o n f i g u r a t i o n s with super heavy domain w a l l s are marked with c i r c l e s , the c o n f i g u r a t i o n s with heavy domain w a l l s are marked with squares.  q  B  in  q  El  in  B  B B  < B  •  • *4 • *  d  •  •  •  •  *  .•.*•/.'•*•*•'*  .*.*.• «««....«.  q d.  mm---.-:-.-.,-«6'6'6|' 6 1 o • o *|  1.5  2.0  2.5  3.0  3.5  % Misfit  4.0  4.5  5.0  123  Figure 5(a). Energy per adatom of a n o n r o t a t e d Incommensurate m o n o l a y e r a s a f u n c t i o n o f i t s m i s f i t when t h e substrate c o r r u g a t i o n i s 3.0 K. The d a s h e d l i n e shows t h e potential e n e r g y p e r a d a t o m ; t h e s o l i d l i n e shows t h e p o t e n t i a l a n d z e r o p o i n t e n e r g y per adatom as i s d i s c u s s e d i n c h a p t e r 3.  -300-1  % Misfit  124  125  Figure 5(c).  As l n f i g u r e 5(a) but with V  -300-1  % Misfit  = 7 . 0 K.  126  F i g u r e 5(d).  As i n f i g u r e 5(a) but with V„ = 8.0 K.  -350-1  % Misfit  127  Figure 5(e).  As i n f i g u r e 5(a) but with V  = 9.0 K. 9  -300  n  % Misfit  128  Figure 6(a). Contour p l o t showing the d i f f e r e n c e i n the p o t e n t i a l energy per adatom between the r o t a t e d and nonrotated configurations. The angle of r o t a t i o n i s g i v e n i n degrees. The s o l i d contours show increments of 0.5 K. The d o t t e d and dashed contours show decrements of 0.02 K. The s u b s t r a t e c o r r u g a t i o n f o r t h i s p l o t i s 5.0 K.  o  % Misfit  129  Figure 6(b). As l n f i g u r e 6(a) c o r r u g a t i o n of 8.0 K.  but with a  o  % Misfit  substrate  130  Figure 6(c). As i n f i g u r e 6(a) c o r r u g a t i o n of 10.0 K.  but with a  o  % Misfit  substrate  131  Figure 7(a). A comparison of the adatom p a i r p o t e n t i a l #(r) ( s o l i d l i n e ) and the polynomial f i t a r + b r + c r + d (dashed l i n e ) f o r v a r i o u s values of r . 3  2  -154 -i  Distance between adatoms (A)  132  Figure 7(b). A comparison of the d e r i v a t i v e s of the curves i n f i g u r e 7(a) f o r v a r i o u s values of r . The s o l i d l i n e i s £'(r), the dashed l i n e i s 3ar +2br+c. 2  150-1  Distance between adatoms (A)  4.3  133  Figure 7(c). A comparison of 0 " ( r ) ( s o l i d l i n e ) and polynomial f i t 6ar+2b (dotted l i n e ) f o r v a r i o u s values  1200  n  the of r .  134  Figure 7(d). A comparison of ^ " ' ( r ) ( s o l i d l i n e ) and the polynomial f i t 6a (dotted l i n e ) f o r v a r i o u s values of r .  -1500-1  -5500 H 4  1 4.1  1 4.2  Distance between adatoms (A)  1 4.3  135  F i g u r e 8. Energy per adatom as i n f i g u r e 5(c) except i n the c a l c u l a t i o n , the p a i r p o t e n t i a l £(r) i s r e p l a c e d by the polynomial f i t ar^+br +cr+d shown i n f i g u r e 7 ( a ) . 2  - 3 5 0 -i  % Misfit  136  Figure 9 . Phonon energies as a f u n c t i o n of wavevector f o r the nonrotated incommensurate monolayer with m i s f i t 2 . 2 2 % , and substrate corrugation 6 . 0 K. r i s at the center of the hexagonal B r i l l o u i n zone, M i s a t the middle of an edge and K is at a corner.  0.35-,  0.30-  0.25 H Q>  0.20 H  P  CD C LU C  coo _c  CL  0.15 H  0.10 0.05  0.00  H J  M  T  K M  Misfit 2.22% Angle 0.0°  137  Figure 10(a). M o t i o n o £ t h e d o m a i n w a l l s f o r t h e p h o n o n mode a t the p o i n t M of f i g u r e 9, h a v i n g an e n e r g y of O . O l l m e V . In t h i s p i c t u r e , the d i r e c t i o n of the wavevector i s a s shown i n the B r i l l o u i n zone s u b d i a g r a m . The d a s h e d l i n e s i n d i c a t e the d e f o r m a t i o n of the w a l l s about t h e i r c e n t r a l p o s i t i o n s (solid lines).  Phonon  Energy=0.01 lmeV  Misf it=2.22 %  Rotation=0.00°  Substrate Corrugation V =6.0K Q  Wave Vector  138  Figure 10(b). As i n f i g u r e 1 0 ( a ) but f o r a phonon mode with energy 0 . 0 2 1 m e V .  Phonon Energy=0.021meV Misfit=2.22 %  Rotation=0.00  0  Substrate Corrugation V =6.0K a  Wave Vector  139  Figure 10(c). As i n f i g u r e 10(a) but f o r a phonon mode with energy 0.027meV.  Phonon Energy=0.027meV Misf it=2.22 %  Rotation=0.00  0  Substrate Corrugation V =6.0K a  Wave Vector  140  F i g u r e 11. E n e r g i e s of the phonon modes as i n f i g u r e 9 but f o r a nonrotated monolayer with m i s f i t 1.75% . At t h i s value of m i s f i t the t r i a d groupings of the domain w a l l s are apparent  0.35 - i  0.30-  0.25-  0.20C LaJ C  o c o  0.15-  0.10-  V =6.0K g  0.05-  0.00  J  M  r  K  Misfit 1.75% Angle 0.0°  M  141  Figure 12(a). Motion of the domain w a l l s as i n f i g u r e 10(a) but f o r a phonon mode with wavevector a t the r p o i n t and energy 0.037meV. From f i g u r e 9 t h i s mode i s the t h i r d lowest i n energy and i s i d e n t i f i e d as being the t h i r d mode of the f i r s t triad.  Scale Y  10  nM  Phonon Energy=0.037meV Misf it=2.22 %  Rotation=0.00 •  Substrate Corrugation Vg=6.0K  Wave Vector  142  F i g u r e 12(b). As i n f i g u r e 12(a) but f o r a phonon mode with energy 0.315meV. This mode i s the s i x t h lowest i n energy and corresponds to the t h i r d mode of the second t r i a d .  \  /  /  /  Scale r-  10 nM  Phonon Energy=0.315 meV Misfit=2.22%  Rotation=0.00°  Substrate Corrugation Vg=6.0K  Wave Vector  143  Figure 13(a). Motion o£ the domain w a l l s as l n f i g u r e 10(a) but f o r a phonon mode having energy 0;201meV. From f i g u r e 9 t h i s mode i s the f i r s t mode of the second t r i a d .  Phonon Energy=0.201 meV Misf it=2.22 %  Rotation=0.00  0  Substrate Corrugation V =6.0K g  W  a  v  e  V  e  c  t  o  r  144  F i g u r e 13(b). As i n f i g u r e 13(a) but showing motion of the mode when the wavevector i s a t r .  Scale h  10 nM  Phonon Energy=0.201 meV Misfit=2.22%  Rototion=0.00°  Substrate Corrugation Vg=6.0K  Wave Vector  145  F i g u r e 14(a). Phonon energies as a f u n c t i o n of wavevector as i n f i g u r e 9 but f o r a nonrotated monolayer with m i s f i t 3.33% when the s u b s t r a t e c o r r u g a t i o n i s 8.0 K. The energy s c a l e i s extended t o show imaginary v a l u e s . The monolayer has an unstable mode.  0.35 n « - .  0.30-  0.25-  .E, P?  CD C LU C  o c o  JCCL  0.20-  V =8.0K  0.15-  g  0.10-  0.05. « •  0.00-  i0.05  M  K • M  J  Misfit 3.33% Angle 0.0*  146  F i g u r e 14(b). As i n f i g u r e 14(a) but f o r a r o t a t e d monolayer with s l i g h t l y s m a l l e r m i s f i t . The modes are not u n s t a b l e .  0.35->  0.30-  0.25-  0.20-  p? CD  c c o c o  LU  V =8.0K  0.15-  g  0.10•  *  • •  0.05-  0.00  J  M  r  K M  Misfit 3.29% Angle 0.225"  147  F i g u r e 15. Motion of the domain w a l l s f o r the unstable phonon mode of f i g u r e 14(a) with wavevector a t the M p o i n t .  Phonon Energy= 10.026 meV Misfit=3.33%  Rototion=0.00°  Substrate Corrugation V =8.0K Q  Wave Vector  148  F i g u r e 16. Phonon e n e r g i e s as a f u n c t i o n of wavevector as i n f i g u r e 14(a), but f o r a monolayer with a m i s f i t value of 3.7% ,  0.35-.  0.30-  0.25-  .E, 0.20<D C LU C  o c o  V =8.0K  0.15-  g  0.10. • •  0.05-  0.00  J  M  T  K M  Misfit 3.70% Angle 0.0°  149  REFERENCES L.W.  Bruch, Surf. S c i . 125. 194  (1983).  For a review see R. Marx, Phys. Rep.  125./ p8-9  (1985).  A. Thorny and X. Duval, J . Chim. Physique 66_, 1966 (1969) A. Thorny and X. Duval, J . Chim. Physique 67., 286 (1970). A. Thorny and X. Duval, J . Chim. Physique 6_7_, 1101  (1970)  H.M. Kramer and J . Suzanne, S u r f . S c i . 54, 659 (1976). C. M a r t i , B. Croset, P. T h o r e l and J.P. Coulomb, S u r f . S c i . 65_, 532 (1977). M.D. Chinn and S.C. F a i n , J r . , Phys. Rev. L e t t . 39_, 146 (1977) . T. Ceva and C. M a r t i , J . 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