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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 NEIL D. SHRIMPTON B. Sc., P h y s i c s , U n i v e r s i t y of V i c t o r i a 1978 M. Sc., The U n i v e r s i t y of B r i t i s h Columbia 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1987 © N e i l Douglas Shrimpton, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date H~ 2<T- ? 7 DE-6G/81) 11 ABSTRACT The low tem p e r a t u r e b e h a v i o u r of a system of incommensurate k r y p t o n monolayers on g r a p h i t e was a n a l y s e d . The f r e e energy was c a l c u l a t e d f o r a v a r i e t y of monolayer c o n f i g u r a t i o n s and the m i s f i t and o r i e n t a t i o n of the minimum energy c o n f i g u r a t i o n d e t e r m i n e d as a f u n c t i o n of the temperature and c h e m i c a l p o t e n t i a l of the system. The f r e e energy d i d not v a r y s i g n i f i c a n t l y over the temperature range from 0 K t o 4 K. The z e r o p o i n t 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 t o the f r e e energy and c o u l d not be n e g l e c t e d . The l o w e s t energy 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 motion of the domain 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 the 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 groups of t h r e e ( t r i a d s ) . In the l o w e s t energy t r i a d , the l o w e s t energy mode was a c o m p r e s s i o n a l mode. The second and t h i r d l o w e s t modes were s h e a r i n g modes. These t h r e e modes d e s c r i b e the fundamental forms of domain w a l l motion. The modes of the h i g h e r energy t r i a d s a r e more e n e r g e t i c forms of the l o w e s t energy t r i a d . i i i TABLE OF CONTENTS TITLE PAGE i ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES '. v i LIST OF FIGURES v i i ACKNOWLEDGEMENT X 1. INTRODUCTION 1 (I-) MOTIVATION ( I I ) FORMALISM 10 2. POTENTIAL ENERGY 14 ( I ) MICROSCOPIC CONSIDERATIONS 14 ( I I ) SIMPLIFICATIONS 23 ( I I I ) METHOD OF SOLUTION 30 (IV) NUMERICAL APPROXIMATIONS 34 (V) CALCULATIONAL PROCEEDURE 4 3 (VI) RESULTS AND DISCUSSION 47 i v 3. DYNAMICAL ASPECTS 54 ( I ) BACKGROUND 54 ( I I ) HARMONIC APPROXIMATION 59 ( I I I ) ZERO POINT ENERGY 65 (IV) CALCULATION 69 (V) RESULTS AND DISCUSSION 73 4. LOW ENERGY PHONON MODES 7 7 ( I ) HELMHOLTZ FREE ENERGY 7 7 ( I I ) NORMAL MODE CALCULATION ..79 ( I I I ) REVIEW OF PREVIOUS WORK 84 (IV) QUANTIZED NATURE OF WALL MOTION 86 (V) DOMAIN WALL INSTABILITIES 88 (VI) FREE ENERGY CALCULATION 91 ( V I I ) RESULTS AND DISCUSSION 94 5. CONCLUSION 101 V APPENDIX A: GEOMETRICAL PROPERTIES 106 APPENDIX B: SOLITON EQUATIONS 109 APPENDIX C: SUBSTRATE ENERGY UNCERTAINTY 112 APPENDIX D: NUMERICAL INTEGRATION METHOD 116 FIGURES 119 REFERENCES 149 v i LIST OF TABLES Table 1. E T v a l u e s f o r a m i s f i t of 3.7% 95 Table 2. E T v a l u e s f o r a m i s f i t of 3.0% 95 Table 3. E T v a l u e s f o r a m i s f i t of 2.8% 96 v i i LIST 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 F i g u r e 2. Types of domain w a l l s 120 F i g u r e 3. T y p i c a l adatom p o s i t i o n s an incommensurate monolayer w i t h hexagonal symmetry 121 F i g u r e 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 F i g u r e 5a. Energy per adatom when Vg = 3. OK 123 F i g u r e 5b. Energy per adatom when V g = 5.OK 124 F i g u r e 5c. Energy per adatom when Vg = 7. OK 125 F i g u r e 5d. Energy per adatom when Vg = 8.OK 126 F i g u r e 5e. Energy per adatom when V g = 9. OK 127 F i g u r e 6a. Energy 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 w i t h r o t a t i o n V g = 8.OK ..129 F i g u r e 6c. Energy 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 .130 F i g u r e 7a. A comparison of the adatom p a i r p o t e n t i a l <Mr) and the p o l y n o m i a l f i t a r 3 + b r 2 + c r + d ...131 v i i i F i g u r e 7b. A comparison of 0 ' ( r ) and 3 a r 2 + 2br+c 132 F i g u r e 7c. A comparison of # 1 ' ( r ) and 6ar + 2b 133 F i g u r e 7d. A comparison of #'''(r) and 6a 134 F i g u r e 8. Energy per adatom when V q = 7.OK when 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 p o l y n o m i a l a r 3 + b r 2 + c r + d 135 F i g u r e 9. D i s p e r s i o n c u r v e s V g = 6.OK, M% = 2.22% ....136 F i g u r e 10a. Domain w a l l motion w i t h energy O.OllmeV ....137 F i g u r e 10b. Domain w a l l motion w i t h energy 0.021meV ....138 F i g u r e 10c. Domain w a l l motion w i t h energy 0.027meV ....139 F i g u r e 11. D i s p e r s i o n c u r v e s V g = 6.OK, M% = 1.75% ....140 F i g u r e 12a. Domain w a l l motion w i t h energy 0.037meV ....141 F i g u r e 12b. Domain w a l l motion w i t h energy 0.315meV ....142 F i g u r e 13a. Domain w a l l motion w i t h energy 0.201meV ....143 F i g u r e 13b. Domain w a l l motion w i t h energy 0.201meV ....144 F i g u r e 14a. D i s p e r s i o n c u r v e s V g = 8.OK, M% = 3.33% ....145 F i g u r e 14b. 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 I X F i g u r e 15. Domain w a l l motion of u n s t a b l e mode 147 F i g u r e 16. D i s p e r s i o n c u r v e s V g = 8.OK , M% = 3.7% ....148 X ACKNOWLEDGEMENTS I would l i k e t o thank B i r g e r 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 the p e r i o d of t h i s t h e s i s . The programs f o r the t h e s i s were d e v e l o p e d on the computer systems a t both U.B.C. and the U n i v e r s i t y of Ottawa, and I am g r a t e f u l f o r the f i n a n c i a l s u p p o r t p r o v i d e d t h r o u g h NSERC a t both 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 produced u s i n g a word p r o c e s s i n g program de v e l o p e d by B i l l Unruh and I am g r a t e f u l t o him f o r making the program a v a i l a b l e t o me. I would a l s o l i k e 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 the y e a r s t a k e n t o produde t h i s t h e s i s . 1 1. INTRODUCTION ( I ) MOTIVATION Rare gas p h y s i s o r b e d systems have been e x t e n s i v e l y s t u d i e d because of the two d i m e n s i o n a l (2D) n a t u r e of the s u r f a c e f i l m . E x p e r i m e n t a l r e s u l t s have i n d i c a t e d a l a r g e number of p o s s i b l e 2D phases and t h i s has prompted c o n s i d e r a b l e i n t e r e s t . One such system i s k r y p t o n on g r a p h i t e . At low t e m p e r a t u r e s , g r a p h i t e 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. In a d d i t i o n t o the gas and l i q u i d phases of the monolayer, a commensurate and an incommensurate s o l i d phase o c c u r s . The purpose of t h i s work i s t o s t u d y the incommensurate phase of k r y p t o n on g r a p h i t e . K r y p t o n i s a r a r e gas 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 weakly w i t h i t s s u r r o u n d i n g environment. The i n t e r a c t i o n s r e s u l t from f l u c t u a t i o n s of the a t o m i c moments (London - van der Waals f o r c e s ) . S i n c e no charge exchange o c c u r s , t h r e e body e f f e c t s can be t r e a t e d as p e r t u r b a t i o n s . The f o r c e s p r e s e n t a t the at o m i c ( m i c r o s c o p i c ) l e v e l can thus be a s c r i b e d e i t h e r t o the s u b s t r a t e or t o the i n d i v i d u a l n e i g h b o u r i n g k r y p t o n atoms. Moreover, because r a r e gas atoms a r e s p h e r i c a l , from a t h e o r e t i c a l p o i n t of view, the number of parameters r e q u i r e d t o d e s c r i b e the m i c r o s c o p i c n a t u r e of the system i s reduced. An 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 p r e s e n t i n a p h y s i s o r b e d system i s p r o v i d e d by B r u c h 1 . 2 E x p e r i m e n t a l l y , e x f o l i a t e d g r a p h i t e s u b s t r a t e s w i t h many l a r g e , d e f e c t f r e e s u r f a c e s can be p r e p a r e d 2 . Large s u r f a c e a r e a s m i n i m i z e the impact of f i n i t e s i z e on the s t u d y of monolayer phase t r a n s i t i o n s . The g r e a t number of such s u r f a c e s i n e x f o l i a t e d g r a p h i t e m a g n i f i e s the measurable p r o p e r t i e s of the f i l m s . F u r t h e r m o r e , by c o n t r o l l i n g the t e m p e r a t u r e of the system and the vapour p r e s s u r e of the gas above the f i l m s the monolayer can be d r i v e n between i t s v a r i o u s phases. V o l u m e t r i c s t u d i e s 3 observed a 2D gas phase, a 2D l i q u i d phase, and I n d i c a t e d the p o s s i b i l i t y of two 2D s o l i d phases'*. E a r l y d i f f r a c t i o n s t u d i e s 5 found o n l y one s o l i d phase where the adatoms formed a t r i a n g u l a r l a t t i c e w i t h s p a c i n g of 4.26 A. T h i s s p a c i n g i s not f a v o u r e d by k r y p t o n atoms as t h e i r i n t e r a c t i o n energy i s m i n i m i z e d f o r a t r i a n g u l a r l a t t i c e w i t h a s m a l l e r s p a c i n g . (From the form of the adatom i n t e r a c t i o n used i n t h i s t h e s i s the s p a c i n g i s c a l c u l a t e d t o be 4.02 A.) A s p a c i n g of 4.26 A r e f l e c t s the f a c t t h a t the s u b s t r a t e does not e x e r t a u n i f o r m a t t r a c t i o n over i t s s u r f a c e . The v a r i a t i o n i n 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 s i t e s on the g r a p h i t e t o be f a v o u r e d ( F i g u r e 1 ) . The bond l e n g t h of the carbon atoms i n the g r a p h i t e p l a n e i s 1.42 A. S i n c e the a d s o r p t i o n s i t e s a r e a t the c e n t e r s of the hexagons formed by the c a r b o n bonds, t h e y form a t r i a n g u l a r l a t t i c e w i t h s p a c i n g / 3 x l . 4 2 A. The observed s p a c i n g of the monolayer r e f l e c t s the f a c t t h a t ^3x1.42 A i s t o o c l o s e a p a c k i n g f o r the Kr atoms t o f i l l e v e r y a d s o r p t i o n s i t e . I n s t e a d , the atoms choose t o f i l l e v e r y t h i r d 3 a d s o r p t i o n s i t e so t h a t t h e i r p o s i t i o n s form a t r i a n g u l a r s u b l a t t i c e w i t h s p a c i n g 4.26 A. T h i s phase, which i s commensurate w i t h the s u b s t r a t e , i s known as the commensurate or the / 3 X / 3 r e g i s t e r e d phase. F u r t h e r e x p e r i m e n t a l w o r k 6 , however, r e v e a l e d a second s o l i d phase which i s c h a r a c t e r i z e d by a s h i f t of the main d i f f r a c t i o n peak away from i t s commensurate l o c a t i o n . The s h i f t r e s u l t s from a d i s c r e p a n c y , o t h e r w i s e known as the m i s f i t , between the averaged adatom s p a c i n g i n t h i s incommensurate phase and t h a t of the commensurate phase. The p r e c i s e d e f i n i t i o n of the m i s f i t i s p r o v i d e d by e q u a t i o n 2.8. The incommensurate phase i s more dense t h a n the commensurate phase. The incommensurate . monolayer can 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 the commensurate c o n f i g u r a t i o n 7 ; the degree of r o t a t i o n i s termed the Q o r i e n t a t i o n a l e p i t a x y 0 . S a t e l l i t e peaks are observed and t h e i r p o s i t i o n s i n d i c a t e t h a t the adatoms w i t h i n the monolayer have a hexagonal s u p e r l a t t i c e s t r u c t u r e . The monolayer can be d r i v e n from the commensurate phase t o the incommensurate phase by i n c r e a s i n g the vapour p r e s s u r e of the gas above the monolayer. The m i s f i t , M%, i s observed t o v a r y w i t h the c h e m i c a l p o t e n t i a l M as M% oc ( M - P c ) * (1.1) where £ - 1/3 and M c i s the c h e m i c a l p o t e n t i a l of the q commensurate phase . T h i s r e l a t i o n s h i p i s found t o be 4 independent of the temp e r a t u r e of the system . For s m a l l m i s f i t s the monolayer i s not r o t a t e d w i t h r e s p e c t t o the s u b s t r a t e ; however, f o r m i s f i t s g r e a t e r than 3.5% the monolayer i s r o t a t e d 1 0 . 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 the b e h a v i o r of the incommensurate monolayer. A c a l c u l a t i o n was performed by Novaco and McTague® f o r a system where the s u b s t r a t e i n t e r a c t i o n i s n e a r l y u n i f o r m a c r o s s i t s s u r f a c e . For t h i s system, the s u b s t r a t e can be c o n s i d e r e d t o be a p e r t u r b i n g i n f l u e n c e on the p o s i t i o n s of the adatoms and the 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 of t h i s c a l c u l a t i o n i n d i c a t e t h a t the incommensurate monolayer w i l l be r o t a t e d w i t h r e s p e c t t o the s u b s t r a t e . For k r y p t o n on g r a p h i t e , however, the s u b s t r a t e i n t e r a c t i o n i s s i g n i f i c a n t l y modulated and as shown 11 by S h i b a x , i t s i n f l u e n c e can not be r e g a r d e d as b e i n g l i n e a r . S h i b a ' s c a l c u l a t i o n s p r e d i c t e d t h a t the monolayer would form a u n i a x i a l phase as the d e n s i t y i n c r e a s e s from i t s commensurate v a l u e . With a f u r t h e r i n c r e a s e i n the d e n s i t y , the monolayer w i l l form a hexagonal phase t h a t be a l i g n e d and then w i l l s t a r t t o r o t a t e w i t h r e s p e c t t o the s u b s t r a t e when the d e n s i t y exceeds a t h r e s h o l d v a l u e . A l t h o u g h the r o t a t i o n a l b e h a v i o u r p r e d i c t e d by S h i b a matched the observed m i s f i t - d e p e n d e n t o r i e n t a t i o n of the incommensurate monolayer- 1- 0, the u n i a x i a l phase has never been observed e x p e r i m e n t a l l y . S h i b a ' s t h e o r y c o n s i d e r e d the adatom 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 the r e s u l t s 1 2 . 5 C a l c u l a t i o n s f o r the energy of the s t a t i c c o n f i g u r a t i o n t h a t do not r e q u i r e a p p r o x i m a t i o n s t o the adatom p a i r i n t e r a c t i o n 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 p o s i t i o n s of the adatoms a r e s h i f t e d u n t i l a s t a t i c f o r c e - f r e e c o n f i g u r a t i o n i s a c h i e v e d . Gooding e t a l . 1 3 have performed r e l a x a t i o n s t u d i e s f o r the n o n - r o t a t e d c o n f i g u r a t i o n s of k r y p t o n monolayers and c o n c l u d e t h a t the c o n f i g u r a t i o n w i t h l o w e s t energy per adatom i s dependent on the magnitude of the s u b s t r a t e c o r r u g a t i o n . 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 1 1 . 0 K, the monolayer w i l l be r e g i s t e r e d w i t h the s u b s t r a t e , o t h e r w i s e the c o n f i g u r a t i o n w i l l be incommensurate. The incommensurate c o n f i g u r a t i o n has r e g i o n s where the adatoms ar e r e g i s t e r e d w i t h the s u b s t r a t e t h a t a r e s e p a r a t e d by h i g h e r d e n s i t y domain w a l l s . These c o n f i g u r a t i o n s can have a hexagonal or u n i a x i a l ( s t r i p e d ) symmetry. For v a l u e s of the s u b s t r a t e c o r r u g a t i o n l a r g e enough f o r the monolayer t o be r e g i s t e r e d a t z e r o t e m p e r a t u r e , the monolayer upon c o m p r e s s i o n w i l l form a s t r i p e d p h a s e x . The s t r i p e d phase i s not observed e x p e r i m e n t a l l y , t h e r e f o r e the h i g h e r e n t r o p y of the domain w a l l s i n the hexagonal c o n f i g u r a t i o n ^ ^ has been e x p e c t e d t o be s i g n i f i c a n t enough t o 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 of the monolayer. M o l e c u l a r dynamic c a l c u l a t i o n s by Abraham e t a l . ^ - 5 - 1 - 0 ' d e s c r i b e the c l a s s i c a l b e h a v i o u r of the system a t f i n i t e t e m p e r a t u r e s . The p e r i o d i c boundary c o n d i t i o n s were chosen so t h a t the monolayer c o u l d form e i t h e r a u n i a x i a l or a hexagonal 6 s u p e r l a t t i c e s t r u c t u r e . No s t r i p e d phases 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 not be r u l e d out f o r c o n f i g u r a t i o n s c l o s e t o the commensurate t r a n s i t i o n 1 7 . The monolayer was found t o form a honeycomb domain w a l l network t h a t approached 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 c o v e r a g e s 1 7 . The r e l a x a t i o n and m o l e c u l a r dynamics c a l c u l a t i o n s are l i m i t e d 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 hence cannot e v a l u a t e the b e h a v i o r of the monolayer t h r o u g h the t r a n s i t i o n t o a commensurate phase. The r e s u l t s of t h e s e s t u d i e s can, however, be extended t o the commensurate limit 1® by u s i n g a p h e n o m e n o l o g i c a l d e s c r i p t i o n of the system based on the r e n o r m a l i z e d system 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 of the monolayer i n terms of domain w a l l s t r u c t u r e s f o l l o w e d from e a r l y work on s o l i t o n systems by F r e n k e l and K o n t o r o v a 2 0 . I n t e r a c t i o n s p r e s e n t w i t h i n the monolayer can p r o v i d e a system of e q u a t i o n s which d e t e r m i n e the p o s i t i o n s of the adatoms (Appendix B ) . o 1 Frank and van der Merwe^ x r e c o g n i z e d t h a t , f o r systems where the adatom seeks t o have a s p a c i n g d i f f e r e n t from t h a t 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 can have s o l i t o n s o l u t i o n s . These s o l i t o n s produce domain w a l l s which s e p a r a t e r e g i o n s where the adatoms a r e r e g i s t e r e d . The energy of the c o n f i g u r a t i o n can be d e t e r m i n e d e n t i r e l y from the s e p a r a t i o n of the domain 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 of the system based on the p o s i t i o n s of the domain w a l l s i s p o s s i b l e . When 7 t h i s r e n o r m a l i z e d d e s c r i p t i o n was extended p h e n o m e n o l o g i c a l l y t o i n c l u d e w a l l i n t e r s e c t i o n s 1 ^ , h e x a g o n a l l y symmetric phases where the domain w a l l s form a honeycomb network c o u l d a l s o be a n a l y z e d . An e x c e l l e n t r e v i e w of t h i s t h e o r y i s p r o v i d e d by V i l l a i n and G o r d o n 2 2 . The monolayer i s e x p e c t e d t o form a s t r i p e d incommensurate phase i f the i n t e r s e c t i o n of the w a l l s i s e n e r g e t i c a l l y u n f a v o u r a b l e . The monolayer w i l l t hen have a c o n t i n u o u s t r a n s i t i o n from the commensurate phase t o a s t r i p e d incommensurate phase, which i s f o l l o w e d by a hexagonal phase as the d e n s i t y i s i n c r e a s e d . I f w a l l i n t e r s e c t i o n s a r e 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 w i l l not form a s t r i p e d phase and the t r a n s i t i o n from the commensurate phase t o the hexagonal incommensurate phase w i l l be f i r s t o r d e r . As noted above, the s t r i p e d phase has never been o b s e r v e d . E x p e r i m e n t a l e v i d e n c e by N i e l s e n e t a l . J s u g g e s t s t h a t the t r a n s i t i o n i s f i r s t o r d e r , a l t h o u g h t h i s 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 2 ^ s u g g e s t s t h a t the monolayer has a c o n t i n u o u s t r a n s i t i o n from a commensurate t o a hexagonal incommensurate c o n f i g u r a t i o n , a l t h o u g h near the t r a n s i t i o n the domain w a l l s t r u c t u r e s appear t o become d i s o r d e r e d 2 5 . T h i s d i s o r d e r i n g of the domain w a l l s t r u c t u r e s has been d i s c u s s e d by Coppersmith e t a l . 2 ^ . T h e i r c a l c u l a t i o n s i n d i c a t e t h a t b oth the s t r i p e d and honeycomb domain w a l l networks a r e s u c c e p t i b l e t o 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 s t r u c t u r e . T h e o r i e s have been d e v e l o p e d 2 7 which d e s c r i b e c o n f i g u r a t i o n s 8 w i t h domain w a l l s a r r a n g e d l n p a t t e r n s o t h e r than s t r i p e d or honeycomb w a l l n etworks. Because each r e g i s t e r e d domain has the 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 s i t e , the domains can be i d e n t i f i e d as b e i n g an a, b, or c, S3 x - ' 3 s u b l a t t i c e of the s u b s t r a t e . The domain w a l l s c o r r e s p o n d t o the boundary between domains of d i f f e r i n g s u b l a t t i c e s w i t h the s i m p l e s t type of w a l l i n v o l v i n g a s h i f t of o n l y one a d s o r p t i o n s i t e between domains. Assuming t h a t the w a l l s a r e c o n s t r a i n e d t o l i e a l o n g the 0, +/-120 0 symmetry d i r e c t i o n s of the s u b s t r a t e , o n l y two t y p e s of heavy w a l l s (and l i g h t w a l l s ) are p o s s i b l e ( F i g u r e 2 ). A system so c o n s t r a i n e d can be d e s c r i b e d as a, b, and c p a t c h e s of adatoms, and i t s energy w i l l depend s o l e l y on the t y p e s of domain w a l l s which s e p a r a t e the p a t c h e s . T h i s s i t u a t i o n can be r e n o r m a l i z e d t o a 3 s t a t e P o t t s model where the patches a r e t r e a t e d as p a r t i c l e s and the domain w a l l s r e p r e s e n t t h e i r i n t e r a c t i o n s . P o t t s model c a l c u l a t i o n s a r e u s e f u l f o r e x amining the v a r i o u s phases, and the t r a n s i t i o n s between them, t h a t the monolayer can have. In p a r t i c u l a r , near the commensurate phase t r a n s i t i o n , C a f l i s c h e t a l . 2 8 p r e d i c t t h a t the domain w a l l s w i l l become f l u i d . The i n t e n t of t h i s t h e s i s i s t o c a l c u l a t e the f r e e energy of the system of an incommensurate k r y p t o n monolayer adsorbed on g r a p h i t e and t o d etermine the impact of the monolayer's dynamics on the b e h a v i o u r of the system a t low t e m p e r a t u r e s . The c a l c u l a t i o n i s based on the m i c r o s c o p i c i n t e r a c t i o n s of the adatoms w i t h i n the monolayer and t a k e s i n t o account the 9 q u a n t i z e d n a t u r e o£ the adatom moti o n . T h i s c a l c u l a t i o n d i f f e r s 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 m o l e c u l a r dynamic c a l c u l a t i o n s assume t h a t the motion of the adatoms i s c l a s s i c a l . The v i b r a t i o n a l modes a s s o c i a t e d w i t h the motion of the domain w a l l s a re s t u d i e d , and t h e i r c o n t r i b u t i o n t o the f r e e energy of the monolayer i s d e t e r m i n e d . The thermodynamics of the system i s d i s c u s s e d i n the f o l l o w i n g s e c t i o n . Chapter 2 c o n s i d e r s the energy of the s t a t i c c o n f i g u r a t i o n . The m i c r o s c o p i c i n t e r a c t i o n s p r e s e n t w i t h i n the monolayer are d i s c u s s e d . The average p o t e n t i a l energy per adatom i s c a l c u l a t e d and i t s dependence on the c o n f i g u r a t i o n of the monolayer i s shown. Chapter 3 c o n s i d e r s the dynamic a s p e c t s of the system. The d y n a m i c a l m a t r i x f o r the motion of the 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 of the a p p r o x i m a t i o n s r e q u i r e d i n i t s d e r i v a t i o n . The z e r o p o i n t energy 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 the c o n f i g u r a t i o n of the monolayer i s a n a l y s e d . Chapter 4 c o n s i d e r s the low energy v i b r a t i o n a l modes of the domain w a l l s . The b e h a v i o u r of these modes i s d i s c u s s e d and t h e i r i n f l u e n c e a t f i n i t e t e m p e r a t u r e s on the f r e e energy of the monolayer i s examined. Chapter 5 p r e s e n t s a c o n c l u d i n g d i s c u s s i o n t h a t t i e s the r e s u l t s of the p r e v i o u s s e c t i o n s t o g e t h e r and s t a t e s the o r i g i n a l c o n t r i b u t i o n of the a u t h o r of t h i s t h e s i s . 10 ( I I ) FORMALISM The b e h a v i o u r of the monolayer must be c o n s i d e r e d i n the c o n t e x t of the t o t a l s y stem of k r y p t o n vapour, g r a p h i t e s u b s t r a t e and condensed k r y p t o n f i l m . 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 i s the c o n f i g u r a t i o n t h a t m i n i m i z e s the t o t a l f r e e 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 f r e e energy which c o n s i d e r s a l l a s p e c t s of the system s i m u l t a n e o u s l y 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 can be made which s e p a r a t e the f r e e energy i n t o s e v e r a l independent components thus r e d u c i n g the c o m p l e x i t y of t h e problem. T h i s 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 and G o r d o n 2 2 and t h e i r arguments a re 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 i s t a k e n t o be a s e a l e d chamber of volume V c o n t a i n i n g N T k r y p t o n atoms and a g r a p h i t e s u b s t r a t e . The temperature of the system i s m a i n t a i n e d a t T. Under a p p r o p r i a t e c o n d i t i o n s the k r y p t o n w i l l condense onto the s u b s t r a t e t o form a s o l i d monolayer. The c o n f i g u r a t i o n of the monolayer w i l l depend on the temperature and pressure, of the gas above the s u b s t r a t e . C o n s i d e r the system, a t te m p e r a t u r e T, when N k r y p t o n atoms are adsorbed onto the g r a p h i t e l e a v i n g N r p - N atoms as vapour. In o r d e r t o dete r m i n e the p r o p e r t i e s of the system i t i s n e c e s s a r y t o c o n s t r u c t the p a r t i t i o n f u n c t i o n 11 Z = £ e " ^ E s ( 1 . 2 ) s where s ranges over a l l p o s s i b l e s t a t e s of the system ( g i v e n the c o n s t r a i n t s V,N T) , E s i s the energy of the s t a t e , and £=l/k BT. When the vapour i s d i l u t e , the vapour and condensed atoms a r e weakly i n t e r a c t i n g so t h a t the energy can be s p l i t i n t o two components E S ( N ) = E v + E c (1.3) where Ey i s the energy of the vapour and E c i s the energy of the condensed k r y p t o n and the g r a p h i t e s u b s t r a t e . T h i s s e p a r a t e s the p a r t i t i o n f u n c t i o n i n t o c o n t r i b u t i o n s due t o the vapour and t o the condensed system. By u s i n g the c h e m i c a l p o t e n t i a l of the gas M t o d e s c r i b e the c o n t r i b u t i o n of the vapour, the p a r t i t i o n can be g i v e n as z = | T e-0P(N T-N) £ e - 0 E c ( N ) N = 0 s c In p r a c t i c e , the chamber i s t a k e n t o be s u f f i c i e n t l y l a r g e t h a t the c h e m i c a l p o t e n t i a l of the gas i s c o n s t a n t over the range i n N r e l e v a n t t o the f o r m a t i o n of the s o l i d monolayer phases. T h i s a l l o w s the p a r t i t i o n f u n c t i o n t o f a c t o r i z e as 12 where Z — Z-y ZQ Z„ = e " ^ N T (1.5) ,T N=0 s c 'v Nn and Z c = e " ^ = £ £ e ^ ( E ^ M N ) Because the monolayer i s l a r g e , o n l y one c o n f i g u r a t i o n w i l l be s i g n i f i c a n t and the summation over N can be n e g l e c t e d . The e q u i l i b r i u m s t a t e w i l l t hen be the s t a t e t h a t m i n i m i z e s the f r e e energy A C , A c = F c - MN (1.6) where F c i s the H e l m h o l t z f r e e energy of the condensed phase. The s t u d y i s t h e r e f o r e r e s t r i c t e d t o an e x a m i n a t i o n of the condensed system. The e f f e c t of the vapour on the t o t a l system i s p r o v i d e d by the c h e m i c a l p o t e n t i a l M which i s t r e a t e d as a parameter. The energy of the condensed system w i l l i n v o l v e b oth p o t e n t i a l and k i n e t i c e n e r g i e s . Because the components of the system a r e q u i t e m a s s i v e , the dynamic a s p e c t s of the condensed phase a r e , a t low t e m p e r a t u r e s , e x p e c t e d 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 of the s t a t i c p r o p e r t i e s . The p o t e n t i a l energy of the condensed system i s c o n s i d e r e d f i r s t and the d y n a m i c a l c o n t r i b u t i o n s a r e d i s c u s s e d l a t e r . 14 2 . POTENTIAL ENERGY (I ) MICROSCOPIC CONSIDERATIONS The p o t e n t i a l energy of the condensed system can be d i v i d e d i n t o two p a r t s . The f i r s t i s the p o t e n t i a l energy of the bare s u b s t r a t e ; the second i s the energy change when k r y p t o n i s adsorbed onto the s u r f a c e . The c a l c u l a t i o n can be r e s t r i c t e d t o c o n s i d e r o n l y the p o t e n t i a l energy change due t o the monolayer because c o n s t a n t terms have no impact 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 . 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 i n t e r a c t i o n s w i t h n e i g h b o u r i n g k r y p t o n atoms and the i n t e r a c t i o n w i t h the s u b s t r a t e . The i n t e r a c t i o n between k r y p t o n atoms i s due t o f l u c t u a t i o n s i n the m u l t i p o l e moments which, on av e r a g e , c r e a t e an a t t r a c t i v e p o t e n t i a l . 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 can be d e t e r m i n e d from the b u l k p r o p e r t i e s of the r a r e gas* . S i n a n o g l u and P i t z e r noted t h a t the s u b s t r a t e , because 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 the d i p o l e moments of the 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 which m o d i f i e s the i n t e r a c t i o n between k r y p t o n atoms i n the monolayer. A more e x a c t form f o r t h i s m o d i f y i n g e f f e c t has been d e t e r m i n e d by M c L a c h l a n ^ 1 . A d d i t i o n a l i n f l u e n c e s have been d i s c u s s e d i n d e t a i l by B r u c h x w i t h the c o n c l u s i o n t h a t t h e y do have an impact on the 15 c o n f i g u r a t i o n of the monolayer. Thus, the I n t e r a c t i o n energy of an adatom w i t h i t s n e i g h b o u r s can be d e t e r m i n e d by a summation of p a i r p o t e n t i a l s . In t h i s s t u d y , the p a i r p o t e n t i a l i s t a k e n t o be the b u l k p o t e n t i a l of A z i z ^ 9 and the parameters of Rauber e t a l . 3 2 have been used i n the M c L a c h l a n ^ l form f o r the s u b s t r a t e m o d i f i c a t i o n . The s u b s t r a t e i n t e r a c t i o n , which i n v o l v e s the a t t r a c t i o n between s u b s t r a t e carbon atoms and the adsorbed k r y p t o n , i s however, not as e a s i l y d e s c r i b e d . The charge d i s t r i b u t i o n w i t h i n the s u b s t r a t e , due t o c h e m i c a l bonding between carbon atoms, c o m p l i c a t e s the i n t e r a c t i o n t o the e x t e n t t h a t i t has not y e t been m o d e l l e d r e l i a b l y . Because of t h i s , a p a r a m e t e r i z e d form of the s u b s t r a t e p o t e n t i a l t h a t r e f l e c t s the symmetry of the s u b s t r a t e i s used i n the c a l c u l a t i o n . As shown i n f i g u r e 1, the s u b s t r a t e has a B r a v a i s l a t t i c e w i t h b a s i s v e c t o r s 1.42 x ^ i . 42 y Bj / 3 1.42 y (2.1) where the u n i t s are Angstroms. I f we d e f i n e g t o be a r e c i p r o c a l l a t t i c e v e c t o r of the l a t t i c e , the s u b s t r a t e p o t e n t i a l has the form 16 V ( r , z ) = V a ( z ) (2.2) where z i s the h e i g h t of the k r y p t o n atom above the s u b s t r a t e and r i s a v e c t o r c o n f i n e d t o the p l a n e of the s u b s t r a t e h a v i n g i t s o r i g i n a t an a d s o r p t i o n s i t e . The summation proceeds over a l l 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 . The dominant term i n ( 2 . 2 ) , V a ( z ) , i s the s u r f a c e average of the p o t e n t i a l energy V ( r , z ) . The r e m a i n i n g terms i n (2.2) p r o v i d e the l a t e r a l 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 . As shown by S t e e l e 3 3 , the l a t e r a l v a r i a t i o n i n the s u b s t r a t e p o t e n t i a l i n f l u e n c e s the adsorbed atoms i n two ways: F i r s t l y , the s u b s t r a t e can e x e r t a l a t e r a l f o r c e on the adatoms, and s e c o n d l y , the h e i g h t z a t which each atoms s i t s v a r i e s as a f u n c t i o n of i t s l a t e r a l p o s i t i o n r . For the commensurate monolayer, each atom s i t s above an e q u i v a l e n t s i t e on the s u b s t r a t e and the monolayer i s p l a n a r . For the incommensurate monolayers, however, the adatoms a r e not a l l s i t u a t e d a t e q u i v a l e n t s i t e s of the s u b s t r a t e and the monolayer w i l l t h e r e f o r e not be p l a n a r . Gooding e t a l . * 3 have c o n s i d e r e d the v a r i a t i o n i n the h e i g h t of the adatoms i n the monolayer. They f i n d t h a t because the p o s i t i o n s of the adatoms ( r , z ) a re smoothly modulated, the d i f f e r e n c e i n h e i g h t s between an adatom and i t s n e i g h b o u r s w i l l be n e g l i g i b l e . The i n t e r a c t i o n s between the adatoms can t h e r e f o r e be c a l c u l a t e d as though the monolayer was p l a n a r ; the 17 h e i g h t s of the adatoms a r e dependent o n l y on the s u b s t r a t e i n t e r a c t i o n . Because the s u b s t r a t e p o t e n t i a l d e t e r m i n e s the h e i g h t s z of the adatoms as a f u n c t i o n z ( r ) of the t h e i r p o s i t i o n r , the monolayer's c o n f i g u r a t i o n i s d e t e r m i n e d e n t i r e l y from the l a t e r a l p o s i t i o n s r of the adatoms. T h i s does not i m p l y t h a t the monolayer can be assumed p l a n a r , r a t h e r t h a t the adatoms l i e on the s u r f a c e d e f i n e d by the f u n c t i o n z ( r ) . On t h i s s u r f a c e the p o t e n t i a l of the s u b s t r a t e can be e x p r e s s e d < e x c l u s i v e l y as a f u n c t i o n of r , w i t h the r e s u l t i n g p o t e n t i a l V ( r ) h a v i n g a form s i m i l a r t o ( 2 . 2 ) . The v a r i a t i o n i n the s u b s t r a t e p o t e n t i a l i s so smooth on t h i s s u r f a c e , however, t h a t summation over r e c i p r o c a l l a t t i c e v e c t o r s can be t r u n c a t e d a f t e r the f i r s t s h e l l - ^ . T h i s l e a d s t o a s i m p l i f i e d form V ( r ) = V Q + V g V ( i - e 1 ? ' 1 ) (2.3) g where the summation i n v o l v e s o n l y the f i r s t s h e l l r e c i p r o c a l l a t t i c e v e c t o r s . Vg i s a parameter, known as the s u b s t r a t e c o r r u g a t i o n , which p r o v i d e s the l a t e r a l v a r i a t i o n i n the s u b s t r a t e i n t e r a c t i o n . Because of the symmetry of the i n t e r a c t i o n , Vg i s a c o n s t a n t t h a t can be f a c t o r e d from the summat i o n . G i v e n the p a i r p o t e n t i a l <Mr), which d e s c r i b e s the i n t e r a c t i o n between k r y p t o n atoms i n the monolayer, and the 18 s u b s t r a t e p o t e n t i a l V ( r ) , which g i v e s the i n t e r a c t i o n between a k r y p t o n atom and the s u b s t r a t e , the p o t e n t i a l energy of the monolayer can be c a l c u l a t e d . E = T Y * ( r ' - r ) + V V ( r ) (2.4) r , r 1 r where r and r ' a r e p o s i t i o n s of the adatoms w i t h i n the monolayer, and the f i r s t summation e x c l u d e s the p o s s i b i l i t y t h a t b oth p o s i t i o n s c o i n c i d e . The p o t e n t i a l energy i s m i n i m i z e d f o r c o n f i g u r a t i o n s of the monolayer i n which the p o s i t i o n s of the adatoms s a t i s f y the c o n d i t i o n r> -» -» -» -» -» 0 = > W ( r ' - r ) + W ( r ) (2.5) r ' As b e f o r e , the summation e x c l u d e s the p o s s i b i l i t y of c o i n c i d e n c e . E q u a t i o n (2.5) r e f l e c t s the f a c t t h a t the s t a t i c f o r c e s f e l t by the adatoms a r e z e r o when the p o s i t i o n s of the adatoms m i n i m i z e the p o t e n t i a l energy. More g e n e r a l l y , the r i g h t hand s i d e of t h i s e q u a t i o n d e s c r i b e s the f o r c e f e l t by an adatom a t r due t o i t s n e i g h b o u r i n g adatoms and the s u b s t r a t e . T h i s f o r c e , i f non-zero, w i l l a c c e l e r a t e the adatoms and the r e s u l t i n g b e h a v i o u r of the monolayer can be a n a l y s e d by m o l e c u l a r dynamics c a l c u l a t i o n s . 19 The d e n s i t y m o d u l a t i o n s due t o the domain w a l l s a r e not as s h a r p as shown i n f i g u r e 2; i n s t e a d , the r e g i s t e r e d r e g i o n s s m o o t h l y b l e n d i n t o the h i g h e r d e n s i t y w a l l s 1 3 . The adatoms i n the monolayer always m a i n t a i n a hexagonal p a c k i n g c o n f i g u r a t i o n . 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 u from the averaged t r i a n g u l a r l a t t i c e of v e c t o r s R* r = R + u(R) (2.6) The p r i m i t i v e l a t t i c e v e c t o r s , cif^ and d 2 , f o r the t r i a n g u l a r l a t t i c e of v e c t o r s R*, have t h e i r l e n g t h d e t e r m i n e d by the averaged s p a c i n g of the adatoms i n the monolayer. For the commensurate phase u(R*) = 0, and the b a s i s v e c t o r s a re the same as the s u b s t r a t e v e c t o r s and D*2, d e f i n e d from (2.1) t o be Di = 2Bi + B 2 (2.7) D 2 = - B i + B 2 The e x p e r i m e n t a l l y measured q u a n t i t i e s of m i s f i t M% and o r i e n t a t i o n 9 a r e r e l a t e d t o t h e s e b a s i s v e c t o r s by M% = [ l - ) * 100 (2.8) 20 w i t h 6 b e i n g the a n g l e between the two v e c t o r s cfj, and D\ . These 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 the s i z e and arrangement of the domain 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 the monolayer forms a honeycomb network of domain w a l l s a t low 1 7 t e m p e r a t u r e s . The domain w a l l s a r e not p i n n e d t o the s u b s t r a t e 3 4 and move e a s i l y . For systems w i t h f i x e d c o v e r a g e , t h i s b e h a v i o u r can be d e s c r i b e d i n terms of domain w a l l v i b r a t i o n s about a p e r f e c t honeycomb. The l o w e s t energy d e v i a t i o n of the domain w a l l s from the p e r f e c t honeycomb c o n f i g u r a t i o n i s a b r e a t h i n g d i s t o r t i o n which v a r i e s the s i z e of the r e g i s t e r e d domains w i t h o u t changing the t o t a l l e n g t h of the domain w a l l s i n the m o n o l a y e r 1 4 . I f energy i s not r e q u i r e d t o 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 can be c a l c u l a t e d from the c o n f i g u r a t i o n t h a t i s p e r i o d i c . I f the b r e a t h i n g motion r e q u i r e s 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 be the c o n f i g u r a t i o n of minimum energy. I f the p e r i o d i c c o n f i g u r a t i o n 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 , t h e n a n o t h e r c o n f i g u r a t i o n w i l l be lower i n energy. Thus, the energy of the monolayer a t a g i v e n m i s f i t and o r i e n t a t i o n can be c a l c u l a t e d w i t h the a s s u m p t i o n t h a t the c o n f i g u r a t i o n i s p e r i o d i c . T h i s a s s u m p t i o n i s t e s t e d when the v i b r a t i o n a l modes of the system ar e c a l c u l a t e d . Assuming the s t a t i c incommensurate monolayer has s u p e r l a t t i c e p e r i o d i c i t y , f i g u r e 3 shows t h a t an adatom i n one 21 r e g i s t e r e d r e g i o n w i l l be s e p a r a t e d from i t s I d e n t i c a l c o u n t e r p a r t i n another r e g i s t e r e d r e g i o n by a d i s t a n c e -4 -4 -4 - 4 - 4 R i = nDi+mD 2 + £Bi+kB 2 (2.9a) nD*i+mB2 i s the p o s i t i o n of the c o u n t e r p a r t adatom i n the absence of th e i n t e r v e n i n g w a l l ; S.fi+kB+2 i s the s h i f t i n i t s p o s i t i o n due t o the domain w a l l . G i v e n the p e r i o d i c n a t u r e of the system, t h e r e w i l l be many p o s s i b l e 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^ can be t a k e n t o be one of the l a t t i c e v e c t o r s which generate t h i s group. By symmetry, the o t h e r p r i m i t i v e l a t t i c e v e c t o r R*2 w i l l be R 2 = -mDi + (n-m)D 2 - k i \ + (S,-k)B £ (2.9b) 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 r e l a t e d t o the p r i m i t i v e l a t t i c e v e c t o r s f o r the averaged adatom p o s i t i o n s by Ri -t R 2 The i n t e g e r s n and m s p e c i f y the s i z e of the domains and % and k i n d i c a t e the type of w a l l s t h a t s e p a r a t e the domains. T h i s t h e s i s 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 -4 -4 nd i + md 2 -4 -4 -md( + (n-m)d; ( 2 . 1 0 ) 22 c o n f i g u r a t i o n s w i t h w a l l s t h a t c o r r e s p o n d t o s h i f t s of o n l y one s u b s t r a t e a d s o r p t i o n s i t e . The d i s c r e t e s e t 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 of the monolayers t h a t conform t o t h i s r e s t r i c t i o n i s g i v e n i n f i g u r e 4. S e t t i n g &=1 and k=0 does not reduce the a b i l i t y t o g e n e r a t e t h i s s e t . T h i s s e t of m i s f i t s and o r i e n t a t i o n s can be augmented by c o n s i d e r i n g v a l u e s of % and k which c o r r e s p o n d t o s h i f t s of more th a n one a d s o r p t i o n p o i n t . The e x t e n s i o n t o an a r b i t r a r y v a l u e of m i s f i t and r o t a t i o n can be o b t a i n e d by a procedure analagous t o the e x t e n s i o n of a sequence of r a t i o n a l numbers t o an i r r a t i o n a l l i m i t . However, a t low m i s f i t s and s m a l l r o t a t i o n a n g l e s , r e s t r i c t i n g the c a l c u l a t i o n t o c o n f i g u r a t i o n s f o r which 8, = 1 and k = 0 ( f i g u r e 4) p r o v i d e s a dense enough s a m p l i n g t h a t the g e n e r a l b e h a v i o u r of the monolayer can be d e t e r m i n e d . Because f u r t h e r work w i l l i n v o l v e summations of v a r i o u s q u a n t i t i e s ( a l l of which r e q u i r e summation i n d i c e s ) , the i n t e g e r v a r i a b l e s Sl, k, m and n w i l l be used i n d i f f e r e n t c o n t e x t s . To remove c o n f u s i o n , a l l f u r t h e r mention of the 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 and o r i e n t a t i o n ; the i n t e g e r s %, k, m and n ( a s i d e from appendix A) w i l l not be a s s o c i a t e d w i t h the c o n f i g u r a t i o n of the system. The r e l a t i o n s h i p between the v a r i o u s g e o m e t r i c p r o p e r t i e s of the monolayer and t h e s e i n t e g e r v a r i a b l e s i s d e s c r i b e d i n Appendix A. 23 ( I I ) SIMPLIFICATIONS From (2.2) the adatoms w i l l be f o r c e d t o l i e on a s u r f a c e z ( r ) . The p o s i t i o n s of the adatoms on t h i s s u r f a c e w i l l be det e r m i n e d by the s u b s t r a t e ' s c o r r u g a t i o n , which imposes a l a t e r a l f o r c e on the adatoms, and the i n t e r a c t i o n s between the adatoms. The v a r i a t i o n i n h e i g h t z of the adatoms i s so s l i g h t , t h a t the adatom p a i r i n t e r a c t i o n s can be c a l c u l a t e d as i f the monolayer was p l a n a r 1 3 . A l s o , w i t h the form of the s u b s t r a t e i n t e r a c t i o n (2.3) the l a t e r a l p o s i t i o n s r of the adatoms can be c a l c u l a t e d as though the monolayer was p l a n a r . The h e i g h t s of the adatoms can be s u b s e q u e n t l y c a l c u l a t e d from ( 2 . 2 ) . T h i s s i m p l i f i e s the c a l c u l a t i o n . In 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 the r e g i s t e r e d domains a r e s e p a r a t e d by domain w a l l s which s h i f t 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 o r d e r t o det e r m i n e the v a r i a t i o n of the monolayer's p o t e n t i a l energy as a f u n c t i o n of i t s m i s f i t and o r i e n t a t i o n . T h i s reduces the c a l c u l a t i o n , 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 examining o n l y the adatoms i n one of the p e r i o d i c domains. F u r t h e r s i m p l i f i c a t i o n s t o the c a l c u l a t i o n a r e p o s s i b l e , because the d i s p l a c e m e n t s of the adatoms u are smoothly modulated and the d i s t a n c e s between adatoms remain c l o s e t o the averaged s p a c i n g s R* determined from the m i s f i t . T h i s a l l o w s the adatom p a i r p o t e n t i a l s used i n ( 2 . 4 ) and (2.5) t o be expanded as a T a y l o r s e r i e s about the averaged s p a c i n g . 24 Gi v e n ( 2 . 6 ) , t h i s c o n t r i b u t i o n t o the p o t e n t i a l of an adatom a t -4 -» r ( R ) can be e x p r e s s e d as -4 1 r> f - 4 - 4 - + 1 -4 -» -» , E a ( R ) = £ ) [1 + (u-u')-V + i . ( (u-u') • v " ) + , < 2 - 1 1 } + ••• J * ( R - R « ) where the summation e x c l u d e s t h e p o s s i b i l i t y t h a t R=R'. From -4 -4 —) -4 (2.6) u and u 1 a r e f u n c t i o n s of R and R' r e s p e c t i v e l y , and the g r a d i e n t o p e r a t o r works on the argument R* of the f u n c t i o n The 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 f o r c e f e l t by a g i v e n adatom a t r i s l i k e w i s e F a ( R ) = "X l. 1 + (u-u')-V + j{ ( u - u 1 ) - V ) 2 + ..-)W(R-R') R ' (2.12) These s e r i e s can be t r u n c a t e d a f t e r the f i r s t few terms because the d i s t a n c e s between the adatoms remain hear the s p a c i n g s of the t r i a n g u l a r l a t t i c e . The most f r e q u e n t l y used t r u n c a t i o n f o r a n a l y t i c s t u d i e s e x c l u d e s the anharmonic terms. W i t h i n the continuum a p p r o x i m a t i o n (appendix B ) , the d i s p l a c e m e n t s of the adatoms must s o l v e a second o r d e r n o n l i n e a r s ystem of e q u a t i o n s . These a p p r o x i m a t i o n s have been d i s c u s s e d i n a i o p r e v i o u s work J- i w i t h the c o n c l u s i o n t h a t the e x p a n s i o n of (2.11) must i n c l u d e a t l e a s t the c u b i c terms t o produce 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 systems have c o n s i d e r e d the q u a r t i c term and have c o n c l u d e d t h a t t r u n c a t i n g (2.11) a t the c u b i c i s adequate. T h e r e f o r e , t o a l l o w the i n f l u e n c e of 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 p a s t the c u b i c a r e n e g l e c t e d . Another b e n e f i t of the smooth m o d u l a t i o n i n the adatom p o s i t i o n s i s t h a t the amount of i n f o r m a t i o n r e q u i r e d t o d e s c r i b e the c o n f i g u r a t i o n of the monolayer can be r e d u c e d 1 1 . From the p e r i o d i c i t y of the monolayer, the d i s p l a c e m e n t s of adatoms u (R*) can be e x p r e s s e d as a F o u r i e r s u m 1 1 ' J D s u p e r l a t t i c e ( A . S ) . The F o u r i e r s e r i e s can be t r u n c a t e d a f t e r the f i r s t few l o n g w avelength s h e l l s because the d i s p l a c e m e n t s have such a smooth s p a t i a l m o d u l a t i o n . The e x a c t c r i t e r i a f o r the t r u n c a t i o n i s e s t a b l i s h e d i n s e c t i o n IV of t h i s c h a p t e r . The r e s u l t i n g r e d u c t i o n i n the amount of i n f o r m a t i o n r e q u i r e d t o d e s c r i b e t h e system makes the computations f o r the d i s p l a c e m e n t s u more e f f i c i e n t . When the e x p a n s i o n (2.13) i s i n s e r t e d i n t o ( 2 . 1 2 ) , the f o r c e due t o the adatom i n t e r a c t i o n s becomes (2.13) where q a m = lqL + mq 2 i s a r e c i p r o c a l l a t t i c e v e c t o r of the 26 'a<R> = £ f L M e i q L M ' R where (2.14) a L, M i j' n F L , M " " UL , M ' Q L , M + 7 ^ " L - f c , : ( R L - * . , - R L , M &,m M-m M-m and Q L M = £ 2 s i n 2 ( q L M - h ) 7 W ( h ) ' n - V -r ->->-> "T w i t h R L / M = / - s i n ( q L M - h ) W W ( h ) ' h ' the h4 summations range over the t r i a n g u l a r l a t t i c e g e n e r a t e d by the b a s i s v e c t o r s dfx and d?2 ( A . 2 ) . S i m i l a r l y , when (2.13) i s i n s e r t e d i n t o ( 2 . 1 1 ) , the average energy per adatom due t o the 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 i s r - i f - i - * -» = (2.15) Ua = Uo + l\ \ u L / M u :QL M + L , M . ' tt UL,M L UZ-L, u - i l , - m :l. R L - S l , ~ RL,M + Ra,,mJ 6 ' %,m m-M' M-m ' ' > where U 0 = ^ #(h) i s a summation of the p a i r p o t e n t i a l s over the averaged t r i a n g u l a r l a t t i c e . From ( 2 . 3 ) , the f o r c e e x e r t e d on a g i v e n adatom due t o the s u b s t r a t e i s 27 -i -i -i F S ( R ) = -vV<2) = V g ^ - i 9 j k e i g : k ' R e ^ k ' " * * ' (2.16) 3 f k ~* ~* ~> where S j ] ^ j g A + k g 2 (A.4) and the summation over j and k i s r e s t r i c t e d t o the f i r s t s h e l l 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 . Because of the s u p e r l a t t i c e p e r i o d i c i t y , f u n c t i o n s of u(R*) can be e x p r e s s e d as e i g j k - u ( R ) = I . 3 k e i 5 L M - R ( 2 . i 7 ) L,M 7 E q u a t i o n s (2.17) and (A.7) a l l o w (2.16) t o have the form F S(K> = - I ( i V g l 9 j k M _ R ) e ^ L M ' S - ( 2 . 1 8 ) L,M j , k where the summation over j and k m a i n t a i n s the r e s t r i c t i o n s of (2.16) . S i m i l a r l y , the average p o t e n t i a l per adatom due t o the s u b s t r a t e becomes "s = V o + 6 V g " V g 1 A ^ (2.19) D,k I f t he d i s p l a c e m e n t s of the adatoms a r e known, the sum of (2.19) and (2.15) w i l l p r o v i d e the t o t a l p o t e n t i a l energy per 28 adatom. The d i s p l a c e m e n t s a r e d e t e r m i n e d from the f o r c e - f r e e s o l u t i o n s of ( 2 . 5 ) . From the r e s u l t s of (2.14) and ( 2 . 1 8 ) , the f o r c e - f r e e s o l u t i o n r e q u i r e s t h a t . -* -* F(R) = - £ F L M e i q L M ' R = o where (2.20a) L,M *LM = "L,M*5LM + TI 4,m *L-l,:[*L-l,- RL,M + h,m) %,m M-m M-m v -» j k (2.20b) + V \ 9jk A L _ j / M - k J / K G i v e n t h a t v a l u e s of u ^ m and A ^ k L M a r e d e f i n e d over the f i r s t B r i l l o u i n zone, the summations over Sim and j k i n (2.20b) w i l l e x t e n d the v a l u e s of F L M p a s t the zone edge. As a consequence, the f o r c e - f r e e c o n d i t i o n (2.20a) does not i m p l y F*LM=fj f o r a l l LM, r a t h e r the summation of F*^ over the s e t of s i m i l a r p o i n t s LM of a l l B r i l l o u i n zones i s z e r o . From (2.20b), t h i s r e q u i r e m e n t w i l l p r o v i d e enough e q u a t i o n s t o match the number of v a r i a b l e s u ^ m and the system of e q u a t i o n s can, i n t h e o r y , be s o l v e d . However, c o n s i d e r i n g t h a t the monolayer s t r u c t u r e i s so smooth s p a t i a l l y , the h i g h e r s h e l l c o e f f i e n t s of u ^ m and A ^ L M are n e g l i g i b l e and the c o e f f i c i e n t s of F* L M which extend beyond the f i r s t B r i l l o u i n zone can be i g n o r e d . Thus F* L M i s t a k e n t o , be z e r o i n (2.20b). T h i s p r o v i d e s a s i m p l e r system of e q u a t i o n s , the s o l u t i o n of which w i l l p r o v i d e , t o a l l i n t e n t s and p u r p o s e s , the g e n e r a l f o r c e - f r e e c o n f i g u r a t i o n . 29 The f o l l o w i n g t h r e e s e c t i o n s r e l a t e p u r e l y t o the n u m e r i c a l 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 the method f o r c a l c u l a t i n g the v a l u e s of the c o e f f i c i e n t s u<j m. G i v e n t h e s e v a l u e s the energy 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 from ( 2 . 1 5 and 2 . 1 9 ) . The d i s c u s s i o n of the p h y s i c a l p r o p e r t i e s of the system resumes i n s e c t i o n VI of t h i s c h a p t e r . 30 ( I I I ) METHOD OF SOLUTION A Newton s t e p method i s chosen t o s o l v e the n o n l i n e a r system of e q u a t i o n s ( 2 . 2 0 b ) . T h i s method i s an i t e r a t i v e p r o c e s s where the v a l u e s of u ^ m approach the s o l u t i o n i n a s e r i e s of s t e p s . An i n i t i a l e s t i m a t e i s made f o r the v a l u e s of u ^ m , and the f o r c e s F L M a r e c a l c u l a t e d from ( 2 . 2 0 b ) . The response of the F^ Lj^ j t o a s m a l l p e r t u r b a t i o n ^ u ^ m i s then c o n s i d e r e d . By expanding the f o r c e 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 i n terms of A u £ m and r e t a i n i n g o n l y the l i n e a r terms F L M ( u + Au) = F L M ( u ) + X 5 r M , • AUfcm ( 2 . 2 1 ) 1 , m Sim where °LM, = ~§LM S(L-Sl)S(M-m) - \ uL_t : ( R L _ < 1 / - R L / M + Sim M-m M-m + V > g j k g j k A ^ . ^ 3 ' k M-m-k and advantage i s t a k e n of the f a c t t h a t L,M 51,m M-m R5l,m.) ( 2 . 2 2 ) 31 The p e r t u r b a t i o n s A u ? m a r e t a k e n t o be the v a l u e s f o r which F* L M(u+Au)=0 (2.21) and t h e r e f o r e must s o l v e the l i n e a r s e t of e q u a t i o n s F L M ( u ) = - I B L M / • A u ^ (2.24) £,m Sim These p e r t u r b a t i o n s a r e added t o the u ^ m v a l u e s and the p r o c e s s of (2.20b), (2.22) and (2.24) i s then used t o d e t e r m i n e , from t h e s e new v a l u e s , the next s e t of p e r t u r b a t i o n s . T h i s p r o c e s s i s r e p e a t e d u n t i l the change i n the energy, AU d e t e r m i n e d from A U = E F <3fcm) • A u L M (2.25) L,M f a l l s below a s p e c i f i e d v a l u e . The minimum energy (or f o r c e - f r e e ) c o n f i g u r a t i o n can thus be d e t e r m i n e d as a c c u r a t e l y as r e q u i r e d . 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. However, the s ystem under c o n s i d e r a t i o n i s not q u i t e so e a s i l y h a n d l e d . As has been d i s c u s s e d p r e v i o u s l y , the adatoms w i l l have e x t r e m e l y s o f t phonon modes, i n f a c t two modes ar e a c o u s t i c , due t o motion of the domain w a l l s . S i n c e the m a t r i x D i n (2.24) i s s i m p l y the d y n a m i c a l m a t r i x of the system c a l c u l a t e d a t the B r i l l o u i n zone c e n t e r , the m a t r i x i s s i n g u l a r and (2.24) does not have a unique s o l u t i o n . T h i s causes the Newton s t e p method t o f a i l . 32 To r e s o l v e t h i s problem, a s p e c i f i c c o n f i g u r a t i o n must be s e l e c t e d and the Newton s t e p method r e s t r i c t e d so t h a t i t seeks o n l y t h i s s o l u t i o n . The c o n f i g u r a t i o n chosen has an adatom s i t u a t e d a t the c e n t e r of the hexagonal domains formed by the 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 r e g i s t e r e d , the c e n t r a l adatoms are a t a d s o r p t i o n s i t e s of the g r a p h i t e . G i v e n the hexagonal symmetry of the s u p e r l a t t i c e and the symmetry of the s u b s t r a t e , the minimum energy c o n f i g u r a t i o n s h o u l d be i n v a r i a n t under r o t a t i o n s of 60° about t h e s e c e n t r a l adatoms. I t i s t h e r e f o r e r e a s o n a b l e t o impose i n v a r i a n c e under r o t a t i o n s of 180° (or i n v e r s i o n symmetry) on the s o l u t i o n s t o ( 2 . 2 4 ) . T h i s r e s t r i c t i o n f o r c e s the v a l u e s of u Q O and F* Q O t o be z e r o and t h e i r c o n t r i b u t i o n s t o the system of e q u a t i o n s i n (2.24) can be n e g l e c t e d . The r e s u l t i n g reduced m a t r i x 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 of imposing i n v e r s i o n symmetry i s t h a t the u 0 ^ m v a l u e s a r e t o t a l l y i m a g i n a r y and u^ n\=-u*-!l,-m • * s i m i l a r r e q u i r e m e n t a p p l i e s t o the F* L M , and o n l y one q u a r t e r of the m a t r i x D need be c o n s i d e r e d i n (2.24) t o o b t a i n a f u l l s o l u t i o n . T h e r e f o r e , i n v e r s i o n symmetry i s imposed t o ease the 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 of (2.24) t o be o b t a i n e d , i t a l s o s i g n i f i c a n t l y r e duces the amount of c o m p u t a t i o n n e c e s s a r y . The Newton s t e p method was s e l e c t e d because, w i t h an a c c u r a t e s t a r t i n g p o i n t , i t converges r a p i d l y . A l l n o n l i n e a r 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 of the F* L M v a l u e s ; a 33 n u m e r i c a l l y t e d i o u s p r o c e s s . T h e r e f o r e , r e p e a t e d computations s h o u l d be m i n i m i z e d . T h i s c r i t e r i a e l i m i n a t e s the o t h e r commonly used s e c a n t and g r a d i e n t methods because the system has low energy phonon modes a s s o c i a t e d w i t h domain w a l l m o t i o n . 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 the a c o u s t i c modes, the low energy o p t i c a l modes s t i l l 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 the d i s p l a c e m e n t s of the adatoms w i t h o u t s i g n i f i c a n t impact upon the c a l c u l a t e d energy of the c o n f i g u r a t i o n . 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 which t a k e s t e p s a c c o r d i n g t o the d i r e c t i o n and magnitude of the c u r r e n t f o r c e , and can cause s t a b i l i t y problems i n s e c a n t methods. M o d i f i e d s e c a n t methods ar e s t a b l e , but each s t e p r e q u i r e s more c o m p u t a t i o n than those of the Newton s t e p method. 34 (IV) NUMERICAL APPROXIMATIONS The f o l l o w i n g s e c t i o n i s concerned with approximations made to f a c i l i t a t e e f f i c i e n t c a l c u l a t i o n of the adatom displacements. These approximations w i l l i n t r o d u c e e r r o r s i n t o the c a l c u l a t e d q u a n t i t i e s . However, these e r r o r s w i l l be shown to be c o n t r o l l a b l e . The f i r s t numerical approximation concerns the number of adatom s h e l l s i n c l u d e d i n the summation of (2.11). Since the adatom i n t e r a c t i o n i s s h o r t range, the summation can be t r u n c a t e d to i n v o l v e o n l y the f i r s t few nearest neighbour s h e l l s . T h i s i s e q u i v a l e n t to s e t t i n g bounds on h used i n the c a l c u l a t i o n f o r and R*L/M I N (2.14) and f o r U Q i n (2.15). Once the m i s f i t , M%, f o r the c o n f i g u r a t i o n i s s p e c i f i e d , U Q can be c a l c u l a t e d and the e f f e c t of the t r u n c a t i o n examined. Because the e r r o r i n t r u n c a t i n g the c a l c u l a t i o n f o r U Q r e f l e c t s the e r r o r i n t r u n c a t i n g (2.11), t h i s provides a means of e s t a b l i s h i n g bounds f o r the summation i n (2.11) and c o r r e s p o n d i n g l y , the summations over h r e q u i r e d i n (2.14). Another numerical approximation concerns equation (2.13) where the displacements of the adatoms are d e s c r i b e d by a F o u r i e r s e r i e s summation. From the preceeding s e c t i o n a method of d e t e r m i n i n g the values of U £ M has been e s t a b l i s h e d . Because the displacements are smoothly modulated, the f i r s t few long wavelength terms are s u f f i c i e n t to d e s c r i b e the system. E x a c t l y how many terms are needed depends on the accuracy 35 required for the calculated values of the energy (2.15 and 2.19). A t r i a l and error method of determining the number of terms is i n e f f i c i e n t , and therefore, an a n a l y t i c method is sought. Truncation of the Fourier series r e s t r i c t s the a b i l i t y to describe sharp structures in the monolayer. Apart from the ve r t i c e s , the sharpest structures in the monolayer w i l l be the domain walls. The e f f e c t of truncating the Fourier series can be examined by considering how the calculated energy of the domain wall changes as the truncation is changed. For t h i s analysis the simpler system of uniaxial domain walls is suitable. Corrections for the vertices w i l l be discussed l a t e r . The t h e o r e t i c a l work by Frank and van der Merwe 2 1 provides an a n a l y t i c description of a uniaxial system of domain walls. This is reviewed in appendix B to establish the equations required in the context of t h i s thesis. For the case where the monolayer has a single domain wall, displacements of the adatoms w i l l be given as (B.3 and B.6) -4 u(R) = 2?f 4 t a n _ 1 ( e* s ) (2.26) -> 4 -> where S = Bj. »R and B i is a basis vector of the substrate. For the solution (2.26) the energy of a given adatom (B.4) can be reduced to 36 E ( R ) = E 0 + a ( 9 ' ( S ) ) 2 + b 6 ' ( S ) ( 2 . 2 7 ) where 9 ( S ) i s d e f i n e d b y ( B . 6 ) . T h i s p r o v i d e s an a n a l y t i c f o r m f o r t h e e n e r g y a s s o c i a t e d w i t h a d o m a i n w a l l , a n d t h e i m p a c t o f t r u n c a t i n g t h e F o u r i e r s e r i e s ( 2 . 1 3 ) c a n be t e s t e d . G i v e n t h e d e f i n i t i o n o f 9 ( S ) i n ( B . 6 ) , i n t h e l i m i t Q -+ oo , t h e f u n c t i o n 9'(s) c a n be e x p r e s s e d a s - J 9» (S) = J f (q) e 1 ^ dq (2 . 2 8 ) -Q where f ( q ) i s lot e " q T t / 2 * f ( q ) = ^7 — -T77J ( 2 . 2 9 ) The e r r o r due t o t r u n c a t i n g t h e F o u r i e r s e r i e s i s t h e n d e t e r m i n e d b y c o n s i d e r i n g t h e i m p a c t o f f i n i t e v a l u e s o f Q on t h e t o t a l e n e r g y o f t h e s y s t e m d e t e r m i n e d f r o m ( 2 . 2 7 ) . The p o r t i o n o f t h e t o t a l e n e r g y w h i c h i s d e p e n d e n t on t h e v a l u e o f Q i s oo , E(Q) = aj ( 6 ' ( S ) ) 2 dS ( 2 . 3 0 ) where 9'(S) v a r i e s w i t h Q a s d e f i n e d b y ( 2 . 2 8 ) . 37 From (2.29) E(Q) = a ( i - 2e -Qlt/ot -Qit/ot (2.31) 1 + e The r e l a t i v e 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 v a l u e of Q i s then The Q s e n s i t i v e terms p r e s e n t i n the c a l c u l a t i o n of the energy w i l l be the harmonic and c u b i c terms of (2.15) and the s u b s t r a t e c o r r u g a t i o n terms of ( 2 . 1 9 ) . S i n c e the d e r i v a t i o n of (2.32) has n e g l e c t e d the v e r t i c e s p r e s e n t i n the monolayer's s t r u c t u r e , the r e s u l t i n g e s t i m a t e of the e r r o r may not be a c c u r a t e . To t e s t the e s t i m a t e p r o v i d e d by ( 2 . 3 2 ) , sample c a l c u l a t i o n s were performed t o o b t a i n the energy per adatom when the summation (2.13) was t r u n c a t e d t o i n c l u d e o n l y terms w i t h I q ^ m I = Q. For c o n f i g u r a t i o n s w i t h l a r g e m i s f i t s , because 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 g^ and q 2 a r e l a r g e o n l y a few r e c i p r o c a l l a t t i c e v e c t o r s q^m a r e r e q u i r e d , f o r c o n f i g u r a t i o n s w i t h s m a l l e r m i s f i t v a l u e s , however, 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 a r e spaced c l o s e r t o g e t h e r and more are r e q u i r e d . A v a r i e t y of 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 ; the c a l c u l a t i o n was r e p e a t e d 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 v a l u e s f o r Q. In a l l c a s e s , e x p r e s s i o n (2.32) a c c u r a t e l y %S(Q) = 2 (2.32) 1 + e Qrf/ot 38 e s t i m a t e d 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 Q. Thus, the e s t i m a t e of (2.32) a l l o w s bounds t o be s e t , i n advance, f o r the number of c o e f f i c i e n t s u ^ m r e q u i r e d t o c a l c u l a t e the energy of the monolayer t o a s p e c i f i e d a c c u r a c y . The f i n a l n u m e r i c a l a p p r o x i m a t i o n r e q u i r e d i s f o r the e v a l u a t i o n of A 3 ' k L M i n ( 2 . 1 7 ) . A n a l y t i c a l l y , t h e s e c o e f f i c i e n t s can be o b t a i n e d by t a k i n g sums and p r o d u c t s of B e s s e l f u n c t i o n s 3 6 . From (2.13) -4 -4 -* -» -» • ~* ~* iqo m-R (2.33) The c o e f f i c i e n t s u ^ m a r e s m a l l , and the summation over B e s s e l f u n c t i o n s can t h e r e f o r e be t r u n c a t e d a f t e r the f i r s t few terms. Moreover, f o r the h i g h e r s h e l l terms, the B e s s e l summation can be t r u n c a t e d a f t e r the f i r s t term because the magnitude of the c o e f f i c i e n t s U £ m d e c r e a s e s r a p i d l y f o r i n c r e a s i n g I q ^ m I. T h i s i s e q u i v a l e n t t o r e p l a c i n g the u ^ m i n (2.13) by the t r u n c a t e d s e t which i n v o l v e s o n l y the s i g n i f i c a n t low o r d e r s h e l l s . T h i s l a c k of s h a r p s t r u c t u r e i s a l s o r e f l e c t e d i n the v a l u e s of A ^ p ^ which can be assumed t o be z e r o f o r the h i g h e r o r d e r s h e l l s . With t h i s r e s t r i c t i o n the number of c o e f f i c i e n t s A ^ ^ and the c o m p u t a t i o n r e q u i r e d t o 39 de t e r m i n e them can be reduced. However, even w i t h t h e s e r e d u c t i o n s , the n u m e r i c a l e v a l u a t i o n of (2.33) f o r A 3 k g , m becomes i n c r e a s i n g l y t e d i o u s as the number of c o e f f i c i e n t s u ^ m i s i n c r e a s e d . The number of o p e r a t i o n s r e q u i r e d i n the c a l c u l a t i o n i n c r e a s e s e x p o n e n t i a l l y w i t h the number of c o e f f i c i e n t s u ^ m . S i n c e a l a r g e number of c o e f f i c i e n t s may be r e q u i r e d i n the c a l c u l a t i o n , t h i s method i s not s u i t a b l e . A b e t t e r method of e v a l u a t i n g (2.17) can be d e r i v e d from the f a c t t h a t a domain w a l l i n d u c e s a s h i f t of i n the phase of an adatom w i t h r e s p e c t t o the s u b s t r a t e 2 1 . - Because the d i s p l a c e m e n t s , u of the adatoms a r e measured r e l a t i v e t o the averaged s p a c i n g of the monolayer, the maximum v a l u e of g j k ' u ( R ) i s !4rt. T h i s bound on the v a l u e s of g j k " u (R*) a l l o w s the e x p a n s i o n of e z = £ i-j z n where z = i g j k - u ( R ) (2.34) n > t o be t r u n c a t e d a f t e r the f i r s t few terms. S i n c e t h i s e x p r e s s i o n i n v o l v e s powers of z, the c o e f f i c i e n t s A ^ £ m can be e v a l u a t e d by a s h o r t summation i n v o l v i n g r e p e a t e d c o n v o l u t i o n s of the c o e f f i e n t s U £ m . T h i s can be f u r t h e r improved by u t i l i z i n g s e l f c o n v o l u t i o n s , which r e q u i r e h a l f as many o p e r a t i o n s as the g e n e r a l c a s e . From 40 e z = ( A n ) 2 n where eZ//^n = ^ n + g n ( 2 . 3 5 ) A n i s o b t a i n e d f r o m a t r u n c a t e d T a y l o r s e r i e s e x p a n s i o n ( a s i n 2.34) w i t h t h e r e s u l t i n g e r r o r S n . S i n c e t h e a r g u m e n t w i l l be much s m a l l e r t h a n !4Tt f o r l a r g e v a l u e s o f n, t h e e x p a n s i o n c a n be t r u n c a t e d a f t e r f e w e r t e r m s t h a n ( 2 . 3 4 ) . e z i s t h e n o b t a i n e d b y p e r f o r m i n g n r e p e a t e d s q u a r e s o f t h e s u m m a t i o n ' s r e s u l t . The e r r o r i n e z due t o t h i s method c a n be c a l c u l a t e d f r o m ( A n ± S n ) An n Sn A n ) ( 2 . 3 6 ) ( 1 ± 2 n S n ) s i n c e A n -* 1 I f t h e s e r i e s f o r A n i s t r u n c a t e d a f t e r m t e r m s , t h e e r r o r S n w i l l be m ( 2 . 3 7 ) w i t h t h e r e l a t i v e e r r o r %S i n t h e v a l u e f o r e z due t o t h i s method o f c a l c u l a t i o n b e i n g 41 The c o m p u t a t i o n a l e f f o r t f o r o b t a i n i n g t h e s e t A £ m u s i n g t h e a b o v e scheme w i l l i n v o l v e m g e n e r a l c o n v o l u t i o n s o f t h e s e t U £ m and n s e l f c o n v o l u t i o n s . B e c a u s e t h e d i s p l a c e m e n t s a r e s m o o t h l y m o d u l a t e d , t h e s e t o f U £ m and t h e s e t o f A ^ k 0 m a r e r e s t r i c t e d s o t h a t o n l y t h e l o n g w a v e l e n g t h c o e f f i c i e n t s a r e n o n - z e r o . G i v e n t h e s e c o n d i t i o n s , t h e number o f o p e r a t i o n s r e q u i r e d t o c a l c u l a t e t h e v a l u e s A ^ k £ m v a r i e s a s t h e s q u a r e o f t h e number o f c o e f f i c i e n t s . T h i s r a t e o f i n c r e a s e i s more r e a s o n a b l e t h a n t h e e x p o n e n t i a l i n c r e a s e o f t h e B e s s e l s e r i e s a p p r o a c h . F r o m t h e e x p r e s s i o n f o r %S i n ( 2 . 3 8 ) , i t w o u l d a p p e a r t h a t t h e l e a s t amount o f c o m p u t a t i o n w o u l d r e s u l t i f t h e number o f s e l f c o n v o l u t i o n s i s g r e a t e r t h a n t h e number o f g e n e r a l c o n v o l u t i o n s ( i . e . m=2 and n l a r g e ). However, due t o m a c h i n e r o u n d i n g , t h e e r r o r i s more s i g n i f i c a n t f o r l a r g e v a l u e s o f n t h a n i t i s f o r l a r g e v a l u e s o f m. B e c a u s e o f t h i s , t h e a c t u a l e r r o r s due t o t h i s method were c a l c u l a t e d f o r s e v e r a l r e p r e s e n t a t i v e c o n f i g u r a t i o n s o f t h e m o n o l a y e r ; t h e g e n e r a l r u l e was e s t a b l i s h e d t h a t t h e b e s t a c c u r a c y a n d t h e l e a s t work o c c u r s when m=4 and n i s v a r i e d t o s u i t t h e s p e c i f i e d e r r o r bound %S. W i t h t h e c o n d i t i o n t h a t _z£Yirt, and f o r t h e s p e c i f i e d e r r o r bound %S, t h e v a l u e o f n must be t h e s m a l l e s t i n t e g e r t h a t s a t i s i f i e s l o g <*/2> n h ^- 41 %S } ( 2 . 3 9 ) l o g ( 8 ) 42 This value of n can be larger than necessary, since some smoothly modulated configurations have maximum values of z that are well below !4rt. Thus a c r i t e r i o n that also takes into account the configuration of the monolayer provides the most e f f i c i e n t scheme. This is discussed in appendix C. 43 (V) CALCULATIONAL PROCEDURE The p o t e n t i a l energy of the monolayer c o n f i g u r a t i o n s a re c a l c u l a t e d f o r v a r i o u s v a l u e s V g of the s u b s t r a t e c o r r u g a t i o n a t the p o i n t s of m i s f i t and 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 the Newton s t e p method r e q u i r e s an a c c u r a t e i n i t i a l c o n f i g u r a t i o n t o s t a r t , a means of e s t i m a t i n g t h i s c o n f i g u r a t i o n must be found. F u r t h e r m o r e , the c a l c u l a t i o n r e q u i r e s e r r o r bounds t o be s e t f o r the v a r i o u s n u m e r i c a l a p p r o x i m a t i o n s used i n the c a l c u l a t i o n . The e r r o r bounds used t o t r u n c a t e the summation (2.13) and t o c a l c u l a t e the A^ k LM i n (2.17) must be e x p r e s s e d as 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 an e s t i m a t e of the f i n a l r e s u l t . A procedure whereby a c c u r a t e e s t i m a t e s of the c o n f i g u r a t i o n a r e o b t a i n e d p r i o r t o c a l c u l a t i o n i s thus r e q u i r e d . From r e l a x a t i o n s t u d i e s , the s t r u c t u r e of the monolayer changes s m o o t h l y w i t h m i s f i t . As a r e s u l t , the u*£m f o r one c o n f i g u r a t i o n c l o s e l y a p p r o x i m a t e s the u ^ m of the c o n f i g u r a t i o n w i t h a s l i g h t l y s m a l l e r m i s f i t . For l a r g e m i s f i t c o n f i g u r a t i o n s the domain w a l l s b l e n d t o g e t h e r and the d i s p l a c e m e n t s of the adatoms a r e s l i g h t . For such a c o n f i g u r a t i o n the i n i t i a l s t a t e can be ta k e n t o be the t r i a n g u l a r l a t t i c e of the averaged p o s i t i o n s ( i . e . u 0 j i n = 0 ), and the e r r o r bounds (2.32) and (2.38) can be made more s t r i n g e n t w i t h o u t s i g n i f i c a n t c o m p u t a t i o n a l consequences. T h i s 44 establishes a procedure whereby a large m i s f i t configuration is calculated, so that the smaller m i s f i t can be obtained sequentially using the results of one c a l c u l a t i o n to estimate the next configuration of smaller m i s f i t . This is the same type of procedure as the bootstrap procedure used by Novaco 3 5. A possible danger of t h i s procedure is that the solutions may converge to metastable states. Such configurations w i l l be force-free solutions to (2.20b), but they w i l l not be the minimum energy configuration. For non-rotated configurations, such metastable states were sought by relaxation c a l c u l a t i o n s 1 3 . Regardless of the i n i t i a l state, the monolayer always relaxed to the configuration which was concluded to be the minimum energy configuration. Because the present c a l c u l a t i o n a l procedure matches the results of these relaxation studies, the method is considered to be accurate. For rotated states, however, relaxation results were not available to test the c a l c u l a t i o n . Instead, tests for metastable states were made by sequentially solving for solutions along a variety of paths through the set of points in figure 4. More than one force-free solution was never found and i t is therefore believed that the method finds the configuration of minimum energy. However, with increased rotation, there appears to be a sudden f l i p between domain wall systems with highly sheared domain walls and very smoothly modulated systems where commensurate domains are absent. This behaviour may be a r e s u l t of stretching the cub^ic approximation 45 of (2.11) beyond v a l i d i t y . F o r t u n a t e l y t h e s e c o n f i g u r a t i o n s , which a r e o u t s i d e the range of v a l u e s shown i n f i g u r e 4, a r e q u i t e h i g h i n energy and do not i n f l u e n c e the c h o i c e of e q u i l i b r i u m c o n f i g u r a t i o n . They can t h e r e f o r e be d i s r e g a r d e d . The c o n f i g u r a t i o n s r e l e v a n t t o t h i s s t u d y have m i s f i t s and o r i e n t a t i o n s w i t h i n the range of f i g u r e 4. W i t h i n t h i s r e g i o n the p r o c e d u r e f o r c a l c u l a t i n g the c o n f i g u r a t i o n s i s a c c u r a t e . For the r e s u l t s p r e s e n t e d i n t h i s c h a p t e r , the e n e r g i e s were c a l c u l a t e d t o an a c c u r a c y of 0.005 K. Because t h e r e a r e f o u r s o u r c e s of e r r o r t o be c o n t r o l l e d w i t h i n the c a l c u l a t i o n , the t o t a l of the f o u r e s t i m a t e d u n c e r t a i n t i e s was s e t t o be 0.005 K. By p o r t i o n i n g out the e r r o r , the summation over n e a r e s t neighbour adatoms i n (2.11) i s r e q u i r e d t o be a c c u r a t e t o 0.0005 K. The bounds a r e s e t so t h a t the e r r o r due t o t r u n c a t i n g the F o u r i e r e x p a n s i o n i n (2.13) i s l e s s than 0.003 K, the e r r o r i n v o l v e d i n c a l c u l a t i n g the i n f l u e n c e of the s u b s t r a t e c o r r u g a t i o n (C.8) i s l e s s than 0.0005 K, and the e r r o r (2.26) a s s o c i a t e d w i t h l i m i t i n g the number of Newton s t e p s i s l e s s than 0.001 K. I t i s r e c o g n i z e d t h a t t h e s e e r r o r s are i n t e r r e l a t e d , and t h a t s e p a r a t i n g them out as i n d i c a t e d can u n d e r e s t i m a t e the f i n a l 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 . However, p e r f o r m i n g sample c a l c u l a t i o n s w i t h a v a r i e t y of e r r o r bounds i n d i c a t e s t h a t p o r t i o n i n g out e r r o r s as shown p r o v i d e s a v a l u e of the energy a c c u r a t e t o 0.005 K w i t h a minimum of c o m p u t a t i o n a l e f f o r t . For a more a c c u r a t e c a l c u l a t i o n of the energy, the 46 i error bounds can be uniformly scaled smaller. 47 (VI) RESULTS AND DISCUSSION The monolayer's energy is found to vary primarily with the m i s f i t . In comparison, the v a r i a t i o n in energy for d i f f e r e n t orientations of the monolayer is quite minor. The lowest energy configuration is found to 'be non-rotated for small m i s f i t s , and rotated when the m i s f i t exceeds a value which varies with the substrate corrugation Vg. However, when the energy of the rotated configuration is compared to the energy of the non-rotated configuration of same m i s f i t , the difference is extremely s l i g h t . Thus, the change with m i s f i t in the energy of the monolayer can be determined by considering only the non-rotated configurations. Plots of energy versus m i s f i t for the non-rotated configurations are shown in figures 5a-f . The s o l i d l i n e s indicate the potential energy per adatom in the monolayer. The calculations have been performed for substrate corrugations ranging up to 11.0 K and the most representative plots have been shown. Since rotating the monolayer does not cause any s i g n i f i c a n t decrease in the potential energy, the general resu l t remains that the monolayer forms incommensurate islands i f the substrate corrugation is less than 9.8 K x 2. This value d i f f e r s from the value of 11.0 K quoted by Gooding et a l . 1 3 because a d i f f e r e n t form of the substrate screening is used in the c a l c u l a t i o n 1 2 . As noted in the introduction (1.1), the equilibrium 48 « configuration of the monolayer depends on the chemical potential l-i of the vapour above i t s surface. The c r i t i c a l exponent & of (1.1) is found experimentally to be approximately 0.3 9 . The chemical potential P of the vapour, which is also the chemical potential of the monolayer, can be determined by taking derivatives of the t o t a l energy of the monolayer with respect to the number of adatoms in the monolayer. This is equivalent to seeking the minimum of (1.5). To f i t these calculated values of H to (1.1), because there is no data near the commensurate phase, Mc is not known accurately and both B and P c are treated as parameters. Unfortunately, given the latitude of values for Mc and for deciding which data points should be considered, linear log f i t s could be made for a wide range of B values. B was found to l i e anywhere between 0.28 and 0.33. Furthermore, the value of B is insensitive to the value chosen for the substrate corrugation. This remains true even when points are included that correspond to nonphysical configurations calculated for Vg < 9.8 K which would require negative spreading pressures to ex i s t . The ori e n t a t i o n a l epitaxy has been analysed for three d i f f e r e n t substrate corrugations which represent the general behaviour of the monolayer. Selective data has been calculated for other Vg to confirm the trends i l l u s t r a t e d . As noted above, the energy of the monolayer is r e l a t i v e l y insensitive to i t s orientation. In order to separate out the orientational 49 behaviour from the misfit-dependent behaviour, the results are displayed as energy differences between the rotated configuration and the non-rotated configuration. For the s p e c i f i e d substrate corrugations, the energies are calculated for a l l the points of m i s f i t and orientation shown in figure 4. The energy differences are mapped out as contours on a surface of possible m i s f i t s and orientations (figures 6a-c). When the substrate corrugation has a value Vg = 5.0 K the minimum energy path matches qu a n t i t i v e l y the behaviour described by S h i b a 1 1 . This is expected since harmonic theories are found to be adequate for such a c o r r u g a t i o n 1 2 . However, for larger corrugations the behaviour changes and the monolayer rotates at m i s f i t s much smaller than Shiba's theory would predict. Furthermore, the l i n e shape of the minimum energy configuration rotation 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 scaling of Shiba's parameters is used. To understand th i s behaviour, the structure of the monolayer must be considered. For configurations with substrate corrugations greater than 10.0 K the rotated configurations at small m i s f i t s that would correspond to the higher energy contours of figure 6c did not converge. Structures that would correspond to heavy domain walls (figure 4) could not be produced. This behaviour was also found by Gooding et. a l . 1 3 for striped heavy walls at high substrate corrugations since the relaxation 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 structures are expected 50 to be s i g n i f i c a n t l y higher in energy than the stable configurations that are less rotated. They w i l l therefore not have an influence on the equilibrium configuration of the monolayer. A general analysis of the structure of non-rotated monolayers having small m i s f i t s reveals that the density of the adatoms is constant at the center of the domains, i t increases along the domain walls, and reaches a maximum at the ver t i c e s . This is in agreement with the concept of s o l i t o n density modulations arranged in a honeycomb network. The peak in density at the vertex is due to the focused compression of the domain walls which intersect at the vertex. The compression of the adatoms at the vertices becomes stronger for higher substrate corrugations r e s u l t i n g in the spacings of the adatoms at the vertex being less than 4.02 A which is the spacing favoured by the f l o a t i n g monolayer. For example, when Vg = 8.0 K, the spacing at the vertex is found to be 3.99 A and the monolayer is overcompressed. If the monolayer is allowed to rotate, the domain walls form a staggered intersection and the compression due to the domain walls is diffused. The density of the adatoms at the vertices for rotated configurations is not as high as that for the non-rotated configuration and spacings closer to 4.02 A are found. This decreases the interaction energies of the adatoms at the ver t i c e s , and since the change in the substrate energy due to d i f f u s i n g the vertex is neglible, the rotated configuration 51 w i l l have vertices of lower energy. When the domains are small the monolayer w i l l have many vertices and the rotated configurations w i l l be ene r g e t i c a l l y more favourable. For larger domains the preferred configuration is non-rotated despite lthe overcompression of the v e r t i c e s . This is due to the fact that the number of adatoms at the vertices becomes s i g n i f i c a n t l y less than the number of adatoms on the domain walls. Since the domain walls for rotated configurations must contain a shear as well as a compression, the domain wall adatoms would favour a non-rotated configuration. At smaller m i s f i t s , the adatoms on the domain walls dominate the adatoms at the vertices and the monolayer becomes non-rotated. This describes the orient a t i o n a l behaviour found in figures 6b,c. However, th i s behaviour must be v e r i f i e d as real and not an a r t i f a c t of the constraints of the c a l c u l a t i o n . The configurations examined have inversion symmetry about the domain centers. By breaking t h i s inversion symmetry, the vertices can be diffused. This p o s s i b i l i t y has been examined by Duesbery and J o o s 3 8 who concluded that breaking the inversion symmetry does not change the energy s i g n i f i c a n t l y . The or i e n t a t i o n a l behaviour observed in figure 6 is thus considered to be correct even with symmetry breaking. The consequences to the orient a t i o n a l behaviour of forcing the monolayer to be periodic are discussed in the section on phonon modes. 52 As noted p r e v i o u s l y , the o r i e n t a t i o n a l b e h a v i o u r does not 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 of S h i b a 1 1 . T h i s d i s c r e p a n c y i s d i r e c t l y a t t r i b u t a b l e t o the anharmonic terms p r e s e n t i n the a p p r o x i m a t i o n t o the p a i r p o t e n t i a l ( 2 . 1 1 ) . In harmonic t h e o r i e s the s h a r p p o t e n t i a l minimum of the adatom i n t e r a c t i o n i s not w e l l m o d e l l e d , however, i n c l u d i n g the c u b i c terms improves the f i t c o n s i d e r a b l y . A d d i t i o n a l terms i n the T a y l o r s e r i e s e x p a n s i o n w i l l add t o the a c c u r a c y , but as noted p r e v i o u s l y , t h e i r c o n t r i b u t i o n w i l l be s l i g h t compared t o the impact of i n c l u d i n g the c u b i c term. Thus, f o r a p e r i o d i c s ystem, the o r i e n t a t i o n a l b e h a v i o u r p r e d i c t e d by t h i s c a l c u l a t i o n i s e x p e c t e d t o be r e a l ; a d d i t i o n a l improvements i n the t h e o r y due t o i n c l u d i n g more T a y l o r s e r i e s terms w i l l o n l y cause s l i g h t changes i n the m i s f i t v a l u e f o r the onset of r o t a t i o n . Gordon and Lancon have c o n s i d e r e d the d i f f u s i o n of the v e r t e x i n r e l a x a t i o n c a l c u l a t i o n s . They c o n c l u d e t h a t s p r e a d i n g the v e r t e x does not cause any o b s e r v a b l e change i n the energy. However, as seen i n f i g u r e s 6 the energy d i f f e r e n c e i s v e r y s l i g h t and w i t h i n the a c c u r a c y of t h e i r c a l c u l a t i o n t h e y may not 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 b e h a v i o u r i s observed t o match the r e s u l t s of S h i b a 1 1 . S i n c e the.measurements were t a k e n a t f a i r l y h i g h t e m p e r a t u r e s ( > 30 K ), the t h e r m a l motion of the adatoms s h o u l d d i f f u s e the v e r t i c e s and a l l o w the n o n - r o t a t e d phases t o be f a v o u r e d where the z e r o temperature 53 r e s u l t s shown in figure 6 would indicate otherwise. This is not unreasonable since the domain walls are predicted to broaden with temperature 2 2' 3 9. Furthermore, the thermal broadening of the vertices is suggested by the molecular dynamics study of Schobinger and Abraham 1 8. They find that at f i n i t e temperatures, hexagonal phases are ene r g e t i c a l l y more favourable than striped phases for configurations that at zero temperature are predicted to be s t r i p e d 1 3 . 54 3. DYNAMICAL ASPECTS (I) BACKGROUND The previous chapter has treated the system c l a s s i c a l l y and the configurations of minimum energy assumed that the krypton atoms are stationary. This is an incomplete description of the system since the krypton and carbon atoms vibrate about their equilibrium positions (even at zero temperature) and a ca l c u l a t i o n of the free energy must also consider this motion. For the commensurate configuration of the monolayer de Wette et al. 4*-* have performed extensive calculations, with the harmonic approximation, for the system's dynamics. They found that the in-plane motion of the adatoms is decoupled from the out-of-plane motion and can be calculated using the r i g i d substrate approximation (2.2). The out-of-plane long wavelength modes of the monolayer are heavily coupled to the motion of the substrate, and, for the out-of-plane motion, the r i g i d substrate approximation appears to be inappropriate. The coupling between the out-of-plane motion of the monolayer and substrate, however, has not been observed in any rare gas physisorbed system 4 1. Instead, the experimental results can best be explained (except for Xe on the highly corrugated KCl s u b s t r a t e 4 2 ) by treating the adatoms as independent E i n s t e i n o s c i l l a t o r s acting under the holding potential V a(z) (cf 2.2) of a r i g i d substrate. For graphite 55 t h i s r e s u l t is reasonable since the substrate is quite r i g i d In comparison to the monolayer 4 0' 4 3. If the substrate is assumed to be t o t a l l y r i g i d , the holding potential V a(z) of the substrate so dominates the out-of-plane behaviour that the substrate corrugation and interactions of the surrounding adatoms are of l i t t l e influence. For A g ( l l l ) Hall et a l . 4 4 have performed calculations for an e l a s t i c substrate, which indicate that the monolayer's out-of-plane motion can become hybridized with the substrate Rayleigh modes. Furthermore, hybridization with the continuum of bulk modes of the substrate broadens the frequencies of the long wavelength modes. (Hall et a l . 4 4 refer to t h i s as radiative damping by the substrate.) However, when the dynamics of the substrate and monolayer is compared with that of the bare substrate, the difference is a motion of the monolayer, whose major frequency out of the plane is dispersionless except near the crossing point of the Rayleigh mode. Therefore the out-of-plane motion of the adatoms, r e l a t i v e to the substrate, would appear to be Einstein l i k e . In any case, experimental evidence indicates that the motion s p e c i f i c to the monolayer appears to be less sensitive to the-substrate's dynamics than de Wette et a l . 4 0 suggest. The discrepancy in de Wette's c a l c u l a t i o n may be due to the fact that only 13 sheets of graphite were used to model the substrate, and the monolayers may thus have an undue Influence on dynamics of the t o t a l system. Apart from the fact that a 56 f i n i t e slab c a l c u l a t i o n can not describe radiative damping, i t is expected that a ca l c u l a t i o n involving more sheets of graphite w i l l produce frequencies which cluster closer to the dispersionless Einstein frequency. In summary, i t appears reasonable to separate the in-plane motion from the out-of-plane motion of the monolayer. For in-plane motion the substrate can be treated as r i g i d . For out-of-plane motion, most of the dynamics w i l l be associated with the substrate. Out-of-plane motion s p e c i f i c to the monolayer is expected to be "Einstein l i k e " , although the Rayleigh mode and the long wavelength bulk modes of the substrate w i l l a l t e r the behaviour s l i g h t l y . Because the intent of this work is to determine the equilibrium m i s f i t and orientation of the monolayer, any quantity that does not vary with the monolayer's configuration w i l l be irr e l e v a n t . The short wavelength motion of the monolayer out of the plane is dispersionless, even i f the long wavelength motion may not b e 4 u ' 4 i / 4 4 . This implies that the adatom interactions are i n s i g n i f i c a n t and can be neglected for these modes. For the incommensurate monolayers, having density fluctuations that do not overly compress the adatoms, t h i s conclusion w i l l be v a l i d . Furthermore, S t e e l e ' s 3 3 estimates of the substrate corrugation indicate that the influence of corrugation on the substrate interaction is t o t a l l y overshadowed by the magnitude of the holding potential V a(z) and, for the out-of-plane behaviour, the substrate interaction can be regarded as 57 uniform. Thus, for motion perpendicular to the substrate, the adatoms move independently of each other and without regard to their location on the substrate. This removes any possible d i s t i n c t i o n between out-of-plane motion of adatoms in the commensurate and incommensurate monolayers, and the mode coupling between the monolayer and the substrate is expected to be, on average, the same regardless of the configuration of the monolayer. For very dense monolayers, the mass of krypton atoms on the surface w i l l s t a r t to influence the motion even i f the adatom interactions remain n e g l i g i b l e . However, the intent of t h i s work is to examine incommensurate phases near the commensurate t r a n s i t i o n and for such systems, the out-of-plane dynamics are not expected to influence the equilibrium configuration. The average holding potential of the substrate, V a(z) in (2.2), which dominates the out-of-plane motion does not a f f e c t in-plane behaviour. The behaviour of adatoms within the plane is determined by the substrate corrugation and the interactions between the adatoms. Since the positions of the adatoms change with d i f f e r e n t configurations of the monolayer, the in-plane free energy contribution w i l l be configuration dependant. Thus, the in-plane dynamics of the monolayer must be determined in order to find the equilibrium configuration. The in-plane modes are of considerable inte r e s t , because the dynamics are s i g n i f i c a n t l y d i f f e r e n t for the commensurate and incommensurate monolayers. For the commensurate monolayer, 58 each adatom is at an adsorption s i t e . This locks the monolayer onto the substrate and does not allow v i b r a t i o n a l energies below a given band gap value. The incommensurate phase, however, contains domain walls which have a great deal of mobility and the monolayer w i l l have low energy modes. Indeed, as mentioned in the introduction, V i l l a i n 1 4 has predicted that a domain wall breathing mode could become completely soft with consequences for the commensurate - incommensurate phase t r a n s i t i o n 2 6 . Therefore, for the c a l c u l a t i o n of the dynamics, i t i s assumed that the motion of the monolayer can be s p l i t into in-plane and out-of-plane components. The out-of-plane motion is not expected to have any influence on the equilibrium m i s f i t and orientation of the monolayer and w i l l not be considered. However, the in-plane motion w i l l influence the equilibrium configuration and must be considered. In the following sections, any further mention of dynamics or motion w i l l refer to that within the plane of the monolayer. 59 (II) HARMONIC APPROXIMATION Krypton atoms are quite massive and at low temperatures their v i b r a t i o n a l amplitudes r e l a t i v e to their nearest neighbours w i l l be small. Calculations of the dynamics of fl o a t i n g monolayers indicate that anharmonic ef f e c t s provide only small perturbations to the general results obtained from harmonic treatments 4 5. Thus, at low temperatures, the general behaviour of the adatoms in response to each other can be obtained i f the motion is assumed to be harmonic. For incommensurate monolayers, the adatoms are also influenced by the substrate. The domain wall 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 substrate and the influence of the substrate interaction w i l l be harmonic only for small displacements of the walls. Since the walls move f r e e l y 3 4 , the energy associated with their motion w i l l be small and the amplitude of the i r vibrations w i l l be large even at zero temperature. Thus, the motion of adatoms cannot be calculated within the harmonic approximation. However, th i s does not imply that the mode frequencies cannot be calculated with the harmonic approximation. From a renormalized description of the incommensurate monolayer, the properties of the monolayer can be determined e n t i r e l y from the shape of i t s domain wall structures. The location of these structures on the substrate i s not relevant since the pinning is so small. In the renormalized picture, 60 the domain walls w i l l have v i b r a t i o n a l modes associated with their motion. Since the behaviour of the domain walls is insensitive to their location on the substrate, anharmonic effects w i l l r e s u l t only from the walls c o l l i d i n g . The following c a l c u l a t i o n assumes that these c o l l i s i o n s do not occur. The v i b r a t i o n a l modes can then be determined by considering the behaviour of the system under i n f i n i t e s i m a l perturbations of the domain walls. The consequences of this assumption w i l l be discussed with the r e s u l t s . For harmonic motion of the adatoms, the normal modes of the incommensurate monolayer can be calculated from the dynamical matrix. The frequencies of the modes w i l l be the eigenvalues of the dynamical matrix. The low frequency motion of the domain walls w i l l not, however, be given d i r e c t l y by the eigenvectors. Instead, the form of the domain wall motion must be interpreted from the movement of the adatoms on the domain wa 11 s . The positions of the vib r a t i n g adatoms are defined to be r(R,t) = r(R) + v(R,t) (3.1) where r(R*) is the s t a t i c force free position of (2.6) and v(R*,t) provides the time dependent behaviour. With the assumption of harmonic behaviour, the force f e l t by a given adatom w i l l be l i n e a r l y dependent on the motion of the adatoms in the monolayer. This force w i l l cause the adatom to i 61 accelerate; the re s u l t i n g motion w i l l obey M- 3 2 v(R,t) 3 t 2 D(r(R). -4 -4 r (R» ) ) •v(R,t) (3.2) where Ma is the mass of the adatom, and the summation over R' includes the point R*. The periodic nature of r(R:) allows the matrix D in (3.2) to be expressed as _ -4 -4 B(r(R) /r(R»)) = £ 5 j t m(h) e i q * n t , R (3.3) %m where h* = R*-R*' and q%m i s the reciprocal l a t t i c e vector as defined for (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 pair interaction to only cubic terms as in (2.11), D£m(h*) can be calculated from ^2,m(n>) = SU)S(m) [ Ws&(h) - 5(h) ^  W^(h') ] + h' s ( h ) [ v q Z g j k 3 j k A ^ . / m _ k ] + -4 -4 -* r V - i a o -h' -* -+ 5(h) I I (1-e iq^m n ) u 5 t m-WV^(h') 6 2 -T -7 ( l - e lc3Stm n ) U s i m-WW(h) (3.4) where S(2,) and S(m) are integer delta functions, S(h*) is a vector delta function, and £(Fi) is the adatom pair interaction. For harmonic systems, the equilibrium positions of the adatoms w i l l be the force free positions r(R*) of the s t a t i c configuration. From the p e r i o d i c i t y of the s t a t i c configuration, v(R*,t) must have the form v(R,t) = I I v^ m(k) e ^ + W - K " i"<iot (3.5) k % m where it can be r e s t r i c t e d to the f i r s t B r i l l o u i n zone of the superlattice and q^ m is defined above. For equation (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 (?) in (3.5) must solve Ma o'(ic) v L M ( k ) = I v j ^ k ) • I BL_j, M_ m(I?) eI(Wta»'h' (3.6) This i d e n t i f i e s the dynamical matrix of the incommensurate monolayer gLM,5lm(k) = ^ . ^ ^ ( h ) e 1 (3.7) h 63 The summation over h* involves nearest neighbour s h e l l s of the surrounding adatoms and includes the case when F?=0. Since the adatom interactions are short range, the terms in (3.4) decrease r a p i d l y away from the o r i g i n and the summation can be truncated after the f i r s t few s h e l l s . This is equivalent to assuming that DL_<^M_m( h*) is zero for values of I h* I past a given cut off value. The possible solutions to equation (3.6) can be calculated and provide the in-plane normal modes (CJ s(J<), v^ m(s /k 4)) of the system. The index s is used to i d e n t i f y the mode. Since the system is two dimensional, the number of modes w i l l be twice the number of adatoms per domain. With these normal mode frequencies, the in-plane component of the average Helmholtz free energy per adatom is *c = U c + H ^' Js(k> + NJ I _ m ( l - e - ^ s ( i 0 / 2 ) ( 3 > 8 ) s, k s, k where F c = Nf c for equation-(1.6), and U c is the energy per adatom of the s t a t i c configuration (equations 2.15 plus 2.19). Equation (3.8) has s p l i t the free energy into a s t a t i c energy contribution, a zero point energy, and a temperature dependent contribution. The s t a t i c energy, U c , has been calculated in the previous section. The c a l c u l a t i o n for the zero point energy, E Q , is given in the following section. The c a l c u l a t i o n for the temperature dependent contribution, E T , 64 are presented later In chapter 4 which considers the low energy phonon modes. 65 (III) ZERO POINT ENERGY From (3.8) the zero point energy can be obtained by solving for the eigenvalues of the dynamical matrix and then summing for a l l values of wavevector i< in the B r i l l o u i n zone. For incommensurate monolayers, the number of adatoms per domain increases dramatically as the density of the configuration approaches the commensurate l i m i t . As an example, the non-rotated configuration with a m i s f i t of 2.22% has 675 adatoms per domain and thus 1350 in-plane normal modes. Solving for the eigenvalues of such a large system i s computationally unwieldy. Furthermore, c a l c u l a t i n g the eigenvalues for several vt points compounds the problem. Therefore, 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 Q is required. If the eigenvalues of (3.6) are defined to be xs(k*), the frequency " S ( J 4 ) of the mode can be determined from X g ( k ) = Og(IC) - * 2 ~ * (3.9) The zero point energy can then be expressed as d\ (3.10) where x m and xQ are the upper and lower bounds of >^s(k) 66 r e s p e c t i v e l y , and g(x) i s the d e n s i t y of s t a t e s 9 ( x ) = N I S ( x - x s(k) ) (3.11) s, k The d e n s i t y of s t a t e s , b e i n g a f u n c t i o n of x, can be expanded as a sum of o r t h o g o n a l p o l y n o m i a l s . In the case of two d i m e n s i o n a l systems, g ( x ) i s non-zero a t i t s minimum x Q and maximum x m . F u n c t i o n s of t h i s form a r e w e l l d e s c r i b e d u s i n g Legendre p o l y n o m i a l s 4 6 £ 2 ( x - x a ) 9 ( M = A ^ ? n ( J ( 3 ' 1 2 ) XJKJ + X Q where x a = 5 , and x<3 = x m - x 0 S i n c e (3.12) must s a t i s f y ( 3 . 1 1 ) , a n must have the form a 1 (2n+l) y 2( x s ( k ) - x a ) n N X(j ZJ_J n* X D } (3.13) s, k U s i n g the form (3.12) f o r g ( x ) i n (3.10) produces E o = JTMa J a n c n ( x m , x 0 ) (3.14) where 67 r ,, 2(X"Xa) c n ( x m , x 0 ) = J x X i P n ( x a ) dx (3.15) x 0 Thus E Q can be calculated once the orthogonal polynomial moments a n of the dynamical matrix are determined. While many moments may be required to accurately describe the density of states g(x) ( 3 . 1 2 ) , calculations of smooth averages (3.14) require r e l a t i v e l y few moments 4 6. The method of Wheeler and Blumstein 4 7 allows the summation over s in (3.13) to be expressed as the trace of an orthogonal polynomial matrix. Because of the form of the dynamical matrix in ( 3 . 7 ) , the orthogonal polynomial matrix must have the form LM, 5Cm p(n) L-8.,M-m (h) e i(k <3!lm> •4 h (3.16) The recurrence r e l a t i o n to provide these orthogonal polynomial matrices is ( n + l > ^ ( S ) ( 2 n + 1 ) \ [xd(5L-*,M-m<h,> - x a ! L _ i , M _ m ( h ' > ) 5tm,h' - n P L^ M (h) P<n>(h-h') e i ^ m ^ ' l St, m -i (3.17) 68 = (0) -» = -» =(-1) -* = -» where PrV M(h) = I L / M ( h ) a n d PL,M <n> = °L,M( h) s t a r t the recurrence process. ( I I M(h*) i s defined to be the 2x2 id e n t i t y matrix i f L, M and h are zero, otherwise i t i s the 2x2 n u l l matrix; 0L^M(h*) is defined to be the n u l l matrix for a l l L, M and h*. ) S ince I Y e i ( k + qfc m)-h = S ( n») ( 3 . 1 8 ) k, Sim performing the summations of (3.13) on the trace of the matrix in (3.16) w i l l r e s u l t in a n being given by a n = ^ T r [ P^ o(h=0) ] (3.19) 69 (IV) CALCULATION The method for ca l c u l a t i n g the zero point energy, presented above, allows several approximations to be made which improve the execution speed of the ca l c u l a t i o n without degrading the accuracy of the r e s u l t s . F i r s t l y , the summation in (3.14) can be truncated after the f i r s t few terms because i t converges ra p i d l y given an appropiate choice of X Q and x m . Furthermore, as n increases, the weights c n ( x m , x 0 ) decrease in magnitude and an increasing error in the moments a n can be permitted without s i g n i f i c a n t loss of accuracy in E Q . This increasing tolerance of error in a n allows approximations to be made to the recurrence process (3.17). The recurrence process in (3.17) involves a summation over hi and a summation over £ and m. Advantage can be taken of the -4 form of D^ m(h) in (3.4) which allows the summations to be separated. If each of these summations can be limited, the amount of computation required by the recurrence process is s i g n i f i c a n t l y reduced. From (3.4), the summation over h*' in (3.17) w i l l depend on the magnitude of the adatom pair interaction. Because the pair interaction i s short range the summation can be truncated after the f i r s t few s h e l l s . Furthermore, the r e s u l t i n g values of the P L^ M(Fi) w i l l diminish rapi d l y with i Fx I and, as in (3.7), the summation of (3.16) can be truncated. This is equivalent to the assumption that the values of P L ^(h4) are zero for values 70 of I r i | larger than a given cutoff. With the cutoff value set, the accuracy of (3.16) (and the subsequent accuracy of (3.19)) decreases with increasing n because (3.17) involves a convolution over h* and the successively generated polynomials have values that are s i g n i f i c a n t beyond the cutoff. The recurrence process (3.17) also involves a summation over & and m. Because the monolayer has a smoothly modulated structure, terms in (3.4) w i l l decrease r a p i d l y with I q ^ m I. This behaviour is numerically analogous to the rapid decrease with I h* I in the terms of (3.4), and similar consequences r e s u l t . The summation over Sim in (3.17) can thus be r e s t r i c t e d to the long wavelength s h e l l s , and the values of the re s u l t i n g orthogonal polynomials P L ^(n*) can be taken to be zero for I q L M I larger than a given cutoff. As with the truncation for it, the truncation for Sim w i l l cause increasing inaccuracy in the higher moments. The moments a n decline in accuracy at a rate which is dependent on the values of bounds set for I f? I and I q £ m I in the recurrence process. When the bounds are set low, only a few moments w i l l have s i g n i f i c a n t accuracy. If many accurate moments are required, the bounds must be set higher. Since high bound values increase the amount of computation required by the recurrence process, i t i s desirable to calculate E Q with as few moments as possible. Given that E Q must be calculated to within a spec i f i e d uncertainty, the number of moments required is determined by 71 the convergence o£ the summation in (3.14) and the rate of convergence of (3.14) depends on the values of xQ and xm . The most rapid convergence occurs when xQ and xm are, respectively, the minimum and maximum eigenvalues of the system. Unfortunately, the approximations in the recurrence process due to the truncations for h* and q^ m cause the method to become unstable i f xQ and x m are exactly the mimimum and maximum eigenvalues. This i n s t a b i l i t y arises from the description of the density of states g(x) in (3.12). Should g(x) be non-zero beyond the in t e r v a l defined by xQ and xm , the polynomial moments calculated w i l l diverge to i n f i n i t y . Because the recurrence process has been approximated, the re s u l t i n g density of states w i l l extend beyond the bounds set by the maximum and minimum eigenvalues, and the calculated moments a n w i l l diverge. To s t a b i l i z e the method, values for xQ and xm must be chosen which are outside the range of possible eigenvalues. This w i l l have the consequence that the summation of (3.14) w i l l not converge as quickly as expected, but given a reasonable choice of xQ and xm , the convergence w i l l not be s i g n i f i c a n t l y slower. The system does not have eigenvalues less than zero, and therefore, xQ is chosen to be s l i g h t l y negative. x m should be s l i g h t l y larger than the maximum eigenvalue, but determining t h i s value is numerically tedious and not necessary because only an estimate is required. Instead, xm is set to be twice the maximum frequency of a fl o a t i n g monolayer which has a density equal to the highest 72 density present in the monolayer; the point o£ highest density is at the center of the domain wall v e r t i c e s . This value for x m can be e a s i l y calculated and provides a reasonably rapid convergence for (3.14). Given these values for xQ and xm , E Q can be calculated to within 0.0005 K with only 11 moments. For these moments to have s u f f i c i e n t accuracy, the bounds on h must be set so that I Fi I 4. 25 A. The bounds on q^ m are adequately set i f they are taken to be the same as the bounds used to calculate U c to the 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 larger bounds on fi and q^ m. The values for E Q were confirmed to be accurate to within the prescibed error of 0.0005 K. A more general test for the convergence of the method was performed for the e a s i l y handled commensurate configuration and the bounds indicated above produced the required accuracy when compared to the results of di r e c t summation of the eigenvalues. 73 (V) RESULTS AND DISCUSSION For several d i f f e r e n t values of the substrate corrugation, the zero point energies have been calculated for a variety of m i s f i t s and orientations. The zero point energy is r e l a t i v e l y insensitive to the orientation of the monolayer. Given the accuracy of the c a l c u l a t i o n , E Q appears invariant over the range of orientations in figure 4. Thus the zero point energy of the monolayer w i l l not influence the o r i e n t a t i o n a l behaviour of the monolayer. The zero point energydoes have s i g n i f i c a n t v a r i a t i o n with the m i s f i t of the monolayer. As shown in figures 5a-f, the zero point energy increases with the density of the monolayer. This i s not unexpected because as the adatoms become more c l o s e l y packed, the force constants, and thus the frequencies, increase. Since the zero point energy does not change as the monolayer rotates, i t would appear that 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 influence should duplicate these r e s u l t s . Such a c a l c u l a t i o n was performed and although the general behaviour agreed, the values of E Q could not be made to match. Thus, compression of the monolayer produces the general trend in the zero point energy but the actual values are subject to more rigorous considerations. The most s i g n i f i c a n t consequence of these results is that the zero point energy changes the value of the substrate corrugation at which the monolayer switches from an 74 Incommensurate ground state to a commensurate ground state. From consideration of the s t a t i c energy of the monolayer alone, the c r i t i c a l value of substrate corrugation is 9.8 K. This value is higher than that predicted by theories which calculate the substrate corrugation from the microscopic interactions between the adatom and the s u b s t r a t e 1 8 . V i d a l i and C o l e 4 8 have provided the l a t e s t estimate of Vg = 7.4 K. When the zero point energy is considered, the c r i t i c a l value of substrate corrugation i s reduced to 7.0 K (figure 5c). For a substrate corrugation of 7.4 K, the monolayer should be commensurate unless i t is forced to be incommensurate. This is in agreement with the experimental evidence that the monolayer can be driven from a commensurate s o l i d to an incommensurate s o l i d by increasing the chemical p o t e n t i a l 6 . The potential between the adatoms is approximated by a Taylor series that is truncated past the cubic term. The zero point energy depends on the second derivative of the pair potential and the cubic approximation may not be v a l i d . To test the e f f e c t of this approximation, the adatom pair interaction was replaced by cubic polynomials f i t to the segments over which the nearest neighbours range. This i s an alt e r n a t i v e description to the truncated Taylor series expansion and the differences are displayed in figure 7. For a substrate corrugation of 7.0 K the two calculations produce the same behaviour of the system (figure 8). Though the behaviour of the adatoms can generally be 75 described as harmonic, anharmonic influences w i l l a l t e r the results s l i g h t l y . From self-consistent calculations for the f l o a t i n g monolayer 4 5, the zero temperature motion of the adatoms causes the zero point energy to be 6% higher than the value obtained under the harmonic approximation. The increase in energy can be attributed to the interactions of the adatoms since the force constants increase faster upon compression than they decrease on d i l a t i o n . The incommensurate monolayer is more complex since i t s structure allows many d i f f e r e n t phonon modes. Some of these modes correspond to domain wall motion and move blocks of adatoms as a unit. These modes do not contribute s i g n i f i c a n t l y to the zero point energy since they are low in energy and, as w i l l be seen in the following section, few in number. The modes which do contribute s i g n i f i c a n t l y to the zero point energy are the higher energy modes which involve motion of the adatoms r e l a t i v e to the i r neighbours. For these modes the anharmonic effects w i l l be the same as those for the commensurate configuration. Therefore, the zero point energies in figures 5a-e of the incommensurate configurations near the 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 by 6%. Away from the commensurate t r a n s i t i o n , the density of the monolayer is greater and, from the nature of the adatom interaction, the anharmonic contribution to the zero point energy should decrease. It is anticipated that t h i s decrease w i l l not have a s i g n i f i c a n t impact on the general trend of the zero point 76 energy. As a r e s u l t , where the zero point energy calculated with the harmonic approximation predicts that the c r i t i c a l substrate corrugation value is 7.0 K (figure 5c), the anharmonic contribution to the zero point energy w i l l s h i f t the value to be s l i g h t l y greater than 7.0 K. 77 4. LOW ENERGY PHONON MODES (I) HELMHOLTZ FREE ENERGY The f i r s t two terms in (3.8) for the Helmholtz free energy were determined in the previous chapters, and represent the energy of the monolayer at zero temperature. The t h i r d term, i d e n t i f i e d as E T , provides the temperature dependence of the o free energy. Eip = ^ I ^ l n d - e - ^ u s ( k ) / 2 } ( 4 > 1 ) s, k where N is the number of adatoms in the monolayer and £=l/k BT. At low temperatures only the lowest energy modes of the monolayer w i l l be s i g n i f i c a n t in this expression. The commensurate monolayer with a l l the adatoms locked in adsorption s i t e s w i l l not have any modes with energies below a band gap value. The incommensurate monolayer, on the other hand, has highly mobile domain walls and low energy modes are possible. From (4.1) configurations with extremely soft modes could have a large free energy; the m i s f i t or orientation 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 potential M is kept constant in (1.6). It is therefore interesting to determine the magnitude of E T at di f f e r e n t temperatures for various configurations of the 78 monolayer. E>p can be c a l u l a t e d by the polynomial expansion method which was used to c a l c u l a t e E Q . However, (4.1) i s seen to be a r a p i d l y v a r y i n g f u n c t i o n as u-*0. The c o e f f i c i e n t s c n ( x m / \ Q ) c a l c u l a t e d f o r t h i s e x p r e s s i o n would cause the summation, corres p o n d i n g to (3.14), to converge very s l o w l y . Thus, many moments a n must be c a l c u l a t e d and the v i a b i l i t y of the method i s reduced. F o r t u n a t e l y , the low energy modes, because of t h e i r nature, a l l o w approximations to be made to (3.6) which do not compromise the accuracy of the r e s u l t i n g e i genvalues and e i g e n v e c t o r s . These approximations overcome the p r e v i o u s l y d i s c u s s e d d i f f i c u l t i e s a s s o c i a t e d with the l a r g e s i z e of the dynamical matrix and a l l o w a p r a c t i c a l s o l u t i o n f o r the eigenvalues and e i g e n v e c t o r s of (3.6). Thus (4.1) can be evaluated through a d i r e c t summation of e i g e n v a l u e s . 79 ( I I ) NORMAL MODE C A L C U L A T I O N The s t a t i c i n c o m m e n s u r a t e s t r u c t u r e o f t h e m o n o l a y e r i s known t o be s m o o t h l y m o d u l a t e d 1 3 . I n t h e p r e v i o u s c h a p t e r s t h i s f a c t h a s a l l o w e d a p p r o x i m a t i o n s t o be made t o t h e d e s c r i p t i o n o f t h e a d a t o m p o s i t i o n s ( 2 . 1 3 ) s o t h a t t h e s t a t i c c o n f i g u r a t i o n a n d t h e c o r r e s p o n d i n g e n e r g y p e r a d a t o m c a n be c a l c u l a t e d e f f i c i e n t l y . I t i s t h e r e f o r e a s s u m e d t h a t a c o m p a r a b l e a p p r o x i m a t i o n c a n be made f o r t h e l o w e n e r g y m o t i o n o f t h e a d a t o m s ( 3 . 5 ) . T h e m o t i o n o f t h e a d a t o m s i s d e s c r i b e d b y t h e t r u n c a t e d e x p r e s s i o n The c u t o f f p a r a m e t e r n , d e t e r m i n e s t h e a c c u r a c y o f t h e a p p r o x i m a t i o n , a n d w i l l be d i s c u s s e d b e l o w . I f t h i s a s s u m p t i o n i s c o r r e c t , t h e n o n l y a p o r t i o n o f t h e d y n a m i c a l m a t r i x ( 3 . 7 ) w i l l be o f s i g n i f i c a n c e i n d e t e r m i n i n g t h e l o w e n e r g y m o d e s . The l i n e a r s e t o f e q u a t i o n s ( 3 . 6 ) c a n t h e n be r e d u c e d t o c o n s i d e r o n l y t h e s e r e l e v a n t t e r m s . The s i g n i f i c a n t p o r t i o n o f t h e d y n a m i c a l m a t r i x w i l l be t h e s u b m a t r i x S where ( 4 . 2 ) 80 S T M 8 , m ( k > = 0 F O R I St I , I ml , IX. I or IM| > n ' (4.3) SLM,X,m(k> = *>LM,Slm(k> otherwise T h i s produces the reduced s e t of equations Ma x 2 WL M(k) = I S L M ^ ( ^ . ^ ( k ) (4.4) with w L M ( k ) = 0 i f ILI>n or IMI>n (4.5) The e i g e n v a l u e s x2(j<) and e i g e n v e c t o r s w^ m(ic) of (4.4) can be t r e a t e d as t r i a l s o l u t i o n s to (3.6). The values x 2 of these t r i a l s o l u t i o n s are c l o s e t o being the a c t u a l e i g e n v a l u e s o 2 of (3.6). In f a c t 4 9 , M a I u 2 - x 2 | i s (4.6) where e = \X ^ L M , H m ( k ) - W j t m ( k ) - m X 2 ^ L M ( k ) ] | (4.7) Using (4.4), equation (4.7) can be reduced so t h a t e can be c a l c u l a t e d from 81 1 I I DLM^mfkVwj^ k) I" (4.8) ILI>n IHUn or IMI>n Iml^n From ( 4 . 2 ) , the a c c u r a c y of the approximation should i n c r e a s e as n i s i n c r e a s e d , thus the value of s w i l l depend on the value chosen f o r n. For a g i v e n c h o i c e of n the value of s w i l l a l s o depend on the energy of the phonon mode. From the p r e c e d i n g work, the ground s t a t e c o n f i g u r a t i o n of the monolayer i s known to be smoothly modulated and DLM,Slm(^ ( c a l c u l a t e d from (3.7) and (3.4)) must decrease r a p i d l y i n magnitude f o r i n c r e a s i n g v a l u e s of IL-SU and IM-ml. T h i s has the consequence t h a t e i n (4.8) w i l l depend on how r a p i d l y Wo^fic) decreases f o r i n c r e a s i n g l&l and I ml. The low energy modes of the monolayer correspond to motion of the domain w a l l s , and s i n c e the domain w a l l s are smoothly modulated s t r u c t u r e s , w^fk 4) w i l l decrease r a p i d l y with I SI I and I ml. With i n c r e a s i n g phonon e n e r g i e s , however, the modes w i l l be a s s o c i a t e d with s t r u c t u r a l v a r i a t i o n s t h a t are much sharper and w^ m(ic) w i l l not decrease as r a p i d l y . Thus s i n c r e a s e s as the energy of the phonon mode i n c r e a s e s and the a c c u r a c y of the t r i a l v alues x w i l l d ecrease. F o r t u n a t e l y , o n l y the low energy modes are 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 temperature dependence of the Helmoltz f r e e energy (4.1). For these modes the eigenvalues and e i g e n v e c t o r s can be determined a c c u r a t e l y from (4.4) f o r v a l u e s of n s m a l l enough to make the c a l c u l a t i o n p r a c t i c a l . From the preceding d i s c u s s i o n , the approximating system of 82 e q u a t i o n s ( 4 . 4 ) w i l l p r o d u c e s o l u t i o n s t h a t a r e w i t h i n a d e t e r m i n a b l e e r r o r o f t h e s o l u t i o n s t o ( 3 . 6 ) . T h i s d o e s n o t I m p l y , h o w e v e r , t h a t a l l t h e l o w e n e r g y modes o f t h e s y s t e m c a n be d e t e r m i n e d b y t h i s m e t h o d . To show t h a t t h e t r i a l s o l u t i o n s f r o m ( 4 . 2 ) do i n d e e d p r o v i d e t h e c o m p l e t e s e t o f l o w e n e r g y s o l u t i o n s t o ( 3 . 6 ) i t i s n e c e s s a r y t o show a o n e - t o - o n e c o r r e s p o n d a n c e . The a p p r o x i m a t e ( 4 . 2 ) a n d e x a c t ( 3 . 6 ) s y s t e m o f e q u a t i o n s a g r e e when t h e s u b s t r a t e i s n o t c o r r u g a t e d , i . e . Vg = 0 , a n d a o n e - t o - o n e c o r r e s p o n d e n c e e x i s t s b e t w e e n t h e e x a c t a n d a p p r o x i m a t e m o d e s . S i n c e t h e d i s p e r s i o n c u r v e s o f t h e n o r m a l modes a r e c o n t i n u o u s l y d e f o r m a b l e a s a f u n c t i o n o f c o r r u g a t i o n s t r e n g t h V g , t h i s o n e - t o - o n e c o r r e s p o n d e n c e must be m a i n t a i n e d f o r a l l v a l u e s V g . T h u s a l l f r e q u e n c i e s o f o r t h e l o w e n e r g y modes o f t h e m o n o l a y e r c a n be d e t e r m i n e d t o w i t h i n t h e a c c u r a c y o f ( 4 . 6 ) b y t h i s m e t h o d . The e i g e n v e c t o r s c a n a l s o be c a l c u l a t e d f r o m ( 4 . 2 ) ; h o w e v e r , when t h e mode f r e q u e n c i e s a r e c l o s e t o g e t h e r t h e e i g e n v e c t o r s c a l c u l a t e d may be a l i n e a r c o m b i n a t i o n o f t h e c o r r e s p o n d i n g n e a r l y d e g e n e r a t e modes o f ( 3 . 6 ) . D i s p e r s i o n c u r v e s a r e c a l c u l a t e d f o r a r a n g e o f s u p e r l a t t i c e c o n f i g u r a t i o n s . Th e a c c u r a c y o f t h e v a l u e s c a l c u l a t e d i s d e t e r m i n e d b y v a r y i n g t h e p a r a m e t e r n . I t i s a p p a r e n t t h a t t h e e r r o r b o u n d e i s g e n e r o u s , s i n c e t h e e i g e n v a l u e s c h a n g e v e r y l i t t l e w i t h n f o r v a l u e s o f n much l e s s t h a n t h e v a l u e o f e w o u l d s u g g e s t . T h i s o v e r l y l a r g e e r r o r b o u n d i s a r e s u l t o f 83 the sharp c u t o f f a t n present i n the e i g e n v e c t o r s w L M ( i c ) . A l i n e a r mix of the e i g e n v e c t o r s v L M ( i c ) i s r e q u i r e d t o d e s c r i b e t h i s c u t o f f , r e s u l t i n g i n w L M(ic) being l e s s than a pure match to i t s corre s p o n d i n g e i g e n v e c t o r v L M ( k * ) . T h i s d i s c r e p a n c y between w L M(i<) and v L M ( i c ) would not o r d i n a r i l y be n o t i c e d , but i n e quation (4.6) i t becomes s i g n i f i c a n t . The eigenvalue x, however, can be expected to be a c l o s e r match t o o than the value of s suggests, s i n c e w L M(ic) i s predominantly i n the d i r e c t i o n of v^Cl?). A b e t t e r i n d i c a t i o n of the e r r o r i n x i s obtained by examining the valu e s of the two lowest phonon branches a t the r p o i n t . These v a l u e s are found t o approach zero as n i s i n c r e a s e d and, f o r a giv e n n, t h e i r computed value s g i v e an i n d i c a t i o n of the a b s o l u t e e r r o r i n the eige n v a l u e s of the low energy phonon modes. For the d i s p e r s i o n curves c a l c u l a t e d , the parameter n was chosen so t h a t e i s s m a l l enough t h a t a l l but the three s o f t e s t modes are a c c u r a t e t o b e t t e r than 0.1% . For the three s o f t e s t modes an a n a l y s i s a t the r p o i n t i s necessary t o o b t a i n the e r r o r bounds. The f r e q u e n c i e s are given by the square r o o t of the c a l c u l a t e d e i genvalues and because the a b s o l u t e e r r o r i n the e i g e n v a l u e s i s roughly constant f o r the low energy phonon modes c o n s i d e r e d , the percent e r r o r i n the c a l c u l a t e d f r e q u e n c i e s w i l l be p r o p o r t i o n a l to 1/u f o r f r e q u e n c i e s of u away from z e r o . 84 (III) REVIEW OF PREVIOUS WORK In a previous paper 3 4, the behaviour of the three lowest phonon modes was discussed for monolayer configurations with m i s f i t values between 1.75% and 2.22%. For reasonable values of the substrate corrugation, the domain walls w i l l be c l e a r l y separated at these m i s f i t s . These m i s f i t values are large enough that the c a l c u l a t i o n for the low energy normal modes is p r a c t i c a l , and given the renormalized nature of the domain walls, the information obtained about these domain wall modes w i l l apply to incommensurate monolayers in the l i m i t of the commensurate t r a n s i t i o n . A t y p i c a l dispersion curve for the incommensurate monolayer is shown in figure 9. Of the normal modes possible for the monolayer, three modes are s i g n i f i c a n t l y softer than the re s t . For the wavevector k at M, figures lOa-c i l l u s t r a t e each of the three lowest modes of figure 9; the dashed l i n e s show the deformation of the walls about their equilibrium positions ( s o l i d l i n e s ) . The scale i s provided so that the size of the domains can be determined, although the magnitudes of the deformations are scaled a r b i t r a r i l y to make the modes c l e a r l y v i s i b l e . The value used for the s c a l i n g of the wall deformations i s , however, consistent in a l l of figures 10. From figure 10a the softest mode i s the breathing mode predicted by V i l l a i n 1 4 , with the other accoustic mode and the o p t i c a l mode producing a shearing of the domain walls. 85 When the configuration of the monolayer was rotated the two shear modes were found to increase in energy. The dispersion of the breathing mode, however, did 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 breathing mode became s i g n i f i c a n t l y softer, while the shear modes showed l i t t l e change. 86 ( I V ) QUANTIZED NATURE OF WALL MOTION The p r e v i o u s p a p e r 3 4 c o n c e n t r a t e d on t h e t h r e e l o w e s t e n e r g y modes a s t h e s e modes a r e e x p e c t e d t o be t h e mos t s i g n i f i c a n t i n f l u e n c e on t h e t e m p e r a t u r e d e p e n d a n c e o f t h e H e l m h o l t z F r e e e n e r g y . Th e f o r m s o f t h e h i g h e r e n e r g y modes were n o t c o n s i d e r e d . F i g u r e 11 shows t h e d i s p e r s i o n c u r v e s c a l c u l a t e d a t a m i s f i t o f 1.75% . The d i s p e r s i o n c u r v e s f o r m a t r i a d g r o u p i n g o f m o d e s . Th e s h a p e o f t h e d i s p e r s i o n c u r v e s f o r t h e h i g h e r e n e r g y t r i a d s r e p e a t s t h e s h a p e o f t h e t r i a d f o r t h e l o w e s t e n e r g y mode . The f o r m o f t h e d o m a i n w a l l m o t i o n c a l c u l a t e d a t t h e r p o i n t i s shown i n f i g u r e 12a f o r t h e t h i r d l o w e s t a n d i n f i g u r e 12b f o r t h e s i x t h l o w e s t mode . I t i s c l e a r t h a t t h e h i g h e r e n e r g y mode i s a more e n e r g e t i c v e r s i o n o f t h e l o w e r e n e r g y mode . Where t h e l o w e n e r g y mode c a u s e s t h e d o m a i n w a l l s t o v i b r a t e w i t h two n o d e s ( a t t h e v e r t i c e s ) t h e c o r r e s p o n d i n g mode a t t h e h i g h e r e n e r g y c a u s e s t h e d o m a i n w a l l s t o v i b r a t e w i t h t h r e e n o d e s . F u r t h e r m o r e , t h e n i n t h l o w e s t e n e r g y mode ( n o t shown) i s f o u n d t o h a v e t h e same t y p e o f m o t i o n e x c e p t t h e d o m a i n w a l l s v i b r a t e w i t h f o u r n o d e s . As a n a l t e r n a t e e x a m p l e , t h e m o t i o n o f t h e d o m a i n w a l l s a t t h e M p o i n t f o r t h e l o w e s t e n e r g y mode f i g u r e 10a c a n be c o m p a r e d w i t h t h e m o t i o n o f t h e f o u r t h l o w e s t e n e r g y mode f i g u r e 1 3 a . Th e f o u r t h l o w e s t mode i s c l e a r l y a more e n e r g e t i c f o r m o f t h e l o w e s t e n e r g y mode . B e c a u s e t h i s mode c a u s e s t h e 87 domain walls to vibrate with a wavelength that is twice th e i r length, the frequency of the mode provides the v e l o c i t y at which vibrations w i l l propagate along the domain walls. The form of t h i s mode i s constant through to the r point (figure 13b), and should therefore be dispersionless. Further confirmation for the value of the domain wall's wavevelocity can be obtained by examining the seventh mode (figure 11). This mode Is twice as energetic as the fourth mode and causes the domain walls to vibrate with a wavevector equal to the i r length. Since no contradictions to t h i s quantized behaviour were found, the general conclusion can be made that the domain wall modes are quantized according to the motion of a fundamental t r i a d of modes. The motion of a domain wall mode can be determined according to which of the three t r i a d modes i t corresponds to, and the number of the tr i a d ' s group. This behaviour is expected to break down when the energy of the modes approaches the band gap energy of the commensurate monolayer. At thi s energy the adatoms in the registered regions also vibrate and the monolayer w i l l have v i b r a t i o n a l modes other than domain wall 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 greater than 2.5% the monolayer w i l l be rotated when the substrate corrugation is greater than 8.OK. The non-rotated configuration, 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. This observed rotation, 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 since i t Is possible to diffuse the vert i c e s by breaking the inversion symmetry imposed on the monolayer. This allows the domain walls to s h i f t r e l a t i v e to the substrate so that the domain walls no longer meet at a point. Should the monolayer seek to form a less energetic configuration by s h i f t i n g the domain walls, the t r a n s l a t i o n a l modes calculated for the monolayer with inversion symmetry should be unstable. Within the accuracy of the c a l c u l a t i o n , unstable 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. Figure 14a shows the dispersion curves calculated for the non-rotated monolayer with m i s f i t 3.33% and substrate corrugation 8.0 K. It i s apparent that one of the modes becomes unstable at the M point. This mode i s shown in figure 15 and corresponds to a shearing of the domain walls. Since the energy of the monolayer w i l l decrease with rotation (figure 6b), t h i s suggests that the monolayer i s t r y i n g to rotate. Alternately, the i n s t a b i l i t y may indicate that the 89 domain wall p e r i o d i c i t y imposed on the monolayer w i l l be broken so that the monolayer can stay non-rotated. However, t h i s i s not l i k e l y since the unstable mode (figure 15) indicates that the domain walls wish to be sheared. Because the rotated configuration imposes a shear on the domain walls the non-rotated configuration can not r e s i s t being rotated. This i s contrary to the behaviour of the monolayer at m i s f i t s below 2.5%. At these m i s f i t s the shear mode is not unstable and the domain walls r e s i s t shearing. The non-rotated configurations of the monolayer are unstable when the substrate corrugation i s greater than 7.0 K and the m i s f i t i s between 2.5% and 3.5% . This i n s t a b i l i t y i s also found to a lesser degree in some configurations that are s l i g h t l y rotated. For a given m i s f i t the rotated configurations of lowest energy from figure 6 w i l l be stable as w i l l configurations that are rotated further (figure 14b). For mi s f i t s larger than 3.5% , the monolayer no longer has d i s t i n c t domain walls and i t becomes smoothly modulated; th i s eases the lo c a l i z e d overcompression at the vertices and the motion of the monolayer i s s t a b i l i z e d . Despite t h i s fact, the lowest energy configuration w i l l s t i l l be rotated. The dispersion curves for the non-rotated configuration with m i s f i t 3.7% is shown in figure 16. In summary, overcompression of the vertic e s i s expected to be a s i g n i f i c a n t influence, not only on the s t a t i c energy of the monolayer, but also on the s t a b i l i t y of the configuration. 90 The monolayer w i l l seek to ease the overcompression by rotating or breaking the p e r i o d i c i t y imposed on the configuration. At higher temperatures, thermal motion of the adatoms is expected to d i f f u s e the vertices so that the periodic non-rotated configurations become stable. Such an e f f e c t requires a sel f - c o n s i s t e n t phonon c a l c u l a t i o n , and confirmation of t h i s prediction awaits further work. 91 (VI) FREE ENERGY CALCULATION i The expression (4 .1) which determines E T involves a summation over a l l modes of the monolayer. In order to calculate E T i t is necessary to separate (4 .1) as E T = jj £ £ (JO (4 .9) d k where f<JO = j ln( 1 - e-0Ku s(k)/2 ) ( 4 > 1 0 ) N p s e The t o t a l number of adatoms in the monolayer N has been factored into the number of adatoms per domain N p and the number of domains N^. For temperatures near zero only the lowest energy modes of the monolayer need be considered. Since i t is possible to calculate the frequencies of these modes, f(k) can be determined. From ( 3 . 6 ) , f(k) i s a periodic function with rec i p r o c a l l a t t i c e vectors q ^ m being those of the su p e r l a t t l c e (A.5). Equation (4 .9) can be described as an average of f(k) evaluated at a l l points it in the f i r s t B r i l l o u i n zone of the s u p e r l a t t i c e . For a monolayer of i n f i n i t e extent, the summation in (4 .9) Is a c t u a l l y an i n t e g r a l . 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 i n t e g r a t i n g p e r i o d i c f u n c t i o n s t h a t a r e a n a l y t i c . With t h e i r method, e q u a t i o n (4.9) becomes a weighted sum of the f u n c t i o n f ( i ? ) e v a l u a t e d a t a d i s c r e t e s e t of s p e c i a l p o i n t s ic. The s e t 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 t o (4.9) have been d e t e r m i n e d by Cunningham 5 1. The number of p o i n t s i n the s e t and t h e i r p o s i t i o n s v a r y a c c o r d i n g t o the a c c u r a c y r e q u i r e d f o r the n u m e r i c a l i n t e g r a t i o n . T h i s method i s re v i e w e d i n appendix D. U n f o r t u n a t e l y , f(kf) d e t e r m i n e d from (4.10) i s not a n a l y t i c . The a c o u s t i c modes of the monolayer cause 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? approaches a r e c i p r o c a l l a t t i c e v e c t o r q j , m . I n o r d e r t o use the n u m e r i c a l i n t e g r a t i o n scheme d i s c u s s e d , f(£) must be m o d i f i e d so t h a t i t i s a n a l y t i c . The a s y m p t o t i c b e h a v i o u r of t h e a c o u s t i c f r e q u e n c i e s "^(J?) w i t h ic near q x > m i s otf(k) -* ctflk-q^ml (4.11) The mode in d e x cr c o r r e s p o n d s t o e i t h e r the l o n g i t u d i n a l or the t r a n s v e r s e a c c o u s t i c mode. I f f ( i c ) i s e x p r e s s e d as f ( k ) = j j ^ l n s ( 1 - e~0 f i ( Js(io/2 j JtS 9 < 1 - e-^^'k-qjiml/2 ) (4.12) £ I I \ l n ( 1 - e-**ctflk-qVml/2 , N p Sim 5r P 93 the logarithmically divergent portion of f(k) can be separated out and integrated over ic a n a l y t i c a l l y . The remaining portion of f (ic) i s now a n a l y t i c and can be integrated using the Cunningham spe c i a l p o i n t s 5 1 . 94 (VII) RESULTS AND DISCUSSION At low temperatures, the magnitude of E T i s extremely sensitive to errors in the frequencies calculated for the low energy modes. With the present computer resources, when the m i s f i t is small or the substrate corrugation is large, the low frequency values obtained from (4.4) w i l l not be a c c u r a t e 3 4 . Thus the c a l c u l a t i o n must be constrained to configurations with larger m i s f i t values or smaller substrate corrugations. This is unfortunate, because the configurations with small m i s f i t or large substrate corrugation w i l l have the softest phonon modes, and p o t e n t i a l l y the greatest E T . The free energy c a l c u l a t i o n is also not possible i f the underlying s t a t i c configuration i s unstable. When the substrate corrugation i s greater than 7.0 K, the non-rotated configurations become unstable when the m i s f i t i s between 2.5% and 3.5%. Thus the c a l c u l a t i o n for E T i s confined to stable configurations that permit the frequencies to be determinined accurately. The configurations chosen for t h i s study have m i s f i t s of 2.8%, 3.0%, and 3.7%. When the m i s f i t i s 2.8% or 3.0% , only rotated configurations are considered. The substrate corrugation is taken to be 8.OK so that the domain walls are c l e a r l y separated at 2.8% m i s f i t , while the configuration at 3.7% i s a modulated structure. This allows the free energy associated with soft domain wall 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 walls. The values E T for these configurations are as follows: Table r . Values of E T calculated for various temperatures and orientations when M%=3.7% T=0.5K T=1.0K T=2.0K T=4.OK 0. 00° -0.0019K -0.0086K -0.0325K -0.1329K 0. 52° -0.0011K -0.0058K -0.0252K -0.1171K 0. 79° -0.0011K -0.0055K -0.0240K -0.1145K 0. 95° -0.0009K -0.0051K -0.0231K -0.1130K Table 2. Values of Eip calculated for various temperatures and orientations when M%=3.0% T=0.5K T= l.OK T=2.0K T=4.OK 0. 09° -0.0034K -0. 0116K -0.0366K -0.1419K 0. 71° -0.0020K -0. 0073K -0.0260K -0.1192K 96 T a b l e 3 . V a l u e s o f E T c a l c u l a t e d f o r v a r i o u s t e m p e r a t u r e s a n d o r i e n t a t i o n s when M%=2.8% T = 0 . 5 K T = 1 . 0 K T = 2 . 0 K T = 4 . 0 K 0 . 2 1 ° - 0 . 0 0 2 2 K - 0 . 0 0 8 1 K - 0 . 0 2 7 6 K - 0 . 0 9 4 1 K 0 . 6 8 ° - 0 . 0 0 2 4 K - 0 . 0 0 8 0 K - 0 . 0 2 6 5 K - 0 . 0 9 1 6 K The i m m e d i a t e o b s e r v a t i o n c a n be made t h a t a t l o w t e m p e r a t u r e s t h e c o n t r i b u t i o n o f E T t o t h e f r e e e n e r g y w i l l be n e g l i b l e . The o n l y i m p a c t t h a t t h e d o m a i n w a l l modes c a n h a v e i s t o c h a n g e t h e e q u i l i b r i u m o r i e n t a t i o n o f t h e m o n o l a y e r . H o w e v e r , t h e t e m p e r a t u r e s r e q u i r e d t o make E T s i g n i f i c a n t may i n v a l i d a t e t h e h a r m o n i c a p p r o x i m a t i o n . F u r t h e r m o r e , t h e r a p i d s o f t e n i n g o f t h e d o m a i n w a l l modes d o e s n o t c a u s e a n y s i g n i f i c a n t i n c r e a s e i n t h e v a l u e s . The v a l u e s o f E T have n o t b e e n d e t e r m i n e d f o r t e m p e r a t u r e s g r e a t e r t h a n 4 . O K . T h i s l i m i t h a s b e e n i m p o s e d s i n c e no modes a r e c a l c u l a t e d w i t h e n e r g i e s g r e a t e r t h a n t w i c e t h e c o m m e n s u r a t e b a n d g a p v a l u e o f 1 1 . O K . The c o m m e n s u r a t e b a n d gap v a l u e i s t h e l o w e s t p o s s i b l e e n e r g y t h a t t h e modes o f t h e c o m m e n s u r a t e m o n o l a y e r c a n h a v e . B e c a u s e t h e a c c u r a c y o f t h e f r e q u e n c i e s d e t e r i o r a t e s w i t h i n c r e a s i n g e n e r g y , v a l u e s w i t h e n e r g i e s much g r e a t e r t h a n t h e b a n d g a p w i l l be s u s p e c t . F u r t h e r m o r e , a d d i t i o n a l modes may be p r e s e n t i n t h e m o n o l a y e r 97 that are missed by the c a l c u l a t i o n . Thus the temperatures must be kept small so the inaccuracies caused by the approximate equations (4.4) are minimized. The frequencies have been calculated under the assumption of harmonic motion. This approximation w i l l not be v a l i d i f the domain walls c o l l i d e with each other. Since c o l l i s i o n s of the domain walls w i l l increase the energy of wall modes, anharmonic eff e c t s are expected to diminish the values of Eip. Confirmation of this prediction awaits a self - c o n s i s t e n t phonon ca l c u l a t i o n of the incommensurate monolayer. The only mechanism that could soften the domain wall motion is the formation of domain wall d i s l o c a t i o n s 2 6 . Further work based on the e l a s t i c constants of the renormalized domain wall system should be performed. In the absence of domain wall d i s l o c a t i o n s , however, at low temperatures domain wall motion should not influence the m i s f i t and orientation for the equilibrium configuration of the monolayer. To extend the res u l t s to the commensurate l i m i t , i t is necessary to estimate the change ln E T as the m i s f i t decreases. From previous work 3 4, when the domain size i s large the s o f t e s t mode is the breathing mode. At low temperatures this mode w i l l dominate the value of E T and the va r i a t i o n of i t s contribution due to m i s f i t should be examined. The breathing mode i s influenced by the domain wall interaction. The strength of t h i s interaction w i l l decrease as 98 U W(L) = c L t f e _ ) C L (4.13) for Increasing separation L of the domain w a l l s 2 2 . The constants c, c and £ are determined from the calculated energetics of the system. Correspondingly, the frequency of the breathing motion should decrease as u(k) = f(ic,L) e _ , < : L / 2 (4.14) where f(ic,L) i s a function which is periodic in k and algebraic in L. To leading order, the contribution of t h i s mode to (4.1) w i l l decrease as £ ln( e _ , C L / 2 ) (4.15) with increasing L. Since Np varies as L 2 , the contribution w i l l decrease as 1/L. This analysis assumes that the harmonic approximation for motion i s v a l i d . For low energy modes the vib r a t i o n a l amplitudes w i l l be large even at zero temperature, and the domain walls may c o l l i d e with each other. If c o l l i s i o n s occur, the motion w i l l not be harmonic and (4.15) w i l l not be correct. Coppersmith et a l . 2 6 have presented a model which describes the domain walls as being free to move provided that their displacements do not exceed L/4. This ensures that the domain 99 walls move independently of each other, and to some extent, simulates the anharmonic repulsion between the doman walls. They predict that the free energy contribution of the domain wall breathing w i l l decrease as l n ( L ) / L 2 . This is a more r e s t r i c t i n g l i m i t than 1/L predicted by (4.15) and should be v a l i d for configurations approaching the commensurate t r a n s i t i o n . At higher temperatures, other domain wall modes begin to influence the value of E T . More low energy domain wall modes appear as the domain size i n c r e a s e s 3 4 , and the value of E T could be expected to increase as well. However, the number of these modes does not increase as quickly as the t o t a l number of modes N p and, from (4.10), E T is expected to remain small as the domain size increases. The r e s u l t s presented are for configurations where the substrate corrugation is 8.OK. Because the domain wall modes decrease in energy as the value of Vg increases, E T w i l l increase with Vg. For reasonable values of Vg the change in E T can be estimated from the renormalized description based on domain walls. Increasing the substrate corrugation w i l l sharpen the domain walls; from a renormalized description based on the continuum model, the configuration w i l l resemble a configuration that has a lower value of Vg and smaller m i s f i t . Because < in (4.13) varies as 'Vg , scaling considerations predict the r a t i o of the number of adatoms per domain N D for these two configurations i s 100 (4.16) For the configuration with the larger Vg , the domain walls vibrate faster because they contain fewer adatoms and are less massive. The e f f e c t of t h i s on (4.10), however, i s expected to be minor compared to the impact of rescaling Np . Because the substrate corrugation cannot be s i g n i f i c a n t l y greater than 8.OK, the values of E T for these configurations should remain small. Correspondingly, i t i s obvious that decreasing the substrate corrugation w i l l decrease the value of E T . 101 5. CONCLUSION The p u r p o s e o f t h i s t h e s i s i s t o d e t e r m i n e t h e l o w t e m p e r a t u r e b e h a v i o u r o f k r y p t o n m o n o l a y e r s a d s o r b e d on g r a p h i t e . Of s p e c i f i c i n t e r e s t i s t h e e x p e r i m e n t a l l y o b s e r v e d v a r i a t i o n o f t h e m o n o l a y e r ' s m i s f i t a n d o r i e n t a t i o n a s a r e s p o n s e t o t h e c h e m i c a l p o t e n t i a l and t e m p e r a t u r e o f t h e s y s t e m . To u n d e r s t a n d t h i s b e h a v i o u r , t h e v a r i a t i o n o f t h e f r e e e n e r g y w i t h m i s f i t a n d o r i e n t a t i o n must be c a l c u l a t e d . The f r e e e n e r g y o f t h e s y s t e m i s d e p e n d e n t on t h e i n t e r a c t i o n s i n a n d b e t w e e n t h e k r y p t o n v a p o u r , t h e a d s o r b e d k r y p t o n m o n o l a y e r s , and t h e g r a p h i t e s u b s t r a t e . To s i m p l i f y t h e c a l c u l a t i o n , o n l y t h e component o f t h e f r e e e n e r g y t h a t v a r i e s w i t h 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 was c o n s i d e r e d . T h i s component i n v o l v e s t h e i n t e r a c t i o n s b e t w e e n k r y p t o n atoms i n t h e m o n o l a y e r , 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 s e a d a toms 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 b e t w e e n a datoms was m o d e l l e d b y p a i r p o t e n t i a l s . The f o r m o f t h e p a i r p o t e n t i a l was o b t a 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 k r y p t o n and m o d i f i e d t o i n c l u d e t h e s c r e e n i n g o f t h e s u b s t r a t e . B e c a u s e t h e i n t e r a c t i o n b e t w e e n t h e k r y p t o n atoms and t h e s u b s t r a t e c a n n o t be d e t e r m i n e d a c c u r a t e l y , a p a r a m e t e r i z e d f o r m f o r t h e s u b s t r a t e p o t e n t i a l , t h a t r e f l e c t s t h e s y m m e t r y o f t h e s u b s t r a t e , was u s e d i n t h e c a l c u l a t i o n . On a c c o u n t o f t h e n a t u r e o f t h e s u b s t r a t e i n t e r a c t i o n , o n l y one p a r a m e t e r , t h e s u b s t r a t e c o r r u g a t i o n , was 102 necessary to define the substrate p o t e n t i a l . This parameter was varied so that the impact of the substrate corrugation on the properties of the monolayer could be determined. The substrate corrugation influences the way the m i s f i t and orientation of the monolayer varies with the temperature and chemical potential of the system. At constant chemical p o t e n t i a l , over the range of temperatures considered in this thesis ( 0 - 4 K), the m i s f i t does not change with temperature. When the substrate corrugation i s greater than 7.0 K, the monolayer forms a commensurate s o l i d that becomes incommensurate when the chemical potential of the system i s increased. However, when the substrate corrugation i s less than 7.0 K, the monolayer does not form a commensurate s o l i d . The s i g n i f i c a n c e of t h i s r e s u l t is that the substrate corrugation value of 7.4 K calculated by Vida l d i and C o l e 4 8 produces a commensurate s o l i d at zero temperature. This produces agreement between a t h e o r e t i c a l l y derived value for the substrate corrugation and the experimental evidence that the monolayer can form a commensurate s o l i d . The m i s f i t of the most e n e r g e t i c a l l y favourable configuration varies with the chemical potential as shown in (1.1), regardless of the substrate corrugation. This r e s u l t agrees with experimental observations although the information does not provide corroboration that the substrate corrugation should be 7.4 K. Monolayers with small m i s f i t values are not rotated, however when the m i s f i t exceeds a given threshold value, the monolayer 103 is rotated. The r e l a t i o n s h i p between the orientation of the monolayer and i t s m i s f i t i s dependent on the substrate corrugation. For substrate corrugations less than 7.0 K, the ori e n t a t i o n a l behaviour matches that predicted by S h i b a 1 1 . However, when the substrate corrugation is greater than 7.0 K the anharmonic nature of the adatom pair interaction causes Shiba's results to be inaccurate. The monolayer is rotated at m i s f i t values less than those predicted by Shiba because the non-rotated configurations have adatoms that are overcompressed at the vertices of the domain walls. In fact, for a range of m i s f i t values where the rotated configurations are e n e r g e t i c a l l y favourable, the overcompression at the vertices causes the non-rotated configurations to be unstable. The orientation of the monolayer is sensitive to the temperature of the system. As the temperature is increased, configurations that are rotated w i l l become less rotated. The c a l c u l a t i o n for the free energy required s p l i t t i n g the free energy into three components - the potential energy, the zero point energy, and a temperature dependent component. The most s i g n i f i c a n t of these components is the potential energy which, for the range of temperatures considered in t h i s study determines the orientation behaviour of the monolayer. The zero point energy of the monolayer is not n e g l i g i b l e , and influences the r e l a t i o n s h i p between the m i s f i t of the lowest energy configuration and the chemical p o t e n t i a l . Because of t h i s , relaxation studies and low temperature molecular dynamics 104 calculations do not predict the correct ground state of the monolayer. The temperature dependent component of the free energy, which is determined by the low energy v i b r a t i o n a l modes of the monolayer, is not s i g n i f i c a n t in determining the configuration of the monolayer. The low energy modes of the incommensurate monolayer were calculated and correspond to motion of the domain walls. A renormalized model based on domain walls would provide not only the s t a t i c properties of the monolayer, but the dynamical properties as well. The lowest energy mode is the breathing mode predicted by V i l l a i n 1 4 , and though th i s mode and other domain wall modes are quite s o f t , the entropy associated with them does not contribute s i g n i f i c a n t l y to the free energy. The domain wall v i b r a t i o n a l modes are grouped into triads with the motion of a given mode in the t r i a d corresponding to one of three fundamental forms of domain wall motion. By specifying the t r i a d group and the mode of the t r i a d , the motion associated with the mode i s determined. In the course of the work for this thesis two papers were p u b l i s h e d 1 2 ' 3 4 . The f i r s t of these papers, Shrimpton et. a l . 1 2 examined approximations that are made to calculate the s t a t i c properties of the monolayer. The results of thi s paper indicated that for krypton on graphite theories which assumed that the adatom pair interactions were harmonic would not be quan t i t a t i v e l y correct. The paper also established the v a l i d i t y of the techniques used in thi s work to calculate the 105 s t a t i c configuration of the monolayer. The thesis extends the results presented in t h i s paper to cases where the monolayer i s rotated with respect to the substrate. The results indicate that harmonic theories may not describe c o r r e c t l y the rotation verses m i s f i t behaviour of the monolayer either. The second paper published, Shrimpton et. a l . 3 4 , concerned the dynamics of the incommensurate monolayer. Prior to this work there was no quantitative information about the in-plane motion of the adatoms in the incommensurate monolayer. The low energy mode predicted by V i l l a i n 1 4 was v e r i f i e d and several other low energy modes were described. With t h i s information about the low energy modes, and with other information calculated for the zero point energy, the free energy of the incommensurate monolayer was calculated. The results of the c a l c u l a t i o n for the free energy indicate that the low energy modes of the monolayer do not have a catastrophic impact on the extension of zero temperature calculations to f i n i t e temperatures. However, the results also indicate that the zero point energy of the monolayer influences the configuration of the monolayer, and calculations based s o l e l y on the s t a t i c properties of the monolayer may not be correct. 106 APPENDIX A The monolayer c o n f i g u r a t i o n s are determined by the i n t e g e r parameters m and n. From equation (2.9) the s i z e of the s u p e r l a t t i c e domains are determined. However, t h i s i s not the o n l y i n f o r m a t i o n that can be obtained d i r e c t l y from m and n, and t h i s s e c t i o n w i l l c o n s i d e r other geometric q u a n t i t i e s . From equation (2.9), the r e s t r i c t e d case of 51=1 and k = 0 produces a r e l a t i o n s h i p between these parameters and the b a s i s v e c t o r s R\ and R*2 of the s u p e r l a t t i c e p e r i o d i c i t y [ R i , R a ] = [ D i , D 2 ] n -m m n-m + , B £ ] (A.1) This i n t r o d u c e s the n o t a t i o n whereby two v e c t o r s are combined to form a 2x2 matrix. The top row c o n t a i n s the x components and the bottom row c o n t a i n s the y components. From equation (2.10), with the help of equation (2.7), the averaged l a t t i c e v e c t o r s d^ and d*2 can be determined from the commensurate l a t t i c e v e c t o r s 6\ and D*2 to be l_CU / d 2 j " - i -1 - lv "l o" 1 2 -1 n -m 0 1 T 1 1 m n-m (A.2) R e f e r r i n g back to (2.8) the m i s f i t and o r i e n t a t i o n can then a l s o be determined i n terms of m and n. 107 Given the p e r i o d i c i t y o£ the superlattice from (A.l) the set of r e c i p r o c a l l a t t i c e vectors with basis cf i, q 2 are defined by <3i -> <32 L R J RE J = -i -4 -4 -4 q 2 , R i -4 -4 -4 q -4 Ra L-i oj (A.3) The notation for the matrix formed by the q vectors has the l e f t column containing the x components of the vectors and the right column containing the y components. These conventions cause the matrix m u l t i p l i c a t i o n of the R* and q vector sets to produce the indicated dot products between the vectors. Given the R matrix from (A.l) the q matrix (and subsequent qL and q 2) can be determined by simple manipulation of matrices in equation (A.3). By analogy to the previous procedure, the rec i p r o c a l l a t t i c e -4 -4 vectors g A and g 2 for the substrate can be defined so that 92 [Bj , B 2] = 2rt 0 -1 (A.4) from which the rec i p r o c a l l a t t i c e vectors of the substrate for a given m and n can be calculated. -4 q 2 — _ -1 1 1 -m -n " l 0 9 i .-2 1. _n-m -m_ + .0 1. 92 (A.5) 108 One of the quantities that is of interest is the dot product between averaged l a t t i c e vectors of the adatoms and the recip r o c a l l a t t i c e vectors of the substrate. Any vector dot product can be determined from the dot products between the basis vectors of the two l a t t i c e s . Given equation (A.2), with the help of (2.7) and (A.4), 92 [di , d 2 ] = 2rt 0 -1 1 0 n -m m n-m 1-1 2 1 -1 1 (A.6) From (A.3) and (2.10) i t is clear that gi -» -ga [di , d 2] <3i <32 [di , d 2] 2rt 1 -2 (A.7) 109 APPENDIX B Much early interest in the problem of mismatched spacings between an overlayer of adatoms and the underlying substrate stemmed from the fact that the displacements, u, of the adatoms in the minimum energy configuration, given suitable approximations to the microscopic interactions, were obtained by solving a second order nonlinear set of d i f f e r e n t i a l e q uations 2 1. By considering configurations where the displacements occur only along one d i r e c t i o n , the displacements were found to have s o l i t o n v a r i a t i o n . This v a r i a t i o n produces regions where the adatoms are in r e g i s t r y with the substrate, separated by stri p e s of higher (or lower) density domain walls. The minimum energy configuration is determined by finding the c l a s s i c a l force free solution to the microscopic interactions. The adatam interaction $ is assumed to be harmonic and the substrate interaction is assumed to be sinusoidal (cf 2.3). According to these conditions the energy per adatom is E(R) = A L 1 + (u-u')-V + Vz (. (u-u').7.) J *(R-R') R' 3 o -+ -* (B.l) + V D + 2V g £ ( 1 " cos(g k-R) ) k = l g 3 i s defined from g A and g 2 (cf A.4) to be g 3 = ~qL- g 2 The continuum approximation is made which treats the 110 displacements as a continuous function of the averaged position R*. If the displacements are slowly varying, the following approximation can be made u' = u(R') = LI + (R'-R)-v" + 'A. (. (R'-R)-V.) J u(R) (B.2) With the assumption of uniaxial modulations, the displacements correspond to u(R) = ~ 6(S) Bi with S = B\-R (B.3) where B^  is a basis vector for the substrate. Using (B.3), and (B.2) which i s substituted into ( B . l ) , gives E(R) = E(S) = E 0 + tta (§§)' + t> | | + 4 V g ( l - cos8) (B.4a) where E Q = ^ #(h) + V Q h (B.4b) and a = J^jz § ( h - B i ) 2 (Bi-V) 2 *(h) (B.4c) h with b = £ (h-Bi) (Bi-V) 0(h) (B.4d) I l l S i m i l a r i l y , when the force on a given adatom Is calculated the force free condition requires that the displacements s a t i s f y d 28 . „ . 4V q = sin© where oL- = _^ (B.5) The solutions to (B.5) are solitons with the sp e c i a l case of a single s o l i t o n having the form 9 = 4tan~ i(e* s) (B.6) This provides the shape of the domain wall s o l i t o n , with the other possible solutions, where the domain walls are well separated, corresponding to periodic r e p e t i t i o n of the domain walls over the surface of the monolayer. From (B.6), i f ot is large, indicating a strongly corrugated substrate, the domain walls w i l l be quite sharp; conversely i f oc is small, indicating strong interactions between the adatoms, the domain walls w i l l be very broad. 112 APPENDIX C From (2.38), the maximum error in (2.17) r e s u l t i n g from approximations in ca l c u l a t i n g the A^kLjwf can be determined. However, the quantity of interest is the energy of the monolayer and thi s must be calculated to a preset accuracy. The contribution of the substrate to this energy involves an average of a l l the individual adatom-substrate interaction energies. Thus an error bound which r e f l e c t s the uncertainty in the average value of the adatom-substrate energy must be calculated. In order to estimate the error, the displacements u are considered to be anal y t i c functions of the mean positions R*. Two extreme cases for the behaviour of the displacements are considered with the assumption that the actual re s u l t w i l l l i e somewhere in between the two. F i r s t , the domain walls are considered to be step functions. The displacements u vary l i n e a r l y with R*, up to the edge of the domain, which for ease of c a l c u l a t i o n is taken to be c i r c u l a r . u(R) = cR i f IR I = a (C.l) where c is to be determined, and a is the radius of the domain. The quantity of concern to (2.38) is 113 z = ig-u(R) (C.2) Performing the average of <z2> using (C.l) over the area of the c i r c l e r esults in <z2> = j (aclg I) 2 (C.3) The estimate of the error in the energy is <%S>, from (2.38) and (C.l) i t is calculated to be < % 5 > = = k lacLgJl 24 8 n 8 24 8 n with (C.3) - <z'>" (C.4) 12 8 n The other extreme p o s s i b i l t y i s that the displacements of adatoms are so smoothly modulated that z can be given as z = i d g I sin( - IRI ) cos(6) for IRI ^  a (C.5) where 8 is the angle between R* and g. For such a case <z2> is calculated to be <z2> = j (acl g I ) 2 (C.6) 114 and the average error in <%S> for such a system is < % s > = k ( a c l ^ ' n 4 16 24 8 n (C.7) with (C.6) 8 8 n Because the configuration of the monolayer l i e s between the two extreme cases i t is assumed that s e t t i n g the uncertainty to be < % 5 > = 7 ^ <C.8) w i l l overestimate the actual error present in the calculated energy contribution of the substrate. Equation (C.8) requires a value for <z2>. From (2.13) th i s can be calculated by <z2> = - £ (g-u^m* (g-u-a, ) (C9) Sim ' allowing an estimate of <%£> to be determined from (C.8). The parameter value for n used to calculate the values A^ k LM is chosen from the error bound %S to be the smallest integer that s a t i s f i e s 115 n i V- 6 %S } (C.10) log(8) where <z2> is given by (C.9). It i s re a l i z e d that <z4> can be determined d i r e c t l y from (2.13). However, only a quick estimate of the uncertainty is required, and (C.8) with (C.9) is adequate to determine <%S>. Furthermore, the quantity (C.9) can be gleaned from the calculations required to evaluate the A N of (2.34) whereas <z4> would have to be calculated separately. The uncertainty in 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 from.(C.4) and (C.7). This is necessary because the errors in ca l c u l a t i n g the A - ' ^ L M w i l l influence the ove 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 £ m are used to calculate the set of A ^ j ^ , the error w i l l be compounded. Thus (C.4) and (C.7) underestimate the consequences of a s p e c i f i c choice of n. By analysing the observed error in the substrate energy calculated for various n values, (C.8) i s found to be a r e l i a b l e estimate of the uncertainty, and given an error bound %S, (C.10) can be used to set the value of n. 116 APPENDIX D Equation (4.9) requires integrating the periodic function 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 . Chadi and Cohen 5 0 have developed a technique to evaluate such an integral numerically. The technique is reviewed here to establish the terminology. f(k*) i s a periodic function and can be expanded as f(k) = X f£ e i k ' R (D.l) R where R* i s a superlattice vector generated by (2.9). The symmetry of the superlattice allows the c o e f f i c i e n t s f^ to be combined so that (D.l) becomes f(k) = I f m A m(k) (D.2) m where A^k) = ^ £ e l k ' R (D.3) "Rl=^m The values of c m range over a l l possible values of IRI and are ordered to increase with m. If f(k) varies smoothly with k the c o e f f i c i e n t s f m in (D.2) w i l l decrease rapi d l y with m. A weighted summation of the A m(j?) evaluated over a set of 117 points can be performed so that 2 oCi \ i ( k i ) = 0 (D.4) for a l l m in the range 0 < m £ N . The weights oc^  are normalized so that they t o t a l to unity. Given (D.2), performing the same weighted sum on the function f(i?) w i l l r e s u l t in ^ c ^ f ' k j ) = f Q + I f m IoiiAm[ki) (D.5) i m>N i Since f m decreases rapi d l y with m, the right hand side of (D.5) w i l l approach f Q as N is increased. f Q is the average value of f(k*) in the B r i l l o u i n zone. Thus, 2, 04 f ( k i ) = | J f(k) dk (D.6) to increasing accuracy as N is increased. For a given value of N there are many sets of points k^ that w i l l s a t i s f y (D.4). The points in the minimal set are ca l l e d the s p e c i a l points of the B r i l l o u i n zone. As the value of N i s increased, the number of special points increases. Chadi and Cohen 5 0 have devised a procedure whereby a set of special points is generated by point group operations on the points in 118 the two preceding sets. Cunningham 0 1 has calculated the f i r s t few sets of special points appropriate a two dimensional l a t t i c e with hexagonal symmetry. By examining these special points, i t i s clear that when a given set of points is combined with a l l the preceding sets of points, a triangular l a t t i c e is formed. Furthermore, none of the l a t t i c e points belong to more than one set of special points. When the next succesive set of special points is included, a more dense triangular l a t t i c e is formed of which the previous combined set forms a / 3 x / 3 s u b l a t t i c e . For the c a l c u l a t i o n of (4.9), advantage has been taken of th i s f a c t . A triangular l a t t i c e is generated with spacing small enough that the s p e c i a l points derived from i t w i l l provide the required accuracy. The special points are extracted from the triangular l a t t i c e by subtracting from this set a l l points that are common with the ' 3 x / 3 s u b l a t t i c e . The weight oi^ associated with t h i s set can be determined from the successive nature of the sets of special points. The weight for one set w i l l be 1/3 the weight of the previous set. By knowing the number of preceding sets of points, the weight can be calculated. It is remembered that the weight associated with points 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 . D i a g r a m o f t h e g r a p h i t e s u b s t r a t e . The honeycomb n e t w o r k i n d i c a t e s t h e c a r b o n b o n d s , t h e a d s o r p t i o n s i t e s a r e a t t h e c e n t e r s o f t h e h e x a g o n s f o r m e d b y t h e c a r b o n b o n d s . a n d B 2 a r e t^e bas i s^ 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 s i t e s . D A a n d D £ a r e t h e b a s i s v e c t o r s f o r t h e / 3 X / 3 s u b l a t t i c e . H 1-42 A I— 120 Figure 2. Types of domain walls between the '3x/3 l a t t i c e s of krypton atoms. Krypton atoms in an a subla t t i c e are i d e n t i f i e d by •. Atoms in a b subla t t i c e are i d e n t i f i e d by a •. Atoms in a c su b l a t t i c e are i d e n t i f i e d by 1 . 121 Figure 3. Typical positions of adatoms for an incommensurate monolayer with hexagonal symmetry. For t h i s p a r t i c u l a r configuration n=4, m=-4, k = 0, and *, = -! in equation (2.9). 122 Figure 4. Possible m i s f i t and orientation values for hexagonally periodic incommensurate monolayers. The angle of ro t a t i o n is given ln degrees. The monolayers are constrained so that, over the length of the primitive l a t t i c e vectors, the adatoms are s h i f t e d by only one adsorption s i t e r e l a t i v e to the commensurate monolayer. The configurations with super heavy domain walls are marked with c i r c l e s , the configurations with heavy domain walls are marked with squares. q in B q El i n < B B B B • • • d q d. * • • * • 4 * • .•.*•/.'•*•*•'* .*.*.• «««....«. 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 F i g u r e 5 ( a ) . E n e r g y p e r a d a t o m o f a n o n r o t a t e d I n c o m m e n s u r a t e 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 s u b s t r a t e 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 p o t e n t i a l 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 p e r a d a t o m a s 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 ln figure 5(a) but with V = 7 . 0 K. -300-1 % Misfit 126 Figure 5(d). As in figure 5(a) but with V„ = 8.0 K. -350-1 % Misfit 127 Figure 5(e). As in figure 5(a) but with V = 9.0 K. 9 - 3 0 0 n % Misfit 128 Figure 6(a). Contour plot showing the difference in the potent i a l energy per adatom between the rotated and nonrotated configurations. The angle of rotation is given in degrees. The s o l i d contours show increments of 0.5 K. The dotted and dashed contours show decrements of 0.02 K. The substrate corrugation for thi s plot is 5.0 K. o % Misfit 129 Figure 6(b). As ln figure 6(a) but with a substrate corrugation of 8.0 K. o % Misfit 130 Figure 6(c). As in figure 6(a) but with a substrate corrugation of 10.0 K. o % Misfit 131 Figure 7(a). A comparison of the adatom pair potential #(r) ( s o l i d line) and the polynomial f i t ar 3+br 2+cr+d (dashed line) for various values of r. -154 -i Distance between adatoms (A) 132 Figure 7(b). A comparison of the derivatives of the curves in figure 7(a) for various values of r. The s o l i d l i n e i s £'(r), the dashed l i n e is 3ar 2+2br+c. 150-1 4.3 Distance between adatoms (A) 133 Figure 7(c). A comparison of 0 " ( r ) ( s o l i d l ine) and the polynomial f i t 6ar+2b (dotted line) for various values of r. 1200 n 134 Figure 7(d). A comparison of ^ " ' ( r ) ( s o l i d l ine) and the polynomial f i t 6a (dotted line) for various values of r. -1500-1 -5500 H 1 1 1 4 4.1 4.2 4.3 Distance between adatoms (A) 135 Figure 8. Energy per adatom as in figure 5(c) except in the c a l c u l a t i o n , the pair potential £(r) is replaced by the polynomial f i t ar^+br 2+cr+d shown in figure 7(a). - 3 5 0 -i % Misfit 1 3 6 Figure 9 . Phonon energies as a function of wavevector for the nonrotated incommensurate monolayer with m i s f i t 2 . 2 2 % , and substrate corrugation 6 . 0 K. r is at the center of the hexagonal B r i l l o u i n zone, M i s at the middle of an edge and K is at a corner. 0.35-, Q> P CD C LU C o c o _c CL 0.30-0.25 H 0.20 H 0.15 H 0.10 0.05 H 0.00 J M T K M Misfit 2.22% Angle 0.0° 137 F i g u r e 1 0 ( 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 t h e p o i n t M o f f i g u r e 9 , h a v i n g a n e n e r g y o f O . O l l m e V . I n t h i s p i c t u r e , t h e d i r e c t i o n o f t h e w a v e v e c t o r i s a s shown i n t h e 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 t h e d e f o r m a t i o n o f t h e w a l l s a b o u t t h e i r c e n t r a l p o s i t i o n s ( s o l i d l i n e s ) . Phonon Energy=0.01 lmeV Misf it=2.22 % Rotation=0.00° Substrate Corrugation VQ=6.0K Wave Vector 138 Figure 1 0(b). As in figure 10(a) but for 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 a=6.0K Wave Vector 139 Figure 10(c). As in figure 10(a) but for a phonon mode with energy 0.027meV. Phonon Energy=0.027meV Misf it=2.22 % Rotation=0.00 0 Substrate Corrugation V a=6.0K Wave Vector 140 Figure 11. Energies of the phonon modes as in figure 9 but for a nonrotated monolayer with m i s f i t 1.75% . At thi s value of m i s f i t the t r i a d groupings of the domain walls are apparent 0.35 - i 0.30-0.25-C LaJ C o c o 0.20-0.15-0.10-0.05-0.00 J Vg=6.0K M r K M Misfit 1.75% Angle 0.0° 141 Figure 12(a). Motion of the domain walls as in figure 10(a) but for a phonon mode with wavevector at the r point and energy 0.037meV. From figure 9 thi s mode i s the t h i r d lowest in 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 t r i a d . Scale Y 10 nM Phonon Energy=0.037meV Misf it=2.22 % Rotation=0.00 • Substrate Corrugation Vg=6.0K Wave Vector 142 Figure 12(b). As in figure 12(a) but for a phonon mode with energy 0.315meV. This mode i s the sixth lowest in 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 walls as ln figure 10(a) but for a phonon mode having energy 0;201meV. From figure 9 th 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 Vg=6.0K W a v e V e c t o r 144 Figure 13(b). As in figure 13(a) but showing motion of the mode when the wavevector i s at r . Scale h 10 nM Phonon Energy=0.201 meV Misfit=2.22% Rototion=0.00° Substrate Corrugation Vg=6.0K Wave Vector 145 Figure 14(a). Phonon energies as a function of wavevector as in figure 9 but for a nonrotated monolayer with m i s f i t 3.33% when the substrate corrugation is 8.0 K. The energy scale i s extended to show imaginary values. The monolayer has an unstable mode. .E, P? CD C LU C o c o JC CL 0.35 n 0.30-0.25-0.20-0.15-0.10-0.05-0.00-« - . Vg=8.0K . « • M K • M i 0 . 0 5 J Misfit 3.33% Angle 0.0* 146 Figure 14(b). As in figure 14(a) but for a rotated monolayer with s l i g h t l y smaller m i s f i t . The modes are not unstable. 0.35-> 0.30-0.25-p? CD c LU c o c o 0.20-0.15-0.10-0.05-0.00 J Vg=8.0K • * • • M r K M Misfit 3.29% Angle 0.225" 147 Figure 15. Motion of the domain walls for the unstable phonon mode of figure 14(a) with wavevector at the M point. Phonon Energy= 10.026 meV Misfit=3.33% Rototion=0.00° Substrate Corrugation VQ=8.0K Wave Vector 148 Figure 16. 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