UBC Theses and Dissertations

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UBC Theses and Dissertations

LEED crystallographic studies of oxygen adsorbed on the (0001) surface of zirconium Hui, Ka-Chung 1986

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LEED CRYSTALLOGRAPHIC STUDIES OF OXYGEN ADSORBED ON THE (0001) SURFACE OF ZIRCONIUM by KA-CHUNG HUI A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES De p a r t m e n t of C h e m i s t r y We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u n e , © Ka-Chung BRITISH COLUMBIA 1986 H u i , 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e The U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head of my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n ot be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e partment of C h e m i s t r y The U n i v e r s i t y of B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : J u n e , 1986 A b s t r a c t The work i n t h i s t h e s i s i n c l u d e s LEED c r y s t a l l o g r a p h i c i n v e s t i g a t i o n s f o r t h e (2x2) s t r u c t u r e o b t a i n e d by a d s o r b i n g oxygen on t h e (0001) s u r f a c e of z i r c o n i u m a t below 2 5 0 ° C , a s t u d y o f t h e s t a b i l i t y o f t h e i n t e n s i t y o f f r a c t i o n a l o r d e r beams w i t h d i f f e r e n t c o v e r a g e s o f a d s o r b e d s p e c i e s on h c p ( O O O l ) m e t a l s u r f a c e s , a p r e l i m i n a r y s t r u c t u r a l i n v e s t i g a t i o n o f t h e Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 s t r u c t u r e , and t h e s e t t i n g up o f a TV camera s y s t e m f o r f a s t e r a c q u i s i t i o n o f d i f f r a c t e d beam i n t e n s i t i e s . The LEED c r y s t a l l o g r a p h i c s t u d y f o r o xygen a d s o r p t i o n on t h e Z r ( 0 0 0 1 ) s u r f a c e y i e l d s t h e f i r s t s t r u c t u r a l d a t a f o r t h e i n i t i a l s t a g e s o f o x i d a t i o n on any h c p ( 0 0 0 l ) s u r f a c e . The z i r c o n i u m s u r f a c e was c u t f r o m a h i g h p u r i t y s i n g l e c r y s t a l , and c h a r a c t e r i z e d w i t h LEED and Auger e l e c t r o n s p e c t r o s c o p y . I n t e n s i t y v e r s u s e n e r g y ( 1 ( E ) ) c u r v e s were measured by t h e p h o t o g r a p h i c method a t n o r m a l i n c i d e n c e f o r s e v e n and two d i f f r a c t e d beams r e s p e c t i v e l y f o r t h e (2x2) and (1x1) s t r u c t u r e s . T h e o r e t i c a l 1 ( E ) c u r v e s were c a l c u l a t e d u s i n g t h e 'combined s p a c e ' a p p r o a c h t o m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s ( e . g . c o m p o s i t e l a y e r c a l c u l a t i o n s , r e n o r m a l i z e d f o r w a r d s c a t t e r i n g and l a y e r d o u b l i n g ) f o r a r a n g e o f Z r - 0 i n t e r l a y e r s p a c i n g s and many d i f f e r e n t a d s o r p t i o n m o d e l s . L e v e l s o f c o r r e s p o n d e n c e between e x p e r i m e n t a l and c a l c u l a t e d 1(E) c u r v e s were a s s e s s e d w i t h t h e P e n d r y R - f a c t o r . The a n a l y s e s s u g g e s t t h a t 0 atoms oc c u p y o c t a h e d r a l h o l e s between s u c c e s s i v e b u l k Zr l a y e r s i n i i t h e (2x2) s t r u c t u r e , and t h a t t h e s u b s t r a t e Zr l a y e r s u ndergo a f e e t y p e r e c o n s t r u c t i o n w h i c h s p a n s t h e d e p t h of O atoms. The Z r - 0 i n t e r l a y e r s p a c i n g i s f o u n d t o be 1.33 A; t h i s c o r r e s p o n d s t o a Z r - 0 bond l e n g t h o f 2.29 A, and t h e r e f o r e i s i n c l o s e agreement w i t h t h e v a l u e (2.31 A) f o r b u l k ZrO. H a v i n g o n l y two i n d e p e n d e n t beams f o r t h e (1x1) s u r f a c e p r e c l u d e s any d e f i n i t i v e s t r u c t u r a l c o n c l u s i o n s , but t h e p r e l i m i n a r y a n a l y s i s h e r e i n d i c a t e s t h a t t h e f i r s t t h r e e l a y e r s r e s e m b l e (111) l a y e r s of ZrO w i t h t h e Z r - 0 i n t e r l a y e r s p a c i n g e q u a l t o 1.37 A and a c o n s e q u e n t bond l e n g t h o f 2.31 o A. D i f f r a c t e d beam 1( E ) c u r v e s were c a l c u l a t e d a t n o r m a l i n c i d e n c e w i t h t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method f o r (2x1) and (2x2) s t r u c t u r e s i n v o l v i n g a d s o r p t i o n on (0001) s u r f a c e s of t i t a n i u m and z i r c o n i u m , and t h e s e d a t a were compared w i t h t h e R - f a c t o r s o f P e n d r y and o f Z a n a z z i and J o n a . T h i s s t u d y s u p p o r t s an o b s e r v a t i o n by Yang et al. t h a t LEED f r a c t i o n a l o r d e r beam i n t e n s i t i e s may be c l o s e l y c o n s t a n t w i t h c h a n g i n g a d s o r b a t e c o v e r a g e ; a new f e a t u r e f o u n d h e r e i s t h a t t h i s c o n c l u s i o n h o l d s even f o r s t r u c t u r e s where n e i g h b o r i n g a d s o r b e d atoms a r e s e p a r a t e d by j u s t t h e s u b s t r a t e i n t e r a t o m i c d i s t a n c e . T h i s o b s e r v a t i o n c o u l d be e x p l o i t e d i n t h e Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 a n a l y s i s by u s i n g w h i c h e v e r t r a n s l a t i o n a l symmetry t h a t gave t h e e a s i e r c o m p u t a t i o n a l e f f o r t . T h i s work p r o v i d e s f u r t h e r s u p p o r t f o r t h e n e g l e c t - o f - b e a m - s e t a p p r o x i m a t i o n , w h i c h was i n t r o d u c e d by Van Hove et al. f o r making more t r a c t a b l e t h e c a l c u l a t i o n o f L E E D i n t e n s i t i e s f r o m s u r f a c e s w i t h l a r g e u n i t m e s h e s . A l s o e s t a b l i s h e d i n t h i s w o r k i s a n i n t e n s i t y a c q u i s i t i o n s y s t e m , w h i c h u t i l i z e s a s u r v e i l a n c e t y p e T V c a m e r a a n d c o m m e r c i a l v i d e o L E E D a n a l y z e r ( V L A ) . I n i t i a l e x a m i n a t i o n s w e r e g i v e n f o r t w o b a c k g r o u n d i n t e n s i t y s u b t r a c t i o n s c h e m e s . 1 ( E ) d a t a c o l l e c t e d w i t h t h e v i d e o L E E D s y s t e m i n t h e l a t e r s t a g e s o f t h i s w o r k a r e c o m p a r e d s a t i s f a c t o r i l y w i t h t h e i n d e p e n d e n t m e a s u r e m e n t s o b t a i n e d b y t h e p h o t o g r a p h i c m e t h o d . iv T a b l e o f C o n t e n t s A b s t r a c t i i T a b l e of C o n t e n t s v L i s t of T a b l e s i x L i s t o f F i g u r e s x i Acknowledgements x v i i i 1. S t u d i e s o f O r d e r e d A d - l a y e r s on W e l l - C h a r a c t e r i z e d C r y s t a l l o g r a p h i c P l a n e s 1 1.1 H i s t o r i c a l D e v e l o p m e n t 2 1.2 T e c h n i q u e s R e l e v a n t t o A d - l a y e r s S t u d i e s 4 1.2.1 E l e c t r o n S c a t t e r i n g T e c h n i q u e s 5 1.2.2 Ion o r Atom S c a t t e r i n g T e c h n i q u e s 9 1.2.3 P h o t o n Beam T e c h n i q u e s 11 1.3 Some B a s i c Knowledge from A d s o r p t i o n S t u d i e s on M e t a l S u r f a c e s 14 1.3.1 The S t r u c t u r a l P a r a m e t e r s 15 1.3.2 The N a t u r e o f t h e S u r f a c e C h e m i c a l Bond ...16 1.4 Aim o f t h e T h e s i s 18 2. P r i n c i p l e s o f LEED and AES 22 2.1 E l e c t r o n D i f f r a c t i o n f r o m O r d e r e d S u r f a c e s 23 2.1.1 C o n d i t i o n s f o r E l a s t i c D i f f r a c t i o n 23 2.1.2 A p p e a r a n c e of LEED S p o t s : The E w a l d C o n s t r u c t i o n 28 2.1.3 S u p e r l a t t i c e N o t a t i o n 30 2.2 LEED Sp o t I n t e n s i t y - E n e r g y C u r v e 37 2.2.1 The D i f f r a c t i o n Peak P o s i t i o n s 41 2.2.2 Peak W i d t h 44 2.2.3 T h r e e - D i m e n s i o n a l E f f e c t s 45 2.2.4 O v e r l a y e r E f f e c t 47 v 2.3 D i s o r d e r , Domains and I n s t r u m e n t a l R e s p o n s e 49 2.4 Auger E l e c t r o n S p e c t r o s c o p y (AES) 55 2.4.1 The Auger P r o c e s s 55 2.4.2 K i n e t i c E n e r g i e s of Auger E l e c t r o n s 57 2.4.3 AES and S u r f a c e A n a l y s e s 58 2.4.3.1 Q u a l i t a t i v e A n a l y s i s 59 2.4.3.2 Q u a n t i t a t i v e A n a l y s i s 61 3. M u l t i p l e S c a t t e r i n g C a l c u l a t i o n s 64 3.1 I n t r o d u c t i o n 65 3.2 P h y s i c a l P a r a m e t e r s i n LEED C a l c u l a t i o n s 68 3.2.1 The ' M u f f i n - T i n ' A p p r o x i m a t i o n 68 3.2.2 The C o n s t a n t P o t e n t i a l V 0 72 3.2.2.1 The R e a l P o t e n t i a l V 0 r 72 3.2.2.2 The I m a g i n a r y P o t e n t i a l V 0 / 73 3.2.3 Ion C o r e S c a t t e r i n g 74 3.2.3.1 The Ion C o r e P o t e n t i a l V g 74 3.2.3.2 The Phase S h i f t s 6, 76 3.2.4 L a t t i c e M o t i o n 79 3.3 The L a y e r D i f f r a c t i o n M a t r i x 84 3.3.1 B r a v a i s L a t t i c e L a y e r s 87 3.3.2 C o m p o s i t e L a y e r s 89 3.3.2.1 M a t r i x I n v e r s i o n 92 3.3.2.2 R e v e r s e S c a t t e r i n g P e r t u r b a t i o n ...93 3.3.2.3 C o m b i n i n g M a t r i x I n v e r s i o n and RSP 96 3.4 L a y e r S t a c k i n g 97 3.4.1 L a y e r D o u b l i n g 97 3.4.2 R e n o r m a l i z e d F o r w a r d S c a t t e r i n g 100 v i 3.4.2.1 S u b r o u t i n e RFSG 105 3.5 G e n e r a l C o n s i d e r a t i o n s i n 'Combined Space' Method 108 3.5.1 T o t a l Beam R e q u i r e m e n t i n K-Space 108 3.5.2 Use of Symmetry and Beam S e t s 110 3.5.3 S e l e c t i o n o f Methods 113 G e n e r a l E x p e r i m e n t a l A s p e c t s 120 4.1 The UHV Chamber 121 4.2 Sample P r e p a r a t i o n and C l e a n i n g 127 4.2.1 C r y s t a l l o g r a p h i c P l a n e O r i e n t a t i o n 127 4.2.2 S u r f a c e C l e a n i n g i n UHV Chamber 130 4.2.3 S u r f a c e C o m p o s i t i o n by Auger E l e c t r o n S p e c t r o s c o p y 132 4.3 The LEED E x p e r i m e n t 137 4.3.1 LEED O p t i c s 1 37 4.3.2 D i s p l a y o f t h e LEED P a t t e r n 140 4.4 Q u a n t i t a t i v e Measurement of LEED Spot I n t e n s i t i e s 141 4.4.1 P h o t o g r a p h i c M e thod 143 4.4.2 TV Camera Method 146 4.4.3 B a c k g r o u n d S u b t r a c t i o n 155 S t a b i l i t y o f LEED F r a c t i o n a l O r d e r Beam I n t e n s i t i e s 165 5.1 I n t r o d u c t i o n 166 5.2 C o m p a r i s o n o f 1 ( E ) C u r v e s : The R e l i a b i l i t y I n d i c e s 168 5.2.1 Z a n a z z i - J o n a R - f a c t o r 168 5.2.2 P e n d r y R - f a c t o r 170 5.2.3 N o r m a l i z a t i o n o f R - f a c t o r s 171 5.3 (2x1) v e r s u s (2x2) 172 v i i 5.3.1 The C a l c u l a t i o n s • 172 5.3.2 R - f a c t o r A n a l y s e s and D i s c u s s i o n 176 5.4 A p p r o x i m a t e Schemes f o r M u l t i p l e S c a t t e r i n g C a l c u l a t i o n s 185 5.4.1 A p p r o x i m a t i o n s i n L - S p a c e 186 5.4.2 A p p r o x i m a t i o n s i n K-Space 187 5.5 C o n c l u s i o n 190 6. Oxygen A d s o r p t i o n on Z r ( 0 0 0 l ) 192 6.1 I n t r o d u c t i o n ...193 6.2 E x p e r i m e n t a l 196 6.2.1 Sample P r e p a r a t i o n and C l e a n i n g 196 6.2.2 Measurements o f 1 ( E ) C u r v e s 198 6.3 S t r u c t u r e A n a l y s i s o f Zr(0001 ) - ( 2 x 2 ) - 0 204 6.3.1 M u l t i p l e S c a t t e r i n g C a l c u l a t i o n s 204 6.3.2 R e s u l t s and D i s c u s s i o n s 209 6.3.3 S t r u c t u r a l R e f i n e m e n t 221 6.4 S t r u c t u r e A n a l y s i s o f Zr(0001 )-( 1x1 )-0 222 6.4.1 M u l t i p l e S c a t t e r i n g C a l c u l a t i o n s 222 6.4.2 R e s u l t s and D i s c u s s i o n 223 6.4.3 V a r i a t i o n o f Phase S h i f t s 230 6.5 I n t e r p r e t a t i o n o f A d s o r b a t e C o v e r a g e s 233 6.6 C o n c l u s i o n s and F u t u r e Work 238 REFERENCES 241 v i i i L i s t o f T a b l e s 1.1 A d s o r b a t e - s e n s i t i v e t e c h n i q u e s and t h e i r c h a r a c t e r i s t i c s . I n d i v i d u a l e x p e r i m e n t a l d e t a i l c a n be f o u n d i n t h e r e f e r e n c e s q u o t e d under t h e t e c h n i q u e column 6 3.1 The v a r i a t i o n o f t h e number of p l a n e waves r e q u i r e d w i t h i n c i d e n t e n e r g y and t h e s h o r t e s t i n t e r l a y e r d i s t a n c e ( d m ^ n ) f o r t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n o f Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 ; t=0.002. (a) U n s y m m e t r i z e d beams; and (b) Beams s y m m e t r i z e d w i t h r e s p e c t t o a 3 - f o l d r o t a t i o n a x i s and m i r r o r p l a n e ( x z ) symmetry o p e r a t i o n s 109 3.2 R e l a t i v e c o m p u t i n g t i m e s f o r t h e b u i l d i n g b l o c k s i n t h e 'combined s p a c e ' m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s ( A f t e r Van Hove and T o n g [ 5 6 ] ) 115 3.3 F u n c t i o n s o f s e v e r a l i m p o r t a n t and f r e q u e n t l y u s e d s u b r o u t i n e s i n t h e 'combined s p a c e ' m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s p r o v i d e d by Van Hove and T o n g [ 5 6 , 9 7 ] 119 5.1 C o m p a r i s o n s o f c a l c u l a t e d 1 ( E ) c u r v e s f o r i n t e g r a l and f r a c t i o n a l o r d e r beams f o r s u r f a c e s t r u c t u r e s w i t h (2x1) and (2x2) t r a n s l a t i o n a l s y m m e t r i e s on h c p ( 0 0 0 l ) s u r f a c e s a c c o r d i n g t o t h e r e l i a b i l i t y i n d i c e s o f Z a n a z z i - J o n a (R_-:) and P e n d r y (R,_). The Z J p e n e r g y r a n g e o f 1(E) d a t a f o r e a c h c o m p a r i s o n i s i d e n t i f i e d by AE ( i n eV) 177 6.1 Minimum v a l u e s o f m u l t i - b e a m Rp w i t h t h e c o r r e s p o n d i n g Z r - 0 i n t e r l a y e r s p a c i n g s ( d Z r _ 0 ) a n d V 0 r o b t a i n e d f r o m t h e c o m p a r i s o n s o f . e x p e r i m e n t a l and c a l c u l a t e d 1 ( E ) c u r v e s b a s e d on oxygen a d s o r p t i o n m o d e l s l i s t e d i n t h e f i r s t column f o r Zr (0001 )-( 2x2 )-0 a t n o r m a l i n c i d e n c e 216 6.2 Minimum v a l u e s o f m u l t i - b e a m Rp w i t h t h e c o r r e s p o n d i n g Z r - 0 i n t e r l a y e r s p a c i n g s ( d Z r _ 0 ) and V 0 r o b t a i n e d f r o m t h e c o m p a r i s o n s of e x p e r i m e n t a l and c a l c u l a t e d 1 ( E ) c u r v e s b a s e d on oxygen a d s o r p t i o n m o d e l s l i s t e d i n t h e f i r s t column f o r Zr(0001 )-(1x1 )-0 a t n o r m a l i n c i d e n c e 226 ix 6.3 V a r i a t i o n o f ' m u f f i n - t i n ' r a d i i f o r 0 ( T Q ) and Zr ( r Z r ) w i t h i o n i c c h a r g e on 0 ( n e g a t i v e ) and Zr ( p o s i t i v e ) r e s p e c t i v e l y f r o m c a l c u l a t i o n s o f i o n c o r e p o t e n t i a l s f o r t h e two s p e c i e s i n a ZrO c r y s t a l l a t t i c e . The v a l u e s o f t h e p o t e n t i a l a t h t e ' m u f f i n - t i n ' r a d i i f o r e a c h ZrO l a t t i c e w i t h t h e s p e c i f i e d i o n i c c h a r a c t e r f o r t h e two s p e c i e s a r e a l s o g i v e n 231 6.4 Minimum v a l u e s of m u l t i - b e a m R f o r Z r ( 0 0 0 1 ) -( i x l ) - O a t n o r m a l i n c i d e n c e . The c o r r e s p o n d i n g v a l u e s o f ^ z r - 0 a n c ^ v ° r w e r e o b t a i n e d f r o m t h e c o m p a r i s o n s o f e x p e r i m e n t a l and c a l c u l a t e d 1(E) c u r v e s , f o r a d s o r p t i o n m o d els i n w h i c h t h e t h r e e topmost l a y e r s c o r r e s p o n d t o t h e t h r e e (111) l a y e r s o f ZrO l a t t i c e . 0 p h a s e s h i f t s u s e d were d e r i v e d from i o n c o r e p o t e n t i a l s of ( i ) 0 ° , ( i i ) 0 " 1 and ( i i i ) 0 - 2 i n a ZrO l a t t i c e 2 3 2 x L i s t o f F i g u r e s 2.1 Some common r e a l s p a c e s u r f a c e n e t s and t h e i r c o r r e s p o n d i n g r e c i p r o c a l n e t s 26 2.2 E w a l d c o n s t r u c t i o n f o r e l e c t r o n d i f f r a c t i o n . O n l y f o u r r e c i p r o c a l r o d s a r e s e l e c t e d f r o m a l a r g e number o f s u c h r o d s ( i n s e t ) f o r t h e sake o f c l a r i t y 29 2.3 S c h e m a t i c s e t - u p f o r LEED e x p e r i m e n t and t h e r e l a t i o n between t h e s u r f a c e l a t t i c e and t h e d i f f r a c t i o n p a t t e r n 31 2.4 LEED p a t t e r n s f r o m Rh(111) a t 142 eV, n o r m a l i n c i d e n c e : ( a ) C l e a n s u r f a c e ; ( b ) A f t e r e x p o s u r e t o H 2 S . The e x t r a s p o t s a r e l a b e l l e d i n f r a c t i o n s o f t h e ' c l e a n ' r e c i p r o c a l v e c t o r s 32 2.5 Some common s u p e r l a t t i c e s t r u c t u r e s on low M i l l e r i n d e x s u r f a c e s and t h e i r n o m e n c l a t u r e i n b o t h Wood and m a t r i x n o t a t i o n s 35 2.5 ( c o n t i n u e d ) 36 2.6 O r d e r e d a d l a y e r o f b e n z e n e on R h ( 1 1 1 ) . The s t r u c t u r e i s d e s c r i b e d a s R h ( 1 1 1 ) - j i 3|C 6H 6 i n m a t r i x n o t a t i o n ( a f t e r Van Hove et al. [ 5 1 ] ) 38 2.7 1 ( E ) c u r v e f o r (0,0) beam f r o m N i O O O ) a t 6=3°. K i n e m a t i c a l l y e x p e c t e d B r a g g peak p o s i t i o n s a r e i n d i c a t e d by v e r t i c a l b a r s ( a f t e r A n d e r s s o n et al . [ 5 2 ] ) 40 2.8 Some h y p o t h e t i c a l s u r f a c e s and t h e i r c o r r e s p o n d i n g r e c i p r o c a l r o d s . ( a ) P e r f e c t l y o r d e r e d s u r f a c e ; (b) S l i g h t l y d i s l o c a t e d c r y s t a l l o g r a p h i c p l a n e s ; ( c ) I n c r e a s i n g monatomic s t e p s ; and ( d ) A l t e r n a t i n g monatomic s t e p s 50 2.9 T h r e e p o s s i b l e r o t a t i o n a l domains (A,B,C) o f a p ( 2 x 1 ) s u p e r l a t t i c e s t r u c t u r e on a h e x a g o n a l c l o s e - p a c k e d s u r f a c e 53 2.10 S c h e m a t i c r e p r e s e n t a t i o n o f Auger p r o c e s s . K, L and M r e p r e s e n t a t o m i c e n e r g y l e v e l s , and t h e a r r o w s r e p r e s e n t e l e c t r o n s 56 x i 2.11 P r o b a b i l i t y o f Auger e m i s s i o n and o f X - r a y f l u o r e s c e n c e a s a f u n c t i o n o f a t o m i c number f o r a K - s h e l l c o r e h o l e 56 2.12 D e r i v a t i v e Auger s p e c t r a t a k e n from a Z r ( O O O l ) s u r f a c e . ( a ) B e f o r e c l e a n i n g ; and ( b ) A f t e r =*50 h o u r s o f A r * bombardment 60 3.1 L e f t - h a n d e d c o - o r d i n a t e s y s t e m u s e d i n t h e 'combined s p a c e ' m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . H o l l o w and s o l i d c i r c l e s r e p r e s e n t s u b s t r a t e and a d s o r b a t e atoms r e s p e c t i v e l y . B r a v a i s l a t t i c e l a y e r s a r e s e p a r a t e d by a t l e a s t 0.5 A 69 3.2 ' M u f f i n - t i n ' a p p r o x i m a t i o n f o r t h e p o t e n t i a l s o f a s i n g l e row o f i o n c o r e s a l o n g t h e x - a x i s 71 3.3 A p p r o x i m a t i o n f o r V 0 r w i t h known q u a n t i t i e s of F e r m i e n e r g y and work f u n c t i o n <f> o f m e t a l s 71 3.4 U n i t c e l l o f ZrO c r y s t a l l a t t i c e s t r u c t u r e . ( a ) U n i t c e l l d i m e n s i o n s and r e s p e c t i v e h a r d s p h e r e r a d i i f o r Zr° and 0° u s e d i n t h e c a l c u l a t i o n s o f V g f o r b o t h e l e m e n t s . ( b ) A blow-up t o show t h e l o c a l e n v i r o n m e n t of oxygen i n t h e l a t t i c e 77 3.5 C o m p a r i s o n o f oxygen phase s h i f t s d e r i v e d from (a) V s o b t a i n e d f r o m ZrO c r y s t a l l a t t i c e and (b) s u p e r p o s i t i o n p o t e n t i a l s o b t a i n e d by Demuth et al . [98] 80 3.6 C o m p a r i s o n o f z i r c o n i u m p h a s e s h i f t s d e r i v e d from (a) V g o b t a i n e d f r o m ZrO c r y s t a l l a t t i c e and (b) V g f r o m band s t r u c t u r e c a l c u l a t i o n s f 9 6 ] 81 3.7 M u l t i p l e s c a t t e r i n g o f a s e t o f p l a n e waves by a l a y e r o f i o n c o r e s w i t h known d i f f r a c t i o n m a t r i x M~~ 86 3.8 S c h e m a t i c d i a g r a m o f t r a n s m i s s i o n and r e f l e c t i o n m a t r i c e s a t t h e n t h l a y e r . B r o k e n l i n e s a r e midway between c o n s e c u t i v e l a y e r s 86 3.9 E xamples o f c o m p o s i t e l a y e r . ( a ) G r a p h i t i c t y p e oxygen o v e r l a y e r : 2 oxygen atoms p e r u n i t mesh. (b) p ( 2 x 2 ) oxygen u n d e r l a y e r w h i c h i s s e p a r a t e d from t h e t o p z i r c o n i u m l a y e r by =*0.4 A: 4 z i r c o n i u m x i i atoms and 1 oxygen atom p e r u n i t mesh 90 3.10 S c h e m a t i c i l l u s t r a t i o n o f t h e c o n t r i b u t i o n s of s c a t t e r i n g a m p l i t u d e s t o t h e i * " * 1 s u b p l a n e by f o r w a r d and back s c a t t e r i n g . S p h e r i c a l waves a r e t r a v e l l i n g a l o n g ±x d i r e c t i o n s f o r t h e p f c ^ t i m e i n t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n f o r m a l i s m 94 3.11 S c h e m a t i c d i a g r a m o f t h e l a y e r d o u b l i n g method u s e d t o s t a c k 4 i n d i v i d u a l l a y e r s ( w i t h ABAB... r e g i s t r i e s ) i n t o an a s y m m e t r i c s l a b o f 4 l a y e r s ( a f t e r T o n g [ 8 l ]) 99 3.12 S c h e m a t i c d i a g r a m o f t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method. ( a ) E a c h t r i p l e t o f a r r o w s r e p r e s e n t s t h e c o m p l e t e s e t o f p l a n e waves t h a t t r a v e l f r o m l a y e r t o l a y e r , ( b ) I l l u s t r a t i o n o f t h e v e c t o r s w h i c h s t o r e t h e a m p l i t u d e s o f t h e i n w a r d -and o u t w a r d - t r a v e l l i n g waves ( a f t e r Van Hove and T o n g [ 5 6 ] ) 102 3.13 S c h e m a t i c i l l u s t r a t i o n o f some i m p o r t a n t i n p u t p a r a m e t e r s i n s u b r o u t i n e (RFSG) f o r t h e a d s o r p t i o n s y s t e m Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 w i t h 2 oxygen u n d e r l a y e r s o c c u p y i n g o c t a h e d r a l h o l e s . ( a ) S i d e v i e w o f t h e l a y e r a r r a n g e m e n t o f t h e s u r f a c e r e g i o n . ( b ) S e l e c t i o n o f a p p r o p r i a t e d i f f r a c t i o n m a t r i c e s w i t h c o d i n g v e c t o r s NRTNP and NRTP 106 3.14 LEED p a t t e r n f r o m Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 . (a)Symmetry-r e l a t e d beams a r e i n d i c a t e d by t h e same sym b o l s ( a t n o r m a l i n c i d e n c e ) . (b)Beams b e l o n g i n g t o t h e same beam s e t a r e i n d i c a t e d by t h e same number ( i n d e p e n d e n t o f a n g l e of i n c i d e n c e ) 111 4.1 S c h e m a t i c d i a g r a m o f t h e FC12 UHV chamber and some o f i t s i m p o r t a n t a s s e s s o r i e s . AES = A u g e r e l e c t r o n s p e c t r o s c o p y ; CMA = c y l i n d r i c a l m i r r o r a n a l y z e r . ...122 4.2 ( a ) P u m p i n g s y s t e m a s s o c i a t e d w i t h t h e FC12 UHV chamber. ( b ) S t a r t - u p p r o c e d u r e f o r pump-down from a t m o s p h e r i c p r e s s u r e t o UHV 125 4.3 ( a ) S c h e m a t i c d i a g r a m o f l a s e r a l i g n m e n t o f o p t i c a l and c r y s t a l l o g r a p h i c p l a n e s o f a s i n g l e c r y s t a l . ( b ) A blow-up t o show t h e r e l a t i o n s h i p between t h e o p t i c a l and c r y s t a l l o g r a p h i c p l a n e s . A l i g n m e n t i s x i i i a c c e p t a b l e when 8<$° 129 4.4 S c h e m a t i c d i a g r a m of LEED o p t i c s u s e d a s a r e t a r d i n g f i e l d a n a l y z e r f o r AES 133 4.5 S c h e m a t i c d i a g r a m o f t h e e x p e r i m e n t a l s e t - u p f o r AES u s i n g a c y l i n d r i c a l m i r r o r a n a l y z e r and g l a n c i n g i n c i d e n c e e l e c t r o n gun 136 4.6 S c h e m a t i c d i a g r a m o f t h e e l e c t r o n o p t i c s u s e d f o r LEED e x p e r i m e n t s 139 4.7 S c h e m a t i c d i a g r a m o f t h e c o m p u t e r - c o n t r o l l e d a n a l y s i s o f p h o t o g r a p h i c n e g a t i v e s o f LEED p a t t e r n s 145 4.8 S c h e m a t i c d i a g r a m of t h e r e a l - t i m e LEED s p o t i n t e n s i t y a n a l y s i s u s i n g a v i d e o LEED a n a l y z e r (VLA, D a t a - Q u i r e ) 148 4.9 S c h e m a t i c i l l u s t r a t i o n o f t h e number o f p a s s e s r e q u i r e d f o r d i g i t i z a t i o n o f 8 LEED s p o t s i n a s q u a r e n e t 152 4.10 3 - d i m e n s i o n a l d i a g r a m s o f two t y p e s o f i n t e g r a t e d LEED s p o t i n t e n s i t y i n a ( 1 0 - p i x e l x 1 0 - p i x e l ) window. ( a ) H u n d r e d sum ( o r HSUM). (b)N-sum ( o r NSUM) 1 53 4.11 S c h e m a t i c d i a g r a m o f t h e r e a l - t i m e b a c k g r o u n d s u b t r a c t i o n w i t h t h e a i d o f a hardw a r e a d d e r ( a f t e r H e i l m a n n et al . [ 1 3 4 ] ) 156 4.12 LEED s p o t b a c k g r o u n d i n t e n s i t y a p p r o x i m a t e d by HSUM of a window between n e i g h b o r i n g s p o t s 159 4.13 S c h e m a t i c i l l u s t r a t i o n o f t h e use o f HSUM and NSUM f o r t h e e s t i m a t i o n o f b a c k g r o u n d . The sha d e d ' t a i l s ' i n (a) and (b) r e p r e s e n t t h e d i f f e r e n c e (HSUM-NSUM). The a v e r a g e b a c k g r o u n d i s d e f i n e d a s t h i s d i f f e r e n c e d i v i d e d by t h e number o f p i x e l s t h a t make up t h e t a i l s . ( c ) and (d) r e p r e s e n t two t y p e s o f s p o t where HSUM=NSUM, and e q u a t i o n (4.5) i s n o t a p p l i c a b l e 161 4.14 E x p e r i m e n t a l 1(E) c u r v e s f o r (0,1) beam o f c l e a n Rh(111) s u r f a c e ( 0=0°, no s m o o t h i n g ) . (a)No b a c k g r o u n d s u b t r a c t i o n . ( b ) B a c k g r o u n d s u b t r a c t i o n x i v u s i n g e q u a t i o n ( 4 . 4 ) . ( c ) B a c k g r o u n d s u b t r a c t i o n u s i n g e q u a t i o n (4.5) 163 5.1 ( a ) S i d e view and ( b ) t o p v i e w o f t h e two p o s s i b l e d omains r e s u l t i n g f r o m t h e t r u n c a t i o n of t h e b u l k s t r u c t u r e of a hcp(OOOl) s u r f a c e . The two domains a r e r e l a t e d t o e a c h o t h e r by a 180° r o t a t i o n 173 5.2 S u p e r p o s i t i o n of t h e (2x1) r e c i p r o c a l l a t t i c e s i n (a) from t h e t h r e e p o s s i b l e r o t a t i o n a l domains o f a (2x1) s t r u c t u r e on a hc p ( O O O l ) o r f c c ( 1 1 l ) s u r f a c e t o f o r m an a p p a r e n t (2x2) r e c i p r o c a l l a t t i c e i n (b) . S o l i d a n d h o l l o w c i r c l e s r e p r e s e n t i n t e g r a l -and f r a c t i o n a l - o r d e r beams r e s p e c t i v e l y 174 5.3 C o m p a r i s o n o f c a l c u l a t e d 1 ( E ) c u r v e s f o r t h e (1/2,-1/2) beam f o r t h e c o r r e s p o n d i n g (2x1) ( d o t t e d l i n e ) and (2x2) ( s o l i d l i n e ) s t r u c t u r e s f o r a d s o r p t i o n o f (A) 0, (B) T i and (C) Te, a t t h e 3h s i t e on t h e T i ( O O O l ) s u r f a c e . S i n g l e beam Rp a n d R 2 j a r e i n d i c a t e d f o r e a c h p a i r o f 1 ( E ) c u r v e s 179 5.4 Same a s F i g u r e 5.3, e x c e p t f o r t h e (1/2,1/2) beam. .180 5.5 Same a s F i g u r e 5.3, e x c e p t f o r t h e (3/2,-1/2) beam. 181 5.6 Same a s F i g u r e 5.3, e x c e p t f o r t h e (3/2,-3/2) beam. 182 5.7 C o m p a r i s o n o f c a l c u l a t e d 1 ( E ) c u r v e s f o r t h e (1,1) beam f o r t h e c o r r e s p o n d i n g (2x1) ( d o t t e d l i n e ) and (2x2) ( s o l i d l i n e ) s t r u c t u r e s f o r a d s o r p t i o n o f (A) 0 and (B) Z r , a t t h e 3h s i t e on t h e Z r ( 0 0 0 l ) s u r f a c e . S i n g l e beam R_ a n d ' R _ J a r e i n d i c a t e d f o r P ^ j e a c h p a i r o f 1 ( E ) c u r v e s 184 6.1 C o m p a r i s o n o f e x p e r i m e n t a l 1 ( E ) c u r v e s o f s i x ' e q u i v a l e n t ' (1,0) beams ( s o l i d l i n e s ) o b t a i n e d a t n o r m a l i n c i d e n c e f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 w i t h t h e a v e r a g e d c u r v e ( d o t t e d l i n e ) . S i n g l e - b e a m Rp and R _ j a r e a l s o g i v e n f o r s u c h c o m p a r i s o n s 200 z J 6.2 C o m p a r i s o n o f e x p e r i m e n t a l 1 ( E ) c u r v e s measured by M o o r e [ l 3 0 ] ( s o l i d l i n e ) and by t h i s work ( d o t t e d l i n e ) f o r t h e (3/2,-1/2) beam f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 a t n o r m a l i n c i d e n c e 202 6.3 C o m p a r i s o n o f e x p e r i m e n t a l 1 ( E ) c u r v e s m easured w i t h t h e VLA ( s o l i d l i n e ) and w i t h t h e p h o t o g r a p h i c xv method ( d o t t e d l i n e ) f o r t h e (1,1) beam f o r Zr(0001 )-(1x1 )-0 a t n o r m a l i n c i d e n c e 203 6.4 S i d e v i e w of t h e Z r ( 0 0 0 l ) s u r f a c e w i t h (a) hep and (b) r e c o n s t r u c t e d f e e s t a c k i n g s e q u e n c e s t o show some p o s s i b l e o xygen a d s o r p t i o n s i t e s . Upper and l o w e r c a s e l e t t e r s r e p r e s e n t t h e r e g i s t r i e s o f Zr and 0 l a y e r s r e s p e c t i v e l y . E x c e p t f o r b f c and a t , a l l u n d e r l a y e r 0 atoms o c c u p y o c t a h e d r a l h o l e s 206 6.5 C o m p a r i s o n o f t h e e x p e r i m e n t a l 1 ( E ) c u r v e o f t h e (1/2,1/2) beam w i t h t h e ' b e s t ' c u r v e s f r o m s e l e c t e d model c a l c u l a t i o n s f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - O a t n o r m a l i n c i d e n c e 210 6.5 ( c o n t i n u e d ) 211 6.6 C o n t o u r p l o t of m u l t i - b e a m R p f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - O v e r s u s V 0 r and Z r - 0 i n t e r l a y e r s p a c i n g f o r t h e a d s o r p t i o n model d e s i g n a t e d C ( b ) A B . . a t n o r m a l i n c i d e n c e 213 6.7 C o n t o u r p l o t of m u l t i - b e a m R p f o r Z r ( 0 0 0 l ) - ( 2 x 2 ) - 0 v e r s u s V 0 r and Z r - 0 i n t e r l a y e r s p a c i n g f o r t h e a d s o r p t i o n model d e s i g n a t e d A ( c ) B ( a ) C A B . . . a t n o r m a l i n c i d e n c e 214 6.8 C o n t o u r p l o t of m u l t i - b e a m R p f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 v e r s u s V 0 r and Z r - 0 i n t e r l a y e r s p a c i n g f o r t h e a d s o r p t i o n model d e s i g n a t e d A ( c ) B ( a ) C ( b ) a t n o r m a l i n c i d e n c e 215 6.9 C o m p a r i s o n o f e x p e r i m e n t a l 1 ( E ) c u r v e s ( d o t t e d l i n e s ) f o r t h r e e i n t e g r a l and f o u r f r a c t i o n a l o r d e r beams from Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 w i t h t h e c o r r e s p o n d i n g 1 ( E ) c u r v e s c a l c u l a t e d f o r t h e a d s o r p t i o n model A ( c ) B ( a ) C ( b ) a t n o r m a l i n c i d e n c e w i t h d z r - 0 a t e i t h e r 1.33 o r 1.37 A 219 6.9 ( c o n t i n u e d ) 220 6.10 C o m p a r i s o n o f t h e e x p e r i m e n t a l 1 (E) c u r v e o f t h e (1,0) beam w i t h t h e ' b e s t ' c u r v e s f r o m s e l e c t e d model c a l c u l a t i o n s f o r Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 a t n o r m a l i n c i d e n c e 224 6.11 C o m p a r i s o n o f e x p e r i m e n t a l 1 ( E ) c u r v e s ( d o t t e d l i n e s ) f o r two d i f f r a c t e d beams f r o m x v i Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 w i t h t h e c o r r e s p o n d i n g 1 ( E ) c u r v e s c a l c u l a t e d f o r t h e a d s o r p t i o n model A ( c ) C A B . . a t n o r m a l i n c i d e n c e w i t h d Z r_o a t e i t h e r 1.37 o r 1.41 k 228 6.12 C o n t o u r p l o t o f m u l t i - b e a m R p f o r Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 v e r s u s V 0 r and Z r - 0 i n t e r l a y e r s p a c i n g f o r t h e a d s o r p t i o n model d e s i g n a t e d A ( b ) C A B . . a t n o r m a l i n c i d e n c e 229 6.13 A p l o t o f Auger peak r a t i o 0 5 , 0 / Z r i 7 * and e s t i m a t e d oxygen m o n o l a y e r a s a f u n c t i o n o f 0 2 e x p o s u r e ( i n L a n g m u i r , 1 L = 10" 6 t o r r s ) t o t h e (0001) s u r f a c e o f z i r c o n i u m . The oxygen c o v e r a g e s f o r t h e a p p e a r a n c e o f t h e (2x2) and (1x1) p a t t e r n s a r e a l s o g i v e n 234 6.14 A p l o t o f Auger peak r a t i o s O s 1 0 / Z r 1 7 4 and C 2 7 2 / Z r 1 7 « , as a f u n c t i o n o f CO e x p o s u r e t o t h e (0001) s u r f a c e o f z i r c o n i u m . The v a l u e s o f 0 5 1 o / Z r 1 7 n and C 2 7 2 / Z r 1 7 , , a t t h e i n t e r s e c t i o n s o f t h e t a n g e n t s o f t h e s t e e p p a r t and of t h e f l a t p a r t o f t h e c u r v e s a r e u s e d a s r e f e r e n c e f o r h a l f m o n o l a y e r e a c h o f 0 and C r e s p e c t i v e l y ( a f t e r Mooref 130]) 236 x v i i C H A P T E R 1 S T U D I E S OF O R D E R E D A D - L A Y E R S ON W E L L - C H A R A C T E R I Z E D C R Y S T A L L O G R A P H I C P L A N E S 1 2 1.1 HISTORICAL DEVELOPMENT S i n c e t h e f i r s t low e n e r g y e l e c t r o n d i f f r a c t i o n (LEED) e x p e r i m e n t p e r f o r m e d by D a v i s s o n and G e r m e r [ l ] i n 1927, l i t t l e p r o g r e s s was made on t h e s t u d i e s o f w e l l - d e f i n e d s i n g l e - c r y s t a l s u r f a c e s u n t i l t h e 1960's. The main r e a s o n f o r dormancy i n t h i s b r a n c h of s u r f a c e s c i e n c e a r o s e from t h e p r o b l e m o f q u a n t i t y : one s q u a r e c e n t i m e t e r o f a s u r f a c e c o n t a i n s o n l y a b o u t 1 0 1 5 p a r t i c l e s o r a p p r o x i m a t e l y 10" 9 m o l e . I n o t h e r words, t h e c o n v e n t i o n a l b u l k t e c h n i q u e s e m p l o y e d t h e n were n o t s e n s i t i v e enough f o r s u r f a c e s t u d i e s . H i s t o r i c a l l y , t h e d e f i c i e n c y i n i n s t r u m e n t a l s e n s i t i v i t y was c o u n t e r e d by w o r k i n g w i t h m a t e r i a l s w h i c h were e x t r e m e l y p o r o u s o r c o n s i s t e d o f v e r y s m a l l p a r t i c l e s t o e n s u r e a h i g h s u r f a c e / v o l u m e r a t i o . A l t h o u g h some u s e f u l e m p i r i c a l r e l a t i o n s h i p s c a n be d e r i v e d from t h e s e t r a d i t i o n a l s t u d i e s , t h e y n e c e s s a r i l y g i v e a v e r a g e d i n f o r m a t i o n on 'mixed' c r y s t a l l o g r a p h i c p l a n e s . I n o r d e r t o g a i n i n s i g h t s t o t h e p r i m a r y p r o c e s s e s t h a t o c c u r on s u r f a c e s a t t h e a t o m i c l e v e l , t h e ' c l e a n - s u r f a c e ' a p p r o a c h i s f a v o r e d . Such s t u d i e s use w e l l o r i e n t e d c r y s t a l f a c e s h e l d i n c l o s e l y c o n t r o l l e d e n v i r o n m e n t s . As r e c e n t l y a s two d e c a d e s ago t h i s a p p r o a c h was b a r e l y f e a s i b l e b e c a u s e v e r y few s t u d i e s c o u l d answer t h e most f u n d a m e n t a l q u e s t i o n o f whether a s u r f a c e was a t o m i c a l l y c l e a n , l e t a l o n e what t y p e o f f o r e i g n atoms m i g h t have been p r e s e n t . I n t h e l a s t f i f t e e n y e a r s o r so t h e ' c l e a n s u r f a c e ' a p p r o a c h has g a i n e d s t r e n g t h . T h i s h as been e s p e c i a l l y 3 b e c a u s e of t h e d e v e l o p i n g a v a i l a b i l i t y o f c o n v e n i e n t f a c i l i t i e s f o r r e a d i l y o b t a i n i n g u l t r a - h i g h v a c u u m ( U H v t ) ; a l s o many new and r e s u r r e c t e d s u r f a c e s e n s i t i v e t e c h n i q u e s were d e v e l o p e d . I n s p i t e o f s u c h a s h o r t p e r i o d of gro w t h , s u r f a c e s c i e n c e h a s a l r e a d y a c h i e v e d c o n s i d e r a b l e s o p h i s t i c a t i o n . The a n a l y s i s o f s i m p l e c l e a n s u r f a c e s t r u c t u r e s [ 2 ] c a n now be s t r a i g h t f o r w a r d . I n d e e d , c o n s i d e r a b l e knowledge i s now a v a i l a b l e f o r t h e r e l a x a t i o n e f f e c t s a t c l e a n s u r f a c e s , and a s t a r t i s b e i n g made i n s t u d y i n g r e c o n s t r u c t e d s u r f a c e s [ 3 ] . F u r t h e r , s i n c e t h e m i d -1970' s c o n s i d e r a b l e a t t e n t i o n h as been d i r e c t e d t o s y s t e m s i n v o l v i n g o r d e r e d a d s o r b a t e s . Such s t u d i e s may i n v o l v e atoms or s m a l l m o l e c u l e s [ 4 ] s u c h as CO, o r s i m p l e h y d r o c a r b o n s [ 5 ] a d s o r b e d on v a r i o u s m e t a l s u r f a c e s . R e c e n t l y t h e r e h as been a g r o w i n g i n t e r e s t i n s t u d y i n g m e t a l a d s o r b a t e s on s e m i c o n d u c t o r s u r f a c e s [ 6 ] . The l a t t e r s t u d i e s a r e made p o s s i b l e , i n p a r t , by a d v a n c e s i n m o l e c u l a r beam e p i t a x y [ 7 ] . T h r e e major s t i m u l i have e n c o u r a g e d t h i s boom i n s u r f a c e s c i e n c e . They a r e (1) r e c e n t d e v e l o p m e n t s i n m a t e r i a l s s c i e n c e , (2) t h e need f o r new knowledge i n h e t e r o g e n e o u s c a t a l y s i s , s p u r r e d e s p e c i a l l y by s o c i e t y ' s e n e r g y r e q u i r e m e n t s , and (3) e x c i t i n g d e v e l o p m e n t s i n s m a l l s c a l e e l e c t r o n i c s d e v i c e t e c h n o l o g y . F u n d a m e n t a l p r o c e s s e s i n v o l v i n g c r y s t a l g r o w t h , c o r r o s i o n , s u r f a c e s e g r e g a t i o n and e m b r i t t l e m e n t a r e t h e main c o n c e r n s f o r m a t e r i a l s c i e n t i s t s . C h e m i s t s , on t h e o t h e r hand, a r e more i n t e r e s t e d i n t h e tuHV < 1x10" 9 t o r r . 4 e n e r g e t i c s and k i n e t i c s of s u r f a c e p r o c e s s e s , i n c l u d i n g t h e a d s o r p t i o n and d e s o r p t i o n mechanisms. W i t h i t s r e c e n t s p e c t a c u l a r d e v e l o p m e n t s , t h e e l e c t r o n i c s i n d u s t r y n e c e s s a r i l y d e v o t e s much e f f o r t t o s u r f a c e r e s e a r c h , t h e o b j e c t i v e b e i n g t o improve q u a l i t y of t h i n i n t e g r a t e d c i r c u i t s by u t i l i z i n g knowledge o f m e t a l / s e m i c o n d u c t o r i n t e r f a c e s i n f a b r i c a t i o n p r o c e s s e s . T h e i r o b j e c t i v e s may be d i f f e r e n t , b ut t h e t y p e s o f r e s e a r c h on s u r f a c e s c a r r i e d o u t by e a c h o f t h e d i f f e r e n t d i s c i p l i n e s have much i n common, hence t h e r e i s i n e v i t a b l y a s t r o n g i n t e r - d i s c i p l i n a r y component i n modern s u r f a c e s c i e n c e . 1.2 TECHNIQUES RELEVANT TO AD-LAYERS STUDIES S e v e r a l c r i t e r i a a r e i m p o r t a n t i n t h e c o n s t r u c t i o n o f a good a d s o r b a t e - s e n s i t i v e t e c h n i q u e . F i r s t , i t s h o u l d g i v e r e a s o n a b l y s t r o n g s i g n a l s f r o m even low a t o m i c d e n s i t i e s o f a d s o r b a t e m o l e c u l e s . S e c o n d , i t must be a b l e e i t h e r t o a v o i d r e t r i e v i n g i n f o r m a t i o n f r o m t h e b u l k of t h e c r y s t a l , o r t o d i s t i n g u i s h s i g n a l s f r o m t h e b u l k and t h e s u r f a c e . F i n a l l y , i t i s o f t e n p r e f e r r e d t h a t t h e p r o b i n g p a r t i c l e s have minimum d e s t r u c t i v e e f f e c t on t h e a d l a y e r s ( a l t h o u g h o f c o u r s e t h i s i s n o t t h e c a s e i n t h e t h e r m a l , o r p a r t i c l e s t i m u l a t e d d e s o r p t i o n m e t h o d s ) . A s i d e f r o m t e c h n i q u e s d e p e n d e n t on t h e s u p p l y o f t h e r m a l e n e r g y , s u r f a c e methods ca n be c l a s s i f i e d i n t o t h r e e c a t e g o r i e s a c c o r d i n g t o p r o b i n g s o u r c e s : e l e c t r o n s c a t t e r i n g t e c h n i q u e s ; i o n / a t o m beam t e c h n i q u e s and p h o t o n ( i n c l u d i n g s y n c h r o t r o n r a d i a t i o n ) beam 5 t e c h n i q u e s . Most of t h e s e t e c h n i q u e s i n v o l v e s t u d y i n g t h e i n t e r a c t i o n o f t h e p r o b i n g p a r t i c l e s and t h e s u r f a c e r e g i o n o f t h e a d s o r p t i o n s y s t e m t h r o u g h t h e measurement o f t h e a n g u l a r a n d / o r e n e r g y d i s t r i b u t i o n o f t h e s c a t t e r e d or e m i t t e d p a r t i c l e s . S e v e r a l commonly u s e d t e c h n i q u e s and t h e i r acronyms a r e l i s t e d i n T a b l e 1.1; t h e e x p e r i m e n t a l d e t a i l s o f e a c h i n d i v i d u a l t e c h n i q u e c a n be f o u n d i n t h e r e f e r e n c e s g i v e n . However, t h e p r i n c i p l e s i n v o l v e d i n e a c h c a t e g o r y a r e d i s c u s s e d b r i e f l y i n t h e f o l l o w i n g s e c t i o n s . 1.2.1 ELECTRON SCATTERING TECHNIQUES E l e c t r o n s i n t h e e n e r g y r a n g e t o a b o u t 1 keV a r e commonly u s e d a s p r o b e s i n s u r f a c e s c i e n c e . Seah and D e n c h [ 2 5 ] have g i v e n r e c e n t i n f o r m a t i o n o f e l e c t r o n mean f r e e p a t h f o r i n e l a s t i c s c a t t e r i n g i n s o l i d s a s a f u n c t i o n o f t h e e l e c t r o n e n e r g y . F o r m e t a l s , t h e mean f r e e p a t h v e r s u s e n e r g y c u r v e s e x h i b i t b r o a d minima a t between 1C 1 and 1 0 2 eV, w i t h c o r r e s p o n d i n g mean f r e e p a t h s o f a r o u n d 3 t o 10 o A. S i n c e h i g h d e n s i t y l a y e r s i n m e t a l s have i n t e r l a y e r o s p a c i n g s o f a r o u n d 2 t o 3 A, l o w - e n e r g y e l e c t r o n s c a n p r o b e o n l y a v e r y r e s t r i c t e d number o f a t o m i c l a y e r s i f t h e e l e c t r o n s emerge f r o m t h e s u r f a c e w i t h o u t l o s i n g e n e r g y . The mean f r e e p a t h l e n g t h L f o r i n e l a s t i c s c a t t e r i n g i s d e f i n e d by I (/ ) = I 0 e x p ( - / / L ) (1.1) Acronym T e c h n i q u e Probe p a r t i c l e Measured p a r t i c l e I n f o r m a t 1 o n AES A u g e r - e l e c t r o n s p e c t r o s c o p y [ 8 , 9 ] E l e c t r o n E 1 e c t r o n Compos 1 1 i o n HREELS H i g h - r e s o l u t i o n e l e c t r o n ene rgy l o s s s p e c t r o s c o p y ! 1 0 ] E l e c t r o n E 1 e c t r o n V i b r a t i o n a l modes LEED Low energy e l e c t r o n d 1 f f r a c t 1 o n [ 1 1 , 1 2 ] E 1 e c t r o n E 1 e c t r o n Geometry RHEED R e f l e c t i o n h i g h ene rgy e l e c t r o n d 1 f f r a c t 1 o n [ 1 3 ] E 1 e c t r o n E 1 e c t r o n Geometry HEAD H e l i u m - a t o m d 1 f f r a c t 1 o n [ 1 4 ] Atom Atom S u r f a c e p o t e n t i a l map INS Ion n e u t r a l i z a t i o n s p e c t r o s c o p y [ 1 5 ] Ion Atom E l e c t r o n i c s t r u c t u r e L E I S Low energy i o n - s c a t t e r 1 n g [ 1 6 ] Ion Ion A t o m i c p o s i t i o n s RBS R u t h e r f o r d back s c a t t e r 1ng[17] Ion Ion A t o m i c p o s i t i o n s SIMS ' S e c o n d a r y - i o n mass s p e c t r o s c o p y [ 1 8 ] Ion I o n C o m p o s i t i o n p r o f i l e ADPD A n g u l a r - d e p e n d e n t p h o t o e l e c t r o n d i f f r a c t i o n ! 1 9 ] P h o t o n E 1 e c t r o n L o c a l s t r u c t u r e EDPD Energy-dependen t p h o t o e l e c t r o n d 1 f f r a c t 1 o n [ 2 0 ] P h o t o n E l e c t r o n L o c a l s t r u c t u r e NEXAFS Nea r - edge X - r a y a b s o r p t i o n f i n e s t r u c t u r e [ 2 1 ] P h o t o n P h o t o n , e 1 e c t r o n I n t r a m o l e c u l a r b o n d i n g SEXAFS S u r f a c e ex t ended X - r a y a b s o r p t i o n f i n e s t r u c t u r e [ 2 1 , 2 2 ] P h o t o n P h o t o n , e l e c t r o n L o c a l c o o r d i n a t i o n UPS U l t r a - v i o l e t p h o t o e l e c t r o n s p e c t r o s c o p y [ 2 3 ] P h o t o n E l e c t r o n V a l e n c e s t a t e s XD X - r a y d i f f r a c t l o n [ 2 4 ] P h o t o n P h o t o n L o n g - r a n g e s t r u c t u r e XPS X - r a y p h o t o e l e c t r o n s p e c t r o s c o p y [ 9 ] P h o t o n E 1 e c t r o n C h e m i c a l s h i f t s , q u a n t i t a t i v e c o m p o s i t i o n T a b l e 1 .1 : A d s o r b a t e - s e n s l t l v e t e c h n i q u e s and t h e i r c h a r a c t e r i s t i c s , t h e r e f e r e n c e s quo ted under the t e c h n i q u e c o l u m n . Ind1v1dua1 e x p e r i m e n t a l d e t a i l c a n be found i n 7 where I 0 i s t h e i n c i d e n t i n t e n s i t y a t a p a r t i c u l a r e n e r g y , and t h i s i s a t t e n u a t e d t o I ( / ) a f t e r t r a v e l l i n g a d i s t a n c e /. C l e a r l y a low v a l u e o f L c o r r e s p o n d s t o a h i g h p r o b a b i l i t y f o r i n e l a s t i c s c a t t e r i n g ; c o r r e s p o n d i n g l y t h e r e w i l l be a h i g h s u r f a c e s e n s i t i v i t y i n e x p e r i m e n t s i n w h i c h t h e e l e c t r o n s a r e d e t e c t e d a t s p e c i f i c e n e r g y v a l u e s . High-Resolution Electron Energy Loss Spectroscopy (HREELS). E l e c t r o n s s c a t t e r i n g o f f s u r f a c e s c a n l o s e e n e r g y i n v a r i o u s ways. One o f t h e s e i n v o l v e s e x c i t a t i o n o f t h e v i b r a t i o n a l modes o f atoms and m o l e c u l e s on t h e s u r f a c e . W i t h a m o n o c h r o m a t i z e d i n c i d e n t beam, a t e n e r g y f r o m 1 t o 10 eV, HREELS measures s u c h v i b r a t i o n a l modes. T h i s t e c h n i q u e p r o v e s t o be a p o w e r f u l t o o l f o r h y d r o c a r b o n a d s o r p t i o n s t u d i e s s i n c e many o t h e r s u r f a c e t e c h n i q u e s c a n n o t d e t e c t h y d r o g e n . U n l i k e o p t i c a l s p e c t r o s c o p i e s s u c h a s i n f r a - r e d and Raman s p e c t r o s c o p i e s , HREELS c a n measure v i b r a t i o n a l modes b o t h p e r p e n d i c u l a r and p a r a l l e l t o t h e s u r f a c e . Low-Energy Electron Diffraction ( L E E D ) . I n t h i s t e c h n i q u e , t h e e l a s t i c a l l y s c a t t e r e d e l e c t r o n s d i f f r a c t t o p r o v i d e i n f o r m a t i o n a b o u t t h e l o n g - r a n g e p e r i o d i c i t y o f t h e O s u r f a c e s t r u c t u r e s . The de B r o g l i e w a v e l e n g t h , X ( i n A ) , of an e l e c t r o n w i t h e n e r g y E ( i n eV) i s g i v e n by X = / ( 1 5 0 . 4 / E ) (1.2) In t h e e n e r g y r a n g e 10 t o 500 eV, t h e e l e c t r o n w a v e l e n g t h o v a r i e s f r o m 4 t o 0.6 A, and t h e r e f o r e i t i s g e n e r a l l y o f t h e 8 same o r d e r as i n t e r a t o m i c and i n t e r l a y e r d i s t a n c e s i n a d l a y e r s t r u c t u r e s and m e t a l s u b s t r a t e s . The t r a n s l a t i o n a l symmetry o f t h e s u r f a c e d e t e r m i n e s t h e d i f f r a c t i o n p a t t e r n ; t h e m e asured d i f f r a c t e d beam i n t e n s i t i e s , i n c o n j u n c t i o n w i t h m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s , c a n be u s e d t o d e t e r m i n e i n t e r a t o m i c d i s t a n c e s . Reflection High-Energy Electron Diffraction (RHEED) . T h i s t e c h n i q u e i s s i m i l a r i n p r i n c i p l e t o LEED but u t i l i z e s h i g h e r i n c i d e n t beam e n e r g i e s ( l t o 50 k e V ) . The l o n g e r mean f r e e p a t h s a s s o c i a t e d w i t h h i g h e r e n e r g i e s g i v e a r e d u c t i o n i n s u r f a c e s e n s i t i v i t y . In RHEED t h i s i s c o u n t e r e d by u s i n g g r a z i n g a n g l e s o f i n c i d e n c e and emergence. Due t o i t s d e e p e r p e n e t r a t i o n , RHEED i s e s p e c i a l l y s u i t a b l e f o r s t u d i e s o f c h e m i s o r p t i o n w h i c h l e a d t o r e a c t i o n i n t o t h e b u l k . A s i m i l a r t e c h n i q u e e m p l o y i n g i n c i d e n t e l e c t r o n e n e r g i e s between t h o s e f o r LEED and RHEED i s c a l l e d Medium-Energy Electron Di ff r act i on (MEED). Auger Electron Spectroscopy ( A E S ) . When an a t o m i c c o r e l e v e l i s i o n i z e d , t h e s u b s e q u e n t f i l l i n g o f t h e o r b i t a l by an e l e c t r o n f r o m a l e s s - t i g h t l y bound l e v e l may be a c c o m p a n i e d by t h e e m i s s i o n o f a p h o t o n o r a s e c o n d e l e c t r o n . T h i s l a t t e r e l e c t r o n i s t e r m e d an Auger e l e c t r o n a f t e r P. A u g e r [ 2 6 ] who f i r s t i d e n t i f i e d t h e e f f e c t i n a c l o u d chamber e x p e r i m e n t . The k i n e t i c e n e r g y o f t h e Auger e l e c t r o n t h u s c o n t a i n s i n f o r m a t i o n o f t h e e n e r g y l e v e l s i n v o l v e d . A measurement o f t h e k i n e t i c e n e r g i e s o f t h e Auger e l e c t r o n s f r o m one o r more s u c h t r a n s i t i o n s c a n t h e n 9 i d e n t i f y t h e e l e m e n t ( s ) p r e s e n t o n a s u r f a c e . H e n c e A E S i s r o u t i n e l y u s e d f o r e l e m e n t a l a n a l y s i s o f s u r f a c e s . S i n c e L E E D a n d A E S a r e t h e p r i n c i p a l t e c h n i q u e s u s e d i n t h i s l a b o r a t o r y , t h e y w i l l b e d i s c u s s e d i n d e p t h i n t h e n e x t c h a p t e r . 1 . 2 . 2 I O N OR ATOM S C A T T E R I N G T E C H N I Q U E S A t o m s a n d i o n s i n c i d e n t o n s o l i d s u r f a c e s c a n p r o v i d e i n f o r m a t i o n a b o u t t h e l a t t e r t h r o u g h t h e a n a l y s i s o f ( 1 ) t h e b a c k - s c a t t e r e d p r i m a r y p a r t i c l e s ; ( 2 ) p a r t i c l e s s p u t t e r e d o f f f r o m t h e s u r f a c e ; a n d ( 3 ) e l e c t r o n s e m i t t e d f r o m t h e s u r f a c e r e s u l t i n g f r o m e x c i t a t i o n b y t h e p r i m a r y b e a m . T h e s e t e c h n i q u e s t y p i c a l l y u s e b e a m s o f H + o r H e * i n t h e e n e r g y r a n g e o f 1 00 e V t o 3 M e V . I t i s u s e f u l t o d i v i d e t h e e n e r g y i n t o l o w a n d h i g h r e g i m e s c h a r a c t e r i z e d b y i o n v e l o c i t i e s b e l o w a n d a b o v e t h e B o h r v e l o c i t y o f 2 . 2 x l 0 6 m s " 1 . I n t h e h i g h e r e n e r g y r e g i m e , t h e i o n s c a s t a v e r y n a r r o w s h a d o w o c o n e o f r a d i u s a b o u t 0 . 1 A , a n d h e n c e h a v e a n e g l i g i b l e p r o b a b i l i t y o f b e i n g n e u t r a l i z e d . C o n s e q u e n t l y , t h e s c a t t e r i n g p r o c e s s c a n u s u a l l y b e t r e a t e d k i n e m a t i c a l l y , t h u s s i m p l i f y i n g c o m p u t a t i o n s . I n t h e l o w e n e r g y r e g i m e , t h e o s h a d o w c o n e r a d i u s i s a b o u t 1 A a n d t h e p r o b a b i l i t y o f n e u t r a l i z a t i o n i s h i g h . T h e s e i o n s a r e t h e r e f o r e o f t e n u s e d t o s t u d y t h e e l e c t r o n i c s t r u c t u r e s o f t h e s u r f a c e a t o m s . Rut her ford Back Scatteri ng ( R B S ) . H e r e a h i g h e n e r g y ( 2 - 3 M e V ) b e a m o f H + o r H e * i s i n c i d e n t o n a n o r d e r e d s u r f a c e . T h e s e p r i m a r y i o n s a r e s c a t t e r e d b y t h e e x p o s e d 10 s u r f a c e atoms w h i c h 'see' t h e p r i m a r y beam d i r e c t l y . I f t h e b a c k - s c a t t e r e d i o n s a r e e n e r g y a n a l y z e d , t h e a t o m i c w e i g h t ( s ) o f t h e e x p o s e d atoms c an be d e t e r m i n e d t h r o u g h s i m p l e k i n e m a t i c c a l c u l a t i o n s . The i n t e n s i t i e s o f t h e s e e n e r g y p e a k s i d e n t i f y t h e number of e x p o s e d atoms; f u r t h e r , a measurement o f t h e s e i n t e n s i t i e s a s a f u n c t i o n of i n c i d e n c e a n g l e c a n r e v e a l t h e g e o m e t r i c a l a r r a n g e m e n t o f t h e s u r f a c e atoms ( a l t h o u g h c a r e f u l c o r r e c t i o n s a r e r e q u i r e d f o r some p a r a m e t e r s , i n c l u d i n g t h e r m a l v i b r a t i o n s ) . Secondar y-1 on Mass Spect rometry ( S I M S ) . T h i s t e c h n i q u e t y p i c a l l y u s e s an i o n beam i n t h e e n e r g y r a n g e 1 t o 20 keV. S u r f a c e l a y e r s a r e s p u t t e r e d o f f a s atoms and i o n s ( i n c l u d i n g c l u s t e r s ) , and t h e i o n i z e d f r a c t i o n i s i d e n t i f i e d i n a mass s p e c t r o m e t e r . Dynamic SIMS c a n g i v e s u r f a c e e l e m e n t a l c o m p o s i t i o n s i n t h e f o r m o f d e p t h p r o f i l e s ; however, q u a n t i f i c a t i o n i s d i f f i c u l t due t o m a t r i x e f f e c t s and d i f f e r e n t i a l s p u t t e r i n g r a t e s f o r d i f f e r e n t e l e m e n t s . Low-Energy Ion ScatIering ( L E I S ) i s s i m i l a r t o RBS, b u t u s e s i o n s o f e n e r g y a b o u t 1 keV. Due t o t h e l a r g e r r a d i u s o f t h e shadow c o n e , t h i s t e c h n i q u e i s s e n s i t i v e t o t h e e x p o s e d atoms o n l y . A v e r s i o n c a l l e d Impact Collision Ion Scatt eri ng Spectroscopy ( I C I S S [ 2 7 ] ) m e a s u r e s b a c k - s c a t t e r e d i o n s a t a s c a t t e r i n g a n g l e o f 180° o n l y . T h i s e l i m i n a t e s b l o c k i n g e f f e c t s and a l l o w s one t o ded u c e t h e d i s t a n c e between s u r f a c e atoms from t h e sh a d o w i n g e f f e c t a l o n e . Ion Neutralization Sped roscopy ( I N S ) . Lower e n e r g y (=*102 eV) He* i o n s have a h i g h p r o b a b i l i t y of c a p t u r i n g 11 e l e c t r o n s from t h e s u r f a c e atoms and hence becoming n e u t r a l i z e d . The s u b s e q u e n t f i l l i n g o f t h e h o l e t h u s c r e a t e d i n t h e s u r f a c e may r e s u l t i n e j e c t i o n o f a slow e l e c t r o n , a s i n t h e Auger p r o c e s s . T h i s g i v e s i n f o r m a t i o n on o c c u p i e d d e n s i t i e s o f s t a t e s ; however, t h e i n s i g n i f i c a n t p e n e t r a t i o n d e p t h of t h e l o w - e n e r g y i o n s makes INS e x t r e m e l y s u r f a c e s e n s i t i v e . Helium-Atom Diffraction (HEAD). One c a n a l s o p r o b e s u r f a c e s u s i n g l o w - e n e r g y atoms. F o r example, f o r He a t 10 O t o 200 meV, t h e de B r o g l i e w a v e l e n g t h X ( i n A) a t e n e r g y E ( i n eV) i s g i v e n by X = 0.14//E (1.3) Thus He atoms w i t h t h e t h e r m a l e n e r g y o f 200 meV have a o w a v e l e n g t h about 1 A, and so c a n r e a d i l y d i f f r a c t f r o m s u r f a c e s . HEAD me a s u r e s t h e a n g u l a r d i s t r i b u t i o n o f b a c k - s c a t t e r e d He atoms, t h u s r e v e a l i n g t h e 2 - d i m e n s i o n a l c o r r u g a t i o n o f t h e p o t e n t i a l between t h e He atoms and t h e s u r f a c e . 1.2.3 PHOTON BEAM TECHNIQUES S y n c h r o t r o n s o u r c e s a l l o w e x p e r i m e n t e r s a c c e s s t o h i g h l y c o l l i m a t e d and c o n t i n u o u s w a v e l e n g t h s o f r a d i a t i o n t h a t a r e n o t a v a i l a b l e from o t h e r s o u r c e s . The h i g h i n t e n s i t y o f s y n c h r o t r o n r a d i a t i o n g i v e s a h i g h e r s i g n a l t o n o i s e r a t i o t h a n c o n v e n t i o n a l X - r a y g e n e r a t o r s . By way o f 12 c o m p a r i s o n , t h e most p o w e r f u l c o m m e r c i a l l y a v a i l a b l e X - r a y g e n e r a t o r , a 60-kW r o t a t i n g - a n o d e t u b e , d e l i v e r s 10 8 p h o t o n s - s " 1 , whereas a s y n c h r o t r o n s o u r c e c a n d e l i v e r a t h o u s a n d t i m e s t h a t amount. Surf ace-Ext ended X-ray Absorption Fine Structure (SEXAFS) p r o v i d e s one o f t h e most s u c c e s s f u l a p p l i c a t i o n s o f s y n c h r o t r o n r a d i a t i o n t o s u r f a c e s t u d i e s . W i t h an i n c i d e n t beam o f s o f t X - r a y s of i n c r e a s i n g e n e r g y , a p h o t o e l e c t r o n i s e m i t t e d when t h e i n c i d e n t p h o t o n e x c e e d s t h e t h r e s h o l d f o r e x c i t i n g a c o r e e l e c t r o n o f a s u r f a c e atom. I n SEXAFS, t h o s e e j e c t e d p h o t o e l e c t r o n s w i t h k i n e t i c e n e r g i e s f r o m 50 t o 500 eV a r e s t u d i e d . The y i e l d o f t h e e j e c t e d e l e c t r o n s f r o m t h e a b s o r b i n g atoms i s m o d u l a t e d a s a f u n c t i o n o f i n c i d e n t p h o t o n e n e r g y due t o t h e i n t e r f e r e n c e between t h e o u t g o i n g e l e c t r o n s and t h o s e b a c k s c a t t e r e d f r o m n e i g h b o r i n g atoms. By F o u r i e r a n a l y s i s o f t h i s m o d u l a t i o n , t h e d i s t a n c e between t h e a b s o r b i n g atom and i t s n e i g h b o r s c an be d e t e r m i n e d . M o r e o v e r , i f t h e e x c i t a t i o n s o u r c e i s s y n c h r o t r o n r a d i a t i o n ( w h i c h i s p o l a r i z e d ) , c o m p a r i s o n o f t h e a m p l i t u d e o f t h e i n t e r f e r e n c e t e r m s y i e l d s i n f o r m a t i o n on t h e a d s o r p t i o n s i t e [ 2 8 ] r a t h e r t h a n j u s t t h e bond l e n g t h . Near Edge X-ray Absorption Fine Structure (NEXAFS) i s a t e c h n i q u e v e r y s i m i l a r t o SEXAFS. However, NEXAFS s t u d i e s t h o s e e j e c t e d p h o t o e l e c t r o n s w i t h k i n e t i c e n e r g i e s n e a r t h e t h r e s h o l d of e x c i t a t i o n ( e . g . w i t h i n 50 eV o f t h e t h r e s h o l d ) . F o r m o l e c u l a r a d s o r p t i o n s y s t e m s , NEXAFS r e s u l t s a r e d o m i n a t e d by i n t r a m o l e c u l a r r e s o n a n c e s ! 2 1 ] , w h i c h 13 i n v o l v e t r a n s i t i o n s i n t o m o l e c u l a r o r b i t a l s t a t e s , s u c h as o and 7r , whose wavef u n c t i o n s have a m p l i t u d e s t h a t a r e l a r g e l y l o c a l i z e d w i t h i n t h e m o l e c u l e . As a r e s u l t , NEXAFS i s p a r t i c u l a r y s e n s i t i v e t o t h e i n t r a m o l e c u l a r bond l e n g t h s and t h e o r i e n t a t i o n of t h e s e bonds r e l a t i v e t o t h e s u r f a c e . Energy-Dependent and Angular-Dependent Photoelectron Diffraction (EDPD and ADPD) a r e t e c h n i q u e s c l o s e l y r e l a t e d t o SEXAFS and NEXAFS. Here t h e number o f p h o t o e l e c t r o n s e m i t t e d f r o m a g i v e n c o r e l e v e l i s measured d i r e c t l y a s a f u n c t i o n o f p h o t o n beam e n e r g y o r a n g l e . By m e a s u r i n g t h e p h o t o e l e c t r o n s coming f r o m a s p e c i f i c l e v e l , t h e i n t e r f e r i n g a b s o r p t i o n e d g e s seen i n SEXAFS c a n be a v o i d e d . L i k e SEXAFS, t h e s e two t e c h n i q u e s p r o v i d e i n f o r m a t i o n on l o c a l g e o m e t r i c a l s t r u c t u r e . Ul t raviolet Photoelectron Spectroscopy (UPS) and X-ray Photoelectron Sped r os copy (XPS o r ES C A ) . B e s i d e s h e l p i n g d e v e l o p new s u r f a c e t e c h n i q u e s , s y n c h r o t r o n r a d i a t i o n g r e a t l y e n h a n c e s t h e e f f e c t i v e n e s s o f s e v e r a l t r a d i t i o n a l s p e c t r o s c o p i e s f o r s u r f a c e s t u d i e s . The l a t t e r i n c l u d e UPS and XPS. D e p e n d i n g on t h e i n c i d e n t p h o t o n e n e r g y , e l e c t r o n s c a n be e j e c t e d f r o m d i f f e r e n t e l e c t r o n i c l e v e l s o f a s u r f a c e atom. B o t h UPS and XPS i n v o l v e m e a s u r i n g k i n e t i c e n e r g i e s of su c h e m i t t e d p h o t o e l e c t r o n s . UPS i s u s e d t o s t u d y t h e v a l e n c e e l e c t r o n i c s t r u c t u r e of t h e s u r f a c e s p e c i e s ; w h i l e XPS p r o b e s c o r e l e v e l s and p r o v i d e s c h e m i c a l s h i f t i n f o r m a t i o n a s w e l l as q u a n t i t a t i v e s u r f a c e c o m p o s i t i o n . 14 X-ray Di ff ract i on(XD) i s a w e l l - d e v e l o p e d t e c h n i q u e f o r s t u d y i n g s t r u c t u r e i n b u l k s o l i d s , b u t a p p l i c a t i o n s t o s u r f a c e s t r u c t u r e a r e s t a r t i n g t o be d e v e l o p e d . B e c a u se X - r a y s p e n e t r a t e d e e p l y i n t o a s o l i d , t h e s u r f a c e s i g n a l i s n e c e s s a r i l y s u p e r i m p o s e d on a l a r g e s i g n a l f r o m t h e b u l k , a l t h o u g h t h e f o r m e r may be m a x i m i z e d w i t h g r a z i n g d i r e c t i o n s o f i n c i d e n c e . X - r a y d i f f r a c t i o n has been u s e d t o a n a l y z e s u r f a c e n e t s w i t h a d i f f e r e n t p e r i o d i c i t y f r o m t h e b u l k . A p o t e n t i a l a d v a n t a g e o f XD f o r s u r f a c e s c i e n c e stems from t h e a p p l i c a t i o n of s i n g l e ( i . e . k i n e m a t i c a l ) s c a t t e r i n g t h e o r i e s . 1.3 SOME BASIC KNOWLEDGE FROM ADSORPTION STUDIES ON METAL  SURFACES The a t o m i c g e o m e t r y o f a s u r f a c e o r i n t e r f a c e i s , i n c e r t a i n r e s p e c t s , i t s most f u n d a m e n t a l p r o p e r t y . From a c h e m i s t ' s s t a n d p o i n t t h i s p r o p e r t y d e t e r m i n e s e l e c t r o n i c s t r u c t u r e , and u l t i m a t e l y a l l o t h e r p r o p e r t i e s , i n c l u d i n g , a s one example, r e a c t i v i t y a nd s p e c i f i c i t y i n c a t a l y s i s . S t u d i e s w i t h LEED, SEXAFS, p h o t o e l e c t r o n d i f f r a c t i o n and i o n s c a t t e r i n g t e c h n i q u e s a r e s t a r t i n g t o p r o v i d e d a t a on a t o m i c g e o m e t r i c a l a r r a n g e m e n t s f o r o r d e r e d a d l a y e r s on w e l l - d e f i n e d s i n g l e c r y s t a l s u r f a c e s . The p r e s e n t s i t u a t i o n i s t h a t some c o n s i s t e n c y i n s t r u c t u r a l c o n c l u s i o n s c a n be r e a c h e d when d i f f e r e n t t e c h n i q u e s a r e a p p l i e d t o t h e same s u r f a c e ( e . g . c u r r e n t s t a t u s f o r 0 a d s o r b e d on A l ( 1 1 1 ) [ 2 9 ] o r CO on N i ( 1 0 0 ) [ 3 0 ] ) . 1 5 AES, HREELS, INS, thermal d e s o r p t i o n [ 3 1 ] and pho toemiss ion s p e c t r o s c o p i e s now o f f e r r e s e a r c h e r s p o s s i b i l i t i e s to study the na ture of the s u r f a c e chemica l bond at the atomic l e v e l . T h i s knowledge i s e s s e n t i a l to the unders tand ing of su r f a ce r e a c t i v i t y and r e a c t i o n mechanism. S t ud i e s of g e o m e t r i c a l arrangements f o r meta l s adsorbed on s u r f a c e s of o ther meta ls are so fa r r e s t r i c t e d to LEED a n a l y s e s , and the t o t a l i n f o r m a t i o n i s very l i m i t e d . More expe r imen ta l r e s u l t s in t h i s f i e l d are necessa ry to a s s i s t m a t e r i a l s s c i e n t i s t s to unders tand the s o l i d - s o l i d i n t e r f a c e . 1.3.1 THE STRUCTURAL PARAMETERS From the o v e r l a y e r - s u b s t r a t e systems s t u d i e d so f a r , some gene ra l p a t t e r n s r ega rd ing s t r u c t u r a l parameters are s t a r t i n g to emerge. One of these i s that the adsorbed atoms tend to occupy s i t e s where they are in c o n t a c t w i th the maximum number of s u b s t r a t e atoms ( l a r g e s t c o o r d i n a t i o n number) . T h i s s i t e i s o f t e n the one tha t the bu lk atoms would occupy in o rder to con t i nue the bulk l a t t i c e i n t o the o v e r l a y e r ( e . g . bABAB . . . and c A B C A B C . . . , where the lower case l e t t e r s r ep resen t the atom p o s i t i o n s fo r the o v e r l a y e r ) . In o the r words, adsorbed atoms tend to hea l broken bonds at a s u r f a c e . Us ing ICISS, Aono et al. [32] have determined that these s i t e s can co r respond to l o c a t i o n s of h igh su r f a ce e l e c t r o n i c d e n s i t y . The f r e q u e n t l y observed c o n t r a c t i o n of the topmost i n t e r l a y e r spac ing at a c l e an meta l s u r f a c e may 16 r e s u l t f r o m e l e c t r o s t a t i c a t t r a c t i o n s a s s o c i a t e d w i t h c h a r g e p o l a r i z a t i o n s a t t h e i n t e r f a c e . Sometimes a d s o r b e d l a y e r s may a f f e c t p o s i t i o n s o f atoms i n t h e u n d e r l y i n g s u r f a c e . T h i s may i n v o l v e r e l a x a t i o n s , o f e i t h e r v e r t i c a l o r h o r i z o n t a l t y p e s , i n a b a s i c a l l y u n r e c o n s t r u c t e d s u r f a c e . A l t e r n a t i v e l y an a d s o r b e d l a y e r may i n d u c e r e c o n s t r u c t i o n i n an o t h e r w i s e u n r e c o n s t r u c t e d s u r f a c e , o r may c a u s e t h e b a s i c s u r f a c e t o r e v e r t t o t h e e s s e n t i a l l y u n r e c o n s t r u c t e d f o r m even t h o u g h i t i s r e c o n s t r u c t e d i n t h e most s t a b l e f o r m o f t h e c l e a n s u r f a c e . The s t u d y o f s u r f a c e c r y s t a l l o g r a p h y has t h e i m p o r t a n t o b j e c t i v e o f p r o v i d i n g p r e c i s e i n f o r m a t i o n f o r a l l s u c h e f f e c t s . As w e l l t h i s i n f o r m a t i o n c a n e n c o u r a g e t h e d e v e l o p m e n t o f models and u n d e r s t a n d i n g s s p e c i f i c a l l y f o r t h e s u r f a c e c h e m i c a l bond. 1.3.2 THE NATURE OF THE SURFACE CHEMICAL BOND A l t h o u g h a f u l l u n d e r s t a n d i n g o f t h e s u r f a c e c h e m i c a l bond i s n o t y e t p o s s i b l e , s e v e r a l e m p i r i c a l c h a r a c t e r i s t i c s have been o b s e r v e d w i t h a d s o r b a t e - s e n s i t i v e t e c h n i q u e s , s u c h as i n p a r t i c u l a r HREELS, AES, NEXAFS, UPS and XPS. I n t u r n s u c h o b s e r v a t i o n s have e n c o u r a g e d m a t h e m a t i c a l f o r m u l a -t i o n s [ 3 3 , 3 4 ] , b u t so f a r t h e d e g r e e o f match between t h e o r y and e x p e r i m e n t i s s t i l l g e n e r a l l y i n f e r i o r t o t h o s e c u r r e n t l y o b t a i n a b l e f o r i n d i v i d u a l m o l e c u l e s o r f o r b u l k s o l i d s . 17 W i t h v a r i o u s s u r f a c e s p e c t r o s c o p i e s , i t has been d e m o n s t r a t e d t h a t even on t h e more a t o m i c a l l y homogeneous l o w - M i l l e r - i n d e x s u r f a c e s , s e v e r a l b i n d i n g s i t e s a r e o f t e n d i s t i n g u i s h a b l e by t h e i r s t r u c t u r e and b i n d i n g s t r e n g t h . As a r e s u l t , a s e q u e n t i a l f i l l i n g o f b i n d i n g s i t e s may o c c u r w i t h i n c r e a s i n g a d s o r b a t e c o v e r a g e . W i t h a p a r t i c u l a r a d s o r b a t e , t h e s e b i n d i n g s i t e s and b i n d i n g s t r e n g t h s v a r y w i t h c r y s t a l f a c e . T h e s e v a r i a t i o n s add t o t h e c o m p l e x i t i e s w h i c h make a d s o r p t i o n s t u d i e s so c h a l l e n g i n g . One n o v e l f e a t u r e o f t h e s u r f a c e c h e m i c a l bond i s i t s d e p e n d e n c e on t e m p e r a t u r e . A t low t e m p e r a t u r e n e a r t h e b o i l i n g p o i n t s o f t h e a d s o r b a t e atoms o r m o l e c u l e s , a d s o r p t i o n d e c r e a s e s w i t h i n c r e a s e i n t e m p e r a t u r e a t a g i v e n p r e s s u r e . T h i s t y p e o f a d s o r p t i o n i s known as p h y s i c a l a d s o r p t i o n w h i c h i s c h a r a c t e r i z e d by van d e r Waals o r s i m p l e e l e c t r o s t a t i c i n t e r a c t i o n s and low h e a t s o f a d s o r p t i o n (<15 k c a l ' m o l " 1 ) . F o r some s m a l l m o l e c u l e s s u c h a s CO and 0 2 on m e t a l s , t h e amount t h a t c a n be a d s o r b e d s t a r t s t o r i s e a g a i n a s t h e t e m p e r a t u r e goes up f u r t h e r . T h i s t y p e o f a d s o r p t i o n i s g e n e r a l l y c a l l e d c h e m i s o r p t i o n ( o r ' a c t i v a t e d a d s o r p t i o n ' s i n c e an e n e r g y o f a c t i v a t i o n i s r e q u i r e d ) . In t h e c o n t e x t o f t h e t r a n s i t i o n f r o m p h y s i c a l a d s o r p t i o n t o c h e m i s o r p t i o n , s t u d i e s o f h y d r o c a r b o n m o l e c u l e s a d s o r b e d on some m e t a l s u r f a c e s have i l l u s t r a t e d s e q u e n t i a l bond b r e a k i n g w i t h i n t h e a d s o r b a t e m o l e c u l e w i t h i n c r e a s e i n t e m p e r a t u r e [ 3 5 , 3 6 ] . In a d d i t i o n , t h e d e t a i l s a r e most v a r i e d on h e t e r o g e n e o u s s u r f a c e s . F o r example, on s t e p p e d s u r f a c e s , t h e r e i s now 18 a b u n d a n t e v i d e n c e [ 3 7 ] t h a t d i s s o c i a t i o n p r e f e r e n t i a l l y o c c u r s a t s t e p s o r k i n k s . C h e m i s o r p t i o n i n v o l v e s c o v a l e n t bond f o r m a t i o n . In p r i n c i p l e t h e r e f o r e c h e m i s o r p t i o n bond l e n g t h s s h o u l d r e l a t e t o t h o s e i n c o n v e n t i o n a l s o l i d s t a t e c h e m i s t r y . A n a l y s e s of s u r f a c e s t r u c t u r a l i n f o r m a t i o n , f r o m t h i s p o i n t o f vi e w , have been g i v e n by M i t c h e l l and o t h e r members o f our g r o u p f 38 ] . 1.4 AIM OF THE THESIS The work d e s c r i b e d i n t h i s t h e s i s r e p r e s e n t s a c o n t r i b u t i o n t o t h e d e v e l o p m e n t o f t h e s u b j e c t o f LEED c r y s t a l l o g r a p h y . S p e c i f i c s t u d i e s have been made f o r two t y p e s o f o r d e r e d oxygen a d s o r p t i o n on t h e (0001) s u r f a c e o f z i r c o n i u m . The f i r s t g i v e s t h e same number o f d i f f r a c t e d beams as t h e c l e a n Zr ( 0 0 0 l ) s u r f a c e ; w h i l e t h e s e c o n d l e a d s t o f o u r t i m e s a s many d i f f r a c t e d beams as t h e c l e a n s u r f a c e . A t h e o r e t i c a l s t u d y w h i c h e v o l v e d f r o m t h e (2x2) a d s o r p t i o n was a l s o made. The m o t i v a t i o n f o r s t u d y i n g t h e Zr ( 0 0 0 l ) s u r f a c e i s t h a t s t r u c t u r a l p a r a m e t e r s a r e g e n e r a l l y n o t w e l l known f o r a d s o r b a t e s on hcp (OOOl) s u r f a c e s . In f a c t , p r e v i o u s LEED s t u d i e s i n v o l v i n g s u c h a s u r f a c e a r e r e s t r i c t e d m a i n l y t o t h e a d s o r p t i o n o f cadmium [ 3 9 ] , c a r b o n m o n o x i d e f 4 0 ] , and n i t r o g e n [ 4 l ] on T i ( 0 0 0 l ) . One i n t e r e s t i n g r e s u l t f r o m t h e s e s t u d i e s [ 4 l ] i s t h e f o r m a t i o n o f a n i t r o g e n u n d e r l a y e r on T i ( 0 0 0 l ) . S i n c e b o t h t i t a n i u m and z i r c o n i u m b e l o n g t o g r o u p 19 4 t i n t h e p e r i o d i c t a b l e , t h e i r c h e m i c a l p r o p e r t i e s may be v e r y s i m i l a r . A d s o r p t i o n s t u d i e s on Z r ( O O O l ) c o u l d t h e r e f o r e s h e d f u r t h e r l i g h t on t h e phenomenon o f u n d e r l a y e r a d s o r p t i o n . D e s p i t e t h e s i m i l a r i t y w i t h t i t a n i u m , and t h e f a c t t h a t z i r c o n i u m has numerous p o t e n t i a l a p p l i c a t i o n s [ 4 2 ] , s t u d i e s u t i l i z i n g t h e ' c l e a n s u r f a c e ' a p p r o a c h on Z r ( O O O l ) a r e so f a r v e r y l i m i t e d . I n d e e d s t r u c t u r a l i n f o r m a t i o n i s r e s t r i c t e d t o t h e p r e v i o u s LEED a n a l y s i s on t h e c l e a n s u r f a c e f r o m t h i s l a b o r a t o r y [ 4 3 ] . F a c t o r s w h i c h have p o s s i b l y i n h i b i t e d a c t i v e s t u d y i n c l u d e (1) t h e h i g h r e a c t i v i t y o f z i r c o n i u m w h i c h forms s t a b l e o x i d e s , c a r b i d e s and n i t r i d e s i n a i r ; (2) t h e d i f f i c u l t y i n o b t a i n i n g a s i n g l e c r y s t a l o f z i r c o n i u m ( n o t c o m m e r c i a l l y a v a i l a b l e ) ; and (3) a ph a s e t r a n s i t i o n f r o m hep t o bec a t a r a t h e r low , t e m p e r a t u r e ( 8 6 5 ° C ) . The l a s t f a c t o r r e s t r i c t s a n n e a l i n g t e m p e r a t u r e s t o below 800°C i n p r a c t i c e , and t h i s l i m i t s p r o c e d u r e f o r o b t a i n i n g o r d e r e d s u r f a c e s . Oxygen i s a common a d s o r b a t e i n c h e m i s o r p t i o n s t u d i e s , n o t l e a s t b e c a u s e o f i t s i m p o r t a n t r o l e i n f u n d a m e n t a l c h e m i c a l and b i o l o g i c a l r e a c t i o n s . In a d d i t i o n , t h e e f f e c t o f c h e m i s o r b e d oxygen on s i n g l e c r y s t a l s u r f a c e s i s b e i n g s c r u t i n i z e d c l o s e l y by t h e e l e c t r o n i c s i n d u s t r y b e c a u s e sample w a f e r s a r e i n c o n t a c t e i t h e r w i t h a i r o r t r a c e amounts o f oxygen c o n s t a n t l y i n t h e f a b r i c a t i o n p r o c e s s . So f a r t h e s t r u c t u r a l , d e t a i l s f o r oxygen c h e m i s o r b e d on many t i U P A C n o t a t i o n : 1-18. 20 w e l l c h a r a c t e r i z e d s i n g l e c r y s t a l p l a n e s a r e s t i l l unknown i n most c a s e s , a l t h o u g h s e v e r a l LEED s t u d i e s have shown t h a t oxygen o c c u p i e s ' u n e x p e c t e d ' s i t e s more o f t e n t h a n o t h e r g a s e s [ 4 4 ] . T h r o u g h t h e s t u d y of oxygen c h e m i s o r b e d on Z r ( 0 0 0 l ) , i t i s hoped t o add t o o u r knowledge a b o u t t h e p r o p e r t i e s of g r o u p 4 m e t a l s t o w a r d s o x y g e n , and t h e r e b y s t a r t t o g e n e r a t e i n f o r m a t i o n a b o u t t h e i n i t i a l s t a g e s o f o x i d a t i o n . More b r o a d l y , t h i s s h o u l d h e l p t o e l u c i d a t e m o d e ls o f t h e o x i d a t i o n p r o c e s s o f o t h e r e l e c t r o p o s i t i v e e l e m e n t s . I n c i d e n t a l l y , one o f t h e m o s t - s t u d i e d r e l a t e d s y s t e m s i s f o r oxygen on A l ( 1 1 1 ) [ 2 9 ] . The l a y o u t o f t h e t h e s i s i s d e s c r i b e d a s f o l l o w s . C h a p t e r 2 o u t l i n e s t h e p h y s i c s of low e n e r g y e l e c t r o n d i f f r a c t i o n a s w e l l a s t h e Auger p r o c e s s . The f a c t o r s c o n t r i b u t i n g t o d i f f r a c t e d beam i n t e n s i t i e s w i l l be d i s c u s s e d i n a s e m i - q u a n t i t a t i v e a p p r o a c h . C h a p t e r 3 d i s c u s s e s t h e 'combined s p a c e ' a p p r o a c h i n m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s f o r t h e d i f f r a c t e d beam i n t e n s i t i e s , and t h e i m p o r t a n t n o n - g e o m e t r i c a l p a r a m e t e r s r e q u i r e d . A g e n e r a l s t r a t e g y f o r s e t t i n g up an a f f o r d a b l e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n i s g i v e n . C h a p t e r 4 e x a m i n e s some g e n e r a l e x p e r i m e n t a l a s p e c t s i n LEED/AES s t u d i e s . T o p i c s i n c l u d e UHV, sample p r e p a r a t i o n , m o n i t o r i n g s u r f a c e c l e a n l i n e s s by AES, and d a t a a c q u i s i t i o n o f LEED s p o t i n t e n s i t i e s . F o r t h e l a s t t o p i c , s p e c i a l a t t e n t i o n i s f o c u s s e d on t h e o n - l i n e TV camera method w h i c h 21 i s new t o t h i s l a b o r a t o r y , and was d e v e l o p e d d u r i n g t h i s work. C h a p t e r 5 compares t h e c a l c u l a t e d 1 ( E ) c u r v e s o f p ( 2 x l ) a n d p ( 2 x 2 ) s u r f a c e s t r u c t u r e s on two h c p ( O O O l ) s u r f a c e s . M a t h e m a t i c a l methods f o r 1(E) c u r v e c o m p a r i s o n s u c h as t h e Z a n a z z i - J o n a and P e n d r y R - f a c t o r s a r e p r e s e n t e d . S e v e r a l a p p r o x i m a t e schemes o f m u l t i p l e s c a t t e r i n g c a l c u l a t i o n a s i m p l i e d by t h e r e s u l t s h e r e a r e d i s c u s s e d . C h a p t e r 6 r e p o r t s t h e a d s o r p t i o n s t u d i e s o f oxygen on Z r ( 0 0 0 1 ) by LEED and AES. A n a l y s i s o f d i f f r a c t e d beam i n t e n s i t i e s i s p e r f o r m e d f o r b o t h t h e (1x1) and t h e (2x2) s t r u c t u r e s . CHAPTER 2 PRINCIPLES OF LEED AND AES 23 2.1 ELECTRON DIFFRACTION FROM ORDERED SURFACES As m e n t i o n e d i n S e c t i o n 1.2.1, e l e c t r o n s w i t h w a v e l e n g t h s of t h e o r d e r o f i n t e r a t o m i c d i s t a n c e s i n an o r d e r e d s u r f a c e can be d i f f r a c t e d a n a l o g o u s l y t o t h e much b e t t e r known X - r a y d i f f r a c t i o n . The major d i f f e r e n c e between t h e s e two t y p e s of d i f f r a c t i o n i s t h a t X - r a y d i f f r a c t i o n i s b a s i c a l l y k i n e m a t i c a l whereas e l e c t r o n d i f f r a c t i o n i n v o l v e s m u l t i p l e s c a t t e r i n g between t h e i n c i d e n t e l e c t r o n s and t h e atoms o f t h e c r y s t a l l a t t i c e . N e v e r t h e l e s s , t h e LEED p a t t e r n f o r m e d by d i f f r a c t e d beams on a c o l l e c t i n g s c r e e n i s a d i r e c t c o n s e q u e n c e o f t h e t r a n s l a t i o n a l symmetry o f t h e s u r f a c e r e g i o n . The d i p e r i o d i c i t y o f t h e l a t t e r r e g i o n i s c o n v e n i e n t l y r e p r e s e n t e d by a 2 - d i m e n s i o n a l n e t , f o r w h i c h t h e u n i t b a s i s v e c t o r s a r e d e s i g n a t e d h e r e a s s t and s 2 . 2.1.1 CONDITIONS FOR ELASTIC DIFFRACTION O u t s i d e t h e i n f l u e n c e o f t h e c r y s t a l , b o t h i n c i d e n t and d i f f r a c t e d e l e c t r o n s n o r m a l l y e x p e r i e n c e a f i e l d f r e e r e g i o n where t h e y a r e c o n v e n i e n t l y r e p r e s e n t e d by p l a n e waves * k ( r ) = e x p ( / k - r ) . (2.1) In e q u a t i o n (2.1) t h e wave v e c t o r k s p e c i f i e s t h e d i r e c t i o n of t h e beam, and i t s m a g n i t u d e ( | jk J = 27r/X) r e l a t e s t o e n e r g y t h r o u g h 24 E = h 2 | k | 2 / ( 2 m ) , (2.2) where m i s t h e e l e c t r o n ' s mass and n i s P l a n c k ' s c o n s t a n t d i v i d e d by 27r. F o r i n c i d e n t and d i f f r a c t e d beams o f wave v e c t o r s k 0 and k' r e s p e c t i v e l y , t h e d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n r e s u l t i n g f r o m t h e i n t e r a c t i o n w i t h t h e s o l i d c a n be e x p r e s s e d q u i t e g e n e r a l l y a s da / d f l = ( m / 2 i r h 2 ) 2 |<k' | f | k 0 > | 2 , (2.3) where f i s an a p p r o p r i a t e t r a n s i t i o n o p e r a t o r [ 4 5 , 4 6 ] . F u r t h e r m o r e , when t h e H a m i l t o n i a n o f t h e e l e c t r o n i s i n v a r i a n t under a symmetry o p e r a t i o n S a p p l i e d t o t h e c r y s t a l , t h e m a t r i x e l e m e n t s i n (2.3) s a t i s f y ( 4 5 , 4 6 ] <JS' l TlJSo> = |S" 1 T S | k 0 > = <Sk' | f | S k 0 > . (2.4) I f § r e p r e s e n t s t r a n s l a t i o n by a s u r f a c e n e t v e c t o r t , (2.4) becomes <iV|T|ko> = e x p [ / ( k 0 - k ' ) - t ] < k ' |T|k 0>, (2.5) w h i c h i m m e d i a t e l y i m p l i e s t h a t e i t h e r t h e t r i v i a l s o l u t i o n <k'|T|k 0> = 0 (2.6) h o l d s , o r a l t e r n a t i v e l y t h e more i n t e r e s t i n g c o n d i t i o n 25 exp[i ( ko-JS' ) - t ] = 1 . (2.7) Since the s u r f a c e net ve c t o r t has the form t = ms, + n s 2 (2.8) fo r i n t e g e r s m and n, equation (2.7) w i l l be s a t i s f i e d whenever the i n c i d e n t and d i f f r a c t e d wave v e c t o r s d i f f e r by a v e c t o r of the r e c i p r o c a l net g = k 0 - k' = hs* + ks*, (2.9) where h and k are i n t e g e r s . The r e c i p r o c a l net b a s i s v e c t o r s s, and s 2 must s a t i s f y the c o n d i t i o n s it * 5,'S, = s 2 « s 2 = 2ir, (2.10a) * * s,«s 2 = S j - s , = 0, (2.10b) and they are r e l a t e d t o the u n i t mesh v e c t o r s of the r e a l s u r f a c e net by s* = s i n 2 # ( s i - s 2 c o s 0 ) , (2.11a) ~ s 1 * 7 IT s 2 = 1^  s i n 2 # ( s 2 - s ^ o s t * ) , (2.11b) where the S: (i=1,2) represent magnitudes of the b a s i s 26 R e a l R e c i p r o c a l s q u a r e e . g . f c c O O O ) b c c ( 1 0 0 ) r e c t a n g u l a r e . g . f c c ( l i O ) h e x a g o n a l e . g . f e e ( 1 1 1 ) h c p ( 0 0 0 l ) b c c ( 1 1 1 ) o b l i q u e e . g . b c c ( l l O ) * § 2 ; F i g u r e 2 .1: S o m e c o m m o n r e a l s p a c e s u r f a c e n e t s a n d t h e i r c o r r e s p o n d i n g r e c i p r o c a l n e t s . 27 v e c t o r s s^ , and t h e s^ e q u a l 5^/s^ ( i . e . r e p r e s e n t u n i t v e c t o r s g i v i n g t h e d i r e c t i o n of s ^ ) ; and 4> i s t h e a n g l e * * between s, and s 2 . W i t h t h e s e c h o i c e s i n s t and s 2 , i t c a n e a s i l y be shown t h a t g - t = 2NTT, (2.12) where N i s an i n t e g e r . Hence, t h e l e f t hand s i d e o f e q u a t i o n (2.7) i s a l s o u n i t y . F i g u r e 2.1 shows some common examples of s u r f a c e n e t s i n r e a l s p a c e and t h e i r c o r r e s p o n d i n g r e c i p r o c a l n e t s . C o n d i t i o n s f o r e l a s t i c d i f f r a c t i o n c a n now be summarized i n t e r m s o f c o n s e r v a t i o n o f e n e r g y |k~| 2 - I )S o I 2 , (2.13) and c o n s e r v a t i o n o f momentum p a r a l l e l t o t h e s u r f a c e ~£l| = + 9 ( h k ) * ( 2 , 1 4 ) I n e q u a t i o n s (2.13) and ( 2 . 1 4 ) , t h e d i f f e r e n t d i f f r a c t e d beams a r e d i s t i n g u i s h e d by s u b s c r i p t s w h i c h i d e n t i f y a p p r o p r i a t e r e c i p r o c a l n e t v e c t o r s ; t h e s u p e r s c r i p t s +/-s p e c i f y t h e d i r e c t i o n s i n t o / o u t o f t h e c r y s t a l r e s p e c t i v e l y . No r e s t r i c t i o n s have been p l a c e d on e n e r g y , and t h e r e f o r e t h e s e c o n d i t i o n s a p p l y e q u a l l y t o RHEED. 28 2.1.2 APPEARANCE OF LEED SPOTS: THE EWALD CONSTRUCTION The g e o m e t r i c a l c o n d i t i o n s f o r t h e a p p e a r a n c e o f LEED s p o t s have a l r e a d y been s p e c i f i e d by e q u a t i o n s (2.13) and ( 2 . 1 4 ) . However, i t i s o f t e n h e l p f u l t o v i s u a l i z e t h e s i t u a t i o n g r a p h i c a l l y u s i n g an a d a p t a t i o n o f t h e E w a l d c o n s t r u c t i o n u s e d f o r X - r a y d i f f r a c t i o n [ 4 7 ] . F o r an e l e c t r o n w a v e l e n g t h X, t h e m a g n i t u d e o f t h e i n c i d e n t w a v e v e c t o r k 0 i s 27r/X. In t h e E w a l d c o n s t r u c t i o n f o r LEED k 0 i s drawn w i t h a p p r o p r i a t e m a g n i t u d e and d i r e c t i o n t o end a t t h e p o i n t c h o s e n a s t h e o r i g i n O o f t h e r e c i p r o c a l l a t t i c e . The o t h e r end o f k 0 d e f i n e s t h e c e n t e r o f t h e E w a l d s p h e r e o f r a d i u s 27r/X a s shown i n F i g u r e 2.2. The r e c i p r o c a l s p a c e c o n s t r u c t i o n f o r d i p e r i o d i c d i f f r a c t i o n i n v o l v e s a s e t o f r e c i p r o c a l r o d s e a c h o f w h i c h i s p e r p e n d i c u l a r t o t h e sample s u r f a c e and p a s s e s t h r o u g h a p o i n t on t h e r e c i p r o c a l n e t . I n F i g u r e 2.2, o n l y a s i n g l e row o f s u c h r o d s i s c o n s i d e r e d f o r t h e s a k e o f c l a r i t y ( s e e i n s e t ) . E a c h o f t h e r e c i p r o c a l r o d s c a n t h e n be s p e c i f i e d by t h e p a i r o f i n t e g r a l i n d i c e s (hk) w h i c h a r e u s e d f o r d e f i n i n g t h e a p p r o p r i a t e r e c i p r o c a l n e t p o i n t a c c o r d i n g t o e q u a t i o n ( 2 . 9 ) . A l l p o s s i b l e d i f f r a c t e d v e c t o r s k^, w h i c h s a t i s f y e q u a t i o n s (2.13) and ( 2 . 1 4 ) , o r i g i n a t e f r o m t h e c e n t e r o f t h e E w a l d s p h e r e and t e r m i n a t e a t t h e i n t e r s e c t i o n o f t h e E w a l d s p h e r e w i t h a ' r e c i p r o c a l r o d ' . I n a LEED e x p e r i m e n t , o n l y t h e i n t e r s e c t i o n s a r i s i n g f r o m b a c k s c a t t e r e d k^ c a n be s e e n , and t h e n o n l y when t h e y a r e w i t h i n t h e a c c e p t a n c e a n g l e o f t h e a p p a r a t u s . 29 (1,0) beam v Ewald sphere reciprocal rods Figure 2.2: Ewald construction for electron d i f f r a c t i o n . Only four r e c i p r o c a l rods are selected from a large number of such rods (inset) for the sake of c l a r i t y . 30 As t h e e n e r g y o f t h e i n c i d e n t e l e c t r o n i s i n c r e a s e d , X d e c r e a s e s and t h e r e f o r e t h e E w a l d s p h e r e becomes l a r g e r . The f i r s t e f f e c t i s t h a t more s p o t s a r e o b s e r v e d b e c a u s e more r e c i p r o c a l r o d s i n t e r s e c t t h e s u r f a c e o f t h e s p h e r e . The s e c o n d e f f e c t i s t h a t t h e a n g l e <t> between k and k 0 ( F i g u r e 2.2) d e c r e a s e s . S i n c e t h e s p e c u l a r o r (0,0) beam i n v o l v e s no t r a n s f e r o f momentum p a r a l l e l t o t h e s u r f a c e , t h e a n g l e 6 i s un c h a n g e d . T h e r e f o r e when t h e e n e r g y o f t h e i n c i d e n t beam i s c o n t i n u o u s l y i n c r e a s e d , t h e (0,0) beam w i l l r e m a i n s t a t i o n a r y on t h e d e t e c t i n g d e v i c e ( p r o v i d e d o f c o u r s e t h a t t h e e l e c t r o n s move i n f i e l d f r e e s p a c e o u t s i d e o f t h e c r y s t a l ) ; jthe n o n - s p e c u l a r beams however w i l l move t o w a r d s t h e (0,0) beam. The d e t e c t i n g d e v i c e o f LEED beams i s t r a d i t i o n a l l y a h e m i s p h e r i c a l f l u o r e s c e n t s c r e e n whose c e n t e r o f c u r v a t u r e c o i n c i d e s w i t h t h e sample. F i g u r e 2.3 shows a s c h e m a t i c s e t u p f o r d e t e c t i n g LEED s p o t s and how t h e l a t t e r a r e l a b e l l e d ( i n t h i s c a s e , f o r a r e c t a n g u l a r r e c i p r o c a l n e t ) . 2.1.3 SUPERLATTICE NOTATION The LEED p a t t e r n as s e e n on a d e t e c t i n g s c r e e n r e p r e s e n t s t h e r e c i p r o c a l s p a c e image o f t h e d i p e r i o d i c t r a n s l a t i o n a l symmetry of t h e s u r f a c e . The p r e s e n c e o f o r d e r e d a d l a y e r s g e n e r a l l y adds more t r a n s l a t i o n a l symmetry t o t h e s u r f a c e ( e x c e p t when t h e y have t h e same s u r f a c e n e t as t h e c l e a n s u r f a c e ) , t h e r e s u l t o f w h i c h i s an i n c r e a s e i n d i f f r a c t i o n beams i n r e c i p r o c a l s p a c e . Such a d l a y e r s a r e 31 2 viewing window Figure 2.3: Schematic set-up for LEED experiment and the r e l a t i o n between the surface l a t t i c e and the d i f f r a c t i o n pattern. 32 (b) F i g u r e 2.4: LEED p a t t e r n s from Rh ( 111 ) a t 142 eV, n o r m a l i n c i d e n c e : ( a ) C l e a n s u r f a c e ; ( b ) A f t e r e x p o s u r e t o H 2 S . The e x t r a s p o t s a r e l a b e l l e d i n f r a c t i o n s o f t h e ' c l e a n ' r e c i p r o c a l v e c t o r s . 33 c a l l e d s u p e r l a t t i c e s t r u c t u r e s . F u r t h e r , t h e y a r e s a i d t o be commensurate i f t h e t h e p o s i t i o n s o f t h e e x t r a beams ca n be r e p r e s e n t e d by f r a c t i o n a l c o m b i n a t i o n s of t h e s u b s t r a t e r e c i p r o c a l n e t v e c t o r s , a s i l l u s t r a t e d by t h e example shown i n F i g u r e 2.4 f o r t h e a d s o r p t i o n o f s u l f u r on t h e s u r f a c e of R h ( 1 1 1 ) . In s u c h c a s e s , t h e e x t r a beams a r e o f t e n r e f e r r e d t o as f r a c t i o n a l o r d e r beams. In s h o r t , t h e s i z e o f t h e r e c i p r o c a l n e t of a commensurate s u p e r l a t t i c e s t r u c t u r e i s a f r a c t i o n 1/S (S an i n t e g e r ) o f t h a t o f t h e s u b s t r a t e , o r a l t e r n a t e l y t h e r e a l s p a c e s u r f a c e n e t o f t h e s u p e r l a t t i c e i s S t i m e s as l a r g e a s t h a t o f t h e s u b s t r a t e . I n LEED c r y s t a l l o g r a p h y t h e r e a l s p a c e s u p e r l a t t i c e s t r u c t u r e s a r e u s u a l l y named w i t h t h e s u b s t r a t e u n i t c e l l v e c t o r s a s r e f e r e n c e . Two s u c h s y s t e m s o f n o m e n c l a t u r e commonly u s e d a r e t h e Wood and t h e m a t r i x n o t a t i o n s . The Wood n o t a t i o n was o r i g i n a l l y p r o p o s e d by Wood[48], w h i c h r e q u i r e s t h a t t h e a n g l e between t h e a d s o r b a t e u n i t c e l l v e c t o r s a , and a 2 be t h e same a s t h a t between t h e s u b s t r a t e u n i t c e l l v e c t o r s s, and s 2 . When t h i s r e q u i r e m e n t i s s a t i s f i e d , t h e s u r f a c e s t r u c t u r e i s named i n t h e g e n e r a l f o r m p ( n x m ) R 0 ° o r c ( n x m ) R 0 ° d e p e n d i n g on whether t h e u n i t c e l l o f t h e a d s o r b a t e i s p r i m i t i v e o r c e n t e r e d , 6 i s t h e s m a l l e s t a n g l e w h i c h has t o be r o t a t e d t o l i n e up a^ and s n ( o r a 2 and s 2 ) . n and m a r e numbers t o s p e c i f y t h e l e n g t h s o f a, and a 2 i n m u l t i p l e s of s, and s 2 r e s p e c t i v e l y , t h a t i s I a i I = n | s , | , (2.15a) 34 | a 2 | = m j s 2 | . (2.15b) When 6 i s z e r o , t h e R0° i s d r o p p e d from t h e n o t a t i o n . The r a t i o o f t h e s i z e s o f t h e u n i t c e l l of t h e s u p e r l a t t i c e and o f t h e s u b s t r a t e i s g i v e n by mn i f t h e s u p e r l a t t i c e i s p r i m i t i v e , and mn/2 i f t h e s u p e r l a t t i c e i s c e n t e r e d . A g e n e r a l i z e d f o r m o f n o m e n c l a t u r e i s p r o v i d e d by m a t r i x n o t a t i o n . T h i s n o t a t i o n was o r i g i n a l l y d e v e l o p e d by P a r k and M a d d e n [ 4 9 ] , and f u r t h e r d i s c u s s e d by E s t r u p and M c R a e [ 5 0 ] , H e r e , t h e a d l a y e r b a s i s v e c t o r s a^ a n d a 2 a r e e x p r e s s e d as l i n e a r c o m b i n a t i o n s o f t h e s u b s t r a t e b a s i s v e c t o r s s, and s 2 i n t h e f o r m (2.16) = AS. S u p e r l a t t i c e s t r u c t u r e s a r e t h e n d e s c r i b e d by t h e m a t r i x A. The d e t e r m i n a n t o f A d e n o t e s t h e r a t i o of t h e s i z e s o f t h e u n i t c e l l of t h e s u p e r l a t t i c e and o f t h e s u b s t r a t e . An e x p e r i m e n t a l LEED p a t t e r n g i v e s t h e r e l a t i o n s h i p between t h e r e c i p r o c a l n e t v e c t o r s o f t h e s u b s t r a t e and of t h e a d l a y e r . T h i s r e l a t i o n s h i p c a n be r e p r e s e n t e d c o n v e n i e n t l y by a m a t r i x A w h i c h i s d e f i n e d by [ a , a 2 ] = [ s , s 2 ] A * , (2.17) where s u p e r s c r i p t * d e n o t e s r e c i p r o c a l s p a c e . The e l e m e n t s a 1 a 2 a 1 1 a 1 2 a 2 1 a 2 2 s, § 2 35 Substrate + Superlattice Superlattice unit c e l l Wood notation Matrix notation bcc(100) Figure 2.5: Some common superlattice structures on low M i l l e r index surfaces and the i r nomenclature in both Wood and matrix notations. f c c ( 1 1 0 ) ooeeeooo ooeeeooo oo©ee©oo OO0GG0OO b c c ( 1 1 0 ) f c c ( m ) , h c p ( O O O l ) p(2x1 ) c ( 2 x 2 ) p ( 3 x 1 ) p(2x1 ) p ( 2 x 1 ) (/3x/3)R30° p ( 2 x 2 ) F i g u r e 2.5: ( c o n t i n u e d ) 37 o f t h e m a t r i x A c a n be o b t a i n e d from measurement of t h e S B o b s e r v e d LEED p a t t e r n . The e l e m e n t s of A a r e t h e n c a l c u l a t e d b y [ 4 9 ] = 1 A = -~ I A* 1 S B * * a 2 2 " a 1 2 * * _ a 21 a T T (2.18) it ^ where |A | i s t h e d e t e r m i n a n t o f t h e m a t r i x A . U s i n g e q u a t i o n ( 2 . 1 8 ) , t h e s u p e r l a t t i c e s t r u c t u r e w h i c h g i v e s r i s e t o t h e o b s e r v e d LEED p a t t e r n shown i n F i g u r e 2.4 can be d e s c r i b e d a s [.} 2 ] i n m a t r i x n o t a t i o n . The f u l l d e s c r i p t i o n o f t h e a d s o r p t i o n s y s t e m i s e i t h e r R h ( 1 1 1 ) - [ . { 2 ] S i n m a t r i x n o t a t i o n o r R h ( 1 1 1 ) - ( / 3 x / 3 ) R 3 0 ° - S i n t h e Wood n o t a t i o n . F i g u r e 2.5 shows s e v e r a l common s u p e r l a t t i c e s t r u c t u r e s on some low M i l l e r i n d e x s u r f a c e s and t h e i r c o r r e s p o n d i n g n o m e n c l a t u r e s i n b o t h s y s t e m s . A l t h o u g h t h e Wood n o t a t i o n p r o v i d e s an i n t u i t i v e p i c t u r e f o r an o r d e r e d a d l a y e r r e l a t i v e t o t h e s u b s t r a t e , i t s use i s l i m i t e d t o s i m p l e c a s e s . On t h e o t h e r hand, t h e m a t r i x n o t a t i o n c a n be u s e d f o r any complex p a t t e r n s . The l a t t e r i n c l u d e many h y d r o c a r b o n a d s o r p t i o n s y s t e m s on m e t a l s u r f a c e s . One s u c h example i s t h e o r d e r e d s t r u c t u r e o f benzene on R h ( ! 1 l ) s u r f a c e w h i c h i s d e p i c t e d i n F i g u r e 2.6. 2.2 LEED SPOT INTENSITY-ENERGY CURVE The u n i t mesh v e c t o r s f o r t h e s u b s t r a t e and f o r any a d s o r b a t e p r e s e n t c a n u s u a l l y be o b t a i n e d by i n s p e c t i o n o f t h e LEED p a t t e r n f r o m an o r d e r e d s u r f a c e . However, i n o r d e r 38 Rh • C OH Figure 2.6: Ordered adlayer of benzene on R h ( m ) . Th structure i s described as Rh(111)-|? J|C 6H 6 in matri notation (after Van Hove et al. [ 5 1 ] ) . 39 t o d e t e r m i n e a t o m i c p o s i t i o n s , t h e i n t e n s i t i e s o f LEED s p o t s have t o be a n a l y z e d . F o r t h i s p u r p o s e , t h e i n t e n s i t i e s a r e n e a r l y a l w a y s measured a s a f u n c t i o n o f i n c i d e n t e n e r g y . T h i s method o f measurement g i v e s t h e s o - c a l l e d i n t e n s i t y v e r s u s e n e r g y , o r 1 ( E ) , c u r v e s , an example o f w h i c h i s shown i n F i g u r e 2.7. The p r e f e r e n c e f o r m e a s u r i n g i n t e n s i t y a s a f u n c t i o n of i n c i d e n t e n e r g y r a t h e r t h a n i n c i d e n t a n g l e i s due, i n l a r g e p a r t , t o t h e f o l l o w i n g f a c t o r s : 1. E l e c t r o n e n e r g y c a n be v a r i e d and measured s t r a i g h t f o r w a r d l y , b u t a b s o l u t e measurements o f a n g l e of i n c i d e n c e a r e g e n e r a l l y h a r d e r t o a c c o m p l i s h ! 5 3 ] , 2. When t h e i n c i d e n t beam i s c o n t a i n e d i n a symmetry e l e m e n t , e s p e c i a l l y when p e r p e n d i c u l a r t o t h e s u r f a c e , u n c e r t a i n t i e s i n a n g l e measurement c a n be r e d u c e d by a v e r a g i n g n e a r l y - e q u i v a l e n t s e t s of s p o t s [ 5 4 ] , and 3. M u l t i p l e s c a t t e r i n g c a l c u l a t i o n s a r e more a f f o r d a b l e when t h e i n c i d e n t beam i s c h o s e n t o c o i n c i d e w i t h symmetry e l e m e n t ( s ) ( S e c t i o n 3 . 5 ) . A t f i r s t g l a n c e , e x p e r i m e n t a l 1 ( E ) c u r v e s a p p e a r t o be v e r y c o m p l i c a t e d . The r a t h e r s i m p l e k i n e m a t i c f o r m u l a t i o n f o r X - r a y c r y s t a l l o g r a p h y o f t e n f a i l s t o p r e d i c t t h e p o s i t i o n s o f maxima and minima i n a t y p i c a l 1 ( E ) c u r v e s u c h a s t h e one shown i n F i g u r e 2.7. T h i s i s due t o t h e f a c t t h a t t h e i n t e n s i t i e s o f t h e d i f f r a c t e d beams depend n o t o n l y on t h e g e o m e t r i c a l a r r a n g e m e n t of t h e atoms, but a l s o on t h e m u l t i p l e s c a t t e r i n g o f t h e LEED e l e c t r o n s by t h e atoms. The l a t t e r i s d e p e n d e n t e s p e c i a l l y on t h e c o r e p o t e n t i a l o f t h e Figure 2.7: 1(E) curve for (0,0) beam from Ni(lOO) at (9=3°. Kinematically expected Bragg peak positions are indicated by v e r t i c a l bars (after Andersson et al.[52]). 41 a t o m i c s p e c i e s i n v o l v e d . A l t h o u g h numerous m a t h e m a t i c a l t r e a t m e n t s f 1 1 , 5 5 ] o f 1 ( E ) c u r v e i n t e n s i t i e s a r e a v a i l a b l e i n t h e l i t e r a t u r e , a s e m i - q u a n t i t a t i v e a p p r o a c h t a k e n by Van Hove and T o n g [ 5 6 ] t o e x p l a i n t h e f e a t u r e s i n an 1 ( E ) c u r v e seems h e l p f u l t o u n d e r s t a n d t h e p h y s i c a l p r o c e s s e s i n v o l v e d i n e l e c t r o n d i f f r a c t i o n by a c r y s t a l l a t t i c e . T h i s s i m p l i f i e d a p p r o a c h b r i n g s o u t most o f t h e i m p o r t a n t i n g r e d i e n t s r e q u i r e d i n a s u c c e s s f u l m u l t i p l e s c a t t e r i n g c a l c u l a t i o n . The m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s w i l l be d i s c u s s e d i n more d e t a i l i n t h e n e x t c h a p t e r . 2.2.1 THE DIFFRACTION PEAK POSITIONS To s i m p l i f y m a t t e r s , t h e d i s c u s s i o n w i l l s t a r t w i t h a h y p o t h e t i c a l 1 - d i m e n s i o n a l c l e a n c r y s t a l w i t h a l a t t i c e c o n s t a n t a . I t w i l l t h e n be g e n e r a l i z e d t o 3 - d i m e n s i o n s , and f i n a l l y o v e r l a y e r s w i l l be i n t r o d u c e d . In 1 - d i m e n s i o n , i n c i d e n t and d i f f r a c t e d e l e c t r o n s c a n be r e p r e s e n t e d by w a v e f u n c t i o n s e x p ( / k x ) and e x p ( - / k x ) r e s p e c t i v e l y (where k 2=2E i n a t o m i c u n i t s ) . The manner i n wh i c h i n d i v i d u a l s c a t t e r e r s o f t h e c r y s t a l s c a t t e r e l e c t r o n s c a n be summarized by two complex c o e f f i c i e n t s : t t h e a m p l i t u d e w i t h w h i c h a s c a t t e r e r t r a n s m i t s e l e c t r o n s , and r t h e a m p l i t u d e w i t h w h i c h a s c a t t e r e r r e f l e c t s e l e c t r o n s . C o n s e r v a t i o n o f c u r r e n t , i n t h e a b s e n c e o f damping ( i n e l a s t i c s c a t t e r i n g ) , r e q u i r e s t h a t 42 | t | 2 + | r | 2 = 1. (2.19) C o l l e c t i v e s c a t t e r i n g o f t h e i n c i d e n t w a v e f u n c t i o n e x p ( / k x ) by an i n f i n i t e number of i d e n t i c a l s c a t t e r e r s of t h i s hypo-t h e t i c a l c r y s t a l r e s u l t s i n a r e f l e c t e d w a v e f u n c t i o n whose a m p l i t u d e i s g i v e n b y [ 5 6 ] R = ^ r t 2 ^ e x p ( / k 2 a j ) = . (2.20) j=0 1 - t 2 e x p ( / 2 k a ) The summation o v e r t 2 ^ t a k e s i n t o a c c o u n t t h a t any e l e c t r o n ( e x c e p t t h e one c l o s e s t t o vacuum) w h i c h c o n t r i b u t e s t o t h e t o t a l r e f l e c t e d a m p l i t u d e has t o be t r a n s m i t t e d t w i c e a t e a c h i n t e r v e n i n g s c a t t e r e r : once p e n e t r a t i n g i n t o and- once e m e r g i n g out of t h e c r y s t a l . I f t h e t r a n s m i s s i o n c o e f f i c i e n t t were r e a l , r e f l e c t i o n maxima wo u l d o c c u r when t h e B r a g g c o n d i t i o n f o r X - r a y d i f f r a c t i o n 2ka=n27r (n i n t e g e r ) i s s a t i s f i e d . However, t h e use o f a complex t g e n e r a l i z e s t h e c o n d i t i o n f o r r e f l e c t i o n maxima t o 2 [ k a + a r g ( t ) ] = n27r, (2.21) where a r g ( t ) r e p r e s e n t s t h e argument of t h e i m a g i n a r y p a r t ( o r p h a s e ) of t h e t r a n s m i s s i o n c o e f f i c i e n t t . A c c o r d i n g t o e q u a t i o n ( 2 . 2 1 ) , t h e c o n d i t i o n f o r r e f l e c t i o n maxima a p p e a r s t o be i n d e p e n d e n t o f t h e r e f l e c t i o n c o e f f i c i e n t r , a l t h o u g h t h e l a t t e r a c t u a l l y e x e r t s i n d i r e c t ( b u t g e n e r a l l y s m a l l ) i n f l u e n c e t h r o u g h m u l t i p l e s c a t t e r i n g ( a f t e r a l l r and t a r e 43 r e l a t e d by e q u a t i o n ( 2 . 1 9 ) ) . E q u a t i o n (2.21) i s s t i l l v a l i d f o r X - r a y d i f f r a c t i o n , where |t|=1 and a r g ( t ) v a n i s h e s . In e l e c t r o n d i f f r a c t i o n , t h e g e n e r a l l y n o n - z e r o a r g ( t ) o r i g i n a t e s f r o m t h e i n t e r a c t i o n of t h e e l e c t r o n w i t h t h e a t o m i c p o t e n t i a l o f e a c h s c a t t e r e r upon t r a n s m i s s i o n , and i s r e s p o n s i b l e f o r two commonly o b s e r v e d ' a n o m a l i e s ' i n 1 ( E ) c u r v e s . To s i m p l i f y t h e d i s c u s s i o n , t h e a t o m i c p o t e n t i a l can c o n c e p t u a l l y be s p l i t i n t o two components: a c o n s t a n t p o t e n t i a l ( i n n e r p o t e n t i a l ) and a s p h e r i c a l l y s y m m e t r i c p o t e n t i a l w h i c h i s l a r g e l y e n e r g y d e p e n d e n t . F o r an a t t r a c t i v e a t o m i c p o t e n t i a l s u c h a s t h e one i n m e t a l s , t h e e l e c t r o n , i s s p e e d e d up a t e a c h t r a n s m i s s i o n (due l a r g e l y t o t h e i n n e r p o t e n t i a l ) . Some o f t h e p o t e n t i a l e n e r g y o f t h e e l e c t r o n i s u s e d up i n t h i s t e m p o r a r y s p e e d i n g - u p , w h i c h r e s u l t s i n a s h o r t e r k and an a c c o r d i n g l y l o w e r E . T h e r e f o r e , t h e p o s i t i o n s of r e f l e c t i o n maxima w i l l be s h i f t e d t o w a r d s l o w e r e n e r g y s i d e s o f t h e k i n e m a t i c a l l y e x p e c t e d p e a k s . T h i s s h i f t i s r e l a t i v e l y c o n s t a n t f o r a l l t h e r e f l e c t i o n maxima ( i n s o f a r a s t h e s l i g h t e n e r g y d e p e n d e n c e o f i n n e r p o t e n t i a l i s i g n o r e d ) , and i s between 5 and 15 eV d e p e n d i n g on t h e t y p e o f c r y s t a l . The s p h e r i c a l l y s y m m e t r i c p o t e n t i a l o f an atom i n t h i s h y p o t h e t i c a l c r y s t a l p r o v i d e s a ' p o t e n t i a l w e l l ' i n w h i c h an e l e c t r o n w i t h a p p r o p r i a t e k c a n r e s o n a t e and g i v e r i s e t o s t a n d i n g waves. C o n s t r u c t i v e i n t e r f e r e n c e ( o r r e s o n a n c e ) from t h i s t y p e o f a t o m i c s c a t t e r i n g o f t e n l e a d s t o t h e a p p e a r a n c e o f ' u n e x p e c t e d ' p e a k s s c a t t e r e d a r o u n d t h e 44 a l r e a d y - s h i f t e d B r a g g peaks.- T h i s e f f e c t i s most p r o m i n e n t where a b s o r p t i o n i s weak ( e . g . a t low e n e r g i e s i n L E E D ) . I n s u c h i n s t a n c e s i n r e a l d a t a , i t may be d i f f i c u l t t o d i s t i n g u i s h a B r a g g peak from r e s o n a n c e p e a k s . I n a m u l t i p l e s c a t t e r i n g c a l c u l a t i o n , t h e e f f e c t of a t o m i c s c a t t e r i n g i s c o n t a i n e d i n t h e p a r t i a l wave 'phase s h i f t s ' 6^ . 2.2.2 PEAK WIDTH The p e a k s i n an e x p e r i m e n t a l 1 ( E ) c u r v e a l m o s t a l w a y s show l i f e t i m e b r o a d e n i n g from f i n i t e e l e c t r o n p e n e t r a t i o n d e p t h ( w h i c h r e s t r i c t s t h e number o f s c a t t e r e r s an e l e c t r o n ' s e e s ' ) . The l a t t e r i s a r e s u l t o f t h e f i n i t e ( a l b e i t s m a l l ) v a l u e o f | r | , w h i c h i s t y p i c a l l y a r o u n d 0.1 i n t h e e n e r g y r a n g e c o n s i d e r e d i n LEED. S t i l l a s s u m i n g no a b s o r p t i o n and i g n o r i n g t h e a t t e n t u a t i o n a t e a c h s c a t t e r e r , c o n s e r v a t i o n o f c u r r e n t l i m i t s t h e number o f s c a t t e r e r s t o a p p r o x i m a t e l y | r |" 1 , hence a maximum p e n e t r a t i o n d e p t h a r o u n d a | r j ~ 1 (where a i s t h e i n t e r a t o m i c d i s t a n c e ) . The a c t u a l p e n e t r a t i o n d e p t h may be even s h o r t e r due t o i n e l a s t i c s c a t t e r i n g . D i f f r a c t i o n f r o m | rr J ~ 1 s u c h s c a t t e r e r s w i l l p r o d u c e d i f f r a c t i o n maxima whose w i d t h s a r e g i v e n b y [ 5 7 ] 2Ak = 7 r|r|/2a (2.22) i n r e c i p r o c a l s p a c e . D i f f e r e n t i a t i o n o f k 2 =2E and s u b s t i t u t i n g t h e r e s u l t i n t o e q u a t i o n (2.22) t h e n y i e l d s 45 2AE <* 7 r k|r|/2a, (2.23) w h i c h a r e t h e c o r r e s p o n d i n g e n e r g y w i d t h s o f t h e d i f f r a c t i o n maxima. | r | i s o n l y s l i g h t l y e n e r g y d e p e n d e n t i n t h e ra n g e 30 t o s e v e r a l h u n d r e d eV, t h e r e f o r e AE i s m a i n l y d e t e r m i n e d by k w h i c h i s p r o p o r t i o n a l t o E l / 2 . Not s u r p r i s i n g l y , s i m i l a r r e s u l t s were o b t a i n e d f r o m e l e c t r o n i n e l a s t i c mean f r e e p a t h and l i f e t i m e c o n s i d e r a t i o n by S t e r n et al . [ 5 8 ] . T h i s e x p l a i n s t h e i n c r e a s i n g l y b r o a d e r d i f f r a c t i o n maxima a t h i g h e r e n e r g i e s . AE r a n g e from 5 t o 15 eV i n a t y p i c a l 1 ( E ) c u r v e . A t v e r y low e n e r g i e s ( t y p i c a l l y E ^ l O e V ) , mean f r e e p a t h l e n g t h s f o r i n e l a s t i c s c a t t e r i n g i n s o l i d s a r e g e n e r a l l y l a r g e . S t r o n g e l a s t i c s c a t t e r i n g may open up wide band g a p s , and t h e e l e c t r o n s have a h i g h e r p r o b a b i l i t y o f b e i n g t o t a l l y r e f l e c t e d t o t h e vacuum t h a n b e i n g a b s o r b e d . I n o t h e r words, | r | becomes a p p r e c i a b l e , and t h e w i d t h s o f r e f l e c t i o n maxima become l a r g e r a c c o r d i n g l y . R e s o n a n c e p e a k s a r e g e n e r a l l y n a r r o w e r t h a n B r a g g p e a k s b e c a u s e r e s o n a n c e p r o c e s s e s a r e n o r m a l l y a s s o c i a t e d w i t h l o n g e r n a t u r a l l i f e t i m e s . The l a t t e r r e d u c e s l i f e t i m e b r o a d e n i n g t o some e x t e n t . 2.2.3 THREE-DIMENSIONAL EFFECTS An immediate c o n s e q u e n c e of c o n v e r s i o n from a 1 - d i m e n s i o n a l c r y s t a l t o a 3 - d i m e n s i o n a l one i s t h e i n t r o d u c t i o n o f more beams i n w h i c h t h e e l a s t i c a l l y 46 d i f f r a c t e d e l e c t r o n s c an t r a v e l . E a c h l a y e r may d i f f r a c t a beam <j i n t o beams £' w i t h r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s r<jg' a n c ^ to to t' r e s P e c t i v e l y • E a c h beam w i l l have i t s own w a v e v e c t o r k^. L e t k ^ d e n o t e t h e component of k^ p e r p e n d i c u l a r t o t h e c r y s t a l s u r f a c e , and a t h e i n t e r l a y e r s e p a r a t i o n , t h e n e q u a t i o n (2.21) c a n be m o d i f i e d t o [ 5 6 ] k g j a + a r g ( t a 9 . ) + k ^ a + a r g U ^ g ) = n27r (2.24) a s t h e c o n d i t i o n f o r maximum r e f l e c t i o n f r o m beam £ i n t o beam . S t r i c t l y s p e a k i n g , e q u a t i o n (2.24) i s v a l i d o n l y f o r s i n g l e s c a t t e r i n g . N e v e r t h e l e s s , i t i s a v e r y good a p p r o x i m a t i o n f o r m u l t i p l e s c a t t e r i n g a s l o n g a s t h e z e r o - a n g l e f o r w a r d s c a t t e r e d beam i s d o m i n a n t . I t i s i n d e e d a l m o s t a l w a y s t r u e f o r LEED c a l c u l a t i o n s a c c o r d i n g t o Van H o v e [ 5 7 ] . T y p i c a l l y t h e z e r o - a n g l e f o r w a r d s c a t t e r e d beam c a r r i e s a s much a s 40% o f t h e i n c i d e n t f l u x w h i l e o t h e r beams c a r r y o n l y a b o u t 1%. E q u a t i o n (2.24) a g a i n r e q u i r e s p e a k s i n 1 ( E ) c u r v e s t o be s h i f t e d f r o m t h e i r r e s p e c t i v e k i n e m a t i c p o s i t i o n s by amounts w h i c h a r e e n e r g y - d e p e n d e n t . H e r e , a r g f t ^ . ) and a r g ( t a . a ) i m p l i c i t l y c o n t a i n i n f o r m a t i o n on t h e r a t h e r r i g i d s h i f t of B r a g g p e a k s by t h e i n n e r p o t e n t i a l , and t h e r e s o n a n t s c a t t e r i n g p r o p e r t i e s o f t h e s p h e r i c a l l y s y m m e t r i c a t o m i c p o t e n t i a l . I n t r a l a y e r m u l t i p l e s c a t t e r i n g i s a n o t h e r new f e a t u r e i n t h i s 3 - d i m e n s i o n a l c r y s t a l . The c o n t r i b u t i o n t o 47 r e f l e c t i o n a m p l i t u d e f r o m i n t r a l a y e r m u l t i p l e s c a t t e r i n g may be s m a l l , e s p e c i a l l y a t o r n e a r n o r m a l e l e c t r o n i n c i d e n c e where a t l e a s t two l a r g e a n g l e s c a t t e r i n g e v e n t s a r e r e q u i r e d f o r i n t r a l a y e r b u t j u s t one f o r i n t e r l a y e r m u l t i p l e s c a t t e r i n g . When a b s o r p t i o n i s s m a l l ( e . g . a t low e n e r g i e s i n LEED) a n d / o r a n g l e of i n c i d e n c e i s s h a l l o w , i n t r a l a y e r m u l t i p l e s c a t t e r i n g may become i m p o r t a n t . 2.2.4 OVERLAYER EFFECT The f u l l 3 - d i m e n s i o n a l t r e a t m e n t has been g i v e n by A n d e r s s o n and P e n d r y [ 5 9 ] , H e r e , t h e 1 - d i m e n s i o n a l c r y s t a l i s i n v o k e d a g a i n , and t h e ' c l e a n ' c r y s t a l c a n now be v i s u a l i z e d a s a column w i t h t o t a l r e f l e c t i o n c o e f f i c i e n t R g. The o v e r l a y e r i s now a p o i n t s c a t t e r e r w i t h t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s t Q a n d r Q r e s p e c t i v e l y , and i s assumed t o be s e p a r a t e d from t h e s u b s t r a t e by a d i s t a n c e d. An e x p r e s s i o n s i m i l a r t o e q u a t i o n (2.20) b ut i n v o l v i n g t Q , r Q , R s and kd c a n be o b t a i n e d f o r t h e t o t a l r e f l e c t i o n a m p l i t u d e f o r t h i s c o m b i n e d c o l u m n [ 5 7 ] . T r a n s f o r m a t i o n from t h i s e x p r e s s i o n i n t o t h e c o n d i t i o n f o r r e f l e c t i o n maxima i s no t s t r a i g h t f o r w a r d . N e v e r t h e l e s s , a s i m p l e e x p r e s s i o n f o r t h e c o n d i t i o n f o r r e f l e c t i o n maxima i s p o s s i b l e i f s i n g l e s c a t t e r i n g i s assumed f i r s t , a n d m u l t i p l e s c a t t e r i n g e f f e c t i s i n t r o d u c e d q u a l i t a t i v e l y i n l a t e r s t a g e s . In t h i s c a s e , r e f l e c t i o n maxima o c c u r when 48 2kd + 2 a r g ( t Q ) + a r g ( R g ) - a r g ( r Q ) = n2ir. (2.25) The i m p l i c a t i o n s o f e q u a t i o n (2.25) c a n be e x p l a i n e d as f o l l o w s . In an o v e r l a y e r s y s t e m , t h e p e a k s i n t h e 1 ( E ) c u r v e o r i g i n a t e m a i n l y f r o m B r a g g r e f l e c t i o n s f r o m t h e s u b s t r a t e ( i m p l i c i t l y c a r r i e d by t h e t e r m a r g ( R )) b e c a u s e i t has s u f f i c i e n t s c a t t e r i n g power MRSI t y p i c a l l y 0.5 t o 0.1) a v a i l a b l e t o t u r n a r o u n d a s u b s t a n t i a l number of e l e c t r o n s . N e v e r t h e l e s s , t h e o v e r l a y e r e x e r t s i t s i n f l u e n c e by c h a n g i n g t h e a m p l i t u d e s o f e x c i t a t i o n o f t h e w a v e v e c t o r s i n t h e s u b s t r a t e , and by f u r t h e r d i f f r a c t i n g beams r e f l e c t e d f r o m t h e s u b s t r a t e . The t e r m - a r g ( r Q ) may a l s o c a u s e a n n i h i l a t i o n o f t h e B r a g g c o n d i t i o n f o r r e f l e c t i o n maxima. E a c h o f t h e s e p r o c e s s e s i s d e p e n d e n t on d e t a i l s s u c h a s t h e s p a c i n g d, and t h e g e o m e t r y o f t h e o v e r l a y e r ( i n 3 - d i m e n s i o n a l c a s e ) . T h e s e a r e t h e r e a s o n s why 1 ( E ) c u r v e s a r e so s e n s i t i v e t o t h e s u r f a c e l a y e r p o s i t i o n . A t v e r y low e n e r g i e s , where a b s o r p t i o n i s m i n i m a l , t h e t e r m a r g ( r Q ) becomes a p p r e c i a b l e . T h i s has a s t r o n g i n f l u e n c e on 1(E) c u r v e s e i t h e r t h r o u g h i n t e r f e r e n c e o f t h e beams d i r e c t l y r e f l e c t e d f r o m t h e o v e r l a y e r w i t h t h o s e r e f l e c t e d f r o m t h e s u b s t r a t e , o r t h r o u g h t h e r e f l e c t i o n a t t h e i n t e r n a l s u r f a c e o f t h e o v e r l a y e r t o change t h e w a v e f i e l d s i n c i d e n t on t h e s u b s t r a t e . I n t h e e v e n t t h a t r Q i s c o m p a r a b l e t o R g, t h e e l e c t r o n s c a n be r e f l e c t e d many t i m e s between t h e o v e r l a y e r and s u b s t r a t e somewhat a s f o r l i g h t i n an i n t e r f e r o m e t e r . A c o n s e q u e n c e o f t h i s e f f e c t i s 49 t h e a p p e a r a n c e o f i n t e r f e r e n c e f r i n g e s a r o u n d a s i n g l e s t r o n g B r a g g peak i n t h e 1(E) c u r v e . The p o s i t i o n s o f t h e s e f r i n g e s a r e s t r o n g l y d e p e n d e n t on t h e i n t e r l a y e r s p a c i n g d. 2.3 DISORDER, DOMAINS AND INSTRUMENTAL RESPONSE In t h e d i s c u s s i o n o f t h e f o r m a t i o n o f LEED p a t t e r n s , two i m p l i c i t a s s u m p t i o n s have been made. The f i r s t i s t h a t t h e d i f f r a c t i o n t a k e s p l a c e f r o m a p e r f e c t l y o r d e r e d s u r f a c e . The s e c o n d i s t h a t t h e i n c i d e n t beam and t h e i n d i v i d u a l d i f f r a c t e d beams c a n e a c h be r e p r e s e n t e d by a p r e c i s e l y known s i n g l e w a v e v e c t o r . In r e a l i t y , t h e s e two a s s u m p t i o n s c a n n e v e r be s a t i s f i e d . The aim o f t h i s s e c t i o n i s t o d i s c u s s b r i e f l y how t h e d e v i a t i o n s f r o m t h e s e a s s u m p t i o n s a f f e c t t h e a p p e a r a n c e o f t h e LEED p a t t e r n and t h e m e a s u r e d LEED s p o t i n t e n s i t i e s . A p e r f e c t l y o r d e r e d c r y s t a l s h o u l d g i v e r i s e t o i n f i n i t e s i m a l l y narrow r e c i p r o c a l r o d s . T h e r e f o r e a LEED p a t t e r n f r o m t h i s i d e a l s u r f a c e s h o u l d c o n s i s t o f ' i d e a l l y s h a r p ' s p o t s ( i . e . t h e i n t e n s i t y p r o f i l e s a c r o s s t h e s e s p o t s c o u l d be d e s c r i b e d by a 8 - f u n c t i o n [ 6 0 ] ) . R e a l s u r f a c e s , however, i n e v i t a b l y have d e f e c t s ; t h e p r e s e n c e of s t e p s , k i n k s and c r y s t a l p l a n e d i s l o c a t i o n s i s w e l l k n o w n [ 6 l ] . M i c r o s c o p i c a l l y , a r e a l s u r f a c e c a n be r e p r e s e n t e d by a l a r g e number o f s m a l l i s l a n d s ( o r domains) o f ' i d e a l s u r f a c e ' , e a c h o f w h i c h p o s s e s s e s t h e same d i p e r i o d i c i t y . The l a t t e r s t i l l g o v e r n s t h e g e o m e t r i c a l c o n d i t i o n s f o r d i f f r a c t i o n , b u t t h e f i n i t e s i z e s o f t h e domains c a u s e a R e a l s p a c e R e c i p r o c a l Rods (a) cocoocoocxxo ( 1 0 ) ( 0 0 ) ( 1 0 ) ( 1 0 ) ( 0 0 ) ( 1 0 ) Figure 2.8: Some h y p o t h e t i c a l s u r f a c e s and t h e i r c o r r e s p o n d i n g r e c i p r o c a l r o d s . ( a ) P e r f e c t l y o r d e r e d s u r f a c e ; (b) S l i g h t l y d i s l o c a t e d c r y s t a l l o g r a p h i c p l a n e s ; ( c ) I n c r e a s i n g monatomic s t e p s ; and ( d ) A l t e r n a t i n g monatomic s t e p s . 51 b r o a d e n i n g o f t h e r e c i p r o c a l r o d s [ 6 0 ] . T h i s i s one o f t h e r e a s o n s why t h e o b s e r v e d LEED s p o t s have f i n i t e w i d t h s . In g e n e r a l , t h e l a r g e r t h e a v e r a g e domain s i z e , t h e n a r r o w e r t h e s p o t w i d t h . Randomly d i s t r i b u t e d domains u s u a l l y g i v e r i s e t o a g e n e r a l b a c k g r o u n d i n a LEED p a t t e r n . I f t h e d omains p o s s e s s some d e g r e e o f o r d e r among t h e m s e l v e s , a d d i t i o n a l ( u s u a l l y v e r y t h i n ) r e c i p r o c a l r o d s c a n be p r o d u c e d . S e v e r a l s u c h e x a m p l e s a r e shown i n F i g u r e 2.8. In s u c h c a s e s , t h e b a c k g r o u n d i s m o d u l a t e d due t o t h e non-random d i s t r i b u t i o n [ 6 2 ] . A d l a y e r s can a l s o be t r e a t e d i n t h e same manner a s d e f e c t s a t a c l e a n s u r f a c e . A d l a y e r i s l a n d s a r e e x p e c t e d i f t h e a d s o r b e d s p e c i e s e x e r c i s e s u f f i c i e n t l y s t r o n g a t t r a c t i v e i n t e r a c t i o n s , and t h e y may f o r m even a t low o v e r a l l c o v e r a g e s . G e n e r a l l y t h e LEED s p o t s a r e b r o a d e r and more d i f f u s e f o r s m a l l i s l a n d s i z e s t h a n f o r l a r g e r i s l a n d s . In a d d i t i o n , some s p e c i a l e f f e c t s may o c c u r f o r a d l a y e r s . F i r s t , t h e r e i s a p o s s i b i l i t y o f d i f f e r e n t a d s o r p t i o n s i t e s i n d i f f e r e n t d o m a i n s . T h i s i s u s u a l l y a r e s u l t o f s l o w o r d e r i n g k i n e t i c s i n t h e a d l a y e r i t s e l f . F o r example, an a d l a y e r o c c u p y i n g t h e 3 - f o l d s i t e s o f a f c c ( l l l ) s u r f a c e may f o r m a domain w i t h bABCABC... r e g i s t r y and a n o t h e r domain w i t h cABCABC... r e g i s t r y i f t h e two t y p e s o f 3 - f o l d a d s o r p t i o n s i t e s a r e n e a r l y e q u a l l y f a v o r a b l e e n e r g e t i c a l l y ( i n t h i s n o t a t i o n , t h e l o w e r and upper c a s e s r e p r e s e n t t h e r e g i s t r i e s o f t h e a d s o r b e d and s u b s t r a t e l a y e r s r e s p e c t i v e l y ) . S e c o n d l y , s u p e r l a t t i c e s t r u c t u r e s , whose 52 r o t a t i o n a l symmetry i s l o w e r t h a n t h a t o f t h e s u b s t r a t e , i n v a r i a b l y show r o t a t i o n a l l y r e l a t e d d o m a i n s . An example o f t h i s k i n d i s e n c o u n t e r e d i n C h a p t e r 5 f o r oxygen a d s o r p t i o n on Z r ( 0 0 0 1 ) . At low c o v e r a g e , a p ( 2 x 2 ) LEED p a t t e r n i s o b s e r v e d , but t h e same p a t t e r n c o u l d a r i s e from t h r e e s e t s of p ( 2 x l ) domains r e l a t e d by 120° r o t a t i o n s , a s d e p i c t e d i n F i g u r e 2.9. S i m p l e v i s u a l o b s e r v a t i o n o f LEED p a t t e r n s i s n o t a l w a y s a good g u i d e t o t h e p r e s e n c e of domains and d i s o r d e r . F o r example, some s t u d i e s have shown t h a t LEED p a t t e r n s may a p p e a r e s s e n t i a l l y u n changed even a s up t o 30% o f t h e s u r f a c e atoms become d i s t r i b u t e d a s random s t e p s [ 6 3 ] . A s s u m i n g a p e r f e c t i n s t r u m e n t f o r t h e LEED e x p e r i m e n t , t h e f o r e g o i n g d i s c u s s i o n s u g g e s t s t h a t i n f o r m a t i o n on s u r f a c e d omains and d i s o r d e r c a n be o b t a i n e d r a t h e r a c c u r a t e l y by a n a l y z i n g t h e a n g u l a r s p o t p r o f i l e . However, a r e a l LEED d i f f r a c t o m e t e r p u t s some r e s t r i c t i o n on t h e l a t t e r a n a l y s i s . An e l e c t r o n gun u s u a l l y p r o d u c e s a beam a b o u t 1 mm i n d i a m e t e r , w h i c h c o n s i s t s o f e l e c t r o n s w i t h s l i g h t l y d i f f e r i n g v e l o c i t i e s and d i r e c t i o n s a s s o c i a t e d w i t h t h e f i n i t e s i z e and h i g h t e m p e r a t u r e o f t h e s o u r c e . A l s o t h e r e a r e u n c e r t a i n t i e s i n t h e d e t e c t i o n p r o c e s s . Commonly i n LEED r e p e l l i n g g r i d s a r e e mployed t o f i l t e r o u t b a c k s c a t t e r e d e l e c t r o n s w h i c h have l o s t e n e r g y on s c a t t e r i n g f r o m t h e c r y s t a l . The f i n i t e d i s t a n c e t h a t t h e d i f f r a c t e d e l e c t r o n wave p a c k e t s have t o t r a v e l between t h e sample and t h e d e t e c t o r i n e v i t a b l y l e a d s t o some a n g u l a r s p r e a d o f momentum. In s h o r t , l i m i t a t i o n s on t h e LEED d i f f r a c t o m e t e r 53 Figure 2.9: Three possible r o t a t i o n a l domains (A,B,C) of a p(2xl) superlattice structure on a hexagonal close-packed surface. 54 i n e v i t a b l y o b s c u r e t h e i n f o r m a t i o n t h a t c a n be o b t a i n e d from a s u r f a c e . Assuming an i d e a l s u r f a c e , Comsa[64] showed t h a t t h e t o t a l a n g u l a r b r o a d e n i n g by a LEED i n s t r u m e n t c o r r e l a t e s t o a l e n g t h on t h e s u r f a c e , t h e t r a n s f e r w i d t h , w h i c h i s o a r o u n d 100 A. The t r a n s f e r w i d t h c o r r e s p o n d s r o u g h l y t o t h e l a r g e s t p e r i o d i c i t y o f g r a t i n g w h i c h c a n be r e s o l v e d s t r a i g h t f o r w a r d l y . However, L a g a l l y [ 6 5 ] s u g g e s t e d t h a t t h e s i z e o f domains o r i s l a n d s w h i c h c a n be s t u d i e d by s p o t a n g u l a r p r o f i l e a n a l y s i s c a n be l a r g e r t h a n t h e t r a n s f e r w i d t h , and i s d e p e n d e n t on e x p e r i m e n t a l s e t u p a s w e l l as t h e t h e t y p e o f i s l a n d s u nder i n v e s t i g a t i o n . T h e s e f a c t o r s a r e t r a n s l a t e d i n t o a f u n c t i o n c a l l e d t h e r e s o l v i n g power of LEED i n s t r u m e n t w h i c h i s a measure o f t h e s i z e o f t h e domain o t h a t c a n be r e s o l v e d . The l a t t e r r a n g e s from 100 t o 500 A. As f a r a s 1(E) c u r v e s a r e c o n c e r n e d , t h e l a r g e c r o s s s e c t i o n a l a r e a o f t h e i n c i d e n t beam means t h a t i n e v i t a b l y many domains on t h e s u r f a c e a r e s a m p l e d . The o b s e r v e d 1(E) c u r v e s t h u s c o n t a i n c o n t r i b u t i o n s f r o m t h e s e d o m a i n s . The r e s u l t i n g LEED p a t t e r n (and 1 ( E ) c u r v e s ) f r o m t h e r o t a t i o n a l d o m ains shown i n F i g u r e 2.9 w i l l t h e r e f o r e i n d i c a t e a h i g h e r symmetry t h a n from any one o f t h e domains a l o n e . I n LEED c r y s t a l l o g r a p h i c s t u d i e s f o r s u c h s i t u a t i o n s , 1(E) c u r v e s a r e c a l c u l a t e d f o r i n d i v i d u a l domains and a p p r o p r i a t e beam a v e r a g e s a r e t a k e n b e f o r e c o m p a r i s o n s a r e made w i t h t h e e x p e r i m e n t a l c u r v e s . 55 2.4 AUGER ELECTRON SPECTROSCOPY (AES) 2.4.1 THE AUGER PROCESS When a c o r e e l e c t r o n i s e j e c t e d f r o m an atom, t h e r e s u l t i n g i o n i s i n a h i g h l y e x c i t e d s t a t e . A number o f p o s s i b l e p r o c e s s e s may c o n t r i b u t e t o r e d u c i n g t h e e n e r g y o f t h e e x c i t e d s t a t e . One of t h e s e p r o c e s s e s i s t h e e m i s s i o n o f a s e c o n d a r y e l e c t r o n , t h e p r o c e s s b e i n g known as t h e Auger e f f e c t . A u g e r e m i s s i o n i s c a u s e d when an e l e c t r o n d r o p s f r o m a h i g h e r e n e r g y l e v e l i n t o t h e c o r e v a c a n c y , w i t h t h e e x c e s s e n e r g y b e i n g t r a n s f e r r e d t o an e j e c t e d s e c o n d , o r Auger (named a f t e r P. A u g e r [ 2 6 ] ) e l e c t r o n . An e n e r g y r e p r e s e n t a t i o n o f t h i s p r o c e s s i s g i v e n i n F i g u r e 2.10. I t has t o be e m p h a s i z e d t h a t s t e p s (b) and ( c ) a r e s e p a r a t e d c o n c e p t u a l l y o n l y f o r t h e sake o f c l a r i t y . In t h e a c t u a l p r o c e s s t h e f i l l i n g o f t h e c o r e h o l e and t h e e j e c t i o n o f t h e Auger e l e c t r o n o c c u r s i m u l t a n e o u s l y ; t h e p r o c e s s a r i s e s f r o m a C o u l o m b i c r e a r r a n g e m e n t o f t h e s e two e l e c t r o n s . A l t h o u g h F i g u r e 2.10 shows t h e two e l e c t r o n s c o n c e r n e d i n t h e Auger p r o c e s s a s o r i g i n a t i n g i n two d i f f e r e n t a t o m i c l e v e l s , t h i s d i s t r i b u t i o n i s by no means e s s e n t i a l . O t h e r i m p o r t a n t p o s s i b i l i t i e s a r e ( i ) t h a t b o t h e l e c t r o n s come fr o m t h e same l e v e l and ( i i ) t h a t one or b o t h o f them come from t h e v a l e n c e l e v e l s . A s y s t e m of n o m e n c l a t u r e i s c l e a r l y n e e d e d t o i d e n t i f y t h e d i f f e r e n t p o s s i b i l i t i e s . The c o n v e n t i o n a c c e p t e d by most s u r f a c e s c i e n t i s t s i s t h a t e l e c t r o n s o r i g i n a t i n g i n t h e ( 1 s ) s h e l l a r e l a b e l l e d K, t h e 56 PRIMARY er (NOT ANALYZED) VACUUM LEVEL ' or e" T 1 1 1 t 1 l i (a) CREATION OF •VALENCE flL BAND •M L K CORE ELECTRON HOLE AUGER • (ENERGY e" ! ANALYZED) -f—i M •f—? L K (b) FILLING OF HOLE AND RELEASE OF ENERGY 4+ -M L K (c) EJECTION OF AUGER ELECTRON FROM ENERGY RELEASED IN (b) Figure 2.10: Schematic representation of Auger process. K, L and M represent atomic energy l e v e l s , and the arrows represent electrons. atomic number Figure 2 .11: P r o b a b i l i t y of Auger emission and of X-ray fluorescence as a function of atomic number for a K-shell core hole. 57 ( 2 s ) a r e L 1 , t h e (2p) a r e and L3 and so on. V a l e n c e s h e l l e l e c t r o n s a r e c a l l e d V. To i d e n t i f y a t r a n s i t i o n t h e f o l l o w i n g s e quence of l e t t e r s i s u s e d : 1. t h e c o r e h o l e ; 2. t h e h o l e g e n e r a t e d by an e l e c t r o n d r o p p i n g i n t o t h e c o r e h o l e ; 3. t h e h o l e g e n e r a t e d by t h e e j e c t e d e l e c t r o n . Thus i n F i g u r e 2.10 t h e Auger e l e c t r o n w o u l d be d e s i g n a t e d a s (KLM). A r e l a x a t i o n p r o c e s s w h i c h competes w i t h A u g e r e m i s s i o n i s X - r a y f l u o r e s c e n c e . I n t h i s t h e e n e r g y c hange a s s o c i a t e d w i t h t h e d e s c e n t o f t h e f i r s t e l e c t r o n i n t o t h e c o r e h o l e i s r e l e a s e d a s a p h o t o n o f a p p r o p r i a t e e n e r g y . F o r a K - s h e l l c o r e h o l e , t h e l a t t e r p r o c e s s becomes more p r o b a b l e as t h e a t o m i c number i n c r e a s e s , a s shown i n F i g u r e 2.11. I n g e n e r a l , f o r e l e m e n t s w i t h a t o m i c numbers below 30, Auger e m i s s i o n i s more p r o b a b l e t h a n X - r a y f l u o r e s c e n c e r e g a r d l e s s o f e x c i t a t i o n e n e r g y . I n h e a v i e r e l e m e n t s , Auger e m i s s i o n g e n e r a l l y s t i l l d o m i n a t e s when t h e i n i t i a l e x c i t a t i o n e n e r g y i s l e s s t h a n a b o u t 2 keV. 2.4.2 KINETIC ENERGIES OF AUGER ELECTRONS A c c o r d i n g t o F i g u r e 2.10, t h e k i n e t i c e n e r g y of t h e e j e c t e d e l e c t r o n i s g i v e n by E A " E K " E L " E M ( 2 ' 2 6 ) 58 I t c a n be seen t h a t t h e k i n e t i c e n e r g y o f t h e Auger e l e c t r o n i s c h a r a c t e r i s t i c of t h e e n e r g y l e v e l s o f t h e atom and i n d e p e n d e n t o f t h e e n e r g y of t h e e x c i t i n g r a d i a t i o n . As a c o n s e q u e n c e , i t i s n o t n e c e s s a r y t o m o n o c h r o m a t i z e t h e r a d i a t i o n s o u r c e : t h i s i s i n f a c t an e x p e r i m e n t a l c o n v e n i e n c e w h i c h makes t h e e l e c t r o n - e x c i t e d AES more p o p u l a r (and more e c o n o m i c a l ) t h a n X - r a y - e x c i t e d AES. E q u a t i o n (2.26) o n l y g i v e s a c r u d e a p p r o x i m a t i o n t o t h e k i n e t i c e n e r g y o f an Auger e l e c t r o n b e c a u s e i t d o e s n o t t a k e i n t o a c c o u n t t h e d i f f e r e n t d e g r e e s o f i o n i z a t i o n o f t h e atom. E x t r a e n e r g y i s r e q u i r e d t o remove t h e s e c o n d e l e c t r o n f r o m a p o s i t i v e l y c h a r g e d i o n , and Chung and J e n k i n s [ 6 6 ] p r o p o s e d t h e e x p r e s s i o n E A ( Z ) = E R ( Z ) - ^ [ E L ( Z ) + E L ( Z + 1 ) ] - 2 [ E M ( Z ) + E M ( Z + 1 ) ] (2.27) w h i c h d epends on t h e a t o m i c number Z f o r t h e atom o f i n t e r e s t . The k i n e t i c e n e r g i e s c a l c u l a t e d u s i n g t h i s f o r m u l a a r e a c c u r a t e t o a b o u t 5 eV. More d e t a i l e d c o r r e c t i o n s a r e p o s s i b l e , b ut f o r our p u r p o s e s i t i s s u f f i c i e n t t o e s t a b l i s h t h a t t o w i t h i n t h i s s m a l l u n c e r t a i n t y m e asured Auger e n e r g y v a l u e s a r e c h a r a c t e r i s t i c o f p a r t i c u l a r atoms. 2.4.3 AES AND SURFACE ANALYSES When t h e e x c i t a t i o n s o u r c e i s an e l e c t r o n beam, t h e Auger e l e c t r o n s from s u r f a c e atoms a r e o b s c u r e d by l a r g e 59 numbers o f s e c o n d a r y e l e c t r o n s e m i t t e d i n t h e same e n e r g y r a n g e where t h e Auger t r a n s i t i o n s o c c u r [ 6 7 ] . T h i s g e n e r a l b a c k g r o u n d e m i s s i o n , w h i c h i n c r e a s e s s l o w l y w i t h e n e r g y , p o s e s a p r o b l e m f o r q u a n t i t a t i v e d e t e c t i o n o f t h e Auger e l e c t r o n s . The t r a d i t i o n a l p r a c t i c e t o overcome t h i s p r o b l e m i s t o employ t h e e l e c t r o n i c t e c h n i q u e o f p h a s e s e n s i t i v e d e t e c t i o n ( C h a p t e r 4 ) . As a r e s u l t , Auger p e a k s a r e commonly r e c o r d e d a s d N ( E ) / d E v e r s u s E c u r v e s . A t y p i c a l f i r s t d e r i v a t i v e Auger s p e c t r u m i s shown i n F i g u r e 2.12. I t i s c l e a r t h a t i m p u r i t y e l e m e n t s a r e e a s i l y d e t e c t e d by t h i s a p p r o a c h . I n d e e d , i m p u r i t y e l e m e n t s on m e t a l s u r f a c e s c a n o f t e n be d e t e c t e d a t l e v e l s o f t h e o r d e r o f 1% m o n o l a y e r [ 6 8 ] . As n o t e d b e f o r e , t h e e n e r g y o f an Auger t r a n s i t i o n i s m a i n l y d e t e r m i n e d by t h e e n e r g y l e v e l s o f a p a r t i c u l a r e l e m e n t . A l t h o u g h t h e c h e m i c a l e n v i r o n m e n t c a n sometimes c o n t r i b u t e t o e n e r g y s h i f t s o f t h e o r d e r o f a few e V [ 6 9 , 7 0 ] , t h e s e a r e a p p r e c i a b l y l e s s t h a n t h e d i f f e r e n c e s between d i f f e r e n t e l e m e n t s f o r a g i v e n t y p e o f Auger t r a n s i t i o n . T h e r e f o r e AES i s most s u i t a b l e f o r e l e m e n t a l a n a l y s i s of s u r f a c e s . F o r t h i s p u r p o s e , t h e main i n t e r e s t i s i n Auger e l e c t r o n s w i t h e n e r g i e s i n t h e a p p r o x i m a t e r a n g e 20-700 eV w h i c h a r e c h a r a c t e r i z e d by s h o r t mean f r e e p a t h l e n g t h s i n t h e s o l i d s . 2.4.3.1 Q u a l i t a t i v e A n a l y s i s The use o f AES i n q u a l i t a t i v e s u r f a c e a n a l y s i s d e p e nds on t h e a b i l i t y t o a s s i g n p e a k s i n an Auger 60 Figure 2 .12 : D e r i v a t i v e Auger s p e c t r a t a k e n f r o m a Zr(000l) s u r f a c e . ( a ) B e f o r e c l e a n i n g ; and ( b ) A f t e r *50 h o u r s of A r " bombardment. 61 s p e c t r u m t o p a r t i c u l a r e l e m e n t s . A l t h o u g h Auger p e a k s from d i f f e r e n t e l e m e n t s do o v e r l a p o c c a s i o n a l l y , t h e e x i s t e n c e o f more t h a n one c h a r a c t e r i s t i c Auger t r a n s i t i o n f o r most e l e m e n t s u s u a l l y e l i m i n a t e s any a m b i g u i t y i n a s s i g n m e n t . I n r o u t i n e a n a l y s i s , t h i s i s o f t e n a c c o m p l i s h e d by c o m p a r i n g t h e e x p e r i m e n t a l s p e c t r u m w i t h c a t a l o g s o f r e f e r e n c e d a t a [ 7 1 , 7 2 ] . T h i s a s p e c t o f AES a p p l i c a t i o n i s most commonly u s e d w i t h LEED f o r m o n i t o r i n g o f s u r f a c e c l e a n l i n e s s . 2.4.3.2 Q u a n t i t a t i v e A n a l y s i s A n o t h e r i m p o r t a n t a p p l i c a t i o n of AES i s t h e q u a n t i t a t i v e a s s e s s m e n t o f t h e a t o m i c d e n s i t y (N) o f i m p u r i t i e s o r s t r u c t u r e d a d l a y e r s on a s u r f a c e . A t a p r i m a r y beam e n e r g y E 0 , t h e Auger e m i s s i o n c u r r e n t I ( E A , E 0 ) a t a c h a r a c t e r i s t i c Auger t r a n s i t i o n e n e r g y E A n o r m a l i z e d t o p r i m a r y beam c u r r e n t i s a c o n v e n i e n t measure o f N. In a d d i t i o n , I ( E A , E 0 ) i s a f u n c t i o n of s e v e r a l o t h e r p a r a m e t e r s c h a r a c t e r i s t i c o f t h e p a r t i c u l a r s u r f a c e atom and i t s e n v i r o n m e n t . Some of t h e more i m p o r t a n t ones i n c l u d e [ 7 3 ] i o n i z a t i o n c r o s s s e c t i o n , Auger t r a n s i t i o n p r o b a b i l i t y , Auger e l e c t r o n e s c a p e d e p t h , r o u g h n e s s f a c t o r o f t h e s u r f a c e and i n s t r u m e n t a l f a c t o r . T h i s i s by no means an e x h a u s t i v e l i s t , b u t i t s e r v e s t o i n d i c a t e t h e c o m p l e x i t y i n v o l v e d i n c a l c u l a t i n g N d i r e c t l y f r o m measured I ( E A , E 0 ) . A l t h o u g h a t t e m p t s have been made t o c a l c u l a t e some of t h e s e p a r a m e t e r s f r o m f i r s t p r i n c i p l e s [ 7 4 ] , and hence 62 make d i r e c t q u a n t i t a t i v e a n a l y s e s , some of t h e a s s u m p t i o n s made a r e s t i l l d e b a t a b l e [ 7 5 ] . P r e s e n t l y , most o f t h e q u a n t i t a t i v e s t u d i e s o f a t o m i c d e n s i t i e s w i t h A u g e r e l e c t r o n s p e c t r o s c o p y a r e done i n c o n j u n c t i o n w i t h c a l i b r a t i o n methods. The l a t t e r i n c l u d e s t h e use o f r a d i o a c t i v e c o u n t i n g [ 7 6 ] , n u c l e a r m i c r o a n a l y s i s [ 7 7 ] , and measurements o f d e p o s i t i o n c u r r e n t s ! 7 8 ] . In s u c h s t u d i e s , t h e p e a k - t o - p e a k h e i g h t o f an Auger peak i n a d e r i v a t i v e s p e c t r u m i s o f t e n t a k e n a s p r o p o r t i o n a l t o I ( E A , E 0 ) , a l t h o u g h t h i s a s s u m p t i o n i s v a l i d o n l y f o r Auger p e a k s w h i c h e x h i b i t c o n s t a n t shape i n t h e N(E) v e r s u s E s p e c t r u m . I f o n l y t h e a t o m i c f r a c t i o n s , o r c o v e r a g e s , o f v a r i o u s e l e m e n t s on a s u r f a c e a r e o f i n t e r e s t t o an e x p e r i m e n t e r , P a l m b e r g ' s r e l a t i v e a t o m i c s e n s i t i v i t y f a c t o r s [ 7 2 ] (S^) c a n be u s e d . I n t h i s scheme I , t h e p e a k - t o - p e a k h e i g h t n o r m a l i z e d t o p r i m a r y beam c u r r e n t , i s u s e d . The a t o m i c f r a c t i o n o f e l e m e n t i on a s u r f a c e i s d e f i n e d a s 6i = ( V L i S i J / ^ l / L ^ , (2.28) v where t h e s u b s c r i p t i d e n o t e s t h e s p e c i e s under i n v e s t i g a t i o n , t h e summation o v e r v c o v e r s a l l a t o m i c s p e c i e s ( i n c l u d i n g i ) p r e s e n t on t h e s u r f a c e , and L a r e t h e i n s t r u m e n t a l s e n s i t i v i t y f a c t o r s . 63 I n a s s o c i a t i o n w i t h LEED, a s t i l l s i m p l e r a p p r o a c h can sometimes be u s e d t o e s t i m a t e t h e s u r f a c e c o v e r a g e o f a p a r t i c u l a r ad-atom. T h i s o c c u r s when t h e a d s o r b a t e g i v e s r i s e t o s h a r p LEED p a t t e r n ( s ) . The Auger p e a k - t o - p e a k h e i g h t r a t i o o f t h e a d s o r b a t e and t h e s u b s t r a t e ( I a c i / I s u b ^ * s t a k e n when t h e LEED p a t t e r n i s most w e l l - d e f i n e d . I n s o f a r a s t h e a p p e a r a n c e o f a s i m p l e LEED p a t t e r n c o r r e s p o n d s t o a d e f i n i t e c o v e r a g e ( e . g . (1x1) f o r 1 m o n o l a y e r , p ( 2 x 1 ) f o r 1/2 m o n o l a y e r , (i/3x/3) f o r 1/3 m o n o l a y e r ) , t h e r a t i o ^3(3/^5^^ c a n t h u s be u s e d as an i n t e r n a l s t a n d a r d f o r c a l i b r a t i o n o f c o v e r a g e s . The most f a v o r a b l e c o n d i t i o n f o r t h i s method a r i s e s when t h e same a d s o r b a t e g i v e s r i s e t o s e v e r a l o f t h e s e s i m p l e LEED p a t t e r n s . Then measurements o f I a d / / ' I s u b ^ o r e a c ^ c a n g i v e a h e l p f u l c a l i b r a t i o n c u r v e . T h i s a p p r o a c h may become ambiguous when s u p e r l a t t i c e s t r u c t u r e s have s e v e r a l atoms p e r u n i t m e s h [ 7 9 , 8 0 ] , and t h e r e f o r e i t must be u s e d w i t h e x t r e m e c a u t i o n . CHAPTER 3 M U L T I P L E S C A T T E R I N G C A L C U L A T I O N S 64 65 3. 1 INTRODUCTION D e t a i l e d d e s c r i p t i o n s o f m u l t i p l e s c a t t e r i n g c a l c u -l a t i o n s f o r LEED i n c l u d e t h e book by P e n d r y [ l l ] , a s w e l l as s e v e r a l o t h e r r e v i e w s f 5 6 , 8 1 , 8 2 ] . The o b j e c t i v e o f t h i s c h a p t e r i s t h e r e f o r e t o p r o v i d e an o v e r v i e w and add p e r s -p e c t i v e i n r e l a t i o n t o t h e s t u d i e s u n d e r t a k e n i n t h i s work. A m u l t i p l e s c a t t e r i n g c a l c u l a t i o n i s o f t e n r e f e r r e d t o as a ' d y n a m i c a l ' c a l c u l a t i o n b e c a u s e i t c o n s i d e r s n o t o n l y t h e g eometry o f t h e c r y s t a l l a t t i c e and t h e ad-atoms, but a l s o t h e i n t e r a c t i o n o f t h e LEED e l e c t r o n w i t h t h e v i b r a t i n g i o n c o r e s o f t h e l a t t i c e , a s w e l l a s t h e m u l t i p l e s c a t t e r i n g e v e n t s between t h e s e i o n c o r e s . Thus t h e p o t e n t i a l s w i t h i n and between t h e s e i o n c o r e s p l a y i m p o r t a n t r o l e s i n a ' d y n a m i c a l ' c a l c u l a t i o n . The p o t e n t i a l o f t h e s o l i d i s u s u a l l y a p p r o x i m a t e d by t h e ' m u f f i n - t i n ' m o d e l , where t h e p o t e n t i a l i s assumed c o n s t a n t between i o n c o r e s but s p h e r i c a l l y symmetric w i t h i n a t o m i c r e g i o n s . The p o t e n t i a l s of t h e l a t t e r a r e d e s c r i b e d by a p p r o p r i a t e s e t s of p h a s e s h i f t s 6 j . D e t a i l s of t h e ' m u f f i n - t i n ' model a r e d i s c u s s e d i n S e c t i o n 3.2. W i t h t h e known s c a t t e r i n g p r o p e r t i e s o f i n d i v i d u a l i o n c o r e s , t h e s c a t t e r i n g by i n d i v i d u a l p l a n e s o f atoms ( i o n c o r e s ) c a n be c o n s t r u c t e d i n angular-momentum, o r L-, s p a c e by u t i l i z i n g t h e s p h e r i c a l symmetry o f t h e i o n c o r e p o t e n t i a l . To b u i l d up t h e s u r f a c e r e g i o n o f a s o l i d , i s o l a t e d p l a n e s have t o be a s s e m b l e d and t h e i n t e r p l a n a r s c a t t e r i n g c o n s i d e r e d . T h i s s t e p c a n be done e x a c t l y or 66 p e r t u r b a t i o n a l l y . E x a c t methods s u c h a s t h e T - m a t r i x method!81 ,83] and t h e B l o c h - w a v e method[84] c a l c u l a t e i n t e r p l a n a r s c a t t e r i n g t o i n f i n i t e o r d e r i n t h e L - s p a c e and r e c i p r o c a l ( o r K-) s p a c e r e p r e s e n t a t i o n s r e s p e c t i v e l y . F o r t h e c a s e t h a t t h e i n d i v i d u a l p l a n e s c o n t a i n j u s t one atom p e r u n i t mesh, t h e T - m a t r i x method i n v o l v e s t h e i n v e r s i o n of a s q u a r e m a t r i x o f d i m e n s i o n N ^ m a x + 1 ) 2 f where N i s t h e number o f a t o m i c p l a n e s r e q u i r e d t o r e p r e s e n t t h e s o l i d and ( Z m a x + 1 ) i s t h e number o f p h a s e s h i f t s r e q u i r e d t o d e s c r i b e t h e i o n c o r e s c a t t e r i n g . The B l o c h - w a v e method, on t h e o t h e r hand, has t o s o l v e an e i g e n v a l u e p r o b l e m w i t h m a t r i x d i m e n s i o n 2n, where n i s t h e number o f p l a n e waves r e q u i r e d t o r e p r e s e n t t h e w a v e f i e l d between t h e a t o m i c p l a n e s . T h i s number i n c r e a s e s r a p i d l y w i t h d e c r e a s i n g i n t e r p l a n a r d i s t a n c e . The e x a c t methods a r e v e r y demanding on b o t h c o m p u t i n g t i m e and s t o r a g e , and i n p r a c t i c e t h e i r use i s l i m i t e d t o r e s e a r c h g r o u p s w i t h ( e s s e n t i a l l y ) u n l i m i t e d c o m p u t i n g b u d g e t s . To r e d u c e t h e c o m p u t i n g r e q u i r e m e n t s t o manageable l e v e l s , a p e r t u r b a t i o n a l scheme c a l l e d t h e 'combined s p a c e ' m ethod[85] was e m p l o y e d i n t h i s work. In t h i s a p p r o a c h , O c l o s e l y - s p a c e d a t o m i c p l a n e s ( e . g . ^0.5 A) a r e g r o u p e d t o g e t h e r t o f o r m a c o m p o s i t e l a y e r i n L - s p a c e ( s i m i l a r t o t h e T - m a t r i x method) w h i l e more w i d e l y - s p a c e d p l a n e s ( e . g . O £0.5 A) a r e c o n s i d e r e d a s s i m p l e ( o r B r a v a i s l a t t i c e ) l a y e r s . The manner i n w h i c h e a c h l a y e r s c a t t e r s p l a n e waves i s d e s c r i b e d by a d i f f r a c t i o n m a t r i x M " ( S e c t i o n 3 . 3 ) . The 67 i n t e r l a y e r s c a t t e r i n g i s t h e n t r e a t e d p e r t u r b a t i o n a l l y i n K - s p a c e ( S e c t i o n 3 . 4 ) . The p r e s e n c e o f i n e l a s t i c s c a t t e r i n g e n s u r e s t h a t a f i n i t e s t a c k o f l a y e r s w i l l be s u f f i c i e n t t o r e p r e s e n t t h e s u r f a c e s c a t t e r i n g . The a d v a n t a g e of t h e 'combined s p a c e ' method i s t h a t i t r e d u c e s N t o s i m p l y t h e number o f s u b p l a n e s i n a c o m p o s i t e l a y e r w h i l e k e e p i n g t h e number of p l a n e waves f o r t h e w a v e f i e l d e x p a n s i o n between l a y e r s t o a manageable l e v e l ( s i n c e h e r e t h e i n t e r l a y e r s e p a r a t i o n i s l a r g e r ) . In g e n e r a l , i t i s m a t h e m a t i c a l l y e a s i e r t o h a n d l e p l a n e waves ( i n K - s p a c e ) t h a n s p h e r i c a l waves ( i n L - s p a c e ) , a n d symmetry c a n be u s e d ( S e c t i o n 3.5) t o r e d u c e t h e number o f p l a n e waves i n p r a c t i c e . T r a n s f o r m a t i o n between t h e wave t y p e s h a s been d i s c u s s e d i n d e t a i l by M a r c u s [ 8 6 ] , The f i n a l s t e p o f a m u l t i p l e s c a t t e r i n g c a l c u l a t i o n i s t o compute e l e c t r o n beam r e f l e c t i v i t i e s f r o m t h e s t a c k e d l a y e r s . The r e s u l t may be w r i t t e n a s R 2 = ( V 7 ^xHCgJ 2 <3.1> where k~ and kt r e p r e s e n t t h e p e r p e n d i c u l a r components of t h e d i f f r a c t e d beam an d t h e i n c i d e n t beam r e s p e c t i v e l y and a r e c o e f f i c i e n t s i n t h e e x p a n s i o n o f t h e t o t a l w a v e f i e l d o u t s i d e t h e c r y s t a l , namely, o u t ( r ) = e x p ( i k o - r ) + ^ T c ^ e x p f i k " • r ) 3 (3.2) 68 w h e r e s u p e r s c r i p t s d e n o t e w a v e s p r o p a g a t i n g i n t h e + x / - x d i r e c t i o n s i n a l e f t - h a n d e d C a r t e s i a n c o - o r d i n a t e s y s t e m ( F i g u r e 3 . 1 ) u s e d i n t h e m u l t i p l e s c a t t e r i n g p r o g r a m s . C a l c u l a t e d b e a m r e f l e c t i v i t i e s c o r r e s p o n d t o m e a s u r e d b e a m i n t e n s i t i e s d e f i n e d i n S e c t i o n 4 . 4 . 3 . 2 P H Y S I C A L P A R A M E T E R S I N L E E D C A L C U L A T I O N S B e s i d e s a s t r u c t u r a l m o d e l , v a r i o u s n o n - g e o m e t r i c a l p a r a m e t e r s a r e r e q u i r e d f o r t h e c a l c u l a t i o n o f 1 ( E ) c u r v e s . N o n - g e o m e t r i c a l p a r a m e t e r s i n c l u d e t h e c r y s t a l p o t e n t i a l a n d t h e v i b r a t i o n p r o p e r t i e s o f t h e c r y s t a l l a t t i c e . T h e m o s t i m p o r t a n t p a r t o f t h e c r y s t a l p o t e n t i a l i s t h e i o n c o r e p o t e n t i a l w h i c h a c c o u n t s f o r t h e m u l t i p l e s c a t t e r i n g e x p e r i e n c e d b y t h e L E E D e l e c t r o n s i n a s o l i d . T h e s c a t t e r i n g c a u s e d b y t h e v a l e n c e - e l e c t r o n r e g i o n s o f t h e s o l i d a r e r e s p o n s i b l e f o r t h e s l i g h t s h i f t s o f B r a g g p e a k s a n d s t r o n g e l e c t r o n d a m p i n g f o u n d i n L E E D . T h e i n e l a s t i c s c a t t e r i n g o f c o u r s e m a k e s L E E D s u r f a c e s e n s i t i v e ; t h i s d a m p i n g r e s u l t s i n t h e b r o a d e n i n g o f p e a k s i n 1 ( E ) c u r v e s . T e m p e r a t u r e a l s o h a s a n e f f e c t o n b e a m i n t e n s i t i e s . T h e l a t t e r d e c r e a s e s a s t e m p e r a t u r e i n c r e a s e s . S u i t a b l e m o d e l s m u s t t h e r e f o r e t r e a t t h e s o l i d a s a v i b r a t i n g l a t t i c e . 3 . 2 . 1 T H E ' M U F F I N - T I N ' A P P R O X I M A T I O N T h e ' m u f f i n - t i n ' m o d e l i s w e l l e s t a b l i s h e d f o r c a l c u l a t i n g b a n d s t r u c t u r e s ! 8 7 ] . I t c o n s i d e r s t h e s o l i d a s a r e g u l a r a r r a y o f h a r d s p h e r e s , e a c h c e n t e r e d o n a n a t o m a n d 69 F i g u r e 3 . 1 : L e f t - h a n d e d c o - o r d i n a t e s y s t e m u s e d i n t h e ' c o m b i n e d s p a c e ' m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . H o l l o w a n d s o l i d c i r c l e s r e p r e s e n t s u b s t r a t e a n d a d s o r b a t e a t o m s r e s p e c t i v e l y . B r a v a i s l a t t i c e l a y e r s a r e s e p a r a t e d b y a t l e a s t 0 . 5 A . 70 separated from one another by an i n t e r s p h e r e region of constant p o t e n t i a l V 0 ( t h i s i s sometimes r e f e r r e d to as the ' m u f f i n - t i n z e r o ' ) . Each sphere can be viewed as an ion core w i t h i n which the p o t e n t i a l V i s s p h e r i c a l l y symmetric. F i g u r e 3.2 r e p r e s e n t s a 1-dimensional p i c t u r e of such a model, with a row of s c a t t e r e r s along the x - a x i s . Here each s c a t t e r e r i s e q u i v a l e n t t o a plane of ion cores i n 3-dimensions. To minimize the volume approximated by the constant p o t e n t i a l , the r a d i i of the spheres are chosen so that they j u s t touch one another. The s c a t t e r i n g by any d i s c o n t i n u i t i e s between V 0 and V*s at the sphere boundaries must be ignored when comparing c a l c u l a t e d 1(E) curves with those from experiment. The p o t e n t i a l runs c o n t i n u o u s l y through the s u r f a c e from vacuum to the s o l i d ' s i n t e r i o r . S c a t t e r i n g by the s u r f a c e b a r r i e r may occur, although t h i s s c a t t e r i n g i s not p a r t i c u l a r l y important at the e n e r g i e s used i n LEED c r y s t a l l o g r a p h y t . m p r a c t i c e , s u r f a c e b a r r i e r e f f e c t s are u s u a l l y ignored i n LEED c r y s t a l l o g r a p h y , and V 0 i s set to zero at a d i s t a n c e r Q from the surface-vacuum i n t e r f a c e on the vacuum s i d e . In the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s , r Q i s u s u a l l y chosen as the hard-sphere r a d i u s of the atomic s p e c i e s at the topmost s u r f a c e r e g i o n ( F i g u r e 3.2). T T h i s s u r f a c e b a r r i e r becomes important only at shallow i n c i d e n t angles and/or low i n c i d e n t e n e r g i e s (^40 eV) where t o t a l r e f l e c t i o n s are p o s s i b l e . 71 surface barrier ion cores +X Figure 3.2: 'Muffin-tin' approximation for the potentials of a single row of ion cores along the x-axis. Energy t Vacuum l e v e l Work function Energy Ej Fermi energy 0 Lowest l e v e l of conduction band Figure 3.3: Approximation for V 0 r with known quantities of Fermi energy E£ and work function <t> of metals. 72 3.2.2 THE CONSTANT POTENTIAL V 0 The main e f f e c t of t h e r e a l p a r t of V 0 i s t o model t h e s p e e d i n g up t h a t an e l e c t r o n e x p e r i e n c e s on e n t e r i n g t h e c r y s t a l . However, V 0 a l s o r e q u i r e s an i m a g i n a r y p a r t , V 0 /. , t o accommodate t h e i n e l a s t i c s c a t t e r i n g e x p e r i e n c e d by t h e LEED e l e c t r o n . V 0 can t h u s be w r i t t e n as V 0 = V 0 j. + i V 0 / , (3.3) where t h e component V 0 r c o r r e s p o n d s t o t h e ' i n n e r p o t e n t i a l ' , and V0i i s d e f i n e d a s a n e g a t i v e number. 3.2.2.1 The R e a l P o t e n t i a l V 0 r T y p i c a l l y V 0 r i s between -5 and -15 eV. T h e o r e t i c a l l y i t i s d e p e n d e n t on e n e r g y , due t o ex c h a n g e and c o r r e l a t i o n e f f e c t s [ 8 7 ] , b u t f o r LEED c r y s t a l l o g r a p h i c s t u d i e s , i t c a n be g e n e r a l l y t r e a t e d a s e n e r g y - i n d e p e n d e n t t 1 1 ] . I n c r e a s i n g j V 0 r | s h i f t s t h e B r a g g p e a k s t o l o w e r e n e r g i e s i n c a l c u l a t e d 1 ( E ) c u r v e s (where E c o r r e s p o n d s t o e n e r g y i n vacuum). A f i r s t a p p r o x i m a t i o n v a l u e f o r V 0 r i s o b t a i n e d by t h e sum o f t h e F e r m i e n e r g y and t h e work f u n c t i o n <j>. F i g u r e 3.3 s c h e m a t i c a l l y shows how V 0 r , <j> and E j a r e r e l a t e d i n m e t a l s . In t h e LEED s t u d i e s on z i r c o n i u m , an i n i t i a l v a l u e o f -10 eV was us e d f o r V 0 r . However, d u r i n g c o m p a r i s o n w i t h e x p e r i m e n t a l 1 ( E ) c u r v e s , t h e c a l c u l a t e d 1 ( E ) c u r v e s a r e g i v e n a r i g i d s h i f t t o p r o v i d e t h e b e s t agreement w i t h t h e f o r m e r . T h i s e n a b l e s 73 a r e f i n e d v a l u e o f V 0 r t o be d e t e r m i n e d . 3.2.2.2 The I m a g i n a r y P o t e n t i a l V 0 / I n e l a s t i c s c a t t e r i n g i s i n t r o d u c e d i n t o m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s by t h e p h e n o m e n o l o g i c a l a p p r o a c h of g i v i n g an i m a g i n a r y component t o V 0 . S l a t e r [ 8 8 ] n o t e d f o r an e l e c t r o n i n a r e g i o n o f c o n s t a n t p o t e n t i a l w i t h r e a l and i m a g i n a r y components a s i n e q u a t i o n (3.3) t h a t i n t e n s i t y d e c a y s w i t h t i m e as e x p ( - 2 V 0 / - t / f i ) . The l i f e t i m e , T , o f an e l e c t r o n t h e n r e l a t e s t o V 0 / a c c o r d i n g t o T = n / 2 V 0 / . (3.4) An i n c r e a s e i n V 0 /- c o r r e s p o n d s t o a d e c r e a s e i n l i f e t i m e and hence t o an i n c r e a s e i n t h e i n e l a s t i c s c a t t e r i n g . In g e n e r a l , t h e s h o r t e r t h e l i f e t i m e t h e g r e a t e r t h e u n c e r t a i n t y i n e n e r g y . U s i n g t h e H e i s e n b e r g u n c e r t a i n t y p r i n c i p l e , P e n d r y [ l l ] d e r i v e d t h e r e l a t i o n s h i p between t h e peak w i d t h A E W i n an 1 ( E ) c u r v e and V 0 /. a s A E W > 2 V 0 / . (3.5) T h e r e f o r e , i n p r a c t i c e , one examines t h e peak w i d t h s o f a s e r i e s o f B r a g g p e a k s i n an 1 ( E ) c u r v e and c o r r e l a t e s t h e i n c i d e n t e n e r g y t o V 0 /. , u s i n g t h e e q u a l s s i g n i n e q u a t i o n ( 3 . 5 ) . An e n e r g y - d e p e n d e n t V 0 /- has o f t e n been u s e d , f o r example w i t h t h e f u n c t i o n a l f o r m [ 8 9 ] 74 V 0 / = - B E 1 / 3 , (3.6) where E i s t h e i n c i d e n t e n e r g y and B i s a p a r a m e t e r t o be c h o s e n f o r e a c h m a t e r i a l . A l t h o u g h e q u a t i o n (3.6) was u s e d q u i t e s u c c e s s f u l l y i n a LEED c r y s t a l l o g r a p h i c a n a l y s i s f o r c l e a n Z r ( O O O l ) , i t was f o u n d i n t h e p r e s e n t work t h a t a c o n s t a n t v a l u e o f -5 eV f o r V 0 /. was e q u a l l y f a v o r a b l e . I n d e e d use of t h e c o n s t a n t v a l u e e l i m i n a t e d some c o n v e r g e n c e p r o b l e m s w h i c h o c c u r r e d a t l o w e r e n e r g i e s when e q u a t i o n (3.6) was us e d i n t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . 3.2.3 ION CORE SCATTERING 3.2.3.1 The Ion C o r e P o t e n t i a l V s The s p h e r i c a l l y s y m m e t r i c a l p a r t s o f t h e ' m u f f i n - t i n ' p o t e n t i a l , V,,, c o r r e s p o n d t o a t o m i c p o t e n t i a l s i n a s o l i d a nd t h e s e c o n t r i b u t i o n s a r e p r i m a r i l y r e s p o n s i b l e f o r t h e back s c a t t e r i n g i n LEED. I d e a l l y V g s h o u l d be o b t a i n e d from a s e l f - c o n s i s t e n t e n e r g y band c a l c u l a t i o n f o r t h e s o l i d s t r u c t u r e o f i n t e r e s t [ 9 0 ] . However, s u c h c a l c u l a t i o n s a r e not g e n e r a l l y a v a i l a b l e f o r a d s o r p t i o n s y s t e m s . A l s o a d e t a i l e d band s t r u c t u r e c a l c u l a t i o n r e q u i r e s knowledge of t h e s u r f a c e g e o m e t r y , w h i c h of c o u r s e i s t h e o b j e c t i v e o f a LEED c r y s t a l l o g r a p h i c s t u d y . F o r f i r s t a n a l y s e s , t h e n , s i m p l e r a p p r o a c h e s a r e r e q u i r e d t o 75 e s t i m a t e V , and t h i s may be done from c h a r g e d i s t r i b u t i o n s . I n t h e s i m p l e s t c a s e t h e s e a r e e s t i m a t e d by a d i r e c t summation o f i n d i v i d u a l a t o m i c c h a r g e d e n s i t i e s , a l t h o u g h a p r e f e r a b l e a p p r o a c h would i n v o l v e some s o r t o f m o l e c u l a r o r b i t a l i n v e s t i g a t i o n . E i t h e r way an a p p r o p r i a t e s t r u c t u r e has t o be assumed, and t h i s may be i n t h e f o r m of a c l u s t e r [ 9 l ] o r a d i p e r i o d i c s l a b [ 9 2 , 9 3 ] t o mimic t h e s u r f a c e s y s t e m . In g e n e r a l , t h e s p h e r i c a l l y s y m m e t r i z e d a t o m i c p o t e n t i a l s e p a r a t e s i n t o c o u l o m b i c and exchange c o n t r i b u t i o n s s u c h t h a t V s ( r ) = V c ( r ) + V e x ( r ) . (3.7) The C o u l o m b i c t e r m V c ( r ) c a n be o b t a i n e d by i n t e g r a t i n g t h e P o i s s o n e q u a t i o n w i t h t h e a p p r o p r i a t e i o n c o r e c h a r g e d e n s i t y d i s t r i b u t i o n s [ 9 4 ] . The exc h a n g e c o r r e l a t i o n t e r m v e x ^ r ^ c a n ^ e c a l c u l a t e d a c c o r d i n g t o [ 9 5 ] V e x ( r ) = - 3 a e 2 [ 3 a s u p ( r ) / 3 2 7 r 2 r 2 ] l / 2 , (3.8) where a S U p ( r ) i s fche c h a r g e d e n s i t y and a depends on t h e exch a n g e a p p r o x i m a t i o n u s e d . Many i n v e s t i g a t i o n s i n d i c a t e t h a t b u l k a t o m i c p o t e n t i a l s c a n be a d e q u a t e f o r d e s c r i b i n g s u r f a c e and n e a r s u r f a c e atoms i n c u r r e n t LEED c r y s t a l l o g r a p h i c a n a l y s e s . Band s t r u c t u r e a t o m i c p o t e n t i a l s f o r many 76 m e t a l s have been t a b u l a t e d by M o r u z z i et al .[96], and a v a r i e t y o f a t o m i c p o t e n t i a l s f o r ad-atoms a r e a v a i l a b l e , f o r example t h e computer c o m p i l a t i o n made by Van H o v e [ 9 7 ] . I d e a l l y , i t i s p r e f e r a b l e t o t e s t s e v e r a l t y p e s of V , and a s s e s s t h e i r m e r i t s by c o m p a r i n g c a l c u l a t e d 1 ( E ) c u r v e s w i t h t h o s e from e x p e r i m e n t . I t i s i n t h i s s p i r i t t h a t t h e c r y s t a l l a t t i c e method[92,93] was t r i e d f o r t h e oxygen a d s o r p t i o n on Zr(OOOV). F i g u r e 3.4 shows t h e c r y s t a l l a t t i c e o f z i r c o n i u m o x i d e (ZrO) us e d i n t h i s p a r t i c u l a r method t o o b t a i n V g f o r oxygen w i t h v a r y i n g n e g a t i v e c h a r g e s . 3.2.3.2 The Phase S h i f t s 8^  G i v e n t h e a t o m i c p o t e n t i a l s , t h e wave f u n c t i o n s f o r an e l e c t r o n i n s i d e a s p h e r e a r e o b t a i n e d by s o l v i n g t h e S c h r o d i n g e r e q u a t i o n [ - ( h 2/2m)V 2 + V s ] ¥ = E¥, (3.9) where E i s t h e e n e r g y w i t h r e s p e c t t o t h e ' m u f f i n - t i n z e r o ' . The a s y m p t o t i c s o l u t i o n t o e q u a t i o n (3.9) has t h e form • k ( r , 0 s ) = e x p ( / k r c o s 0 s ) + t ( k , 0 g ) e x p ( / k r ) / r , (3.10) where t h e t e r m e x p ( i k r ) / r i s a s p h e r i c a l wave a t a d i s t a n c e r from t h e c e n t e r of t h e i o n c o r e and t ( k , 0 s ) i s t h e i o n c o r e s c a t t e r i n g f a c t o r w h i c h c a n be expanded 77 (b) F i g u r e 3.4: U n i t c e l l o f ZrO c r y s t a l l a t t i c e s t r u c t u r e . (a) U n i t c e l l d i m e n s i o n s and r e s p e c t i v e h a r d s p h e r e r a d i i f o r Zr° and 0 ° u s e d i n t h e c a l c u l a t i o n s o f V g f o r b o t h e l e m e n t s . (b) A blown-up t o show t h e l o c a l e n v i r o n m e n t o f oxygen i n t h e l a t t i c e . 78 a s [ 9 8 ] t ( k , 0 s ) = 4TT V (2/+1 ) t , ( k ) P z ( c o s t 9 s ) . (3.11) In e q u a t i o n ( 3 . 1 1 ) , i s t h e L e g e n d r e p o l y n o m i a l a s s o c i a t e d w i t h t h e a n g u l a r momentum quantum number / and t h e t ^ ( k ) a r e d e f i n e d a s t z ( k ) = ( l / 2 k ) e x p ( / 6 [ ) s i n 6 / . (3.12) The q u a n t i t i e s 6^  a p p e a r i n g i n e q u a t i o n (3.12) a r e known as a t o m i c phase s h i f t s ; t h e y d e t e r m i n e t h e t o t a l e l a s t i c s c a t t e r i n g c r o s s s e c t i o n t h r o u g h a e l = U u / k2 ) (2/ + 1 ) s i n 2 6 / . (3.13) /=0 Phase s h i f t s f o r a p a r t i c u l a r i o n c o r e p o t e n t i a l V s a r e f o u n d by s o l v i n g e q u a t i o n (3.9) i n s i d e t h e a t o m i c s p h e r e and, f o r e a c h v a l u e o f /, j o i n i n g s m o o t h l y t h e a s y m p t o t i c form o f t h e s o l u t i o n a t t h e b o u n d a r y o f t h e s p h e r e t o t h e c o r r e s p o n d i n g s o l u t i o n f o r t h e o u t s i d e r e g i o n [ l l ] . From e q u a t i o n ( 3 . 9 ) , i t c a n be u n d e r s t o o d t h a t p h a s e s h i f t s a r e d e p e n d e n t on b o t h V s and t h e e n e r g y o f t h e i n c i d e n t e l e c t r o n . A l t h o u g h , a s s u g g e s t e d by e q u a t i o n ( 3 . 1 1 ) , t h e e x p a n s i o n f o r t h e i o n c o r e s c a t t e r i n g f a c t o r r e q u i r e s an i n f i n i t e number o f phase s h i f t s , i n p r a c t i c e t h e e x p a n s i o n s can be t r u n c a t e d t o a 79 f i n i t e s e t . F o r example, f o r e n e r g i e s l e s s t h a n 250 eV, t h e use o f a maximum o f e i g h t p h a s e s h i f t s ( i . e . ' m a x = 7 ) g e n e r a l l y p r o v i d e s a v e r y a d e q u a t e a p p r o x i m a t i o n i n o r d e r t o r e d u c e t h e c o m p u t i n g e f f o r t i n o b t a i n i n g t h e l a y e r d i f f r a c t i o n m a t r i c e s M " . I n d e e d , f o r weak s c a t t e r e r s , a s m a l l e r number o f phase s h i f t s may be a c c e p t a b l e . F i g u r e 3.5 compares t h e p h a s e s h i f t s of oxygen o b t a i n e d f r o m d i f f e r e n t V g w h i l e F i g u r e 3.6 does the,same f o r Z r . The computer p r o g r a m u s e d c a l c u l a t e s t h e p h a s e s h i f t s a s modulus o f it, so d i s c o n t i n u i t i e s may a p p e a r t o be p r e s e n t i n a pha s e s h i f t v e r s u s e n e r g y p l o t . T h i s i s n o t a p r o b l e m f o r p r e s e n t a t i o n p u r p o s e s , b u t s u c h d i s c o n t i n u i t i e s a r e u n d e s i r a b l e f o r t h e e n e r g y i n t e r p o l a t i o n p r o c e d u r e s u s e d i n a LEED c a l c u l a t i o n . T h e r e f o r e , b e f o r e u s e , t h e d i s c o n t i n u i t i e s must be removed m a n u a l l y by a d d i n g t o o r s u b t r a c t i n g f r o m t h e c a l c u l a t e d phase s h i f t s a v a l u e of n. 3.2.4 L A T T I C E MOTION So f a r t h e i o n c o r e s c a t t e r i n g f a c t o r s t ( k , 0 s ) have been c a l c u l a t e d a s s u m i n g a r i g i d l a t t i c e . T h i s i s t r u e p r o v i d e d t h e t i m e o f i n t e r a c t i o n o f an i n d i v i d u a l e l e c t r o n w i t h a s c a t t e r i n g c e n t e r i s s h o r t . T y p i c a l l y i n LEED t h i s i s a r o u n d 1 0 " 1 6 S , w h i c h i s c o n s i d e r a b l y s m a l l e r t h a n t h e t i m e f o r an a t o m i c v i b r a t i o n ( 1 0 ~ 1 3 S ) . N e v e r t h e l e s s t h e s e t o f e l e c t r o n s c o l l e c t e d d u r i n g a LEED i n t e n s i t y measurement i n e v i t a b l y sample p o s i t i o n s o v e r t h e c o m p l e t e r a n g e o f t h e 0 .0 2 .0 4.0 6 .0 8.0 0 .0 2 . 0 4.0 6 .0 8 .0 Energy (Hartree) Energy (Hartree) Figure 3.5: Comparison of oxygen phase s h i f t s derived from (a) V g obtained from ZrO c r y s t a l l a t t i c e and (b) superposition potentials obtained by Demuth et al . [98]. /=3 I 1 1 1 1 1 1 1 1— I 1 1 1 1 1 \ 1 1 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 Energy (Hartree) Energy (Hartree) Figure 3.6: C o m p a r i s o n of z i r c o n i u m phase s h i f t s d e r i v e d f r o m (a) V s o b t a i n e d f r o m ZrO c r y s t a l l a t t i c e and (b) V g from band s t r u c t u r e c a l c u l a t i o n s [ 9 6 ] . CO 82 a t o m i c v i b r a t i n g m o t i o n s . • T h i s s u g g e s t s t h e use of s c a t t e r i n g f a c t o r s t h a t a r e a v e r a g e d o v e r t h e m o t i o n s of t h e v i b r a t i n g atom. In o t h e r words, t h e r i g i d l a t t i c e i s r e p l a c e d by a l a t t i c e of ' b l u r r e d atoms'. The e f f e c t o f t h e r m a l m o t i o n s o f t h e atoms i n a s o l i d i s most g e n e r a l l y i n c o r p o r a t e d i n t o m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s by a d d i n g an i s o t r o p i c D e b y e - W a l l e r t y p e c o n t r i b u t i o n t o t h e a t o m i c s c a t t e r i n g f a c t o r . J e p s e n et al. [99] showed t h a t f o r i s o t r o p i c v i b r a t i o n s , and n e g l e c t i n g t h e c o r r e l a t i o n s between n e i g h b o r i n g atoms, t h e a t o m i c s c a t t e r i n g f a c t o r t ( k , 0 s ) T f o r s u c h a v i b r a t i n g l a t t i c e c a n be r e l a t e d t o t ( k , 0 ) by where t h e e x p o n e n t i a l t e r m i s t h e D e b y e - W a l l e r f a c t o r and M i s d e f i n e d a s In e q u a t i o n ( 3 . 1 5 ) , Ak r e p r e s e n t s t h e momentum t r a n s f e r f o r t h e d i f f r a c t i o n and < ( A r ) 2 > T r e p r e s e n t s t h e mean s q u a r e v i b r a t i o n a m p l i t u d e a s a f u n c t i o n o f t e m p e r a t u r e , w h i c h has t h e f o l l o w i n g a s y m p t o t i c f o r m s : t ( k , 0 s ) T = t ( k , 0 s ) e x p ( - M ) , (3.14) M = ( 1 / 6 ) | A k | 2 < ( A r ) 2 > T . (3.15) < ( A r ) 2 > T-*co = 9T/mk B0 2; (3.16) 83 and < ( A r ) 2 > T _ > 0 = (9/mk B e D ) (0.25+1 . 6 4 2 T 2 / 0 2 ) , (3.17) where T i s t h e s u r f a c e t e m p e r a t u r e , 0 n i s t h e Debye t e m p e r a t u r e , m i s t h e a t o m i c mass, and kg i s t h e B o l t z m a n n c o n s t a n t . The f i r s t t h r e e v a r i a b l e s a r e i n p u t p a r a m e t e r s i n a LEED c a l c u l a t i o n . A s s u m i n g T e q u a l z e r o i n e q u a t i o n ( 3 . 1 7 ) , Van Hove and T o n g [ 5 6 ] s u g g e s t e d a s i n g l e a p p r o x i m a t e f u n c t i o n a l f o r m f o r <(Ar) 2>,„ s u c h t h a t — i < ( A r ) 2 > T = / ( < ( A r ) 2 > T = 0 ) 2 + (< ( A r ) 2>T_>J 2 . (3.18) The t e r m < ( A r ) 2 > T _ Q i s g e n e r a l l y u n d e s i r a b l e f o r most m a t e r i a l s ( a t t y p i c a l LEED e x p e r i m e n t a l t e m p e r a t u r e s ) , a n d i s u s u a l l y s e t t o z e r o . However i f t h e u s e r so d e s i r e s , i t c a n be f e d i n t o t h e c a l c u l a t i o n t h r o u g h t h e i n p u t l i s t . W i t h t h e new i o n c o r e s c a t t e r i n g f a c t o r s , a s e t of t e m p e r a t u r e - d e p e n d e n t phase s h i f t s 6^ T c a n be d e r i v e d f r o m t ( k , 0 s ) T = - ( R 2 / 4 / m k ) [ e x p ( 2 / 8 / T ) - 1 ] . (3.19) F o r t h e i n t e r e s t e d r e a d e r s , t h e d i r e c t r e l a t i o n s h i p between 6^  and 6^ T c a n be f o u n d i n r e f e r e n c e [ 5 6 ] , From e q u a t i o n ( 3 . 1 9 ) , i t can be d e d u c e d t h a t 5 ^ T a r e complex q u a n t i t i e s . T h e s e t e m p e r a t u r e - d e p e n d e n t p h a s e s h i f t s a r e u s e d i n t h e c a l c u l a t i o n o f i n t r a l a y e r s c a t t e r i n g ( S e c t i o n 3 . 3 ) , a l t h o u g h 84 f o r i n t e r l a y e r s c a t t e r i n g t h e e f f e c t o f t h e r m a l m o t i o n s of atoms c a n be i n c l u d e d by s i m p l y m u l t i p l y i n g t h e l a y e r d i f f r a c t i o n m a t r i c e s by e x p ( - M ) . V i b r a t i o n a l a n i s o t r o p y c an be i n t r o d u c e d i n t o t h e i n t e r l a y e r s c a t t e r i n g by u s i n g an a n i s o t r o p i c M formed from d i f f e r e n t a m p l i t u d e s f o r t h e p a r a l l e l and p e r p e n d i c u l a r components of Ak and Ar i n e q u a t i o n ( 3 . 1 5 ) . D e t a i l s on i m p l e m e n t a t i o n o f t h e s e c o n c e p t s i n t h e c a l c u l a t i o n s a r e d i s c u s s e d i n Van Hove and Tong's book[ 5 6 ] . 3.3 THE LAYER DIFFRACTION MATRIX The s c a t t e r i n g o f p l a n e waves i n c i d e n t on e i t h e r s i d e o f a l a y e r p a r a l l e l t o t h e s u r f a c e c a n be d e s c r i b e d by t h e l a y e r d i f f r a c t i o n m a t r i x M " . A l a y e r i n t h e 'combined s p a c e ' method i s d e f i n e d a s an a s s e m b l y o f one t o N p a r a l l e l s u b p l a n e s o f i o n c o r e s e a c h c o n t a i n i n g o n l y one i o n c o r e p e r u n i t mesh. When t h e r e i s o n l y one s u b p l a n e i n t h e l a y e r , i t i s c a l l e d a B r a v a i s l a t t i c e l a y e r . When t h e l a y e r i s composed o f more t h a n one s u b p l a n e i t i s r e f e r r e d t o a s a c o m p o s i t e l a y e r . E xamples o f t h e s e two t y p e s o f l a y e r s a r e shown i n F i g u r e 3.1. A l t h o u g h a c o m p o s i t e l a y e r i s shown as t h e t o p l a y e r i n t h e f i g u r e , t h a t i s n o t a n e c e s s a r y c o n d i t i o n . In f a c t t h e two t y p e s o f l a y e r s c a n a p p e a r i n any o r d e r p a r a l l e l t o t h e ( y z ) p l a n e . The p a r t i c u l a r e l e m e n t MgTg °f t h e l a y e r d i f f r a c t i o n m a t r i x r e p r e s e n t s t h e a m p l i t u d e o f t h e p l a n e wave w i t h a 85 w a v e v e c t o r k^, w h i c h i s s c a t t e r e d f r o m a l a y e r f o r an i n c i d e n t p l a n e wave o f u n i t a m p l i t u d e and w a v e v e c t o r k^. S u p e r s c r i p t s i g n s s p e c i f y t h e p r o p a g a t i o n d i r e c t i o n s o f t h e p l a n e waves: a '+' and a '-' f o r p r o p a g a t i o n i n t o and o u t of t h e c r y s t a l r e s p e c t i v e l y , and t h e y a r e r e a d f r o m r i g h t t o l e f t . F i g u r e 3.7 s c h e m a t i c a l l y shows t h e s c a t t e r i n g o f a s e t of p l a n e waves by a p l a n e o f i o n c o r e s w i t h known d i f f r a c t i o n m a t r i x M " . In g e n e r a l , t h e l a y e r r e f l e c t i o n ( r ) and t r a n s m i s s i o n ( t ) m a t r i c e s c a n be o b t a i n e d f r o m M " as as as by t h e f o l l o w i n g r e l a t i o n s h i p s : r + " = M +~; r " + = M~ +; t + + = M + ++ I ; and t""= M~~ + I , (3.20) as as ' as as ' ss as as' as as as' where £ r e p r e s e n t s a u n i t m a t r i x . The n o t a t i o n s o f t h e s e m a t r i c e s a r e f u r t h e r i l l u s t r a t e d d i a g r a m a t i c a l l y i n F i g u r e 3.8. F o r a B r a v a i s l a t t i c e l a y e r , t h e r e i s a m i r r o r p l a n e p a r a l l e l t o t h e l a y e r , b i s e c t i n g e v e r y i o n c o r e , hence r + ~ = r and t = t The e v a l u a t i o n o f M " i s r a t h e r c o m p l i c a t e d even f o r t h e B r a v a i s l a t t i c e l a y e r , t h e r e f o r e no a t t e m p t w i l l be made h e r e t o g i v e t h e d e r i v a t i o n . D e t a i l e d m a t h e m a t i c a l t r e a t m e n t s a r e i n t h e 1 i t e r a t u r e [ 1 1 , 5 6 , 8 6 ] . The i m p o r t a n t s t e p s i n v o l v e d c a n be summarized a s f o l l o w s : 1. The p l a n e waves a r e t r a n s f o r m e d i n t o s p h e r i c a l waves. 2. The a m p l i t u d e s o f t h e s p h e r i c a l waves r e s u l t i n g f r o m t h e s c a t t e r i n g by an i s o l a t e d p l a n e o f i o n c o r e s a r e c a l c u l a t e d . T h i s i n f o r m a t i o n i s s t o r e d i n t h e p l a n a r 86 Incident plane waves § 7 t * Ion cores OCQOCXX)??"!^.-^ i 3' Scattered plane waves Figure 3 .7: Multiple scattering of a set of plane waves by a layer of ion cores with known d i f f r a c t i o n matrix M". M +-n t h layer ++ +-Figure 3.8: Schematic diagram of transmission and r e f l e c t i o n matrices at the n t h layer. Broken l i n e s are midway between consecutive layers. 87 s c a t t e r i n g m a t r i x T . 3. In c a s e o f c o m p o s i t e l a y e r , t h e l a s t s t e p i s r e p e a t e d f o r e a c h s u b p l a n e i n t h e l a y e r . As a c o n s e q u e n c e of i n t e r p l a n a r s c a t t e r i n g , t h e i n d i v i d u a l T 1 f o r s u b p l a n e i i s m o d i f i e d t o a t o t a l p l a n a r s c a t t e r i n g m a t r i x T 1 . The wave a m p l i t u d e s a r e t h e n summed from t h e N s u b p l a n e s . 4. The s p h e r i c a l waves a r e t r a n s f o r m e d i n t o p l a n e waves. 3.3.1 BRAVAIS L A T T I C E LAYERS P e n d r y [ l l ] h as d e t a i l e d t h e o v e r a l l t r e a t m e n t w h i c h l e a d s t o t h e f i n a l e x p r e s s i o n f o r t h e m a t r i x e l e m e n t where A i s t h e a r e a of t h e u n i t mesh; L and L' r e p r e s e n t t h e p a i r s of a n g u l a r momentum quantum numbers (/,m) and (/',m') r e s p e c t i v e l y . A l s o i n e q u a t i o n ( 3 . 2 1 ) , t h e Y L ( k ) a r e s p h e r i c a l h a r m o n i c s f o r t h e a n g l e between t h e k and t h e s u r f a c e n o r m a l , '*' i n d i c a t e s complex c o n j u g a t i o n and T L L , ( k 0 ) i s t h e L L ' e l e m e n t o f t h e p l a n a r s c a t t e r i n g m a t r i x r ( k 0 ) . The l a t t e r m a t r i x i s s i m i l a r t o M " e x c e p t t h a t s p h e r i c a l waves r e p r e s e n t e d by L and L' a r e c o n s i d e r e d i n a n g u l a r momentum s p a c e , and c a n be e x p r e s s e d a s [ 8 l ] (3.21 ) r ( k 0 ) = t ( k 0 ) [ l - X ] " 1 , (3.22) 88 where t ( k 0 ) i s t h e d i a g o n a l i o n c o r e t - m a t r i x whose n o n - z e r o e l e m e n t s a r e g i v e n by e q u a t i o n ( 3 . 1 2 ) , t h e i n t r a p l a n a r s c a t t e r i n g m a t r i x X i s t h e p r o d u c t o f a s t r u c t u r a l f a c t o r G ( k o ) and t h e t - m a t r i x s u c h t h a t as ~ X = G ( k o ) t ( k 0 ) . (3.23) S S S S — ' S B G d e s c r i b e s t h e i n t r a p l a n a r p r o p a g a t o r s i n L - s p a c e ; t h e s e p r o p a g a t o r s a r e d e p e n d e n t on damping and t h e a r r a n g e m e n t o f t h e i o n c o r e s b u t i n d e p e n d e n t o f t h e s c a t t e r i n g p r o p e r t i e s o f t h e i o n c o r e s . The d e t a i l s i n t h e e v a l u a t i o n o f X have been d i s c u s s e d by K a m b e [ l 0 0 ] and by J e p s e n et a / . [ l O l ] . The e x p r e s s i o n f o r M** i n e q u a t i o n (3.21) i s v a l i d -i id o n l y i f t h e o r i g i n o f t h e c o - o r d i n a t e s w i t h i n t h e p l a n e i s an i o n c o r e c e n t e r . In o r d e r t o e x p l o i t symmetry i n K - s p a c e ( S e c t i o n 3.5.2), t h e f o r m u l a i s m o d i f i e d s l i g h t l y t o i n c l u d e a l a t e r a l s h i f t s u c h t h a t t h e o r i g i n o f t h e l a y e r c a n be r e l a t e d t o t h e symmetry e l e m e n t ( s ) c o n t a i n i n g t h e i n c i d e n t e l e c t r o n beam. A l s o , t h e t e m p e r a t u r e e f f e c t i s a c c o u n t e d f o r by m u l t i p l y i n g M ± ± by a D e b y e - W a l l e r f a c t o r a s d i s c u s s e d i n S e c t i o n ( 3 . 2 . 4 ) . The s u b r o u t i n e t h a t computes M " i s c a l l e d MSMF and t h e d e t a i l s on i m p l e m e n t a t i o n c a n be f o u n d i n Van Hove and Tong's b o o k [ 5 6 ] . A l l r e f e r e n c e s t o s p e c i f i c s u b r o u t i n e s i n t h i s t h e s i s a r e e i t h e r t o Van Hove and Tong's b o o k [ 5 6 ] o r t o t h e m a g n e t i c t a p e p r o v i d e d by Van H o v e [ 9 7 ] . 89 3.3.2 COMPOSITE LAYERS H e r e , l a y e r s a r e c o n s i d e r e d i n w h i c h t h e r e a r e N (N>1) s u b p l a n e s o f i o n c o r e s c l o s e l y p a c k e d t o g e t h e r ( i n d e e d p o s s i b l y c o p l a n a r ) . By d e f i n i t i o n , a s u b p l a n e c o n t a i n s a s i n g l e k i n d of atom, and has j u s t one atom p e r u n i t mesh. T h e r e i s no r e s t r i c t i o n on t h e t y p e s o f s u b p l a n e s t h a t a c o m p o s i t e l a y e r i s made up o f . T h u s , a c o m p o s i t e l a y e r can e i t h e r be a t o m i c a l l y homogeneous o r o t h e r w i s e . Two s u c h e x a m ples a s s o c i a t e d w i t h t h e a d s o r p t i o n s t u d i e s of oxygen on Z r ( O O O l ) a r e shown i n F i g u r e 3.9. F i g u r e 3.9(a) shows a g r a p h i t i c (2x2) oxygen o v e r l a y e r on Z r ( 0 0 0 1 ) ; h e r e t h e c o m p o s i t e l a y e r has two oxygen atoms p e r u n i t mesh. F i g u r e 3.9(b) i s an example o f an a t o m i c a l l y h e t e r o g e n e o u s c o m p o s i t e l a y e r . S p e c i f i c a l l y a (2x2) oxygen u n d e r l a y e r o c c u p i e s t h e t e t r a h e d r a l h o l e s a b o u t 0.4 A below t h e f i r s t z i r c o n i u m l a y e r . T h e r e f o r e t h i s c o m p o s i t e l a y e r i s made up o f f o u r z i r c o n i u m atoms and one oxygen atom p e r u n i t mesh. In a d d i t i o n t o t h e e v a l u a t i o n o f t h e p l a n a r s c a t t e r i n g m a t r i x r 1 f o r t h e i s o l a t e d s u b p l a n e i , one has t o c a l c u l a t e t h e t o t a l s c a t t e r i n g m a t r i x T 1 f o r t h a t p a r t i c u l a r s u b p l a n e r e s u l t i n g f r o m t h e m u l t i p l e s c a t t e r i n g .with t h e o t h e r ( N - l ) s u b p l a n e s i n s i d e t h e c o m p o s i t e l a y e r . The g e n e r a l e x p r e s s i o n f o r t h e d i f f r a c t i o n m a t r i x e l e m e n t Mg^g °^ a c o m p o s i t e l a y e r w i t h N s u b p l a n e s i s g i v e n b y [ 5 6 , l 0 2 ] Ak 16TT 2 ^ ( R ^ ) - 1 ^ LL (3.24) Composite layers view (a) Zr(000l)-(2x2)-20 (b) Zr(0001)-p(2x2)-0 graphitic overlayer tetrahedral hole underlayer Figure 3 .9: Examples of composite layer. (a)Graphitic type oxygen overlayer: 2 oxygen atoms per unit mesh. (b)p(2x2) oxygen underlayer which i s separated from the top zirconium layer by »0.4 A: 4 zirconium atoms and 1 oxygen atom per unit mesh. 91 Most o f t h e s y m b o l s i n e q u a t i o n (3.24) have been e x p l a i n e d i n e q u a t i o n ( 3 . 2 1 ) . New symbols a p p e a r o n l y a f t e r t h e l a s t summation s i g n where t h e i n d e x i l a b e l s t h e N s u b p l a n e s . The o r i g i n of e a c h s u b p l a n e can be d e f i n e d by t h e c e n t e r o f any one o f t h e i o n c o r e s , w h i c h has a p o s i t i o n v e c t o r r ^ r e l a t i v e t o an a r b i t r a r y o r i g i n o f t h e c o m p o s i t e l a y e r . The q u a n t i t i e s R^* a r e t h e n d e f i n e d as R ^ = e x p ( ± / k ± . £ i ) . (3.25) The q u a n t i t i e s T 1 ^ , ( i . e . t h e L L ' e l e m e n t s o f t h e t o t a l p l a n a r s c a t t e r i n g m a t r i x T 1 ) g i v e t h e s e l f c o n s i s t e n t a m p l i t u d e o f an o u t g o i n g s p h e r i c a l wave l e a v i n g an i o n c o r e o f s u b p l a n e i due t o a s p h e r i c a l wave Y * L i i n c i d e n t on a l l s u b p l a n e s o f t h e c o m p o s i t e l a y e r . The e v a l u a t i o n o f M^T^ * n e q u a t i o n (3.24) i n v o l v e s t h e c a l c u l a t i o n o f t h e t o t a l p l a n a r s c a t t e r i n g m a t r i x T 1 f o r e a c h s u b p l a n e . T h i s c a n be done e x a c t l y , a s i n B e e b y's m a t r i x i n v e r s i o n s c h e m e [ 8 3 ] , o r p e r t u r b a t i o n a l l y as i n t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n m e t h o d [ 8 1 , 1 0 3 ] ; i n a d d i t i o n a c o m b i n a t i o n o f t h e s e a p p r o a c h e s may be u s e d [ 5 6 , 8 l ] . A l l of t h e s e methods c o n s i d e r t h e s c a t t e r i n g of s p h e r i c a l waves i n L - s p a c e . The c o m p l e t e m a t h e m a t i c a l t r e a t m e n t s c a n be f o u n d i n t h e r e f e r e n c e s q u o t e d , but t h e p r i n c i p l e s i n v o l v e d w i l l be d i s c u s s e d v e r y b r i e f l y as f o l l o w s . 92 3.3.2.1 M a t r i x I n v e r s i o n In t h e m a t r i x i n v e r s i o n method, t h e m u l t i p l e s c a t t e r i n g between any p a i r o f s u b p l a n e s i s t r e a t e d e x a c t l y r e g a r d l e s s o f m a g n i t u d e s . F o l l o w i n g B e e b y [ 8 3 ] , t h e m a t r i c e s T 1 c a n be c a l c u l a t e d by s o l v i n g a s e t of l i n e a r e q u a t i o n s w h i c h c a n be e x p r e s s e d a s "Tr ' r 1 " <*> T 2 T 2 • = [ A ] " 1 • • • • •pN • r N (3.26) The m a t r i x A i n t u r n c o n s i s t s o f N 2 s m a l l e r m a t r i c e s c a l l e d A 1 ^ w h i c h a r e d e f i n e d as l I I , and k i j = - T *G*^. (3.27) The s q u a r e m a t r i x G 1^ d e s c r i b e s t h e i n t e r p l a n a r p r o p a g a t o r s from s u b p l a n e j t o s u b p l a n e i , and has a d i m e n s i o n o f ( / m = + 1 ) 2 where / _ _ v i s t h e l a r g e s t v a l u e UlclX IliaX o f / i n c l u d e d i n e x p r e s s i o n s o v e r a n g u l a r momentum. The d i m e n s i o n o f A, w h i c h c o r r e s p o n d s t o t h e number of unknowns i n e q u a t i o n ( 3 . 2 6 ) , i s t h e r e f o r e N ( / _ = + 1 ) 2 . The computer s t o r a g e f o r A, and t h e c o m p u t i n g t i m e r e q u i r e d f o r i t s i n v e r s e l i m i t t h e number N t o a b o u t 5 93 w i t h t h e computer (Amdahl 470 V8) c u r r e n t l y u s e d by t h i s l a b o r a t o r y . F o r c o m p o s i t e l a y e r s w i t h s t r o n g l y s c a t t e r i n g i o n c o r e s o r c l o s e l y - s p a c e d s u b p l a n e s , ' m a t r i x i n v e r s i o n ' has t o be u s e d . One s u c h example i s t h e g r a p h i t i c oxygen l a y e r d e p i c t e d i n F i g u r e 3.9(a) where two c o p l a n a r oxygen s u b p l a n e s a r e p r e s e n t . 3.3.2.2 R e v e r s e S c a t t e r i n g P e r t u r b a t i o n In LEED f o r w a r d s c a t t e r i n g i s much s t r o n g e r t h a n back s c a t t e r i n g , and t h i s opens t h e p o s s i b i l i t y of t r e a t i n g back s c a t t e r i n g as a p e r t u r b a t i o n w h i l e f o r w a r d s c a t t e r i n g i s c o n s i d e r e d e x a c t l y . The r e v e r s e s c a t t e r i n g p e r t u r b a t i o n schemef56,103] i s i t e r a t i v e i n n a t u r e and i s v e r y s i m i l a r t o t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g m e t h o d [ l l ] f o r l a y e r s t a c k i n g ( S e c t i o n 3 . 4 ) . The major d i f f e r e n c e between t h e s e two methods i s t h a t t h e f o r m e r i s f o r m u l a t e d i n L - s p a c e w h i l e t h e l a t t e r i s i n K - s p a c e . In t h e i m p l e m e n t a t i o n o f t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n method, t h e s u b p l a n e s a r e o r d e r e d a c c o r d i n g t o i n c r e a s i n g d i s t a n c e f r o m vacuum. I n c i d e n t p l a n e waves a r e t r a n s f o r m e d i n t o s p h e r i c a l waves w h i c h t r a v e l back and f o r t h t h r o u g h t h e c o m p o s i t e l a y e r o f N s u b p l a n e s , as shown i n F i g u r e 3.10. S p e c i f i c a l l y , when t h e waves a r e p r o p a g a t i n g i n t o t h e c r y s t a l (+x d i r e c t i o n ) f o r t h e p t h t i m e , t h e t o t a l s c a t t e r i n g a m p l i t u d e a r r i v i n g a t p l a n e i ( F i g u r e 3.10) w i l l c o n s i s t o f t h e m u l t i p l e f o r w a r d s c a t t e r i n g a m p l i t u d e F 1 + ( p ) and t h e s i n g l e back s c a t t e r i n g a m p l i t u d e B 1 + ( p ) 94 ( i - 2 ) t h p l a n e ( i - 1 ) t h p l a n e i t h p l a n e ) t h p l a n e . ( i + 2 ) t h p l a n e . S p h e r i c a l waves * p e n e t r a t i o n f o r t h e p1 t i m e \ \ \. \ t \ x B i + (p) W/' F i + ( p ) \ \ S p h e r i c a l waves e m e r g i n g f o r t h e p t h t i m e / / / A I Figure 3.10: S c h e m a t i c i l l u s t r a t i o n o f t h e c o n t r i b u t i o n s of s c a t t e r i n g a m p l i t u d e s t o t h e i t h s u b p l a n e by f o r w a r d and back s c a t t e r i n g . S p h e r i c a l waves a r e t r a v e l l i n g a l o n g ±x d i r e c t i o n s f o r t h e p*"*1 t i m e i n t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n f o r m a l i s m . 95 T 1 T ( p ) = F 1 T ( p ) + B 1 T ( p ) . (3.28) S i m i l a r l y , when t h e waves a r e p r o p a g a t i n g t o w a r d s vacuum (-x d i r e c t i o n ) , T i _ ( p ) = F i _ ( p ) + B i _ ( p ) . (3.29) The i t e r a t i o n c o n v e r g e s when T 1 (p) e q u a l s T 1 (p-1) f o r a l l t h e N s u b p l a n e s . M a t h e m a t i c a l l y , t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n i s e q u i v a l e n t t o t h e G a u s s i a n S e i d e l - A i t k e n i t e r a t i v e a l g o r i t h m f o r m a t r i x i n v e r s i o n . I f one e x a m i n e s e q u a t i o n (3.26) a g a i n , t h e p i c k i n g up of a m p l i t u d e from t h e p r o p a g a t i o n i n t h e +x d i r e c t i o n i s s i m i l a r t o s o l v i n g 2(P} = [ A = l t ] ~ 1 2(P"1)' (3.30) and f o r p r o p a g a t i o n i n t h e -x d i r e c t i o n , i t i s s i m i l a r t o s o l v i n g Inew<P> - t £ u t r 1 T ( p ) , (3.31) where A j t and A u f c r e p r e s e n t t h e l o w e r and upper t r i a n g u l a r m a t r i c e s o f A r e s p e c t i v e l y . T i s t h e column of m a t r i c e s T 1 , p i s t h e i t e r a t i o n number' ( i . e . t h e as ' o r d e r o f p e r t u r b a t i o n ) and t h e s u b s c r i p t 'new' i d e n t i f i e s t h e f i n a l r e s u l t s a f t e r e a c h i t e r a t i o n . To 96 s t a r t t h e i t e r a t i o n , i t i s s u f f i c i e n t t o n o t e t h a t T ( 0 ) i s s i m p l y T . C a l c u l a t e d T ( p ) from t h e l e f t h and s i d e of e q u a t i o n (3.30) i s t h e n s u b s t i t u t e d i n t o t h e r i g h t hand s i d e o f e q u a t i o n (3.31) t o o b t a i n J n e w ( p ) f o r t h e o u t w a r d - t r a v e l l i n g waves. In t h e n e x t o r d e r of p e r t u r b a t i o n , J n e w i s s u b s t i t u t e d i n t o t h e r i g h t hand s i d e o f e q u a t i o n ( 3 . 3 0 ) , and t h e p r o c e d u r e i s r e p e a t e d u n t i l c o n v e r g e n c e o f T n e w i s a c h i e v e d . The i n v e r s i o n o f a t r i a n g u l a r m a t r i x r e q u i r e s much l e s s c o m p u t i n g t i m e t h a n t h a t o f i n v e r t i n g t h e c o r r e s p o n d i n g s q u a r e m a t r i x . T h e r e f o r e t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n scheme i s v e r y e f f i c i e n t e s p e c i a l l y when t h e number o f i t e r a t i o n s r e q u i r e d f o r c o n v e r g e n c e i s s m a l l . T h i s o c c u r s when t h e c o m p o s i t e l a y e r i s made up o f w e a k l y s c a t t e r i n g i o n c o r e s , o r when O t h e d i s t a n c e between s u b p l a n e s i s l a r g e ( e . g . >0.3 A ) . A n o t h e r a d v a n t a g e o f t h i s p e r t u r b a t i o n scheme i s t h e d r a s t i c r e d u c t i o n o f computer memory r e q u i r e d compared w i t h t h e m a t r i x i n v e r s i o n method b e c a u s e T 1 c a n be c o n t i n u o u s l y w r i t t e n o v e r r 1 . 3.3.2.3 C o m b i n i n g M a t r i x I n v e r s i o n and RSP Some s u r f a c e m o d e ls may c o r r e s p o n d t o c o m p o s i t e l a y e r s i n w h i c h one r e g i o n i s composed o f c l o s e l y - s p a c e d p l a n e s of s t r o n g s c a t t e r e r s w h i l e a n o t h e r r e g i o n has o n l y w e a k l y s c a t t e r i n g i o n c o r e s . F o r s u c h l a y e r s , m a t r i x i n v e r s i o n has t o be e mployed t o c a l c u l a t e a r c f o r t h e f o r m e r r e g i o n , and t h e n r e v e r s e s c a t t e r i n g 97 p e r t u r b a t i o n i s a p p l i e d t o ' j o i n ' i t w i t h t h e r e s t o f t h e c o m p o s i t e l a y e r . A g a i n , t h e m a j o r a d v a n t a g e i s t h e s a v i n g o f computer c o r e memory. One s u c h example i s Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 d e p i c t e d i n F i g u r e 3 . 9 ( b ) . H e r e m a t r i x i n v e r s i o n i s a p p l i e d t o t h e f o u r c o p l a n a r Zr s u b p l a n e s , and r e v e r s e s c a t t e r i n g p e r t u r b a t i o n i s u s e d between t h e oxygen s u b p l a n e and t h e ' c o m p o s i t e ' z i r c o n i u m l a y e r . 3.4 LAYER STACKING T h i s s e c t i o n d i s c u s s e s t h e s t a c k i n g o f v a r i o u s l a y e r s w i t h e i t h e r t h e (1x1) t y p e o r t h e s u p e r l a t t i c e t y p e s t r u c t u r e s , but w i t h t h e i r r e s p e c t i v e d i f f r a c t i o n m a t r i c e s a l r e a d y c a l c u l a t e d by one o f t h e methods m e n t i o n e d i n S e c t i o n 3.3. In t h e 'combined s p a c e ' method employed i n t h i s work, o n l y p e r t u r b a t i v e methods f o r s t a c k i n g , namely l a y e r d o u b l i n g and r e n o r m a l i z e d f o r w a r d s c a t t e r i n g , a r e u s e d . T h e s e methods expand t h e t o t a l w a v e f i e l d between l a y e r s a s p l a n e waves, hence o p e r a t e i n K - s p a c e . In g e n e r a l , p e r t u r b a t i v e methods assume s i g n i f i c a n t i n e l a s t i c s c a t t e r i n g i n s i d e t h e c r y s t a l so t h a t t h e s c a t t e r i n g by a s u r f a c e i s w e l l m o d e l l e d by a maximum of 20 l a y e r s . 3.4.1 LAYER DOUBLING In t h e l a y e r d o u b l i n g m e t h o d [ l l ] , t h e b u l k l a y e r p e r i o d i c i t y of t h e c r y s t a l i s e x p l o i t e d . I t i s most e f f i c i e n t f o r t h e s t a c k i n g o f t h e b u l k r e g i o n where t h e l a y e r s a r e e i t h e r of AAAA... o r ABAB... s t a c k i n g s e q u e n c e . 98 I n t h e s e s p e c i a l c a s e s , two l a y e r s o f t h e c r y s t a l a r e c o n s i d e r e d f i r s t , t h e n f o u r , and f o r e a c h i t e r a t i o n t h e number o f l a y e r s i s d o u b l e d . The minimum i n t e r l a y e r s p a c i n g t h a t t h e l a y e r d o u b l i n g method c a n h a n d l e i s a p p r o x i m a t e l y 0.5 A . S t a r t i n g w i t h known r e f l e c t i o n and t r a n s m i s s i o n m a t r i c e s f o r l a y e r A and l a y e r B f r o m e q u a t i o n ( 3 . 2 0 ) , t h e c o r r e s p o n d i n g m a t r i c e s f o r t h e c o m p o s i t e s l a b C c a n be c a l c u l a t e d f r o m [ 5 6 ] (3.32) t:+ = t : + p + [ i - r Y r - v i - ' t r , SB(_ S S D SS SS S S A SS S S D SS S S A r r = r R + t R P r , P [ I - r R P r , P ] ' t R , w L « B M D mi min mi mi mto mi min mi tma tZ" = t 7 " P _ [ I - r ; + P + r t " P " ] " 1 t ; _ , mt\~ min mi <*> « D mi min mi mio where 1^ i s a u n i t m a t r i x ; P + and P~ a r e d i a g o n a l m a t r i c e s w h i c h d e s c r i b e p l a n e waves p r o p a g a t i n g f r o m a r e f e r e n c e p o i n t i n l a y e r A t o a r e f e r e n c e p o i n t i n l a y e r B and v i c e v e r s a . I f r B A r e p r e s e n t s t h e v e c t o r between t h e two r e f e r e n c e p o i n t s , t h e n t h e e l e m e n t s of P * a r e d e f i n e d a s [ 5 6 ] P* - e x p ( ± / k | . r B A ) . (3.33) In t h e n e x t s t e p i n t h e i t e r a t i o n , t h e m a t r i c e s c a l c u l a t e d on t h e l e f t hand s i d e s o f e q u a t i o n (3.32) c a n be u s e d a s i n p u t m a t r i c e s on t h e r i g h t hand s i d e s t o d e t e r m i n e 99 B B (a) Individual layers with known transmission and r e f l e c t i o n matrices. (b) F i r s t doubling, producing asym-metric slabs of 2 layers. (c) Second doubling, producing asym metric slab of 4 layers. Figure 3.11: Schematic diagram of the layer doubling method used to stack 4 i n d i v i d u a l layers (with ABAB... r e g i s t r i e s ) into an asymmetric slab of 4 layers (after Tong [ 8 l ] ) . 100 t h e t r a n s m i s s i o n and r e f l e c t i o n m a t r i c e s f o r t h e 4 - l a y e r s y s t e m . The p r o c e d u r e i s shown s c h e m a t i c a l l y i n F i g u r e 3.11. The d o u b l i n g p r o c e s s i s c o n t i n u e d u n t i l t h e e l e m e n t s o f t h e m a t r i x r + have c o n v e r g e d . F o r most m e t a l s c o n v e r g e n c e i s a c h i e v e d a f t e r 3 t o 4 i t e r a t i o n s . The b u l k t r e a t m e n t j u s t m e n t i o n e d i s i m p l e m e n t e d by t h e s u b r o u t i n e SUBREF i n Van Hove and Tong's p r o g r a m s . O v e r l a y e r s c a n be added o n t o t h e ' c o m p o s i t e s l a b ' one a t a t i m e by c a l l i n g t h e s u b r o u t i n e DBG. A l t e r n a t i v e l y , p a i r s of o v e r l a y e r s c a n be c o m b i n e d f i r s t u s i n g s u b r o u t i n e DBGL, and t h e n a d d e d o n t o t h e s l a b . The l a s t s t e p i n t h e l a y e r d o u b l i n g scheme i s t o c a l c u l a t e beam r e f l e c t i v i t i e s I a = ( k g j / k j j l ra°"l 2. (3.34) 3.4.2 RENORMALIZED FORWARD SCATTERING The s t r o n g e r f o r w a r d s c a t t e r i n g compared w i t h back s c a t t e r i n g i n L E E D [ 5 7 ] has a l r e a d y been e x p l o i t e d i n t h e r e v e r s e s c a t t e r i n g p e r t u r b a t i o n method f o r t h e c a l c u l a t i o n o f d i f f r a c t i o n m a t r i c e s o f c o m p o s i t e l a y e r s i n t h e p r e s e n c e o f i n e l a s t i c s c a t t e r i n g . S i m i l a r l y , f o r t h e i n t e r l a y e r s c a t t e r i n g , t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method t r e a t s t h e f o r w a r d s c a t t e r i n g e x a c t l y , but t r e a t s back s c a t t e r i n g as a p e r t u r b a t i o n . T h i s method was o r i g i n a l l y d e v e l o p e d by P e n d r y [ l 0 4 ] and d i s c u s s e d f u r t h e r by T o n g [ 8 1 , 1 0 5 ] and by Van Hove and T o n g [ 5 6 ] . The d e s c r i p t i o n 101 g i v e n h e r e f o l l o w s t h a t o f t h e l a s t a u t h o r s . The r e n o r m a l i z e d f o r w a r d s c a t t e r i n g f o r m a l i s m i s an i t e r a t i v e method d e p i c t e d s c h e m a t i c a l l y i n F i g u r e 3 . 1 2 ( a ) . The p l a n e waves g e n e r a t e d by t h e i n c i d e n t beam a r e f o r w a r d s c a t t e r e d from l a y e r t o l a y e r u n t i l t h e y a r e damped o u t ( i . e . t h e i r a m p l i t u d e s a r e a l l l e s s t h a n a p r e d e t e r m i n e d f r a c t i o n o f t h e i n c i d e n t beam a m p l i t u d e ) . Then, s t a r t i n g f r o m t h e d e e p e s t l a y e r r e a c h e d by t h e i n w a r d - t r a v e l l i n g (+x d i r e c t i o n ) waves, o u t w a r d - t r a v e l l i n g (-x d i r e c t i o n ) r e f l e c t e d p l a n e waves a r e a l l o w e d t o f o r w a r d s c a t t e r by e a c h l a y e r t o w a r d s t h e vacuum. The a m p l i t u d e s of t h e i n w a r d - t r a v e l l i n g and o u t w a r d - t r a v e l l i n g p l a n e waves j u s t p a s t t h e n f ck l a y e r c a n be c o n v e n i e n t l y s t o r e d i n t h e column v e c t o r s a ^ ( n ) and a>(n) r e s p e c t i v e l y , where t h e s u b s c r i p t i s t a n d s f o r t h e i i t e r a t i o n . As shown i n F i g u r e 3 . 1 2 ( b ) , t h e components i n a ^ ( n ) r e p r e s e n t a m p l i t u d e s o f p l a n e waves between t h e n f c ^ and ( n + 1 ) f c ^ l a y e r s , whereas t h o s e i n a j ( n ) r e p r e s e n t a m p l i t u d e s between t h e n*"*1 and ( n - 1 ) f c ^ l a y e r s . W i t h t h e s e n o t a t i o n s , t h e a m p l i t u d e s of t h e i n c i d e n t beam i n vacuum i s d e f i n e d a s a?<0) = (3.35) where 1 a t t h e t o p o f t h e column v e c t o r on t h e r i g h t s t a n d s 102 Inc i d e n t beam 1 s t o r d e r 2 n d o r d e r l\\^\]f * 3 r d o r d e r Ji N • Mr N, (a) : N,, N 2 , and N 3 d e n o t e t h e d e e p e s t l a y e r r e a c h e d i n t h e 1 s t , 2 n d , and 3 r d p e n e t r a t i o n , r e s p e c t i v e l y , n-1 n+ (b) A m p l i t u d e s o f t h e i n w a r d - t r a v e l l i n g waves (a^) and o u t w a r d - t r a v e l l i n g waves ( a j ) . J ~ I F i g u r e 3.12: S c h e m a t i c d i a g r a m of t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method. ( a ) E a c h t r i p l e t o f a r r o w s r e p r e s e n t s t h e c o m p l e t e s e t o f p l a n e waves t h a t t r a v e l f r o m l a y e r t o l a y e r , ( b ) I l l u s t r a t i o n o f t h e v e c t o r s w h i c h s t o r e t h e a m p l i t u d e s of t h e i n w a r d - and o u t w a r d - t r a v e l l i n g waves ( a f t e r Van Hove and T o n g [ 5 6 ] ) . 103 f o r t h e u n i t a m p l i t u d e o f t h e (0,0) beam. A l l o t h e r a^(0) ( i > 1 ) a r e n e c e s s a r i l y n u l l v e c t o r s s i n c e t h e r e c a n be no i n c i d e n t beams f o r t h e h i g h e r o r d e r o f i t e r a t i o n . The i n w a r d - t r a v e l l i n g wave a m p l i t u d e s j u s t p a s t t h e n1-*1 l a y e r a r i s e f r o m t r a n s m i s s i o n of i n w a r d - t r a v e l l i n g waves and r e f l e c t i o n o f t h e o u t w a r d - t r a v e l l i n g waves a t l a y e r n, and c a n be e x p r e s s e d as a | ( n ) = t + + P + ( n - D a t ( n - l ) + r + " p " ( n + 1 ) a 7 ( n ) (3.36) ~1 t*t mi ~1 mi mt ~1 +* Yi f o r p l a n e waves p e n e t r a t i n g t h e c r y s t a l f o r t h e i t i m e . H e r e n r u n s f r o m 1 t o N j , where N- r e p r e s e n t s the. d e e p e s t l a y e r r e a c h e d i n t h e i f c ^ i t e r a t i o n . A s i m i l a r e x p r e s s i o n c a n be o b t a i n e d f o r t h e e m e r g i n g wave a m p l i t u d e s , namely a T ( n ) = t " " P " ( n + 1 ) a T ( n + 1 ) + r ~ + P + ( n - 1 ) a t ( n ) , (3.37) ~ 1 SB SB ~1 SB SB ~1 where n r u n s f r o m (N^-1) t o 0. The P* a r e p l a n e wave p r o p -a g a t o r s between a p p r o p r i a t e r e f e r e n c e p o i n t s i n n e i g h b o r i n g l a y e r s . The d e f i n i t i o n s o f t h e s e p r o p a g a t o r s have been g i v e n i n (3.33) and a r e a l s o i l l u s t r a t e d d i a g r a m a t i c a l l y i n F i g u r e 3 . 1 2 ( b ) . The i t e r a t i o n s t a r t s w i t h t h e i n p u t o f at(0) i n t o e q u a t i o n ( 3 . 3 6 ) . From F i g u r e 3.12(a) f o r t h e f i r s t p e n e t r a t i o n , t h e a , ( n ) a r e s i m p l y column v e c t o r s o f z e r o ' s b e c a u s e t h e r e a r e no e m e r g i n g waves y e t . When t h e p l a n e waves r e a c h t h e d e e p e s t l a y e r , namely N, f o r t h e f i r s t 104 p e n e t r a t i o n , t h e c a l c u l a t e d v a l u e o f a t ( N , ) i s s u b s t i t u t e d i n t o t h e r i g h t hand s i d e o f e q u a t i o n ( 3 . 3 7 ) t o o b t a i n a ^ n ) . A f t e r t h e f i r s t i t e r a t i o n , t h e r e f l e c t e d wave a m p l i t u d e s i n t o vacuum a r e s t o r e d i n t h e a ^ ( 0 ) . T h i s p r o c e d u r e i s r e p e a t e d t o o b t a i n t h e h i g h e r o r d e r s j a j ( 0 ) , and t h e n t h e t o t a l r e f l e c t e d a m p l i t u d e s c an be e x p r e s s e d a s A~ = a 7 ( 0 ) + a z ( 0 ) + a a ( 0 ) + ( 3 . 3 8 ) T y p i c a l l y , A~ c o n v e r g e s a f t e r a b o u t 3 o r 4 o r d e r s o f i t e r a t i o n . Some 12 t o 15 l a y e r s a r e u s u a l l y r e q u i r e d f o r t h e f i r s t p e n e t r a t i o n , b u t d e c r e a s e s w i t h i n c r e a s i n g o r d e r o f i t e r a t i o n . However, i n t h e p r e s e n c e o f c l o s e l y - s p a c e d l a y e r s , n o n - c o n v e r g e n c e may o c c u r . The e x p e r i e n c e w i t h Zr i s o t h a t a minimum s e p a r a t i o n o f 1.1 A i s r e q u i r e d f o r t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method t o work e f f e c t i v e l y . A t c o n v e r g e n c e , t h e beam r e f l e c t i v i t i e s a r e g i v e n by i g - <V/k°x> l A a l 2 - ( 3 - 3 9 ) C o n v e r g e n c e p r o b l e m s may o c c u r o c c a s i o n a l l y i n t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g , and t h e t o p i c has been d i s c u s s e d by o t h e r a u t h o r s [ 1 0 6 3 . When t h e s e p o o r c o n v e r g e n c e s a r e f o u n d o n l y a t i s o l a t e d e n e r g i e s , t h e most c o n v e n i e n t s o l u t i o n i n p r a c t i c e i s t o p e r f o r m t h e c a l c u l a t i o n a g a i n a t n e i g h b o r i n g e n e r g y p o i n t s and t h e n i n t e r p o l a t e t h e r e s u l t s t o t h e d e s i r e d e n e r g y . I f t h e y 105 a p p e a r a t s e v e r a l c o n s e c u t i v e e n e r g i e s , t h e n l a y e r d o u b l i n g c a n be u s e d t o r e - c a l c u l a t e them. In t h e e v e n t t h a t even l a y e r d o u b l i n g f a i l s t o r e c t i f y t h e s i t u a t i o n , c o m p o s i t e l a y e r c a l c u l a t i o n s must be c a r r i e d o u t . 3.4.2.1 S u b r o u t i n e RFSG The l a t e s t v e r s i o n o f t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g scheme i s i m p l e m e n t e d by t h e s u b r o u t i n e RFSG (Van H o v e [ 9 7 ] , 1 9 8 3 ) . T h i s s u b r o u t i n e has been g e n e r a l i z e d t o h a n d l e a wide v a r i e t y o f s t a c k i n g s e q u e n c e s , and i t r e c e i v e s s p e c i a l m e n t i o n h e r e b e c a u s e i t was u s e d e x t e n s i v e l y i n t h e l a t e r s t a g e s o f t h i s work. In t h i s r e g a r d , F i g u r e 3.13 r e f e r s t o e x a m ples f o r Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 w i t h two oxygen u n d e r l a y e r s o c c u p y i n g o c t a h e d r a l h o l e s . In RFSG, t h e s u r f a c e i s c o n c e p t u a l l y d i v i d e d i n t o two r e g i o n s , w h i c h a r e p e r i o d i c o r n o n - p e r i o d i c i n t h e p e r p e n d i c u l a r d i r e c t i o n ; t h e l a t t e r i s t h e r e f o r e t h e r e g i o n c l o s e s t t o t h e vacuum i n t e r f a c e . E a c h r e g i o n may be composed o f any c o m b i n a t i o n o f l a y e r s w h i c h c o n f o r m t o t h e s u p e r l a t t i c e or t h e p r i m i t i v e (1x1) t y p e s t r u c t u r e s ; t h e s e l a y e r s have known r e f l e c t i o n and t r a n s m i s s i o n m a t r i c e s . A new f e a t u r e of RFSG r e q u i r e s t h a t t h e e l e m e n t s of t h e s e m a t r i c e s be s t o r e d a s 1 - d i m e n s i o n v e c t o r s . I n g e n e r a l , t h e r e a r e f o u r s u c h v e c t o r s t o d e s c r i b e b o t h t y p e s o f l a y e r s , namely, £ q U D ' 106 3 - f o l d r o t . a x i s + m i r r o r p l a n e NRTNP(1 ) -NRTNP(2 )• NRTNP (3)-NRTNP( 4 )-NRTP( 1 )-NRTP(2)-o p„, o o -e ASNP(2). 1——f ASNPOr ASNP(4) A S P O ) ASP(2) (a) vacuum non-• p e r i o d i c r e g i o n p e r i o d i c r e g i o n s e c . 1 t + + t s u p s e c . 1 - + ~ s u p NRTNP(1) NRTNP(2) NRTNP(3) NRTNP(4) NRTP(1) NRTP(2) s e c . 1 s e c . 2 s e c . 1 s e c . 2 (b) t+ + £lx1 - + -1x1 Figure 3.13: S c h e m a t i c i l l u s t r a t i o n o f some i m p o r t a n t i n p u t p a r a m e t e r s i n s u b r o u t i n e (RFSG) f o r t h e a d s o r p t i o n s y s t e m Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 w i t h 2 oxygen u n d e r l a y e r s o c c u p y i n g o c t a h e d r a l h o l e s . ("a)Side v i e w o f t h e l a y e r a r r a n g e m e n t o f t h e s u r f a c e r e g i o n . ( b ) S e l e c t i o n o f a p p r o p r i a t e d i f f r a c t i o n m a t r i c e s w i t h c o d i n g v e c t o r s NRTNP and NRTP. 107 ~ s u p ' ~ s u p a n d ~ s u p ^ o r t * i e s u P e r l a t t i c e t y p e ; and r j x 1 , ~ l x 1 ' ~Tx1 a n d ~Tx1 f o r t h e ( 1 x 1 ) t y p e l a y e r s . The c o r r e s p o n d i n g t r a n s m i s s i o n and r e f l e c t i o n m a t r i c e s o f a l l t h e r e q u i r e d ( b u t n o n - e q u i v a l e n t ) l a y e r s o f t h e s y s t e m i n v e s t i g a t e d a r e t r a n s f e r r e d i n t o c o n s e c u t i v e s e c t i o n s o f t h e s e v e c t o r s . In t h e s t a c k i n g p r o c e d u r e , t h e d i f f r a c t i o n p r o p e r t i e s o f t h e l a y e r s i n t h e n o n - p e r i o d i c and p e r i o d i c r e g i o n s a r e r e t r i e v e d f r o m t h e a p p r o p r i a t e s e c t i o n s o f t h e s e v e c t o r s by two c o d i n g v e c t o r s NRTNP and NRTP r e s p e c t i v e l y ( F i g u r e 3 . 1 3 ( b ) ) . A code number o v e r 100 s i g n a l s (1x1) t y p e l a y e r w h i l e a code number under 100 d e n o t e s a s u p e r l a t t i c e t y p e l a y e r ; t h e l a s t d i g i t o f t h e co d e number r e f e r s t o t h e s e c t i o n number of t h e a p p r o p r i a t e r e f l e c t i o n and t r a n s m i s s i o n v e c t o r s . The i n t e r l a y e r v e c t o r s f o r t h e n o n - p e r i o d i c r e g i o n a r e s t o r e d i n ASNP w h i l e t h o s e f o r t h e p e r i o d i c r e g i o n a r e i n ASP. The u s e r has t o s e t up a l l t h e m e n t i o n e d v e c t o r s i n t h e main program b e f o r e c a l l i n g RFSG. The example i n F i g u r e 3.13 i s i n t e n d e d t o c l a r i f y t h e usage of t h e s e v e c t o r s . S i n c e t h e two oxygen and a l l z i r c o n i u m l a y e r s a r e o f B r a v a i s l a t t i c e t y p e , r + _ = r + and t = t + + . F u r t h e r , a 3 - f o l d r o t a t i o n a l symmetry i s a v a i l a b l e f o r t h i s p a r t i c u l a r c a l c u l a t i o n , and a l l t h e i n t e r l a y e r v e c t o r s a r e c o n t a i n e d i n t h e r o t a t i o n a l a x i s . The o r i g i n s of b o t h oxygen l a y e r s a r e c h o s e n t o c o i n c i d e w i t h t h e r o t a t i o n a l a x i s so t h a t o n l y one r e g i s t r y s h i f t 108 (hence o n l y one s e c t i o n e a c h i n r -+ and t ++ ) i s sup ~ s u p r e q u i r e d . 3.5 GENERAL CONSIDERATIONS IN 'COMBINED SPACE' METHOD 3.5.1 TOTAL BEAM REQUIREMENT IN K-SPACE In t h e l a y e r d o u b l i n g and r e n o r m a l i z e d f o r w a r d s c a t t e r i n g methods, l a y e r s c a t t e r i n g i s d e s c r i b e d by s q u a r e m a t r i c e s M " whose d i m e n s i o n e q u a l s t h e number o f components i n t h e beam e x p a n s i o n . As a minimum, s u c h e x p a n s i o n s s h o u l d i n c l u d e a l l t h e p r o p a g a t i n g p l a n e waves and t h e f i r s t few e v a n e s c e n t waves. However, w i t h s m a l l i n t e r l a y e r d i s t a n c e s , s u c h a s f o r t h e a d s o r p t i o n o f s m a l l atoms deep i n t o h o l l o w s i t e s o f m e t a l s u r f a c e s , a s u b s t a n t i a l l y i n c r e a s e d number o f e v a n e s c e n t waves i s r e q u i r e d f o r c o n v e r g e n c e . By c o n s i d e r i n g t h e d e c a y f a c t o r a s s o c i a t e d w i t h an e v a n e s c e n t wave, s u c h t h a t t h e a m p l i t u d e d e c a y s t o a f r a c t i o n t o v e r d i s t a n c e d, t h e e s t i m a t e d number o f p l a n e waves r e q u i r e d i n a K - s p a c e c a l c u l a t i o n f o r a minimum i n t e r l a y e r d i s t a n c e o f d m ^ n i s g i v e n by Van Hove and T o n g [ 5 6 ] a s n g = ( A / 4 7 T ) [ 2 ( E - V 0 ) + U o g ( t ) / d m i n ) a ] , (3.40) where A i s t h e a p p r o p r i a t e u n i t mesh a r e a and ( E - V 0 ) i s t h e e l e c t r o n e n e r g y i n s i d e t h e c r y s t a l ( i n a t o m i c u n i t s ) . The d e c a y f a c t o r t i s u s e r - s e l e c t e d . A r e a s o n a b l e v a l u e i s a b o u t 109 E n e r g y ( e V ) 0.90 0.70 0.50 50 187 271 493 100 21 1 313 51 1 150 253 349 553 200 301 373 577 250 337 413 619 (a) E n e r g y ( e V ) 0.90 0.70 0.50 50 39 54 93 1 00 43 62 97 150 51 68 104 200 60 72 108 250 66 79 116 (b) T a b l e 3.1: The v a r i a t i o n o f t h e number of p l a n e waves r e q u i r e d w i t h i n c i d e n t e n e r g y and t h e s h o r t e s t i n t e r l a y e r d i s t a n c e ( d m ^ n ) f o r t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n o f Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 ; t=0.002. (a) U n s y m m e t r i z e d beams; and (b) Beams s y m m e t r i z e d w i t h r e s p e c t t o a 3 - f o l d r o t a t i o n a x i s and m i r r o r p l a n e ( x z ) symmetry o p e r a t i o n s . 110 0.002; f o r t h i s v a l u e T a b l e 3.1(a) shows t h e v a r i a t i o n ng w i t h b o t h e n e r g y and d m ^ n f o r t h e a d s o r p t i o n s y s t e m Z r ( 0 0 0 1 ) - p ( 2 x 2 ) - 0 . A l t h o u g h ng i s a f u n c t i o n o f A, E and d m ^ n , t h e l a s t p a r a m e t e r c a n d o m i n a t e b e c a u s e o f t h e d e p e n d e n c e on d m ^ n " 2 . Thus ng i n c r e a s e s r a p i d l y w i t h d e c r e a s i n g i n t e r l a y e r o d i s t a n c e . F o r example, i f d m ^ n = 0.3 A, o v e r 200 p l a n e waves a r e r e q u i r e d a t a s t a r t i n g e n e r g y o f 50 eV and a r e l a t i v e l y o s m a l l u n i t mesh a r e a A o f 10 A 2 . N o r m a l l y , once t h e d i m e n s i o n s o f M " go beyond t h e 10 2 r a n g e , t h e K - s p a c e methods become v e r y t i m e - c o n s u m i n g and n u m e r i c a l l y u n s t a b l e . o In t h i s work, e f f e c t i v e l i m i t s o f d m ^ n a r e s e t a t 0.5 A. F o r s m a l l e r d m ^ n , c o m p o s i t e l a y e r c a l c u l a t i o n s i n L - s p a c e were u s e d . 3.5.2 USE OF SYMMETRY AND BEAM SETS The number o f p l a n e waves r e q u i r e d i n K - space c a l c u l a t i o n s can be r e d u c e d i f t h e i n c i d e n t e l e c t r o n beam c o i n c i d e s w i t h one o r more symmetry e l e m e n t s o f t h e s u r f a c e . Under s u c h c o n d i t i o n s , i n d i v i d u a l beams r e l a t e d by t h e symmetry e l e m e n t s c a n be r e p r e s e n t e d by a s y m m e t r i z e d w a v e f u n c t i o n . In t h e a d s o r p t i o n s y s t e m q u o t e d i n T a b l e 3.1, t h e r e i s a 3 - f o l d r o t a t i o n a l and a m i r r o r p l a n e ( x z ) symmetry a t n o r m a l i n c i d e n c e . The beams r e l a t e d by s u c h symmetry o p e r a t i o n s a r e i n d i c a t e d by t h e same l a b e l s i n t h e r e c i p r o c a l l a t t i c e a s shown i n F i g u r e 3 . 1 4 ( a ) . In t h e c a l c u l a t i o n , o n l y one of t h e s e s y m m e t r y - r e l a t e d beams needs 111 (a) A A A ~ • o o A # A •x X. • O • A A • 1 (b) 2 3 1 4 . J 3 2 4 1 4 2 3 1 4 1 3 2 1 Figure 3.14: LEED pattern from Zr (0001)-p (2x2) -0 . (a)Symmetry-related beams are indicated by the same symbols (at normal incidence). (b)Beams belonging to the same beam set are indicated by the same number (independent of angle of incidence). 1 12 t o be i n p u t . The c o r r e s p o n d i n g number o f s y m m e t r i z e d w a v e f u n c t i o n s u s e d a t v a r i o u s e n e r g i e s a r e l i s t e d i n T a b l e 3 . 1 ( b ) , and a c o m p a r i s o n w i t h T a b l e 3.1(a) d e m o n s t r a t e s t h e a p p r e c i a b l e r e d u c t i o n i n t h e number o f beams r e q u i r e d when symmetry i s u t i l i z e d . The p r e s e n c e o f an a d s o r b e d l a y e r w i t h a u n i t mesh a r e a S t i m e s t h a t o f t h e s u b s t r a t e p r o d u c e s S t i m e s a s many d i f f r a c t e d beams a s t h e s u b s t r a t e a t a g i v e n e n e r g y . S i n c e t h e m u l t i p l e s c a t t e r i n g p r o g r a m s u s e d h e r e c a n n o t d e a l w i t h i n c ommensurate s u p e r l a t t i c e s , S i s assumed t o be an i n t e g e r . T h u s , w i t h an a d s o r b e d l a y e r , i t would a p p e a r t h a t and w i t h d i m e n s i o n s S t i m e s t h a t o f t h e c l e a n s u b s t r a t e ssSUb have t o be c a l c u l a t e d f o r t h e a d l a y e r and t h e s u b s t r a t e r e s p e c t i v e l y . T h i s i s i n d e e d t h e c a s e f o r M | ^ . However, f o r th e s u b s t r a t e l a y e r , one c a n s e p a r a t e t h e beams i n t o S s e t s s u c h t h a t w i t h i n e a c h s e t a l l beams a r e i n t e r r e l a t e d by t h e (1x1) t y p e r e c i p r o c a l l a t t i c e v e c t o r s . One s u c h example i s shown i n F i g u r e 3 . 1 4 ( b ) . An i m p o r t a n t p r o p e r t y o f t h e s e beam s e t s i s t h a t t h e s u b s t r a t e l a y e r s , w i t h t h e i r (1x1) p e r i o d i c i t y , c a n n o t d i f f r a c t a beam from one beam s e t i n t o a beam i n a n o t h e r beam s e t [ 5 6 ] . As a r e s u l t , ^sub b l o c k d i a g o n a l i z e s a c c o r d i n g t o 1 1 3 0 (3.41 ) 0 S-1 where t h e s u b s c r i p t " i n " r e f e r s t o t h e i n t e g r a l beam s e t from t h e s u b s t r a t e atoms, and " f ^ " r e f e r s t o t h e i t h f r a c t i o n a l beam s e t . The c a l c u l a t i o n s o f t h e s m a l l e r m a t r i c e s M " on t h e r i g h t hand s i d e o f e q u a t i o n (3.41) a r e s i m p l e compared t o t h a t o f M | ^ . F u r t h e r m o r e , w i t h t h e use o f symmetry, some of t h e f r a c t i o n a l beam s e t s may merge i n t o a s i n g l e s e t . One example can be f o u n d i n F i g u r e 3 . 1 4 ( b ) , where t h e t h r e e f r a c t i o n a l beam s e t s ( l a b e l l e d 2, 3 and 4 i n t h e f i g u r e ) c a n n o t be d i s t i n g u i s h e d u nder a 3 - f o l d r o t a t i o n a l and a m i r r o r p l a n e symmetry o p e r a t i o n s . So i n t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n f o r t h a t p a r t i c u l a r s y s t e m , o n l y one of t h e s e t h r e e s e t s i s i n p u t t o g e t h e r w i t h t h e i n t e g r a l beam s e t . 3.5.3 SELECTION OF METHODS B e f o r e s e t t i n g up a m u l t i p l e s c a t t e r i n g c a l c u l a t i o n , one has t o c o n s i d e r i t s f e a s i b i l i t y , c o m p u t i n g c o s t and f l e x i b i l i t y . The l a s t two f a c t o r s become i m p o r t a n t when many a d s o r p t i o n m o d e ls r e q u i r e t e s t i n g on a ' t r i a l and e r r o r ' 1 14 b a s i s . F e a s i b i l i t y i n t h e 'combined s p a c e ' method i s m a i n l y d e t e r m i n e d by t h e p e r t u r b a t i v e ( K - s p a c e ) p a r t o f t h e c a l c u l a t i o n . When t h e number o f beams, N, r e q u i r e d a f t e r s y m m e t r i z a t i o n i s o v e r 100, c o n v e r g e n c e may n o t be a s s u r e d . T h i s s i t u a t i o n can o c c u r w i t h l a r g e u n i t c e l l mesh a r e a s o r w i t h s m a l l i n t e r l a y e r s p a c i n g s . The l a t t e r may be c o u n t e r e d by c o n s t r u c t i n g c o m p o s i t e l a y e r s , w h i c h l e a d t o t h e i n v e r s i o n o f m a t r i c e s o f d i m e n s i o n s P L 2 where P i s t h e number o f s u b p l a n e s and L i s t h e number of phase s h i f t s . However i f P L 2 e x c e e d s a b o u t 200, t h e i n v e r s i o n s t e p i s n u m e r i c a l l y u n s t a b l e . T h i s l i m i t c a n be s l i g h t l y h i g h e r when c o m p u t a t i o n s a r e made i n d o u b l e p r e c i s i o n . C u r r e n t l y i n t h i s l a b o r a t o r y , we a r e l i m i t e d t o m o d e l s w i t h N<100 or P L 2<200. The b u i l d i n g b l o c k s of t h e 'combined s p a c e ' method c o n s i s t o f t h e c o n s t r u c t i o n of l a y e r d i f f r a c t i o n m a t r i c e s and t h e s u b s e q u e n t s t a c k i n g of t h e s e l a y e r s . The c o m p u t i n g e f f o r t f o r t h e f o r m e r d epends on N, P and L, w h i l e , f o r t h e l a t t e r , t h e i m p o r t a n t p a r a m e t e r s a r e N and t h e number of l a y e r s (M) r e q u i r e d t o mimic a s u r f a c e r e g i o n f o r a LEED c a l c u l a t i o n . Van Hove and T o n g [ 5 6 ] have s c a l e d t h e c o m p u t i n g t i m e s f o r d i f f e r e n t methods w i t h i n e a c h b u i l d i n g b l o c k and t h e y a r e summarized i n T a b l e 3.2 t o g e t h e r w i t h t h e i r c h a r a c t e r i s t i c s . In g e n e r a l , t h e most c o m p u t a t i o n a l l y e x p e n s i v e s t e p i s t h e e v a l u a t i o n of t h e l a y e r d i f f r a c t i o n m a t r i x f o r a c o m p o s i t e l a y e r . P r i o r i t y i s t h e r e f o r e g i v e n t o B r a v a i s Building block Method Computing time scale Comments Layer di f fraction matrix: Bravais l a t t i c e layer Composite layer by matrix inversion(mi) L 3N 2 L 3P 3N 2 < W 0 - 5 * Exact, large core size required i f P>4 Composite layer by reverse scattering perturbation(rsp) L 2P 2N 2 Possible convergence problem; smaller core size than (mi) Composite layer by combining (mi) and (rsp) Between (mi) and (rsp) core size comparable to (mi) Layer stacking: Renormalized forward scattering N2M 0 d m:„£1.0A; v e r s a t i l e min ' subroutine (RFSG) Layer doubling N 3 InM o d mj n^0.5A; many working matrices required. N = number of beams after symmetrization ( i f any) L = number of phase s h i f t s P = number of subplanes in the composite layer M = number of layers considered in the surface scattering Table 3 . 2 : Relative computing times for the building blocks in the 'combined space' multiple scattering calculations (After Van Hove and Tong[56]). 1 16 l a t t i c e l a y e r s whenever t h e minimum i n t e r l a y e r s p a c i n g d m ^ n o i s £ 0.5 A, a l t h o u g h c a s e s c a n be i d e n t i f i e d where a c o m p o s i t e l a y e r c a l c u l a t i o n i s c o n s i d e r e d e v e n when t h e B r a v a i s l a t t i c e l a y e r c a l c u l a t i o n i s p o s s i b l e . P e r t u r b a t i v e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s between l a y e r s a r e g e n e r a l l y l e s s c o m p u t a t i o n a l l y demanding t h a n t h e c a l c u l a t i o n s o f t h e l a y e r d i f f r a c t i o n m a t r i c e s . However i f many a d s o r p t i o n models i n v o l v i n g d i f f e r e n t i n t e r l a y e r s p a c i n g s between t h e a d s o r b a t e and t h e s u b s t r a t e have t o be t e s t e d , w h i c h i s t r u e f o r LEED c r y s t a l l o g r a p h y , t h e c o s t of s t a c k i n g becomes i m p o r t a n t . I n t h i s r e g a r d t h e f a s t e s t method, namely r e n o r m a l i z e d f o r w a r d s c a t t e r i n g , i s p r e f e r r e d o whenever t h e i n t e r l a y e r s p a c i n g s a r e above 1.1 A. A l s o t h e s u b r o u t i n e RFSG p r o v i d e s more f l e x i b i l i t y on t h e l a y e r p e r i o d i c i t y . L a y e r d o u b l i n g , on t h e o t h e r hand, h a s t o be u s e d t o s t a c k l a y e r s w i t h i n t e r l a y e r s p a c i n g s f r o m 0.5 t o o 1.0 A. However, t h e p r e s e n t l a y e r d o u b l i n g s u b r o u t i n e SUBREF c a n o n l y s t a c k b u l k l a y e r s w i t h 1- o r 2 - l a y e r p e r i o d i c i t i e s . 4 - l a y e r p e r i o d i c i t y (ABCDABCD....) c a n be h a n d l e d by f i r s t c o m b i n i n g AB and CD w i t h s u b r o u t i n e DBGL, and t h e n s t a c k i n g t h e two ' c o m p o s i t e ' l a y e r s w i t h SUBREF. 3 - l a y e r p e r i o d i c i t y , s u c h as shown by t h e f c c ( l 1 1 ) s u r f a c e , p o s e s a p r o b l e m i n t h e l a y e r d o u b l i n g scheme. A s o l u t i o n i n v o l v e s e x c l u d i n g a l l symmetry e x c e p t f o r one m i r r o r p l a n e . In t h i s c a s e , t h e s u r f a c e c a n be v i s u a l i z e d a s h a v i n g 1 - l a y e r p e r i o d i c i t y w i t h t h e same s l o p i n g i n t e r l a y e r v e c t o r ( c o n t a i n e d i n t h e m i r r o r p l a n e ) l i n k i n g a t o m i c c e n t e r s i n 1 17 c o n s e c u t i v e l a y e r s . I f t h e s m a l l e r i n t e r l a y e r s p a c i n g o c c u r s o n l y between t h e a d s o r b a t e and t h e t o p s u b s t r a t e l a y e r , t h e n t h e s y m m e t r i e s can be p r e s e r v e d e i t h e r by d o i n g a c o m p o s i t e l a y e r c a l c u l a t i o n a s m e n t i o n e d i n t h e p r e c e d i n g p a r a g r a p h , o r by l a y e r d o u b l i n g t h e two l a y e r s w i t h t h e s u b r o u t i n e DBGL and s t a c k i n g them w i t h t h e f a s t e r RFSG s u b r o u t i n e . C o n v e r g e n c e p r o b l e m s o c c u r on o c c a s i o n w i t h p e r t u r b a t i o n methods f o r c a s e s of s t r o n g m u l t i p l e s c a t t e r i n g . O b v i o u s n o n - c o n v e r g e n c e c a n u s u a l l y be i n d i c a t e d by t h e a p p e a r a n c e o f s p u r i o u s p e a k s i n s e v e r a l 1 ( E ) c u r v e s from t h e same c a l c u l a t i o n . T h i s i s l a r g e l y due t o t h e l a c k o f c o n s e r v a t i o n of e l e c t r o n c u r r e n t i n p e r t u r b a t i o n methods. T h e s e p r o b l e m s c a n be a v o i d e d by i n c r e a s i n g t h e damping a n d / o r i n c r e a s i n g t h e l a t t i c e v i b r a t i o n a l a m p l i t u d e w h i c h e f f e c t i v e l y r e d u c e s t h e i o n c o r e s c a t t e r i n g f a c t o r . A l t h o u g h t h e s e two p r o c e d u r e s a f f e c t t h e r e l a t i v e i n t e n s i t i e s o f r e f l e c t i o n maxima s l i g h t l y , t h e peak p o s i t i o n s a r e a l m o s t u n c h a n g e d [ 1 0 6 ] ; t h e s e r e s u l t s a r e a c c e p t a b l e f o r s t r u c t u r a l d e t e r m i n a t i o n s i n LEED c r y s t a l l o g r a p h y . F o r i d e n t i f y i n g t h e l e s s o b v i o u s c o n v e r g e n c e p r o b l e m s , Moore et al. [107] s u g g e s t e d p l o t t i n g t h e sum o f t h e emergent beam i n t e n s i t i e s v e r s u s t h e topmost i n t e r l a y e r s p a c i n g a t a p a r t i c u l a r e n e r g y ; any n o n - c o n v e r g e n c e w o u l d l e a d t o a sum d i s p l a c e d f r o m an o t h e r w i s e smooth c u r v e . The p r o b l e m i s t h e n r e c t i f i e d by c a r e f u l r e s e l e c t i o n o f i n p u t beams. The 'combined s p a c e ' a p p r o a c h has e s p e c i a l l y been e m p l o y e d i n t h e s t u d i e s o f oxygen a d s o r p t i o n on Z r ( 0 0 0 l ) 118 ( C h a p t e r 6), and i n t h e t h e o r e t i c a l s t u d y o f t h e s t a b i l i t y o f f r a c t i o n a l o r d e r beam i n t e n s i t i e s from s u p e r l a t t i c e s . W i t h c a r e f u l s e t - u p of t h e main p r o g r a m and m o n i t o r i n g o f o c c a s i o n a l c o n v e r g e n c e p r o b l e m s , t h i s a p p r o a c h p r o v i d e s an a f f o r d a b l e a l t e r n a t i v e t o e x a c t c a l c u l a t i o n s . E q u a l l y i m p o r t a n t i s t h e c o n s i s t e n c y o b t a i n e d from t h i s a p p r o a c h ; d i f f e r e n t s u b r o u t i n e s a p p l i e d t o t h e same a d s o r p t i o n s y s t e m have y i e l d e d a l m o s t i d e n t i c a l r e s u l t s . T a b l e 3.3 summarizes t h e f u n c t i o n s o f some i m p o r t a n t s u b r o u t i n e s f r e q u e n t l y u s e d i n t h i s work. 1 19 Subroutine Function ADREF1 DBG adds an overlayer onto a substrate, by layer doubling, producing a vector of ref l e c t e d amplitudes. combines 2 layers by layer doubling, producing 1 r e f l e c t i o n matrix (usually as the 'new' substrate). DBLG combines 2 layers by layer doubling, producing a l l r e f l e c t i o n and transmission matrices. MPERTI MTINV MSMF RFSG RINT SUBREF generates d i f f r a c t i o n matrices by reverse scattering perturbation and p a r t i a l matrix inversion i f requested. generates d i f f r a c t i o n matrices by matrix inversion for composite layers. generates d i f f r a c t i o n matrices for Bravais l a t t i c e type layers. stacks layers by renormalized forward scattering, producing a vector of ref l e c t e d amplitudes. produces beam i n t e n s i t i e s from beam amplitudes. stacks bulk layers with 1- or 2-layer p e r i o d i c i t y by layer doubling, producing a r e f l e c t i o n matrix only. Table 3 .3: Functions of several important and frequently used subroutines in the 'combined space' multiple scattering c a l c u l a t i o n s provided by Van Hove and Tong[56,97]. CHAPTER 4 GENERAL EXPERIMENTAL ASPECTS 120 121 4.1 THE UHV CHAMBER The s u c c e s s o f t h e ' c l e a n - s u r f a c e a p p r o a c h ' t o s u r f a c e s t u d i e s h as been b r o u g h t a b o u t , t o a l a r g e e x t e n t , by t h e c o m m e r c i a l a v a i l a b i l i t y o f u l t r a - h i g h vacuum (UHV) f a c i l i t i e s . The need f o r u s i n g UHV c a n be a p p r e c i a t e d by r e f e r e n c e t o a r e s u l t f r o m t h e k i n e t i c t h e o r y o f g a s e s . The number of m o l e c u l e s s t r i k i n g a u n i t a r e a o f a s u r f a c e i n u n i t t i m e i s g i v e n b y [ l 0 7 ] n = 3. 5 2 x 1 0 2 2 (MT) - 1 / 2p m o l e c u l e cm."2 s " 1 , (4.1) where P i s t h e p r e s s u r e i n t o r r , M i s t h e m o l e c u l a r w e i g h t o f t h e g a s e o u s m o l e c u l e , and T i s t h e t e m p e r a t u r e i n K. A t y p i c a l low M i l l e r - i n d e x m e t a l s u r f a c e h a s a p p r o x i m a t e l y 1 0 1 5 a d s o r p t i o n s i t e s p e r cm 2, so a gas s u c h a s 0 2 a t room t e m p e r a t u r e c a n f i l l up a l l t h e a d s o r p t i o n s i t e s i n a b o u t one s e c o n d a t 10" 6 t o r r , a s s u m i n g a c o n s t a n t s t i c k i n g c o e f f i c i e n t o f one. However, a t a p r e s s u r e o f 1 0 " 1 0 t o r r , a m o n o l a y e r i s formed i n =*160 m i n u t e s . A l t h o u g h t h e s t i c k i n g c o e f f i c i e n t s f o r most g a s e s a r e below u n i t y , t h e above c o m p a r i s o n d o e s i n d i c a t e how i m p o r t a n t UHV i s t o keep a s u r f a c e i n a w e l l - d e f i n e d s t a t e f o r t h e d u r a t i o n o f an e x p e r i m e n t . The work i n t h i s t h e s i s was c a r r i e d o u t i n a V a r i a n FC12 vacuum chamber. T h i s p a r t i c u l a r chamber i s made of d e m a g n e t i z e d s t a i n l e s s s t e e l and i s e q u i p p e d w i t h numerous p o r t s u t i l i z i n g c o n f l a t f l a n g e s . F i g u r e 4.1 shows t h e l a y o u t 122 LEED OPTICS CMA VIEWING WINDOW Figure 4.1: Schematic diagram of the FC12 UHV chamber and some of • i t s important assessories. AES = Auger electron spectroscopy; CMA = c y l i n d r i c a l mirror analyzer. 1 23 o f t h e UHV chamber and t h e major a n a l y t i c a l t o o l s a v a i l a b l e . The a c c o m p a n y i n g pumping s y s t e m f o r t h e FC12 chamber and t h e f a c i l i t i e s f o r s t a r t - u p pumping a r e s c h e m a t i c a l l y o u t l i n e d i n F i g u r e 4.2. The t o t a l pumping s y s t e m c o n s i s t s o f f o u r t y p e s o f pumps w h i c h i n c l u d e two s o r p t i o n pumps, a d i f f u s i o n pump, an a u x i l i a r y t i t a n i u m s u b l i m a t i o n pump, a c e n t r a l i o n pump (200 L s " 1 ) and a s m a l l i o n pump (20 L s " 1 ) f o r t h e gas i n l e t l i n e . A s o r p t i o n pump c o n s i s t s of z e o l i t e i n a c o n t a i n e r c o o l e d by l i q u i d n i t r o g e n . The z e o l i t e h as l a r g e e f f e c t i v e s u r f a c e a r e a and i t c a n p h y s i c a l l y a b s o r b l a r g e amounts of g a s , t h e r e b y r e d u c i n g t h e p r e s s u r e of t h e chamber t o t h e 10" 3 t o r r r a n g e from a t m o s p h e r i c p r e s s u r e v e r y q u i c k l y . A d i f f u s i o n pump works by e n t r a i n m e n t o f g a s e o u s s p e c i e s n e a r t h e pump i n l e t by h i g h v e l o c i t y s t r e a m s o f v a p o r of a n o n - v o l a t i l e l i q u i d ( e . g . p o l y p h e n y l e t h e r ) e j e c t e d from a b o i l e r . The s t r e a m s o f v a p o r l a t e r c o n d e n s e on w a t e r - c o o l e d w a l l s and f l o w back down t o t h e b o i l e r w h i l e t h e g a s e o u s s p e c i e s a r e pumped away by a r o t a r y pump. A l i q u i d n i t r o g e n c o l d t r a p i s u s u a l l y c o n n e c t e d between t h e UHV chamber and t h e d i f f u s i o n pump t o p r e v e n t any p o s s i b l e back d i f f u s i o n of t h e n o n - v o l a t i l e l i q u i d . B o t h t h e i o n and t i t a n i u m s u b l i m a t i o n pumps a r e examples of ' g e t t e r t y p e ' pumps. Ion pumps depend on a d i s c h a r g e t o p r o d u c e i o n s f r o m t h e gas t o be pumped. Th e s e i o n s a r e t h e n t r a p p e d by m a g n e t i c f i e l d s and d i r e c t e d on t o r e a c t i v e ' g e t t e r ' p l a t e s made of t i t a n i u m w h i c h remove t h e i o n s by t h e f o r m a t i o n of s t a b l e s o l i d 124 compounds. N o b l e g a s e s a r e pumped by b u r i a l , a l t h o u g h t h a t mechanism i s not v e r y e f f e c t i v e . T i t a n i u m s u b l i m a t i o n pumps, on t h e o t h e r hand, s i m p l y e v a p o r a t e f i l m s of t i t a n i u m w h i c h have v e r y h i g h a d s o r p t i o n r a t e s f o r r e a c t i v e g a s e s . The b u r i a l r o u t e i s a g a i n s l o w . Thus ' g e t t e r t y p e ' pumps a r e g e n e r a l l y n o t us e d d i r e c t l y a f t e r a r g o n bombardment ( S e c t i o n 4 . 2 . 2 ) , b u t i n s t e a d t h e d i f f u s i o n pump i s u s e d t o 'rough o u t ' t h e s y s t e m u n t i l t h e p r e s s u r e i s down t o a l e v e l ( e . g . 10" 7 t o r r ) w h i c h t h e i o n pump c a n h a n d l e . I n t h e s t a r t - u p p r o c e d u r e , t h e i n i t i a l pumping i s p e r f o r m e d by one o r b o t h s o r p t i o n pumps t o r e d u c e t h e p r e s s u r e o f t h e chamber t o t h e 10" 3 t o r r r a n g e . A t t h i s s t a g e t h e w e l l - t r a p p e d ( l i q u i d n i t r o g e n ) d i f f u s i o n pump i s u s e d t o pump t h e chamber down t o a r o u n d 10" 7 t o r r . F i n a l l y t h e c e n t r a l i o n pump i s s w i t c h e d on and t h e chamber i s bak e d a t 2 0 0 - 2 5 0 ° C f o r 12-15 h o u r s . The chamber i s t h e n a l l o w e d t o c o o l t o room t e m p e r a t u r e , and i n t h e a b s e n c e o f l e a k s , t h e p r e s s u r e s h o u l d r e a c h t h e 10" 1 0 t o r r r a n g e . D e g a s s i n g a l l t h e f i l a m e n t s and s w i t c h i n g on t h e t i t a n i u m s u b l i m a t i o n pump f o r a b o u t h a l f an hour s h o u l d r e d u c e t h e p r e s s u r e f u r t h e r , e v e n t o t h e m i d - 1 0 " 1 1 t o r r r a n g e . A l t h o u g h i t does not o c c u r v e r y o f t e n , l e a k s c a n o c c u r between t h e s e a l s . When t h e l e a k r e s u l t s i n a p r e s s u r e >10" 7 t o r r , i t c a n u s u a l l y be d e t e c t e d by a h e l i u m d e t e c t o r ( V a r i a n 9 2 5 - 4 0 ) . S m a l l e r l e a k s c a n be d e t e c t e d by s q u i r t i n g m e t h a n o l on t o t h e s l i t between t h e f l a n g e s where t h e l e a k i s s u s p e c t e d ; a sudden d r o p i n p r e s s u r e r e s u l t i n g from a 125 Degas a l l f i l a m e n t s D . P . = d i f f u s i o n pump T . S . P . = t i t a n i u m s u b l i m a t i o n pump 200 I . 1/s P. B a k e - o u t f o r 12 h r s . I.P. = i on pump S . P . = s o r p t i o n pump F i g u r e 4 . 2 : ( a ) P u m p i n g s y s t e m a s s o c i a t e d w i t h t h e FC12 UHV chamber. ( b ) S t a r t - u p p r o c e d u r e f o r pump-down from a t m o s p h e r i c p r e s s u r e t o UHV. 126 t e m p o r a r y f r e e z i n g of t h e m e t h a n o l would i n d i c a t e t h e p r e s e n c e o f a l e a k . D e t a i l e d i n f o r m a t i o n on pumping methods, measurements of p r e s s u r e and r e l a t e d t o p i c s a r e g i v e n i n s e v e r a l r e v i e w s on UHV t e c h n i q u e s ! 1 0 8 - 1 1 0 ] . The f o l l o w i n g f a c i l i t i e s a r e a l s o a v a i l a b l e i n t h e FC12 chamber: a s e t o f 4 - g r i d LEED o p t i c s ( V a r i a n 981-0127); a s i n g l e p a s s c y l i n d r i c a l m i r r o r a n a l y z e r ( V a r i a n 981-2607); a g l a s s v i e w p o r t d i r e c t l y o p p o s i t e t o t h e f l u o r e s c e n t LEED s c r e e n ; an i o n gun ( V a r i a n 981-2043) f o r c l e a n i n g by i o n bombardment; a g l a n c i n g i n c i d e n c e e l e c t r o n gun ( V a r i a n 981-2454) gun f o r Auger e l e c t r o n s p e c t r o s c o p y ; a molydenum sample h o l d e r w i t h a r e s i s t i v e h e a t i n g b l o c k mounted on a sample m a n i p u l a t o r ( V a r i a n 981-2530) w h i c h a l l o w s ( x , y , z ) l i n e a r t r a n s l a t i o n a n d two d e g r e e s o f r o t a t i o n a l f r e e d o m ( s i n c e t h e FC12 chamber i s b u i l t f o r m u l t i - t e c h n i q u e a n a l y s i s , t h e sample i s h e l d on t h e m a n i p u l a t o r 2^" o f f t h e v e r t i c a l a x i s ) ; a q u a d r u p l e mass s p e c t r o m e t e r (EAI 150A) t o m o n i t o r t h e c o m p o s i t i o n o f t h e r e s i d u a l g a s e s ; a v a r i a b l e l e a k v a l v e f o r l e t t i n g i n g a s e s ; and a nude i o n gauge f o r p r e s s u r e measurement. M a g n e t i c f i e l d s i n t h e chamber a r e a n n u l l e d by t h r e e o r t h o g o n a l s e t s o f H e l m o l t z c o i l s , u s i n g a H a l l p r o b e t o s e t z e r o f i e l d a t t h e sample p o s i t i o n . Z e r o f i e l d c an f u r t h e r be c h e c k e d by o b s e r v i n g t h e s p e c u l a r beam i n t h e LEED e x p e r i m e n t : t h e p o s i t i o n o f t h e (0,0) beam w i l l n o t c h a n ge, even a t t h e l o w e s t e n e r g i e s , i f t h e r e g i o n between t h e sample and t h e LEED s c r e e n i s f i e l d - f r e e . 1 27 4.2 SAMPLE PREPARATION AND CLEANING 4.2.1 CRYSTALLOGRAPHIC PLANE ORIENTATION The s u r f a c e s t u d i e d w i t h LEED i n t h i s l a b o r a t o r y i s u s u a l l y i n t h e f o r m o f a t h i n c i r c u l a r d i s k w i t h d i a m e t e r a r o u n d 5 mm and t h i c k n e s s between 1 and 2 mm. The f o l l o w i n g p r o c e d u r e i s t a k e n t o o b t a i n a d e s i r e d c r y s t a l l o g r a p h i c p l a n e . A h i g h - p u r i t y s i n g l e c r y s t a l i s mounted on a g o n i o m e t e r w i t h p l a s t i c cement and aluminum powder. The p u r p o s e o f t h e l a t t e r i s t o make e l e c t r i c a l c o n t a c t between t h e c r y s t a l and t h e g o n i o m e t e r f o r t h e s p a r k e r o s i o n s t e p a f t e r o r i e n t a t i o n . B e t t e r e l e c t r i c a l c o n t a c t c a n be a c h i e v e d by t y i n g t h e c r y s t a l w i t h a c o p p e r w i r e , one end o f w h i c h i s wrapped t o t h e base o f t h e g o n i o m e t e r . The c r y s t a l i s o r i e n t e d t o t h e d e s i r e d p l a n e u s i n g back r e f l e c t i o n Lau6 X - r a y d i f f r a c t o g r a p h y [ 1 1 1 ] . T h i s i n v o l v e s a n a l y z i n g a p r e - o r i e n t e d d i f f r a c t i o n p h o t o g r a p h and s u b s e q u e n t l y c o r r e c t i n g t h e a n g l e t h r o u g h t h e m a n i p u l a t i o n o f t h e g o n i o m e t e r m i c r o m e t e r s ! 1 1 2 ] . A f t e r o r i e n t a t i o n , s l i c e s o f t h e d e s i r e d p l a n e a r e c u t f r o m t h e r o d u s i n g t h e s p a r k e r o s i o n t e c h n i q u e ( ' A g i e t r o n ' , A g i e , S w i t z e r l a n d ) . The c r y s t a l d i s k i s t h e n mounted i n an a c r y l i c r e s i n ('Quickmount', F u l t o n M e t a l l u r g i c a l P r o d u c t s ) , and g l u e d t o a p o l i s h i n g j i g [ l 1 3 ] , w h i c h has a l i g n m e n t m i c r o m e t e r s t h a t a l l o w t h e sample o r i e n t a t i o n t o be a d j u s t e d . The whole a s s e m b l y c a n be f i t t e d on t o t h e t r a c k o f t h e X - r a y d i f f r a c t o m e t e r e n a b l i n g m i n o r a d j u s t m e n t s t o be 128 made t o compensate f o r t h e e r r o r s i n a l i g n m e n t i n t r o d u c e d d u r i n g t h e c u t t i n g s t a g e ( t y p i c a l l y ^ 2 ° ) . A f t e r r e - a d j u s t m e n t , t h e j i g i s p u t on t o a p l a n e t a r y l a p p i n g s y s t e m (DU 172, C a n a d i a n T h i n F i l m s L t d . ) f o r p o l i s h i n g . The c r y s t a l i s m e c h a n i c a l l y p o l i s h e d w i t h p r o g r e s s i v e l y f i n e r d iamond p a s t e , s t a r t i n g u s u a l l y a t 9 u and f i n i s h i n g a t 1 u. The p o l i s h i n g f r o m 3 u down i s done m a n u a l l y w i t h a r t i f i c i a l d e e r s k i n ( M i c r o c l o t h , B u e h l e r 40-7218) t o m i n i m i z e damage t o t h e s u r f a c e . A f t e r t h e \~n p o l i s h , t h e s u r f a c e i s n o r m a l l y smooth enough f o r an o p t i c a l f a c e a l i g n m e n t by back r e f l e c t i o n o f a Ne/He l a s e r f i x e d on an o p t i c a l b e n ch as shown i n F i g u r e 4.3. When t h e r e f l e c t e d l a s e r beam makes an a n g l e l e s s t h a n h a l f a d e g r e e w i t h t h e i n c i d e n t beam, t h e p o l i s h i n g i s c o n s i d e r e d s a t i s f a c t o r y . The whole a s s e m b l y i s t h e n moved back t o t h e X - r a y d i f f T a c t o m e t e r , and a Laue p h o t o g r a p h t a k e n t o e n s u r e t h a t t h e o p t i c a l f a c e c o i n c i d e s w i t h t h e d e s i r e d c r y s t a l l o g r a p h i c p l a n e t o w i t h i n i ° . A t t h i s s t a g e t h e c r y s t a l i s r e t r i e v e d f r o m t h e r e s i n by d i s s o l v i n g t h e l a t t e r i n a c e t o n e . Any s m e a r i n g i n t r o d u c e d by t h e 1-^ p o l i s h can be smoothed o u t by g e n t l e p o l i s h i n g w i t h 0.05 u a l u m i n a on t y p i n g p a p e r f o r 30 t o 60 s e c o n d s . The c r y s t a l i s t h e n r i n s e d w i t h d e - i o n i z e d w ater i n an u l t r a s o n i c b a t h , and t h o r o u g h l y d e g r e a s e d w i t h t r i c h l o r o e t h y l e n e . The c r y s t a l i s o f t e n a c i d - e t c h e d or e l e c t r o p o l i s h e d t o y i e l d a s h i n y s u r f a c e . LASER SOURCE He/Ne (a) > ri identical brass stands paper • micrometers crystal ^iiivTI?ii7iii?iuiiiiiiiiiii)iiiiii}iiiii parallel optical benches desired crystallographic Figure 4 . 3 : (a)Schematic diagram of laser alignment of o p t i c a l and crystallographic planes of a single c r y s t a l . (b)A blow-up to show the relationship between the o p t i c a l and crystallographic planes. Alignment i s acceptable when 9^\°. 130 4 . 2 . 2 S U R F A C E C L E A N I N G I N UHV CHAMBER A r g o n i o n b o m b a r d m e n t i s t h e m o s t c o m m o n m e t h o d f o r c l e a n i n g a c r y s t a l s u r f a c e u n d e r U H V . T h e d a m a g e d o n e b y b o m b a r d m e n t i s r e l i e v e d b y a n n e a l i n g . When t h e c r y s t a l i s f i r s t p l a c e d i n t h e c h a m b e r , t h e m o s t a b u n d a n t i m p u r i t i e s o n t h e s u r f a c e a r e f r e q u e n t l y o x y g e n a n d c a r b o n . T h e s e c a n b e r e m o v e d b y A r * b o m b a r d m e n t , p r e f e r a b l y a t r o o m t e m p e r a t u r e t o a v o i d t h e i r p o s s i b l e d i f f u s i o n i n t o t h e b u l k [ l 1 4 ] a t e l e v a t e d t e m p e r a t u r e s . A r g o n i o n s w i t h e n e r g i e s i n t h e r a n g e o f 1 t o 2 k e V a r e u s e d f o r t h e b o m b a r d m e n t . I n t h e e a r l y s t a g e s , t h e h i g h e n e r g y e n d o f t h e r a n g e i s u s e d t o s p u t t e r o f f t h e i m p u r i t i e s ; t h i s may t a k e a s l o n g a s 3 0 - 5 0 h o u r s , d e p e n d i n g o n t h e t y p e o f c r y s t a l a n d i t s p r o c e s s h i s t o r y . W h e n e v e r p o s s i b l e l o w e n e r g y i o n s a r e p r e f e r r e d t o m i n i m i z e d a m a g e o r p r o f i l i n g o f t h e s u r f a c e . T o c a r r y o u t t h e A r + b o m b a r d m e n t , t h e c e n t r a l i o n p u m p i s s w i t c h e d o f f a n d a r g o n g a s i s l e a k e d i n t o t h e c h a m b e r u n t i l t h e p r e s s u r e i s i n t h e m i d - 1 0 " 5 t o r r r a n g e . T h e i o n c u r r e n t o n t h e s u r f a c e s h o u l d b e a r o u n d 4 t o 6 M A . D u r i n g t h e b o m b a r d m e n t , t h e s u b l i m a t i o n p u m p i s k e p t s w i t c h e d o n . S i n c e t h e s u b l i m a t i o n p u m p i s n o t e f f e c t i v e i n r e m o v i n g a r g o n v i a t h e b u r i a l r o u t e , i t e s s e n t i a l l y p u m p s a w a y i m p u r i t i e s k n o c k e d o f f f r o m t h e s u r f a c e o f t h e s a m p l e i n a d i f f e r e n t i a l m a n n e r . A f t e r s e v e r a l h o u r s o f b o m b a r d m e n t , t h e s a m p l e s h o u l d b e c h e c k e d t o a s s e s s p r o g r e s s i n c l e a n i n g . F o r t h i s t h e w h o l e c h a m b e r m u s t b e p u m p e d d o w n , f i r s t w i t h t h e d i f f u s i o n p u m p a n d t h e n w i t h t h e i o n p u m p . A t t h i s s t a g e a n A u g e r s p e c t r u m i s t a k e n 131 to assess s u r f a c e c l e a n l i n e s s . I f impurity l e v e l s are s t i l l h i g h , Ar* bombardment i s repeated using f r e s h argon gas. E v e n t u a l l y when the Auger spectrum i n d i c a t e s an e s s e n t i a l l y c l e a n s u r f a c e , the sample can be heated to d r i v e i m p u r i t i e s from the bulk to the s u r f a c e , and to anneal out damage to the s u r f a c e caused by the ion bombardment procedures. These are repeated c y c l i c a l l y u n t i l no a p p r e c i a b l e i m p u r i t i e s can be d e t e c t e d by Auger e l e c t r o n spectroscopy and the s u r f a c e i s s u f f i c i e n t l y w e l l - o r d e r e d to show a sharp LEED p a t t e r n . I f the bulk i m p u r i t i e s were not e f f e c t i v e l y d r i v e n out i n the p r e l i m i n a r y stages of c l e a n i n g they would l i k e l y appear d u r i n g the a n n e a l i n g steps p r i o r to the f i r s t i n v e s t i g a t i o n s with LEED. A r e s i s t i v e heater (Varian 981-2058) with tantalum c l i p s i s used f o r a n n e a l i n g and h e a t i n g of the sample. The sample temperature i s measured with a 0.005" alumel-chromel thermocouple spot-welded to the edge of the c r y s t a l d i s k . I t i s important to know the m e l t i n g p o i n t and any phase t r a n s i t i o n temperature of the p a r t i c u l a r c r y s t a l before any hea t i n g procedure i s undertaken. R e s i d u a l hydrocarbon i m p u r i t i e s knocked o f f from the sample may not be pumped q u i c k l y by the ion or s u b l i m a t i o n pumps. However, i t i s found that l e a k i n g oxygen i n t o the chamber at 10" 8 t o r r f o r s e v e r a l minutes can reduce the amount of these r e s i d u a l gases, presumably by c r a c k i n g them down i n t o some r e a c t i v e forms. 1 3 2 4 . 2 . 3 S U R F A C E C O M P O S I T I O N B Y A U G E R E L E C T R O N S P E C T R O S C O P Y S u r f a c e s s t u d i e d w i t h L E E D c a n b e c o n v e n i e n t l y a s s e s s e d w i t h A u g e r e l e c t r o n s p e c t r o s c o p y . T h e l a t t e r , a s a l r e a d y m e n t i o n e d i n S e c t i o n 2 . 3 . 4 , i s u s e f u l n o t o n l y f o r d e t e c t i n g a n d i d e n t i f y i n g i m p u r i t i e s , b u t a l s o f o r e s t i m a t i n g t h e r e l a t i v e c o v e r a g e s o f a d l a y e r s u n d e r i n v e s t i g a t i o n . I n t h i s l a b o r a t o r y , e l e c t r o n b e a m s a r e e m p l o y e d a s t h e e x c i t a t i o n s o u r c e f o r A u g e r e l e c t r o n s p e c t r o s c o p y . A s a r e s u l t , w e a k A u g e r e l e c t r o n s i g n a l s a r e s u p e r i m p o s e d o n a m u c h l a r g e r g e n e r a l b a c k g r o u n d ! 6 7 , 1 1 5 ] . T w o e l e c t r o s t a t i c a n a l y z e r s o f t h e r e t a r d i n g f i e l d a n d d i s p e r s i v e t y p e s ! 1 1 6 ] a r e a v a i l a b l e f o r t h e d e t e c t i o n o f t h e A u g e r s i g n a l s . A r e t a r d i n g f i e l d a n a l y z e r u s e s t h e h e m i s p h e r i c a l g r i d s o f t h e c o n v e n t i o n a l L E E D d i s p l a y s y s t e m a s s h o w n i n F i g u r e 4 . 4 . I n s u c h a s e t - u p , t h e e l e c t r o n g u n , t h e s a m p l e , a n d t h e g r i d s G1 a n d G 4 a r e a l l e a r t h e d , w h i l e t h e s c r e e n i s b i a s e d a t + 3 0 0 V t o c o l l e c t e l e c t r o n s e m i t t e d f r o m t h e s a m p l e . T h e e l e c t r o n g u n d e l i v e r s a n e l e c t r o n b e a m w i t h a t y p i c a l e n e r g y =*2 k e V a n d a c u r r e n t o f » 2 0 M A . A v a r i a b l e r e t a r d i n g v o l t a g e V r i s a p p l i e d t o g r i d s G 2 a n d G 3 . T h i s v o l t a g e i s r a m p e d b y a m u l t i - c h a n n e l a n a l y z e r ( F a b r i t e k 1 0 6 2 ) l i n k e d t o a p r o g r a m m a b l e p o w e r s u p p l y ( K e p c o O P S 2 0 0 0 ) . T h e t o t a l c u r r e n t c o l l e c t e d f r o m t h e s c r e e n i s a f u n c t i o n o f V r . A d e r i v a t i v e s p e c t r u m c a n b e o b t a i n e d b y s u p e r i m p o s i n g o n V r a s m a l l m o d u l a t i o n V m s i n o t ( f r e q u e n c y C J ^1 k H z ) . P r o v i d e d t h a t V m i s s m a l l ( e . g . < 1 0 V ) , t h e a m p l i t u d e o f t h e c o m p o n e n t o f t h e c o l l e c t o r c u r r e n t a t t h e m o d u l a t i n g MULTI-CHANNEL ANALYZER X-Y PLOTTER SCOPE g u r e 4.4: S c h e m a t i c d i a g r a m o f LEED o p t i c s u s e d a t a r d i n g f i e l d a n a l y z e r f o r AES. 134 f r e q u e n c y i s v m ^ d I / d V r ^ ' a n d fc^e a m P l i f c u d e a t t h e s e c o n d h a r m o n i c i s -0 . 2 5 V m ( d 2 1 / d V 2 . ) [ 1 1 7 ] . T h e r e f o r e , by d e t e c t i n g w i t h a l o c k - i n a m p l i f i e r (PAR HR8) t h e c u r r e n t r e c e i v e d by t h e c o l l e c t o r s c r e e n a t t h e a p p r o p r i a t e f r e q u e n c i e s , t h e d e r i v a t i v e s ( d l / d V r ) and ( d 2 I / d V 2 ) a r e r e a d i l y o b t a i n e d as a f u n c t i o n of V r . The s e c o n d d e r i v a t i v e e s s e n t i a l l y e l i m i n a t e s t h e s l o w l y v a r y i n g b a c k g r o u n d c u r r e n t and g i v e s a s p e c t r u m w i t h p r o m i n e n t Auger p e a k s a s shown i n F i g u r e 2.12. The o u t p u t f r o m t h e l o c k - i n a m p l i f i e r a f t e r e a c h sweep c a n be s t o r e d i n t h e F a b r i t e k and s i g n a l a v e r a g e d u n t i l . t h e s i g n a l t o n o i s e r a t i o i s s a t i s f a t o r y . A t y p i c a l s p e c t r u m r e q u i r e s a b o u t 10 s c a n s , and t a k e s 3-5 m i n u t e s . A t u n e d p r e - a m p l i f i e r b a s e d on t h e d e s i g n by Nathan and H o p k i n s [ l l 8 ] i s u s e d t o i s o l a t e t h e h i g h v o l t a g e a p p l i e d t o t h e c o l l e c t o r s c r e e n . The use o f two r e t a r d i n g g r i d s (G2,G3) p r o v i d e s b e t t e r e n e r g y r e s o l u t i o n i n g e n e r a l . The e a r t h e d g r i d (G4) i s u s e d h e r e t o remove t h e c a p a c i t a n c e between t h e r e t a r d i n g g r i d s and t h e c o l l e c t o r s c r e e n w h i c h w o u l d o t h e r w i s e g e n e r a t e unwanted s i g n a l s . The r e t a r d i n g f i e l d a n a l y z e r has been p o p u l a r b e c a u s e i t d i r e c t l y u s e s a 4 - g r i d LEED o p t i c s . In a d d i t i o n , i t has a v e r y h i g h c o l l e c t o r e f f i c i e n c y : between 10-20% o f t h e Auger e l e c t r o n s e m i t t e d f r o m t h e sample c a n r e a c h t h e s c r e e n . The m a j o r d i s a d v a n t a g e o f a r e t a r d i n g f i e l d a n a l y z e r i s t h a t i t s s e n s i t i v i t y i s r e s t r i c t e d by h i g h ' s h o t n o i s e ' . The l a t t e r i s p r o p o r t i o n a l t o / I E + , where I E + r e p r e s e n t s t h e t o t a l c o l l e c t e d c u r r e n t c o n t r i b u t e d by e l e c t r o n s h a v i n g e n e r g i e s 135 g r e a t e r t h a n E, t h e Auger e n e r g y of i n t e r e s t . The h i g h ' s h o t n o i s e ' r e d u c e s t h e s i g n a l t o n o i s e r a t i o b e c a u s e t h e l a t t e r i s d e p e n d e n t on I £ E / V l E + (where 1^, i s t h e c u r r e n t f r o m t h o s e e l e c t r o n s w i t h e n e r g i e s E ± A E t ) . The s i g n a l t o n o i s e r a t i o c a n be e n h a n c e d by up t o 10 2 by u s i n g a d i s p e r s i v e t y p e a n a l y z e r , w h i c h o p e r a t e s v i a v e l o c i t y s e l e c t i o n . W i t h s u c h an a n a l y z e r , o n l y e l e c t r o n s w i t h i n a narrow e n e r g y s p r e a d AE a r e c o l l e c t e d . The s i g n a l t o n o i s e r a t i o now s c a l e s a s I^g/Vl^g, and i s c l e a r l y much more f a v o r a b l e t h a n t h a t f o r a r e t a r d i n g f i e l d a n a l y z e r . The d i s p e r s i v e t y p e a n a l y z e r u s e d i n t h i s l a b o r a t o r y i s a c y l i n d r i c a l m i r r o r a n a l y z e r (CMA). The CMA i s u s e d i n c o n j u n c t i o n w i t h a g l a n c i n g i n c i d e n c e p r i m a r y beam w h i c h c a n improve t h e s u r f a c e s e n s i t i v i t y . F i g u r e 4.5 shows a s c h e m a t i c e x p e r i m e n t a l s e t - u p f o r Auger e l e c t r o n s p e c t r o s c o p y u s i n g t h e CMA and t h e g l a n c i n g a n g l e e l e c t r o n gun. A t y p i c a l p r i m a r y beam h e r e has an e n e r g y a r o u n d 2-3 keV, a c u r r e n t o f 100-200 uh, a n d a c r o s s - s e c t i o n a l a r e a ^1 mm2. The CMA c o n s i s t s b a s i c a l l y o f two c o - a x i a l c y l i n d e r s o f r a d i i r , ( i n n e r ) and r 2 ( o u t e r ) ( F i g u r e 4 . 5 ) , w i t h e n t r a n c e and e x i t g r i d s c u t i n t h e i n n e r c y l i n d e r ! 1 1 9 ] . The i n n e r c y l i n d e r i s g r o u n d e d w h i l e a v a r i a b l e r e p u l s i v e v o l t a g e V r i s a p p l i e d t o t h e o u t e r c y l i n d e r . O n l y a narrow band o f e l e c t r o n s w i t h mean e n e r g y E have t r a j e c t o r i e s w h i c h c a n p a s s t h r o u g h t h e e x i t g r i d s and be d e t e c t e d by t h e c h a n n e l e l e c t r o n m u l t i p l i e r . The r e s t w i l l c o l l i d e w i t h t h e w a l l s o f TAE i s d e t e r m i n e d by t h e i n s t r u m e n t a l r e s o l u t i o n . 1 3 6 SAMPLE GLANCING ANGLE ELECTRON GUN PROGRAMMABLE POWER SUPPLY CYLINDRICAL MIRROR ANALYZER ELECTRON MULTIPLIER RAMP GEN. SIGNAL GEN. LOCK-IN PRE AMP. ISOLATION PRE AMP. SCOPE X-Y PLOTTER F i g u r e 4.5: Schematic diagram of the experimental set-up for AES using a c y l i n d r i c a l mirror analyzer and glancing incidence electron gun. 137 t h e c y l i n d e r s and e v e n t u a l l y d i e o u t . The f r o n t end of t h e e l e c t r o n m u l t i p l i e r i s u s u a l l y e a r t h e d , b u t i s s e t t o a p o t e n t i a l +300V when e l e c t r o n s w i t h k i n e t i c e n e r g i e s l e s s t h a n 50 eV a r e t o be d e t e c t e d . A p r e - a m p l i f i e r i s u s e d t o i s o l a t e t h e h i g h e r v o l t a g e (2-2.5 kV) on t h e e l e c t r o n m u l t i p l i e r [ 1 2 0 ] . In o r d e r t o o b t a i n a d e r i v a t i v e s p e c t r u m of t h e t y p e shown i n F i g u r e 2.12, a s m a l l m o d u l a t i o n v o l t a g e V m s i n a ; t (=*3V a t 5-10 kHz) i s s u p e r i m p o s e d on V r . In c o n t r a s t t o t h e RFA, t h e o u t p u t from t h e l o c k - i n a m p l i f i e r w i t h t h e CMA i s s e t a t t h e m o d u l a t i o n f r e q u e n c y [ 6 7 ] , and i t y i e l d s d i r e c t l y d e r i v a t i v e A u g e r s p e c t r a a s shown i n F i g u r e 2.12. Due t o t h e i m p r o v e d s i g n a l t o n o i s e r a t i o , a s i n g l e s c a n i s a b l e t o g i v e a s p e c t r u m , i n l e s s t h a n 30 s e c o n d s , w h i c h i s s u p e r i o r t o t h o s e o b t a i n e d w i t h t h e r e t a r d i n g f i e l d a n a l y z e r . The i n c r e a s e d s p e e d e n a b l e s t h e e x p e r i m e n t e r t o s c a n d i f f e r e n t p a r t s o f t h e sample s u r f a c e q u i c k l y . In a d s o r p t i o n s t u d i e s , t h e i n c r e a s i n g a d l a y e r c o v e r a g e i n an a d s o r b a t e u p t a k e c u r v e c a n be c o n t i n u o u s l y m o n i t o r e d by t h e g r o w t h o f an Auger p e a k . 4.3 THE LEED EXPERIMENT 4.3.1 LEED OPTICS F i g u r e 4.6 shows t h e e l e c t r o n o p t i c s u s e d i n t h e FC12 chamber. T h e r e a r e two main p a r t s , namely t h e e l e c t r o n gun, w h i c h d i r e c t s e l e c t r o n s w i t h v a r i a b l e e n e r g i e s on t o t h e sample, and t h e d e t e c t o r , w h i c h e n e r g y - a n a l y z e s t h e back 138 s c a t t e r e d e l e c t r o n s . D e t a i l s on LEED o p t i c s and q u a n t i t a t i v e measurement o f LEED beam i n t e n s i t i e s c a n be f o u n d i n a r e v i e w by L a g a l l y and M a r t i n [ l 2 ] . The e l e c t r o n gun ( V a r i a n 981-2125) p r o d u c e s an e l e c t r o n beam by t h e r m i o n i c e m i s s i o n f r o m a t u n g s t e n c a t h o d e ; t h e s e e l e c t r o n s a r e a c c e l e r a t e d and f o c u s s e d t h r o u g h anode p l a t e s . When t h e gun i s w e l l - t u n e d , i t g i v e s a c o l l i m a t e d beam w i t h a c r o s s - s e c t i o n a l d i a m e t e r <0.75 mm i n t h e e n e r g y r a n g e between 10 and 300 eV. The e n e r g y s p r e a d of t h e e l e c t r o n s i s d e t e r m i n e d m a i n l y by t h e f i l a m e n t t e m p e r a t u r e T. F o r a M a x w e l l i a n d i s t r i b u t i o n , t h e w i d t h a t half-maximum i s g i v e n by AE = 2.54kT, (4.2) where k i s t h e B o l t z m a n n c o n s t a n t . A l t h o u g h f i l a m e n t m a t e r i a l s w i t h low o p e r a t i n g t e m p e r a t u r e s c a n be u s e d f o r LEED guns, c o n s i d e r a t i o n s o f c h e m i c a l s t a b i l i t y and l i f e t i m e h ave e n c o u r a g e d t h e use o f t u n g s t e n f i l a m e n t s . T h e s e f i l a m e n t s o p e r a t e a t a r o u n d 2300K, t h u s g i v i n g a AE o f a p p r o x i m a t e l y 0.5 eV. The beam c u r r e n t i n c r e a s e s a l m o s t l i n e a r l y w i t h e n e r g y below a b o u t 100 eV, and t h e n l e v e l s o f f . T h i s v a r i a t i o n has t o be r e c o r d e d f o r n o r m a l i z i n g m e asured LEED s p o t i n t e n s i t i e s ( o t h e r w i s e t h e i n t e n s i t i e s w o u l d a p p e a r a r t i f i c i a l l y r e d u c e d a t low e n e r g i e s ) . GUN CONTROL Figure 4 .6: Schematic diagram of the electron optics used for LEED experiments. 1 40 4.3.2 DISPLAY OF THE LEED PATTERN The e l e c t r o n s back s c a t t e r e d f r o m t h e c r y s t a l s u r f a c e a r e t r a d i t i o n a l l y d i s p l a y e d on a f l u o r e s c e n t s c r e e n w h i c h i s s i t u a t e d b e h i n d t h e 4 - g r i d s y s t e m . The l a t t e r c a n a l s o be u s e d f o r t h e d e t e c t i o n o f Auger e l e c t r o n s ( S e c t i o n 4 . 2 . 3 ) . In a LEED e x p e r i m e n t , t h e sample i s p o s i t i o n e d a t t h e common c e n t e r o f c u r v a t u r e o f t h e c o n c e n t r i c g r i d s and f l u o r e s c e n t s c r e e n . The f i n a l anode p l a t e of t h e e l e c t r o n gun, G1, G4 and t h e sample a r e u s u a l l y g r o u n d e d t o e n s u r e t h a t b o t h t h e i n c i d e n t and r e f l e c t e d e l e c t r o n s t r a v e l i n an e l e c t r o s t a t i c a l l y f i e l d - f r e e s p a c e . The a c c e l e r a t i n g v o l t a g e V p f o r t h e e l e c t r o n gun i s v a r i a b l e , w h i l e t h e p o t e n t i a l f o r b o t h G2 and G3 ( t h e r e p e l l i n g g r i d s ) i s s e t t o -Vp+AV, where AV i s a s m a l l p o s i t i v e v o l t a g e . W i t h s u c h a r e p e l l i n g p o t e n t i a l , o n l y t h e e l a s t i c a l l y s c a t t e r e d e l e c t r o n s and a s m a l l f r a c t i o n o f i n e l a s t i c a l l y s c a t t e r e d e l e c t r o n s , w h i c h have l o s t an amount of e n e r g y l e s s t h a n eAV, c a n p a s s t h r o u g h t h e g r i d s . The l a t t e r . l e a d t o a g e n e r a l b a c k g r o u n d on t h e f l u o r e s c e n t LEED s c r e e n . The b a c k g r o u n d i n t e n s i t y c a n be r e d u c e d by s e t t i n g t h e r e p e l l i n g p o t e n t i a l a t Vp, but i t r e s u l t s i n much l a r g e r d i f f r a c t e d beam w i d t h s [ l 2 l ] . S i n c e t y p i c a l g r i d s have an e l e c t r o n t r a n s p a r e n c y o f a b o u t 80%, t h e use o f two r e p e l l i n g g r i d s r e d u c e s t h e number of e l e c t r o n s r e a c h i n g t h e s c r e e n . However t h e a d v a n t a g e of 4 - g r i d o p t i c s , o v e r a 3 - g r i d s y s t e m w i t h one r e p e l l i n g g r i d , i s e s p e c i a l l y a p p a r e n t when u s e d as a r e t a r d i n g f i e l d a n a l y z e r f o r Auger e l e c t r o n s p e c t r o s c o p y . 141 N e v e r t h e l e s s f o r LEED a l s o , t h e two r e p e l l i n g g r i d s do o f f e r b e t t e r e n e r g y r e s o l u t i o n w h i c h e n h a n c e s t h e c o n t r a s t i n d i f f r a c t i o n p a t t e r n s . V i s u a l d i s p l a y of a LEED p a t t e r n i s a c c o m p l i s h e d f o r t h e e l e c t r o n s w h i c h p a s s t h r o u g h G4. They a r e a c c e l e r a t e d by a p o s i t i v e p o t e n t i a l of a b o u t 5 keV on t o a p h o s p h o r - c o a t e d m e t a l s c r e e n . Thus e a c h e l a s t i c a l l y d i f f r a c t e d beam p r o d u c e s a s p o t on t h e s c r e e n , and t h e whole p a t t e r n c a n be o b s e r v e d t h r o u g h a g l a s s window d i r e c t l y o p p o s i t e t o t h e s c r e e n . P a r t o f t h e LEED p a t t e r n w i l l be b l o c k e d by t h e l e g s of t h e m a n i p u l a t o r and t h e l a r g e sample h o l d e r i n t h i s t r a d i t i o n a l s e t - u p . One s o l u t i o n i n v o l v e s u s i n g an i n c l i n e d m i r r o r [ l 2 2 ] t o v i e w t h e image o f t h e p a t t e r n t h r o u g h a s i d e v i e w p o r t . A l t e r n a t i v e l y , de B e r s u d e r [ 1 2 3 ] r e d u c e d t h i s p r o b l e m by v i e w i n g t h e LEED p a t t e r n f r o m t h e r e a r o f a g l a s s p h o s p h o r s c r e e n ; r e a r - v i e w i n g s y s t e m s a r e now c o m m e r c i a l l y a v a i l a b l e . 4.4 QUANTITATIVE MEASUREMENT OF LEED SPOT INTENSITIES D i f f r a c t e d beam i n t e n s i t y i s c o n v e n i e n t l y d e f i n e d a s t h e e l a s t i c r e f l e c t i v i t y I = i / i o r (4.3) where i a n d i 0 a r e t h e d i f f r a c t e d and i n c i d e n t beam c u r r e n t s r e s p e c t i v e l y . To o b t a i n an 1 ( E ) c u r v e f o r a p a r t i c u l a r beam, I i s me a s u r e d o v e r a ra n g e o f e n e r g y a t a f i x e d d i r e c t i o n o f i n c i d e n c e . 142 In e a r l y LEED s t u d i e s , the d i f f r a c t e d beam c u r r e n t i was o f t e n measured d i r e c t l y by u s i n g a movable Faraday cup c o l l e c t o r [ 1 , 1 2 4 ] . T h i s method of measurement i s very a c cu ra t e and p r o v i d e s h igh s e n s i t i v i t y ( de t e c t ab l e c u r r e n t - 1 0 " 1 4 A, which i s s e v e r a l o rde r s lower than tha t of 4-gr id and sc reen sys tem) . The major d i sadvan tage of t h i s approach i s that i t i s s low, e s p e c i a l l y f o r the non-specu la r beams whose d i r e c t i o n s vary as the energy of the i n c i d e n t beam i s changed. With h e m i s p h e r i c a l g r i d sys tems [125 ] , a v a r i e t y of methods have been deve loped f o r e s t i m a t i n g I from the b r i g h t n e s s of the spots on the f l u o r e s c e n t s c r e e n [ l 2 ] . These i n d i r e c t methods assume in gene r a l that the luminance i s l i n e a r l y p r o p o r t i o n a l to the e l e c t r o n c u r r e n t s t r i k i n g the s c r e e n . The b r i g h t n e s s of the spo ts can be measured d i r e c t l y w i th an e x t e r n a l spot photomete r [126 ] , or i n d i r e c t l y from a pho tog raph i c p l a t e [ l 2 7 ] c o n t a i n i n g the nega t i ve image of the s p o t s . The l a t t e r t echn ique was m o d i f i e d by F r o s t et al. [128] who used a c o m p u t e r - c o n t r o l l e d V i d i c o n camera to d i g i t i z e the darkness of the nega t i ve image of the s p o t . The data a c q u i s i t i o n t ime in the spot photometer techn ique i s comparable to the Faraday Cup method. The pho tograph i c methods, on the o ther hand, can cut down the a c t u a l LEED expe r imen ta l t ime d r a m a t i c a l l y . Records can be taken at 2 eV i n t e r v a l s from 40 to 300 eV in j u s t a few minu tes , thereby g r e a t l y r educ ing p o s s i b i l i t i e s f o r con tamina t ion or beam e f f e c t s o c c u r r i n g wh i l e the s u r f a c e i s be ing s t u d i e d . 143 However t h i s method i s i n c o n v e n i e n t from a n o t h e r p o i n t of v i e w . The d e v e l o p i n g o f t h e f i l m s , and t h e s u b s e q u e n t a n a l y s i s , can t a k e a day o r s o . T h a t means i n p r a c t i c e t h a t t h e e x p e r i m e n t e r would not l e a r n u n t i l p e r h a p s two d a y s l a t e r o f whether t h e d a t a were s a t i s f a c t o r y , o r w hether, f o r example, e r r o r s i n o r i e n t a t i o n had o c c u r r e d t o r e n d e r t h e whole e x e r c i s e m e a n i n g l e s s . R e c e n t a d v a n c e s i n l o w - l i g h t - l e v e l TV c a m e r a s , and f a s t e r m i c r o p r o c e s s o r s , had opened new p o s s i b i l i t i e s f o r d i r e c t s c a n n i n g and d i g i t i z a t i o n o f t h e b r i g h t n e s s d i s t r i b u t i o n on t h e LEED s c r e e n t h r o u g h t h e v i e w i n g window. Such o n - l i n e methods c a n a c c u m u l a t e d a t a i n a m i n u t e o r so, and p r o d u c e 1 ( E ) c u r v e s s h o r t l y a f t e r v i a an o s c i l l o s c o p e or a p r i n t e r . T h e s e s h o r t e r e x p e r i m e n t a l t i m e s a r e v e r y s i g n i f i c a n t f o r l i m i t i n g t h e e f f e c t s o f b e a m - s u r f a c e i n t e r a c t i o n s . In t h i s work b o t h t h e p h o t o g r a p h i c method d e s c r i b e d by F r o s t et al. [128] and an o n - l i n e TV camera method were u s e d . They a r e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s . 4.4.1 PHOTOGRAPHIC METHOD T h i s method u s e s a N i k o n F2 camera (85 mm-fl.8 w i t h K2 e x t e n s i o n r i n g ) p l a c e d i n f r o n t o f t h e v i e w p o r t . Kodak T r i - X 35 mm b l a c k / w h i t e f i l m (ASA 500) i s u s e d . The a p e r t u r e of t h e l e n s i s n o r m a l l y s e t a t f 1 . 8 o r f 2 . 8 w i t h e x p o s u r e t i m e v a r y i n g from 1 t o 4 s e c o n d s d e p e n d i n g on t h e b r i g h t n e s s of t h e s p o t s . The s h o r t e r e x p o s u r e t i m e i s u s u a l l y a d e q u a t e f o r 144 LEED p a t t e r n s f r o m c l e a n m e t a l s u r f a c e s . W i t h o v e r l a y e r s , t h e i n t e n s i t i e s of t h e s p o t s may be r e d u c e d , so t h a t l o n g e r e x p o s u r e t i m e s a r e r e q u i r e d . When t h e s c r e e n i s p h o t o g r a p h e d a t 2 eV i n t e r v a l s from 40 t o 300 eV, t h e t o t a l t i m e r e q u i r e d i s l e s s t h a n 10 m i n u t e s even w i t h 4 - s e c o n d e x p o s u r e s . The p r o c e d u r e r e q u i r e s r e c o r d i n g o f i n c i d e n t beam c u r r e n t a t e a c h e n e r g y p o i n t f o r l a t e r n o r m a l i z a t i o n o f t h e me a s u r e d i n t e n s i t i e s . The f i l m i s d e v e l o p e d i n a t a n k u s i n g A c u f i n e d e v e l o p e r f o r 5 t o 10 m i n u t e s a t 75° t o 80°C w i t h g e n t l e a g i t a t i o n . Some e x p e r i m e n t a t i o n w i t h t i m e and t e m p e r a t u r e i s needed f o r d i f f e r e n t LEED p a t t e r n s . However, o v e r - l o n g d e v e l o p i n g t i m e s w i l l l e a d t o s a t u r a t i o n o f t h e LEED s p o t i m a g es. A f t e r f i x i n g (Kodak r a p i d F i x e r , =5 m i n . ) , t h e f i l m i s immersed i n d i s t i l l e d w a t e r a t room t e m p e r a t u r e f o r a b o u t 10 m i n u t e s and t h e n d r i e d . T h i s p r o v i d e s a permanent r e c o r d o f t h e d i f f r a c t i o n p a t t e r n s . The d i g i t i z a t i o n o f t h e LEED s p o t s i n v o l v e s p l a c i n g t h e p h o t o g r a p h i c n e g a t i v e on a l i g h t t a b l e and s c a n n i n g w i t h t h e V i d i c o n c a m e r a . The p o s i t i o n o f e a c h s p o t i s i d e n t i f i e d by t h e c o - o r d i n a t e s o f a c u r s o r d i s p l a y e d on t h e TV m o n i t o r ; t h e c u r s o r c a n be moved by t h e u s e r v i a a j o y s t i c k . The c i r c u l a r r e g i o n w i t h a u s e r - s e l e c t e d d i a m e t e r c e n t e r i n g on e a c h s p o t i s d i g i t i z e d ( V i d i c o n and D i g i t i z e r , S p a t i a l D a t a System, G a l e n a , C a l i f o r n i a ) and t h e d e n s i t y v a l u e s a r e i n t e g r a t e d s u b j e c t t o a b a c k g r o u n d s u b t r a c t i o n t . The summed T B a c k g r o u n d s u b t r a c t i o n w i l l be d i s c u s s e d i n S e c t i o n 4.4.3. VIDICON CAMERA SCANNER DIGITIZER INTERFACE FILM (J~ LIGHT TABLE T.V. MONITOR PROFILER NOVA 2 COMPUTER CASSETTE DRIVE SCOPE X-Y PLOTTER F i g u r e 4.7: S c h e m a t i c d i a g r a m of t h e p h o t o g r a p h i c n e g a t i v e s of LEED p a t t e r n s . c o m p u t e r - c o n t r o l l e d a n a l y s i s o f U l 1 46 i n t e n s i t i e s a t e a c h e n e r g y a r e d i v i d e d by t h e a p p r o p r i a t e i n c i d e n t beam c u r r e n t . An o p t i o n a l g r i d t r a n s p a r e n c y c o r r e c t i o n [ 1 2 9 , 1 30] c a n be done o f f - l i n e . The 1 ( E ) c u r v e s a r e u s u a l l y smoothed by a 3 - p o i n t smooth r o u t i n e . A s c h e m a t i c d i a g r a m f o r t h e a r r a n g e m e n t o f t h e c o m p u t e r - c o n t r o l l e d d i g i t i z a t i o n p r o c e s s i s shown i n F i g u r e 4.7. The i n t e g r a t e d i n t e n s i t i e s a r e t h e n s t o r e d on c a s s e t t e t a p e and can be t r a n s f e r r e d t o t h e m a i n f r a m e computer (Amdahl 470 V8) v i a p a p e r t a p e . The a p p l i c a b i l i t y o f t h e p h o t o g r a p h i c method depends on two g e n e r a l a s s u m p t i o n s : 1. t h e o p t i c a l d e n s i t y f o r a s p o t on t h e f i l m n e g a t i v e i s p r o p o r t i o n a l t o t h e amount o f l i g h t t h a t c a u s e s t h e d a r k e n i n g ; and 2. t h e i n t e n s i t y o f a s p o t on t h e f l u o r e s c e n t s c r e e n i s p r o p o r t i o n a l t o t h e i m p i n g i n g e l e c t r o n f l u x . S p o t s w i t h c o m p a r a b l e maximum i n t e n s i t y i n t h e e n e r g y range c o n s i d e r e d a r e a n a l y z e d w i t h t h e same s e t t i n g s o f b l a c k and maximum l e v e l s . The a n a l y s i s p r o c e d u r e i s t h e r e f o r e f a s t e r , b u t i t i s a l s o i m p o r t a n t t h a t i n d i v i d u a l s p o t s span a s much as p o s s i b l e of t h e g r e y s c a l e ; t h i s w i l l y i e l d b e t t e r p e a k - a n d - v a l l e y r e s o l u t i o n i n t h e s u b s e q u e n t 1 ( E ) c u r v e s . 4.4.2 TV CAMERA METHOD The TV camera method t o measure r e l a t i v e i n t e n s i t i e s d i r e c t l y f r o m a LEED s c r e e n was f i r s t i n t r o d u c e d by H e i l m a n n et al.[131]. S i n c e t h e n v a r i o u s m o d i f i c a t i o n s i n v o l v i n g 147 v i d e o r e c o r d i n g ! 1 3 2 ] and c l o s e d - c i r c u i t T V [ 1 3 3 ] have been d e s c r i b e d . The t a p e r e c o r d i n g method i s f a s t , r e q u i r i n g o n l y s e v e r a l s e c o n d s t o s t o r e a h u n d r e d ' f r a m e s ' of LEED p a t t e r n w h i c h c a n be a n a l y z e d l a t e r . However, t h e q u a l i t y and r e s o l u t i o n o f r e a s o n a b l y - p r i c e d c o m m e r c i a l l y a v a i l a b l e v i d e o r e c o r d e r s and t a p e s s e t a major l i m i t a t i o n on t h e a p p l i c a b i l i t y o f t h i s a p p r o a c h . In r e c e n t y e a r s , A/D c o n v e r t e r s and m i c r o p r o c e s s o r s have i m p r o v e d so much t h a t H e i l m a n n et al. [134] have d e s c r i b e d t h e s c a n n i n g and d i g i t i z a t i o n o f p a r t i c u l a r LEED beams a t some 500 d a t a p o i n t s i n o n l y a b o u t 10 s e c o n d s . T h i s can t r u l y be s a i d t o d e f i n e an o n - l i n e method. The need f o r f a s t e r LEED d a t a a c q u i s i t i o n a r i s e s b o t h f r o m t h e need t o m i n i m i z e b e a m - s u r f a c e i n t e r a c t i o n s , and t o be a b l e t o t e s t e x p e r i m e n t a l s e t t i n g s q u i c k l y p r i o r t o making measurements ( i . e . t o overcome t h e s e r i o u s p r o b l e m s n o t e d above f o r t h e p h o t o g r a p h i c m e t h o d ) . I was w o r k i n g on t h e d e v e l o p m e n t o f a v i d e o r e c o r d e r t y p e s y s t e m f o r t h i s l a b o r a t o r y when we became aware o f a c o m m e r c i a l v i d e o LEED a n a l y z e r ( D a t a - Q u i r e C o r p . , S t o n y B r o o k , N . Y . ) . Funds were t h e n o b t a i n e d t o p u r c h a s e t h e l a t t e r w h i c h was i n c o r p o r a t e d i n t o our new TV s y s t e m . The LEED measurement f a c i l i t i e s b a s e d on t h i s VLA i s shown s c h e m a t i c a l l y i n F i g u r e 4.8. The most i m p o r t a n t p a r t of t h e s y s t e m i s a 32K m i c r o p r o c e s s o r ( M o t o r o l a 6 8 0 0 ) . T h i s d i g i t i z e s t h e v i d e o s i g n a l s f r o m t h e camera v i a an A/D c o n v e r t e r and a l s o c o n t r o l s t h e LEED gun v o l t a g e v i a a D/A SCOPE X-Y RECORDER MOTOROLA 6800 D/A CONV. LEED CONTROL UNIT INTERFACE TERMINAL Figure 4 .8 : Schematic diagram of the real-time LEED spot intensity analysis using a video LEED analyzer (VLA, Data-Quire). C D 149 c o n v e r t e r . The TV camera u s e d i s of t h e i n t e n s i f i e d s i l i c o n i n t e n s i f i e r t a r g e t t y p e (COHU 4 4 1 0 / I S I T ) . An I S I T camera b a s i c a l l y c o n s i s t s of a p h o t o c a t h o d e t u b e and a s i l i c o n t a r g e t . When l i g h t p a s s e s t h r o u g h t h e camera l e n s on t o t h e p h o t o c a t h o d e t u b e , e l e c t r o n s a r e p r o d u c e d . In c o n v e n t i o n a l V i d i c o n c a m eras, t h e c u r r e n t f r o m t h e s e e l e c t r o n s i s a measure o f t h e l i g h t i n t e n s i t y . I n an I S I T camera t h e s e e l e c t r o n s a r e f u r t h e r a c c e l e r a t e d on t o t h e s i l i c o n t a r g e t , where more s e c o n d a r y e l e c t r o n s a r e p r o d u c e d , t h e r e b y r e s u l t i n g i n h i g h e r c u r r e n t s f o r g i v e n c o n d i t i o n s t h a n can be o b t a i n e d w i t h a V i d i c o n c a m e r a . C o n s e q u e n t l y , an I S I T camera i s some 10 3 t i m e s more s e n s i t i v e t h a n a c o n v e n t i o n a l V i d i c o n c amera. T h i s e n a b l e s t h e i n c i d e n t c u r r e n t t o be k e p t t o low l e v e l s t o m i n i m i z e any h e a t i n g o r e l e c t r o n s t i m u l a t e d d e s o r p t i o n e f f e c t s [ 1 3 5 - 1 3 7 ] . I n a LEED e x p e r i m e n t t h e TV camera i s p o s i t i o n e d i n f r o n t o f t h e LEED f l u o r e s c e n t s c r e e n i n s u c h a way t h a t i t g i v e s a p i c t u r e o f t h e LEED p a t t e r n t h a t c o v e r s a p p r o x i m a t e l y t h e whole a r e a o f t h e v i d e o m o n i t o r s c r e e n . F o r f i x e d v a l u e s of t h e i n c i d e n t beam ( e n e r g y , d i r e c t i o n of i n c i d e n c e ) and f o r f i x e d c o n d i t i o n s o f t h e s u r f a c e , t h e l a t t e r p i c t u r e i s d e f i n e d as a. f r a m e . Such a frame c o n s i s t s of 256x256 p i c t u r e e l e m e n t s ( o r p i x e l s ) . The i n t e n s i t y of e a c h p i x e l can be d i g i t i z e d i n a g r e y s c a l e o f 0-255. In N o r t h A m e r i c a , o n l y 1/60 s e c o n d i s r e q u i r e d f o r a s i n g l e s c a n o f a l l 65,536 p i x e l s of e a c h f r a m e . The s c a n s t a r t s a t 150 t h e upp e r l e f t c o r n e r and moves l e f t t o r i g h t f i r s t a l o n g t h e topmost h o r i z o n t a l l i n e e x a c t l y one p i x e l w ide, and t h e n c o n t i n u e s down t o s c a n a l l 256 h o r i z o n t a l l i n e s i n o r d e r . I t i s p o s s i b l e , a t l e a s t i n p r i n c i p l e , t o d i g i t i z e t h e d e s i r e d LEED s p o t s w i t h i n t h e t i m e t a k e n t o s c a n t h e whole fra m e . T h i s i s i n d e e d t h e c a s e i n t h e l a t e s t s y s t e m u s e d by H e i l m a n n et al. [ 1 3 4 ] . However, t h e d i g i t i z a t i o n r a t e o f t h e c e n t r a l p r o c e s s o r u n i t o f o u r VLA i s much l o w e r t h a n t h e s c a n n i n g r a t e , so a s l i g h t l y d i f f e r e n t a p p r o a c h i s t a k e n t o 'buy t i m e ' f o r t h e d i g i t i z a t i o n p r o c e s s . The d i g i t i z a t i o n i s done c o l u m n w i s e : o n l y one o f t h e 256 p i x e l s on t h e same h o r i z o n t a l l i n e i s d i g i t i z e d i n a s i n g l e s c a n . In o t h e r words, i t t a k e s as many s c a n s as number of p i x e l s t o d i g i t i z e a row o f s u c h p i x e l s , whereas o n l y one s c a n i s r e q u i r e d t o d i g i t i z e a column o f (any number o f ) p i x e l s i n e a c h f r a m e . A frame c a n t h e r e f o r e be v i e w e d a s b e i n g made up of 256 c o l u m n s o f p i x e l s , and t h e c o m p l e t e d i g i t i z a t i o n o f t h e l a t t e r r e q u i r e s 256/60 s e c o n d s . To f u r t h e r r e d u c e t h e d a t a a c q u i s i t i o n t i m e , o n l y t h e u s e r - s e l e c t e d 10x10 p i x e l windows s u p e r i m p o s e d on d e s i r a b l e LEED s p o t s o b s e r v e d on t h e v i d e o m o n i t o r s c r e e n a r e d i g i t i z e d . F o r an i s o l a t e d window, 10 s c a n s ( b e c a u s e t h e r e a r e 10 c o l u m n s o f p i x e l s ) , w h i c h w i l l be c a l l e d a p a s s (10/60 s e c o n d s ) , a r e r e q u i r e d t o d i g i t i z e t h e 100 p i x e l s i n s i d e . In g e n e r a l n s u c h p a s s e s , o r n/6 s e c o n d s , a r e r e q u i r e d t o d i g i t i z e n s e l e c t e d windows on t h e TV m o n i t o r . However, due t o t h e c o l u m n w i s e d i g i t i z a t i o n , windows n o t 151 l y i n g on t h e same h o r i z o n t a l - s c a n n i n g p a t h (10 p i x e l wide) c a n be d i g i t i z e d i n t h e same p a s s . Thus t o s p e e d up t h e d i g i t i z a t i o n p r o c e d u r e , i t i s sometimes n e c e s s a r y t o r o t a t e t h e TV camera i n s u c h a way t h a t t h e d e s i r e d windows l i e on n o n - o v e r l a p p i n g h o r i z o n t a l p a t h s . F i g u r e 4.9 i l l u s t r a t e s how e i g h t windows on a s q u a r e n e t c a n be measured i n one p a s s w i t h t h e a p p r o p r i a t e r o t a t i o n i n s t e a d of t h r e e p a s s e s o t h e r w i s e . On a v e r a g e , t h e d a t a a c q u i s i t i o n t i m e f o r n s p o t s w ould be l67»/n msec. The t i m e - s a v i n g r o u t i n e i s p a r t i c u l a r l y u s e f u l when m u l t i p l e s c a n n i n g i s d e s i r e d t o enhance t h e s i g n a l t o n o i s e r a t i o . C o m p aring t o t h e s y s t e m u s e d by H e i l m a n n et al. [ 1 3 4 ] , t h e measurement o f a s i n g l e s p o t by t h e VLA t a k e s a l m o s t 10 t i m e s a s l o n g . The f o r m e r s y s t e m , however, m e a s u r e s o n l y a maximum o f 4 s p o t s p e r r u n ; whereas t h e VLA c a n measure up t o 49 s p o t s p e r r u n . So when measurements a r e r e q u i r e d f r o m a LEED p a t t e r n o f 25 s p o t s , t h e VLA t a k e s o n l y a b o u t t w i c e t h e t i m e i n d a t a a c q u i s i t i o n as d o e s H e i l m a n n et al. 's s y s t e m . A t e a c h p a s s , t h e p o s i t i o n o f t h e p i x e l w i t h t h e maximum i n t e n s i t y I m a x i n s i d e e a c h window i s n o t e d t o g e t h e r w i t h t h e v a l u e o f I m a x « The sum o f t h e i n t e n s i t i e s of t h e 100 p i x e l s a r e s t o r e d a s t h e 'hundred sum' o r HSUM. In a d d i t i o n , t h e computer a l s o s t o r e s t h e number (N) o f t h o s e p i x e l s w h i c h have i n t e n s i t i e s h i g h e r t h a n a u s e r - s e l e c t e d f r a c t i o n o f I m a X f and t h e summed i n t e n s i t y o f t h e s e p i x e l s i s c a l l e d t h e NSUM. The q u a n t i t i e s I m a x , HSUM, NSUM and N a r e f u r t h e r i l l u s t r a t e d i n F i g u r e 4.10. The HSUM's a r e 1 52 pass 1 pass 2 pass 3 (a) 3 passes required for the d i g i t i z a t i o n of 8 LEED spots on a 'perfect' square net. The p a r t i c u l a r routes chosen by the computer tend to optimize the time-delay between d i g i t i z a t i o n of each spot. pass 1 (b) 1 pass required for the d i g i t i z a t i o n of the same spots as in (a) with a s l i g h t rotation of the TV camera. Figure 4 . 9 : Schematic i l l u s t r a t i o n of the number of passes required for d i g i t i z a t i o n of 8 LEED spots in a square net. 153 (a) The t o t a l volume under the 'dome' represents HSUM which corresponds to the integrated intensity of 100 p i x e l s . (b) The volume of the 'bullet-shaped' structure represents NSUM. This i s the integrated intensity of N pixels each of which has an intensity ^^ r^ Imax ( w h e r e (fr) may be 1/2, 1/4, 1/8 ). Figure 4.10: 3-dimensional diagrams of two types of integrated LEED spot intensity in a (10-pixel x " 10-pixel) window. (a)Hundred sum (or HSUM). (b)N-sum (or NSUM). 1 54 u s u a l l y u s e d t o r e p r e s e n t t h e r e l a t i v e i n t e n s i t i e s i n 1 ( E ) c u r v e s . W i t h a c a r e f u l p r e s e t f r a c t i o n , t h e NSUM's and t h e N's c a n be u t i l i z e d t o e s t i m a t e s u i t a b l e b a c k g r o u n d c o r r e c t i o n s . T h i s p r o c e d u r e w i l l be d i s c u s s e d i n S e c t i o n 4.4.3. Once a frame i s d i g i t i z e d , one o f t h e p a r a m e t e r s o f a LEED e x p e r i m e n t i s c h a n g e d t o g i v e t h e n e x t f r a m e . In t h i s work, t h e i n t e r e s t i s e s p e c i a l l y i n 1 ( E ) c u r v e s , so t h e n o r m a l p r o c e d u r e i s t o change t h e e n e r g y of t h e i n c i d e n t beam ( e . g . by 1 o r 2 e V ) . The change c a n e i t h e r be p o s i t i v e o r n e g a t i v e , but s p o t p o s i t i o n s i n t h e new frame w i l l have c h a n g e d s l i g h t l y f r o m t h e p r e c e d i n g frame ( a s s u m i n g o f c o u r s e t h a t t h e change i n e n e r g y i s s m a l l ) . The VLA p r o v i d e s two t r a c k i n g modes f o r t h e s p o t s , w h i c h a r e a v a i l a b l e a t t h e u s e r ' s d i s c r e t i o n . One o f t h e s e i s t h e c o m p u t e r - c o n t r o l l e d mode: t h e new s p o t p o s i t i o n s on t h e s c r e e n a r e c a l c u l a t e d by t h e computer from i n f o r m a t i o n on u n i t mesh d i m e n s i o n s and d i r e c t i o n s e n t e r e d by t h e u s e r a t t h e b e g i n n i n g o f t h e r u n . The o t h e r t r a c k i n g mode i s r e f e r r e d t o a s t h e 'dynamic c o r r e c t i o n ' mode; i t u s e s t h e p o s i t i o n s o f t h o s e p i x e l s w i t h Imax * n fc^e P r e c e d i n g frame a s t h e c e n t e r s o f t h e new windows. T h e r e i s a u s e r - s e l e c t a b l e t h r e s h o l d i n t e n s i t y v a l u e t o t r i g g e r 'dynamic c o r r e c t i o n ' ; when l m a x i s below t h i s t h r e s h o l d i n t e n s i t y , t h e c a l c u l a t e d mode must be u s e d e x c l u s i v e l y . The 'dynamic c o r r e c t i o n ' mode s h o u l d be u s e d w i t h c a u t i o n when e i t h e r t h e LEED p a t t e r n i s g e n e r a l l y weak, or when b r i g h t s p o t s a r i s i n g f r o m f l u o r e s c e n t s c r e e n b u r n s 155 a r e p r e s e n t . In t h e f i r s t c a s e , t h e f e e d b a c k t r a c k i n g may be d i s t r a c t e d by t h e g e n e r a l b a c k g r o u n d l i g h t , w h i l e i n t h e s e c o n d t h e c u r s o r may l a t c h on t o t h e b r i g h t s p o t s when t h e a c t u a l LEED s p o t s a r e c l o s e t o t h e f o r m e r . S u b s e q u e n t t r e a t m e n t s o f d a t a f o r n o r m a l i z a t i o n , g r i d t r a n s p a r e n c y c o r r e c t i o n and 3 - p o i n t s m o o t h i n g a r e done of f - 1 i n e . 4.4.3 BACKGROUND SUBTRACTION I n t h e p h o t o g r a p h i c method, a s t r a d i t i o n a l l y u s e d i n t h i s l a b o r a t o r y , a f a i r l y e l a b o r a t e b a c k g r o u n d s u b t r a c t i o n scheme c o u l d be i n c l u d e d b e c a u s e a n a l y s i s t i m e i s n o t a c r u c i a l f a c t o r i n t h i s a p p r o a c h . H e r e a c i r c u l a r LEED s p o t i s assumed t o have a r a d i u s r w h i c h i s u s e r - s e l e c t a b l e b u t i s c o n s t a n t f o r a l l t h e s p o t s i n t h e same r u n . The b a c k g r o u n d i n t e n s i t y i s t h e n e s t i m a t e d as t h e a v e r a g e o f t h e i n t e n s i t i e s o f a l l t h e p i x e l s l y i n g on a t h i n a n n u l u s w i t h w i d t h Ar and mean r a d i u s ( r + A r / 2 ) . The p r o d u c t o f t h i s b a c k g r o u n d d e n s i t y and t h e a r e a o f t h e s p o t i s s u b t r a c t e d from t h e s p o t ' s i n t e g r a t e d i n t e n s i t y t o g i v e a ' b a c k g r o u n d f r e e ' v a l u e . Somewhat s i m i l a r l y i n t h e o n - l i n e s y s t e m d e s c r i b e d by H e i l m a n n et al . [ 1 3 4 ] , a window o f nxn p i x e l s i s assumed. A h a r d w a r e a d d e r sums t h e i n t e n s i t i e s o f n p i x e l s on e a c h p a r a l l e l l i n e i n t h e window, and s t o r e s t h e l i n e - s u m L^ ( f o r t h e i * " * 1 l i n e ) . The p r o c e d u r e i s r e p e a t e d f o r n l i n e s i n t h e window, and t h e t o t a l i n t e n s i t y i s t h e n t h e sum o f L-1 56 10 2 n ROWS intensity _rov > (hardware) xi E 3 C c line-sum A LEED spot centered in an (nxn) window (software) u Oi Xl e C 4) C DEFINITION OF BASELINE line-sum Integrated intensity of the above LEED spot appearing as one data point in an 1(E) curve in z ca E -2 / line-sum line (software) (software) BACKGROUND SUBSTRACTION u 0) XI e c c ENERGY line-sum F i g u r e 4 .11: Schematic diagram of the real-time background subtraction with the aid of a hardware adder (after Heilmann et al. [134]). 157 t h r o u g h L p . The b a c k g r o u n d i s d e f i n e d a s n(L,+ L n ) / 2 and i s s u b t r a c t e d from t h e t o t a l i n t e n s i t y . The above p r o c e d u r e i s r e p r e s e n t e d s c h e m a t i c a l l y i n F i g u r e 4.11. W i t h t h e use of t h e h a r d w a r e a d d e r , t h e t i m e r e q u i r e d f o r t h e b a c k g r o u n d s u b t r a c t i o n s t e p i s r e l a t i v e l y s h o r t . The above p r o c e d u r e assumes t h a t t h e s p o t i s w e l l - c e n t e r e d i n t h e window, w h i c h c a n be a c h i e v e d by c a l c u l a t e d a n d / o r dynamic t r a c k i n g modes. Due t o t h e c o m p a r a t i v e l y low d i g i t i z a t i o n r a t e and t h e l a c k o f a hardware a d d e r , t h i s p r o c e d u r e i s n o t a v a i l a b l e w i t h o u r v i d e o LEED a n a l y z e r . A l t h o u g h p r o f i l i n g o f a s p o t c a n be done by s o f t w a r e , b a c k g r o u n d s u b t r a c t i o n w i t h H e i l m a n n et jzl. ' s scheme wo u l d i n e v i t a b l y p u t a s e v e r e s t r a i n on b o t h t h e s p e e d and t h e c o r e s p a c e o f t h e c e n t r a l p r o c e s s o r when t h e VLA i s u s e d f o r m u l t i - b e a m a n a l y s i s . C u r r e n t l y two r a t h e r s i m p l e methods, w h i c h were o r i g i n a l l y p r o p o s e d by S t r o z i e r and J o n a [ l 3 8 ] , a r e u s e d w i t h our VLA f o r e s t i m a t i n g b a c k g r o u n d c o r r e c t i o n s . The p h i l o s o p h y of a p p r o a c h p a r t l y t a k e s n o t e o f t h e f a c t t h a t t h e r e i s no u n i v e r s a l d e f i n i t i o n o f b a c k g r o u n d i n t e n s i t y i n LEED. In any e v e n t , f o r b o t h t h e methods we c u r r e n t l y u s e , t h e s u b t r a c t i o n s t e p i s done o f f - l i n e . The f i r s t a p p r o a c h u s e d w i t h o u r VLA d e f i n e s t h e b a c k g r o u n d B as t h e i n t e n s i t y o f t h e a r e a between two n e i g h b o r i n g LEED s p o t s . In i t s i m p l e m e n t a t i o n , t h e windows a r e moved ( i n i t i a l l y by t h e u s e r , t h e n by t h e c a l c u l a t e d t r a c k i n g mode) t o t h e s e a r e a s a f t e r t h e d i g i t i z a t i o n of t h e LEED s p o t s o f i n t e r e s t f o r t h e a p p r o p r i a t e e n e r g y r a n g e . 158 HSUM o f t h e s e windows i s t a k e n a s B. To r e d u c e t h e t i m e s p e n t on a c q u i s i t i o n of b a c k g r o u n d i n t e n s i t i e s , o n l y a few r e p r e s e n t a t i v e a r e a s need be s c a n n e d . F i g u r e 4.12(a) i l l u s t r a t e s how two s u c h a r e a s may be s e l e c t e d f o r a n i n e s p o t LEED p a t t e r n . In t h i s example, X and Y r e p r e s e n t b a c k g r o u n d between s t r o n g / w e a k and weak/weak p a i r s o f LEED s p o t s r e s p e c t i v e l y . Thus o n l y one b a c k g r o u n d i n t e n s i t y i s r e c o r d e d f o r e a c h t y p e o f r e l a t e d p a i r s o f beams. In t h i s a p p r o a c h , i t i s p r e f e r a b l e t o s e l e c t a r e a s n o t l y i n g on t h e same h o r i z o n t a l p a t h ; t h i s i s t o e x p l o i t t h e c o l u m n w i s e d i g i t i z a t i o n p r o c e s s . An u n s a t i s f a c t o r y f e a t u r e o f t h i s a p p r o a c h a r i s e s when two n e i g h b o r i n g s p o t s have v e r y m a r k e d l y d i f f e r e n t i n t e n s i t i e s . In s u c h c a s e s , t h e i n i t i a l l y - d e t e r m i n e d b a c k g r o u n d c o r r e c t i o n B, s u c h a s f r o m t h e a r e a marked X i n F i g u r e 4 . 1 2 ( a ) , w i l l be d o m i n a t e d by t h e b r i g h t e r s p o t a s shown i n F i g u r e 4 . 1 2 ( b ) . W e i g h t e d b a c k g r o u n d s , s u c h a s b, = B I , / ( I , + I 2 ) and b 2 = B I 2 / ( I , + I 2 ) (4.4) may t h e n be more a p p r o p r i a t e . I n e q u a t i o n ( 4 . 4 ) , I , and I 2 a r e t h e t o t a l i n t e n s i t i e s o f t h e two LEED s p o t s , and b, and b 2 a r e t h e c o r r e c t e d b a c k g r o u n d v a l u e s . T h i s s i m p l e method has t h e l i m i t a t i o n t h a t when t h e s p o t s a r e v e r y c l o s e t o g e t h e r , t h e r e may n o t be enough room t o accommodate t h e window f o r m e a s u r i n g B. When t h a t o c c u r s , a n o t h e r method c a n be u s e d . 159 • • • • © • • (a) X represents background between bright and weak spots. Y represents background between weak spots. A B (b) Intensity p r o f i l e from A to B . Intensity of the region X i s made up largely of the ' t a i l ' of the bright spot. Figure 4.12: LEED spot background intensity approximated by HSUM of a window between neighboring spots. 160 The s e c o n d method u s e s t h e NSUM and N i n t r o d u c e d i n S e c t i o n 4.4.2. W h i l e HSUM r e p r e s e n t s t h e t o t a l i n t e g r a t e d i n t e n s i t y of t h e 100 p i x e l s i n a window, NSUM g i v e s o n l y t h e sum o f t h e i n t e n s i t i e s o f t h o s e N p i x e l s whose i n t e n s i t i e s a r e g r e a t e r t h a n a p r e - d e t e r m i n e d f r a c t i o n o f t h e maximum p i x e l i n t e n s i t y ( l m a x ) i n t h e window. The b a c k g r o u n d f o r a LEED s p o t w i t h t h e s e sums and N i s d e f i n e d a s B = (HSUM - NSUM)/(100-N). (4.5) The n u m e r a t o r i n e q u a t i o n (4.5) c o r r e s p o n d s t o t h e sha d e d a r e a s i n F i g u r e s 4.13(a) and 4 . 1 3 ( b ) . The d e n o m i n a t o r r e p r e s e n t s t h e number o f p i x e l s i n t h e s h a d e d a r e a . In e f f e c t , B g i v e s an a p p r o x i m a t i o n t o t h e a v e r a g e d i n t e n s i t y i n t h e ' t a i l s ' o f t h e i n t e n s i t y d i s t r i b u t i o n c u r v e s shown i n F i g u r e 4.13. I t i s e v i d e n t f r o m t h e same f i g u r e , t h a t t h e a p p l i c a b i l i t y o f t h i s method depends on t h e c h o i c e o f an optimum f r a c t i o n f o r NSUM. The l a t t e r i s u s e r - s e l e c t e d and has v a l u e s o f 1/2, 1/4, 1/8, B e f o r e any d a t a a c q u i s i t i o n , t h e u s e r s h o u l d s c a n t h e whole LEED e n e r g y r a n g e t o d e t e r m i n e t h e a p p r o p r i a t e v a l u e f o r t h e NSUM i f b a c k g r o u n d s u b t r a c t i o n w i t h e q u a t i o n (4.5) i s d e s i r e d . The v a l u e o f B i s u n s t a b l e when N i s v e r y c l o s e t o 100. T h i s o c c u r s when t h e s e l e c t e d f r a c t i o n i s a low v a l u e s u c h as 1/16 o r 1/8. To a v o i d t h i s , a v a l u e o f 1/2 i s g e n e r a l l y u s e d . However, some ' i n j u s t i c e ' w i l l be done t o t h e s h a r p e r s p o t s s u c h a s t h e one shown i n F i g u r e 4 . 1 3 ( a ) . H e r e , t h e 161 1 max (c ) (d) Figure 4.13: S c h e m a t i c i l l u s t r a t i o n of t h e use o f HSUM and NSUM f o r t h e e s t i m a t i o n o f b a c k g r o u n d . The s h a d e d ' t a i l s ' i n (a) and (b) r e p r e s e n t t h e d i f f e r e n c e (HSUM-NSUM). The a v e r a g e b a c k g r o u n d i s d e f i n e d a s t h i s d i f f e r e n c e d i v i d e d by t h e number o f p i x e l s t h a t make up t h e t a i l s . (c) and (d) r e p r e s e n t two t y p e s o f s p o t where HSUM=NSUM, a n d e q u a t i o n (4.5) i s n o t a p p l i c a b l e . 162 s p o t would have an a p p a r e n t l y low i n t e n s i t y s i n c e a s i g n i f i c a n t p a r t o f i t s ' r e a l ' i n t e n s i t y i s t a k e n as t h e t a i l . E q u a t i o n (4.5) i s n o t a p p l i c a b l e when N e q u a l s 100. T h i s i s t h e c a s e when t h e s p o t i s e i t h e r e x t r e m e l y b r i g h t ( F i g u r e 4 . 1 3 ( c ) ) o r has a r e l a t i v e l y f l a t i n t e n s i t y d i s t r i b u t i o n c u r v e ( F i g u r e 4 . 1 2 ( d ) ) . B r i g h t s p o t s c a n be a v o i d e d by l o w e r i n g t h e g a i n of t h e I S I T c a m e r a . U n f o r t u n a t e l y , t h i s w i l l l e a d t o an a c r o s s - t h e - b o a r d r e d u c t i o n i n LEED s p o t i n t e n s i t i e s . As a r e s u l t some weaker s p o t s may be l o s t . A t t h e p r e s e n t t i m e , t h e r e i s no good s o l u t i o n f o r t h e ' f l a t - t o p ' s p o t s , a l t h o u g h t h e l a t t e r were r a r e l y o b s e r v e d i n t h i s work. F i g u r e 4.14 compares t h e (0,1) beam from a c l e a n s u r f a c e o f R h ( l 1 1 ) o b t a i n e d by t h e VLA, w i t h and w i t h o u t b a c k g r o u n d c o r r e c t i o n s . S m o o t h i n g i s d e l i b e r a t e l y o m i t t e d so t h a t any a b n o r m a l f e a t u r e s r e s u l t i n g from t h e two s i m p l e b a c k g r o u n d c o r r e c t i o n schemes may be o b s e r v e d . In g e n e r a l , b a c k g r o u n d c o r r e c t i o n a c c o r d i n g t o e q u a t i o n (4.4) l e v e l s t h e b a s e l i n e s o f 1 ( E ) c u r v e s s a t i s f a c t o r i l y ; b u t i t does n o t g r e a t l y a f f e c t t h e p o s i t i o n s and s h a p e s o f p e a k s . B a c k g r o u n d s u b t r a c t i o n u t i l i z i n g NSUM e n h a n c e s t h e s h a r p n e s s of pe a k s a p p r e c i a b l y ; but t h i s scheme may l e a d t o u n e x p e c t e d a p p e a r a n c e o f p e a k s and v a l l e y s , p o s s i b l y due t o ' u n f a i r ' s u b t r a c t i o n from v e r y b r i g h t s p o t s . One s u c h s u b t r a c t i o n i s e v i d e n t a t t h e 150 eV peak i n F i g u r e 4.14 ( c u r v e c ) , where t h e l a r g e peak t u r n s i n t o a s m a l l d i p . T h e r e f o r e c a u t i o n has t o be t a k e n when u s i n g NSUM f o r e s t i m a t i n g b a c k g r o u n d 1 63 i 1 1 1 1 1 1 1 1 r 1 1 1 1 1 1 1 1 1 1 1 40 80 120 160 200 240 ENERGY CEV) F i g u r e 4.14: E x p e r i m e n t a l 1(E) c u r v e s f o r ( 0 , 1 ) beam of c l e a n R h ( 1 1 1 ) s u r f a c e (6=0°, no s m o o t h i n g ) . (a)No b a c k g r o u n d s u b t r a c t i o n . ( b ) B a c k g r o u n d s u b t r a c t i o n u s i n g e q u a t i o n ( 4 . 4 ) . ( c ) B a c k g r o u n d s u b t r a c t i o n u s i n g e q u a t i o n ( 4 . 5 ) . 164 c o r r e c t i o n s . CHAPTER 5 S T A B I L I T Y OF LEED FRACTIONAL ORDER BEAM INTENSITIES 165 166 5.1 INTRODUCTION W h i l e making m u l t i p l e s c a t t e r i n g LEED i n t e n s i t y c a l c u l a t i o n s f o r 0 a d s o r b e d on t h e Z r ( O O O l ) s u r f a c e , I o b s e r v e d a s t r o n g t e n d e n c y f o r c o r r e s p o n d i n g f r a c t i o n a l o r d e r beam 1( E ) c u r v e s t o be c l o s e l y c o n s t a n t w i t h c h a n g i n g a d l a y e r c o v e r a g e , s p e c i f i c a l l y f r o m q u a r t e r m o n o l a y e r (2x2) s t r u c t u r e s t o h a l f m o n o l a y e r (2x1) s t r u c t u r e s . T h i s a p p e a r s as an i n t e r e s t i n g o b s e r v a t i o n w h i c h a p p a r e n t l y c o n t r a s t e d w i t h an i n d i c a t i o n by S h i h et al . [40] t h a t t h e s e t r a n s l a t i o n a l s y m m e t r i e s c o u l d be d i s t i n g u i s h e d by LEED c r y s t a l l o g r a p h y f o r m o d e l s o f d i s s o c i a t e d CO on t h e (0001) s u r f a c e o f t i t a n i u m . N e v e r t h e l e s s t h e o b s e r v a t i o n h e r e r e i n f o r c e s and e x t e n d s s i m i l a r o b s e r v a t i o n s by Yang et a l . [ 1 3 9 ] , and i t f u r t h e r p r o v i d e s s u p p o r t f o r some a p p r o x i m a t e schemes o f m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s u c h as t h e q u a s i - d y n a m i c a l m e t h o d [ 1 4 0 , 1 4 1 ] , d i a g o n a l d o m i n a n t m e t h o d [ l 4 2 ] , u n i t c e l l r e d u c t i o n method[139,143] and beam s e t n e g l e c t m e t h o d ! 5 1 ] . The l a s t method, w h i c h was i n t r o d u c e d by Van Hove et a l . , i s e s p e c i a l l y u s e f u l i n making t r a c t a b l e t h e c a l c u l a t i o n o f LEED i n t e n s i t i e s f r o m s u r f a c e s t r u c t u r e s w i t h l a r g e u n i t meshes. When 1(E ) c u r v e s f o r r e l a t e d f r a c t i o n a l o r d e r beams a r e c l o s e l y i n d e p e n d e n t o f c o v e r a g e , f o r c o n s t a n t a d s o r p t i o n s i t e , t h e r e i s an i m p l i e d i n s e n s i t i v i t y o f t h e LEED t e c h n i q u e t o s u r f a c e c o v e r a g e . In one s e n s e t h i s may be seen as a d i s a d v a n t a g e , but e q u a l l y , i n s o f a r as t h e s e beams a r e i n d e p e n d e n t o f t h e u n i t mesh a r e a , m u l t i p l e s c a t t e r i n g 167 c a l c u l a t i o n s may be made w i t h s m a l l e r u n i t mesh a r e a s . The l a t t e r would r e d u c e t h e beam r e q u i r e m e n t i n K - s p a c e . Yang et al. [139] c o n c l u d e d t h a t t h e s t a b i l i t y i n t h e f r a c t i o n a l o r d e r 1 ( E ) c u r v e s w i t h c h a n g i n g c o v e r a g e r e q u i r e s a c o n s t a n t s u b s t r a t e e n v i r o n m e n t a r o u n d e a c h a d s o r b e d atom and n e g l i g i b l e s c a t t e r i n g between t h e a d s o r b e d atoms. G i v e n a p o t e n t i a l u s e f u l n e s s f o r t h e s t a b i l i t y i n f r a c t i o n a l o r d e r beam i n t e n s i t i e s a s d i s c u s s e d by Yang et a / . [ 1 3 9 ] , f u r t h e r i n v e s t i g a t i o n o f t h e r a n g e o f i t s a p p l i c a b i l i t y a p p e a r s u s e f u l . T h i s r e q u i r e s c o n s i d e r a t i o n o f t h e e f f e c t s o f c h a n g i n g s c a t t e r i n g s t r e n g t h s f o r t h e a d s o r b a t e and s u b s t r a t e atoms. In g e n e r a l t h e c l o s e r t h e a d s o r b a t e atoms a r e t o one a n o t h e r , t h e more i m p o r t a n t s h o u l d be t h e s c a t t e r i n g w i t h i n t h e a d s o r b e d l a y e r [ 1 0 6 ] , and t h e r e f o r e a c c o r d i n g t o Yang et al. [139] t h e g r e a t e r a r e l i k e l y t o be t h e d i f f e r e n c e s i n c o r r e s p o n d i n g 1 ( E ) c u r v e s w i t h c h a n g i n g c o v e r a g e s . T h i s b r i n g s o u t t h e i m p o r t a n c e o f new o b s e r v a t i o n s f o r t h e a d s o r p t i o n o f 0 on Z r ( O O O l ) s u r f a c e , s i n c e f o r t h r e e - d o m a i n p ( 2 x l ) m o d els n e i g h b o r i n g 0 atoms a r e t h e n r e l a t i v e l y c l o s e ( i . e . s e p a r a t e d by j u s t t h e s u b s t r a t e i n t e r a t o m i c d i s t a n c e ) , and t h e r e f o r e l a r g e r m u l t i p l e s c a t t e r i n g c o n t r i b u t i o n s may be a n t i c i p a t e d . Thus f u r t h e r c o m p a r i s o n s between p ( 2 x l ) and p ( 2 x 2 ) s t r u c t u r e s s h o u l d be h e l p f u l f o r i l l u m i n a t i n g t h e r a n g e o f a p p l i c a b i l i t y o f t h e f r a c t i o n a l o r d e r beam s t a b i l i t y f i r s t d i s c u s s e d by Yang et al. [ 1 3 9 ] . 168 C o m p a r i s o n s between s e t s of c a l c u l a t e d i n t e n s i t y c u r v e s c a n be a s s e s s e d w i t h t h e methods d e v e l o p e d f o r c o m p a r i n g e x p e r i m e n t a l and c a l c u l a t e d 1 ( E ) c u r v e s i n LEED c r y s t a l l o g r a p h y . T h e r e f o r e t h e n e x t s e c t i o n r e v i e w s t h e LEED r e l i a b i l i t y i n d i c e s w h i c h a r e u s e d i n t h i s c h a p t e r , as w e l l a s i n t h e n e x t f o r s t r u c t u r a l a n a l y s e s f o r oxygen a d s o r b e d a t t h e Z r ( O O O l ) s u r f a c e . 5.2 COMPARISON OF 1 ( E ) CURVES: THE R E L I A B I L I T Y INDICES In LEED c r y s t a l l o g r a p h y , 1 ( E ) c u r v e s c a l c u l a t e d w i t h d i f f e r e n t models a r e compared w i t h t h e c o r r e s p o n d i n g e x p e r i m e n t a l 1(E) c u r v e s u n t i l a ' b e s t f i t ' i s o b t a i n e d . T r a d i t i o n a l l y s u c h c o m p a r i s o n s were made by v i s u a l i n s p e c t i o n s , but t h i s a p p r o a c h becomes i n c r e a s i n g l y u n w i e l d y i n g e n e r a l . The need f o r some m a t h e m a t i c a l f u n c t i o n s w h i c h c a n s y s t e m a t i c a l l y make t h e s e c o m p a r i s o n s i s w e l l d o c u m e n t e d ! 1 4 4 ] . The r e l i a b i l i t y i n d i c e s ( o r R - f a c t o r s ) p r o p o s e d by Z a n a z z i and J o n a [ l 4 5 ] and by P e n d r y [ l 4 6 ] a r e d e s c r i b e d i n t h e f o l l o w i n g . 5.2.1 ZANAZZI-JONA R-FACTOR Z a n a z z i and J o n a [ l 4 5 ] have p r o p o s e d a m a t h e m a t i c a l i n d e x w h i c h compares 1 ( E ) c u r v e s w i t h r e g a r d t o a l l f e a t u r e s w h i c h seem i m p o r t a n t from t h e e x p e r i e n c e o f v i s u a l c o m p a r i s o n s . F o r a s i n g l e beam, t h e s e a u t h o r s u s e d 1 6 9 J W ( E ) | i ; - c i ; | d E r = ? 1 , (5.1) / I e d E where I and I f c r e f e r t o e x p e r i m e n t a l and t h e o r e t i c a l i n t e n s i t i e s r e s p e c t i v e l y ; t h e p r i m e s i n d i c a t e f i r s t d e r i v a t i v e s . The i n t e g r a l s a r e o v e r t h e e n e r g y r a n g e where t h e e x p e r i m e n t a l and t h e o r e t i c a l d a t a o v e r l a p , and t h e f u n c t i o n w(E) i s d e f i n e d a s Ue " c I t l w(E) = - - . (5.2) 1 1 ' I + | i • I 1 e 1 1 e'max H e r e d o u b l e p r i m e s i n d i c a t e s e c o n d d e r i v a t i v e s , | I e l m a x i s t h e maximum v a l u e o f | I e | , and c i s a s c a l i n g f a c t o r w h i c h i s t a k e n a s t h e r a t i o o f t h e mean e x p e r i m e n t a l and c a l c u l a t e d i n t e n s i t i e s s u c h t h a t ; i e d E c = - . (5.3) / I t d E F o r a s i n g l e beam, t h e f u n c t i o n r i n e q u a t i o n (5.1) has t h e p r o p e r t y t h a t t h e low e r i t s v a l u e , t h e b e t t e r i s t h e match between t h e e x p e r i m e n t a l and c a l c u l a t e d 1 (E) c u r v e s . The use of t h e s c a l i n g f a c t o r c makes r i n d e p e n d e n t o f t h e a b s o l u t e v a l u e o f i n t e n s i t i e s o f t h e 1 ( E ) c u r v e s . However, r i s not d i m e n s i o n l e s s , and b e c a u s e o f t h i s Z a n a z z i and J o n a p r o p o s e d t h e r e d u c e d i n d e x 170 r z j = r / r 0 r (5.4) where r 0 has been g i v e n t h e n u m e r i c a l v a l u e Of 0.027, w h i c h r e p r e s e n t s an a v e r a g e from a s e t o f random c u r v e s . F o r c o m p a r i n g i n t e n s i t y c u r v e s from s e v e r a l d i f f r a c t e d beams, one R - f a c t o r d i s c u s s e d by Z a n a z z i and J o n a i s where i r u n s o v e r o v e r a l l t h e i n d i v i d u a l beams. T h i s r e p r e s e n t s an a v e r a g e o f t h e r z j v a l u e s o f t h e i n d i v i d u a l beams, w e i g h t e d a c c o r d i n g t o t h e e n e r g y r a n g e (AE) o v e r w h i c h t h e c o m p a r i s o n w i t h e x p e r i m e n t has been made. 5.2.2 PENDRY R-FACTOR P e n d r y [ l 4 6 ] p r o p o s e d an a l t e r n a t i v e m u l t i - b e a m R - f a c t o r w h i c h a v o i d s t h e use o f s e c o n d d e r i v a t i v e s , and a p p a r e n t l y has a c l e a r e r s i g n i f i c a n c e f o r h i g h v a l u e s . T h i s i s R P " X j( Y e " Y J ) 2 d E / ^ /[.(Yg) 2 + ( Y J ) 2 ] d E , (5.6) where a g a i n t h e summations a r e o v e r t h e d i f f e r e n t beams i . The Y a r e f u n c t i o n s of e n e r g y f o r m e d from t h e 1 ( E ) c u r v e s , and t h e y a r e d e f i n e d a s 171 Y = L/(1 + L 2 V 0 / ) , (5.7) where t h e L a r e l o g a r i t h m i c d e r i v a t i v e s L ( E ) = d l n l / d E (5.8) f o r b o t h Yg and Y £ , r e s p e c t i v e l y d e r i v e d f r o m t h e e x p e r i m e n t a l and t h e o r e t i c a l 1(E) c u r v e s . In ( 5 . 7 ) , V 0 /- i s t h e i m a g i n a r y p a r t of t h e c o n s t a n t ' m u f f i n - t i n ' p o t e n t i a l u s e d i n t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . A c o m p a r i s o n o f i d e n t i c a l s e t s o f 1(E) c u r v e s would y i e l d a v a l u e of Rp e q u a l t o z e r o , b u t t h e d e n o m i n a t o r i n e q u a t i o n (5.7) t e n d s t o r e s t r i c t Rp t o a b o u t one when a c o m p a r i s o n o f random s e t s o f 1(E) c u r v e s a r e made ( a s s u m i n g t h e p r o d u c t y g Y t have random s i g n s and m a g n i t u d e s ) . Rp e m p h a s i z e s t h e p o s i t i o n s of p e a k s and t r o u g h s i n t h e c u r v e s b e i n g compared; m o r e o v e r a l l maxima and minima r e c e i v e e s s e n t i a l l y e q u a l w e i g h t s i n t h e c o m p a r i s o n . 5.2.3 NORMALIZATION OF R-FACTORS R z j and Rp have t h e p r o p e r t y t h a t t h e s m a l l e r t h e i r v a l u e , t h e b e t t e r t h e agreement between two s e t s o f 1(E) c u r v e s ; i n c a s e o f i d e n t i c a l s e t s , t h e R - f a c t o r s w o u l d be z e r o . Van Hove and K o e s t n e r [ 1 4 6 ] d i s c u s s e d s e v e r a l o t h e r ( s i m p l e r ) r e l i a b i l i t y i n d i c e s f o r LEED, b u t , f o r a l l , t h e s e a u t h o r s i n t r o d u c e d n o r m a l i z a t i o n c o n s t a n t s t o e n s u r e t h a t t h e i n d i c e s g e n e r a l l y f a l l between 0 and 1. Van Hove and 172 K o e s t n e r p r o p o s e d t h a t a n o r m a l i z a t i o n c o n s t a n t o f 1/2 i s a p p r o p r i a t e f o r R z j and Rp, a n d R - f a c t o r s r e p o r t e d i n t h i s work have a l l been n o r m a l i z e d w i t h t h i s a d d i t i o n a l f a c t o r . 5.3 (2X1) VERSUS (2X2) D u r i n g t h e LEED s t u d y o f oxygen a d s o r p t i o n on Zr (OOO l ) , a d i f f r a c t i o n p a t t e r n c o r r e s p o n d i n g t o a (2x2) s u r f a c e s t r u c t u r e was o b s e r v e d . The LEED p a t t e r n (and t h e 1(E) c u r v e s ) e x h i b i t e d an a p p a r e n t 6 - f o l d symmetry a t n ormal i n c i d e n c e ; t h i s i s due t o t h e e x i s t e n c e o f two s e t s of d omains a r i s i n g from e i t h e r t h e A o r B t e r m i n a t i o n c h a r a c t e r i s t i c o f a hcp (OOOl) s u r f a c e ( F i g u r e 5 . 1 ) . F i g u r e 5.2 d e t a i l s t h a t t h e o b s e r v e d (2x2) d i f f r a c t i o n p a t t e r n may a r i s e f r o m a (2x2) s u r f a c e s t r u c t u r e o r f r o m a s u p e r p o s i t i o n o f p a t t e r n s f r o m t h e t h r e e t y p e s o f r o t a t i o n a l l y r e l a t e d (2x1) domains w h i c h may be p r e s e n t s i m u l t a n e o u s l y on t h e s u r f a c e (models i n r e a l s p a c e a r e s p e c i f i e d i n F i g u r e 2 . 9 ) . I n o r d e r t o a s s e s s w hether LEED c r y s t a l l o g r a p h y a l o n e i s l i k e l y t o be a b l e t o d i s t i n g u i s h (2x1) a n d (2x2) s t r u c t u r e s on a hcp (OOOl) s u r f a c e , a s e r i e s o f m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s were p e r f o r m e d f o r t h e a d s o r p t i o n of s e v e r a l a t o m i c s p e c i e s on t h e T i ( 0 0 0 1 ) and Zr (OOOl) s u r f a c e s . 5.3.1 THE CALCULATIONS 1(E ) c u r v e s were c a l c u l a t e d f o r s u r f a c e m o d e l s w i t h B r a v a i s l a t t i c e l a y e r s ( S e c t i o n 3.3.1) o f a d s o r b a t e and 1 73 (a) (b) O O o o o "O T e r m i n a t i o n A u .p O" ~0~ Monatomic s t e p T e r m i n a t i o n B o o o o o o o o o o o o > o > o o i o i o o i o i o o o o o o o s 2 o o O Topmost l a y e r atoms S e c o n d l a y e r atoms Figure 5.1: ( a ) S i d e v i e w and ( b ) t o p v i e w of t h e two p o s s i b l e domains r e s u l t i n g f r o m t h e t r u n c a t i o n o f t h e b u l k s t r u c t u r e o f a h c p ( O O O l ) s u r f a c e . The two domains a r e r e l a t e d t o e a c h - o t h e r by a 180° r o t a t i o n . (a) (b) O o o o ^ / 2 o (A) + (B) + (O '0,1 0,0 o o •1,0 —V p(2x1) p(2x2) Figure 5.2: Superposition of the (2x1) reciprocal l a t t i c e s in (a) from the three possible rotational domains of a (2x1) structure on a hcp(OOOl) or fcc (111) surface to form an apparent (2x2) reciprocal l a t t i c e in (b). So l i d and hollow c i r c l e s represent integral- and fractional-order beams respectively. 175 s u b s t r a t e atoms, and t h e l a y e r s were s t a c k e d by t h e r e n o r m a l i z e d f o r w a r d s c a t t e r i n g method ( S e c t i o n 3 . 4 . 2 ) . Normal i n c i d e n c e was u s e d i n a l l c a s e s . Thus a 3 - f o l d r o t a t i o n a l and a m i r r o r p l a n e ( x z ) symmetry c o u l d be assumed f o r t h e (2x2) m o d e l s , and j u s t a m i r r o r p l a n e ( x z ) symmetry f o r t h e (2x1) models (beam l a b e l l i n g i s d e t a i l e d i n F i g u r e 5 . 2 ) . F o r (2x1) s t r u c t u r e s , a c a l c u l a t i o n f o r j u s t one domain i s s u f f i c i e n t a t n o r m a l i n c i d e n c e ; t h e d i f f r a c t e d beams o f t h e o t h e r two domains c a n be o b t a i n e d by 120° r o t a t i o n s . In b o t h (2x2) and (2x1) c a l c u l a t i o n s , d i f f r a c t e d beams w h i c h a r e r e l a t e d by a 180° r o t a t i o n were a v e r a g e d w i t h e q u a l w e i g h t i n g s t o c o r r e s p o n d t o t h e s i t u a t i o n t h a t r e a l hcp(OOCM) s u r f a c e s g e n e r a l l y c o n s i s t o f e q u a l p o p u l a t i o n s of domains w i t h A and B t e r m i n a t i o n s . The a d s o r p t i o n s y s t e m s s t u d i e d i n c l u d e d 1. oxygen on T i ( 0 0 0 l ) ; 2. t i t a n i u m on T i ( 0 0 0 l ) ; 3. t e l l u r i u m on T i ( 0 0 0 l ) ; 4. oxygen on Z r ( 0 0 0 l ) ; and 5. z i r c o n i u m on Z r ( 0 0 0 l ) . Two a d s o r p t i o n s i t e s (3h and 3 f ) were c o n s i d e r e d f o r e a c h a d s o r p t i o n s y s t e m ; an a d d i t i o n a l s i t e (6u) was c o n s i d e r e d f o r s y s t e m ( 1 ) . The a d s o r p t i o n s i t e l a b e l s 3h and 3f i d e n t i f y s i t e s o f 3 - f o l d c o o r d i n a t i o n (3h/3f d i s t i n g u i s h e s w h ether e a c h a d s o r b e d atom i s d i r e c t l y above an atom i n t h e s e c o n d s u b s t r a t e l a y e r / a n o c t a h e d r a l h o l e i n t h e s u b s t r a t e ) , w h i l e 6u c o r r e s p o n d s t o t h e model i n w h i c h t h e a d s o r b e d 176 atoms i n c o r p o r a t e i n t o t h e o c t a h e d r a l h o l e s between t h e f i r s t and s e c o n d s u b s t r a t e l a y e r s . The i n t e r l a y e r s p a c i n g s u s e d were Z r - Z r 2.57 A, Z r - 0 1.30 A, T i - T i 2.34 A, T i - 0 1.30 O O A, T i - T e 2.07 A; phase s h i f t s ( t o 1=1) f o r t i t a n i u m and z i r c o n i u m were d e d u c e d f r o m band s t r u c t u r e a t o m i c p o t e n t i a l s [ 9 6 ] , whereas t h o s e f o r oxygen and t e l l u r i u m were fr o m Demuth et al. [ 9 8 ] , The i m a g i n a r y p a r t o f t h e ' m u f f i n - t i n ' p o t e n t i a l (V 0 /. ) was g i v e n a c o n s t a n t v a l u e o f -5 eV f o r a l l s y s t e m s . 5.3.2 R-FACTOR ANALYSES AND DISCUSSION C a l c u l a t e d 1 ( E ) c u r v e s f o r f o u r i n t e g r a l o r d e r beams ( ( 0 , 0 ) , ( 1 , 0 ) , (1,1) and ( 2 , 0 ) ) and s e v e n f r a c t i o n a l o r d e r beams ( ( 1 / 2 , - 1 / 2 ) , ( 1 / 2 , 1 / 2 ) , ( 3 / 2 , - 1 / 2 ) , ( 3 / 2 , - 3 / 2 ) , ( 3 / 2 , 1 / 2 ) , (5/2,-3/2) and ( 5 / 2 , - 1 / 2 ) ) f o r t h e c o r r e s p o n d i n g (2x1) and (2x2) s t r u c t u r e s were compared w i t h t h e n o r m a l i z e d v e r s i o n s o f R „ ^ and R _ [ 1 4 4 ] . The r e s u l t s f o r c o m p a r i s o n s o f ^ J v c o r r e s p o n d i n g (2x1) and (2x2) 1 ( E ) c u r v e s f o r s e v e n s u r f a c e s t r u c t u r e s and t h e i r t o t a l e n e r g y r a n g e s o f t h e 1 ( E ) d a t a a r e summarized i n T a b l e 5.1. F o r c o m p a r i n g t h e 1 ( E ) c u r v e s f r o m c o r r e s p o n d i n g (2x1) a n d (2x2) m o d e l s , t h e Z a n a z z i - J o n a R - f a c t o r (R_-:) may be z j most s u i t a b l e s i n c e i t a t t e m p t s t o compare 1 ( E ) c u r v e s with-r e g a r d t o a l l s i g n i f i c a n t f e a t u r e s . The c o m p a r i s o n s i n T a b l e 5.1 i n d i c a t e v e r y low v a l u e s f o r R z j r and hence c l o s e s i m i l a r i t y between c o r r e s p o n d i n g 1 ( E ) c u r v e s from t h e (2x1) and (2x2) m o d e l s . T h i s s t a t e m e n t h o l d s t r u e f o r d i f f e r e n t Surface Integral beams Fractional-order beams AE R P AE R z j R P Ti(000l)-0 (3h) 714 0.020 0. 105 952 0.010 0.025 Ti(0001)-O (6u) 714 0.011 0.046 978 0.012 0.036 T i ( 0 0 0 D - T i (3h) 352 0.007 0.060 492 0.009 0.049 Ti(0001)-Te (3h) 710 0.026 0.126 868 0.051 0.119 Ti ( 0 0 0 D-Te (3f) 710 0.041 0. 135 868 0.047 0.143 Zr(000l)-O (3h) 364 0.028 0. 120 508 0.005 0.033 Zr ( 000D-Zr (3h) 364 0.013 0. 122 508 0.019 0.130 Table 5.1: Comparisons of calculated 1(E) curves for integral and f r a c t i o n a l -order beams for surface structures with (2x1) and (2x2) t r a n s l a t i o n a l symmetries on hcp (000l) surfaces according to the r e l i a b i l i t y indices of Zanazzi-Jona ( R z j ) and Pendry (R ). The energy range of 1(E) data for each comparison i s i d e n t i f i e d by AE (in eV). 178 s t r u c t u r a l a r r a n g e m e n t s and f o r a v a r i e t y o f i o n c o r e s c a t t e r i n g f a c t o r s . N e v e r t h e l e s s , t h e r e i s a g e n e r a l t e n d e n c y f o r t h e f r a c t i o n a l o r d e r beam R - f a c t o r s t o i n c r e a s e w i t h t h e s c a t t e r i n g s t r e n g t h o f t h e a d s o r b e d s p e c i e s . T h i s t r e n d i s d e m o n s t r a t e d by t h e s e r i e s of 0, T i and Te a d s o r b e d on t h e 3h s i t e of T i ( O O O l ) s u r f a c e . I n s p e c t i o n o f t h e f r a c t i o n a l o r d e r beam 1(E) c u r v e s ( F i g u r e s 5.3 t o 5.6) of c o r r e s p o n d i n g (2x1) and (2x2) s t r u c t u r e s i n d i c a t e s i n c r e a s i n g , a l b e i t s m a l l , d i f f e r e n c e s f r o m oxygen t o t e l l u r i u m . S i n c e t h e i o n c o r e s c a t t e r i n g shows a g e n e r a l i n c r e a s e w i t h a t o m i c number, t e l l u r i u m i s assumed t o have t h e h i g h e s t s c a t t e r i n g s t r e n g t h i n t h i s s e r i e s o f ad-atoms. The f a c t t h a t t e l l u r i u m has a h a r d s p h e r e r a d i u s s l i g h t l y l a r g e r t h a n t h a t o f t i t a n i u m s u g g e s t s t h a t n e i g h b o r i n g Te atoms a r e t o o c l o s e t o g e t h e r f o r a r e a l (2x1) s t r u c t u r e t o be f o rmed; n e v e r t h e l e s s a p p r e c i a b l e m u l t i p l e s c a t t e r i n g i n t h e a d s o r b e d l a y e r may be e x p e c t e d f r o m a model c a l c u l a t i o n . F o r i n t e g r a l beams t h e s i t u a t i o n i s s l i g h t l y d i f f e r e n t . F o r t h e (2x1) and (2x2) m o d e l s o f T i ( o r Z r ) on t h e (0001) s u r f a c e o f T i ( o r Z r ) ( w i t h t h e a p p r o p r i a t e b u l k topmost i n t e r l a y e r s p a c i n g ) , t h e 1 ( E ) c u r v e s o f i n t e g r a l o r d e r beams become v e r y s i m i l a r t o t h e c o r r e s p o n d i n g beams f r o m t h e (1x1) b u l k - l i k e s u r f a c e . As a r e s u l t , f o r example, R • d e c r e a s e s f o r t h e i n t e g r a l beams on r e p l a c i n g 0 on T i ( O O O l ) by T i or r e p l a c i n g 0 on Z r ( 0 0 0 l ) by Z r . F i g u r e 5.7 shows t h e (2x1) v e r s u s (2x2) c o m p a r i s o n f o r t h e (1,1) beam from 0 on Z r ( 0 0 0 l ) , a n d from Zr on Z r ( 0 0 0 l ) . The l a t t e r p a i r o f I ( E ) 1 7 9 ( 1/2,-1/2) BEAM t A ) n _ P p =0.0119 I?2J =0.0076 i r 100 ENERGY (EV) 160 Figure 5 . 3 : Comparison of calculated 1(E) curves for the ( 1 /2 , - 1 /2 ) beam for the corresponding (2x1) (dotted l i n e ) and (2x2) ( s o l i d l i n e ) structures for adsorption of (A) 0, (B) Ti and (C) Te, at the 3h s i t e on the Ti(OOOl) surface. Single beam R and R • are indicated for each pair of 1(E) p z J curves. 180 Figure 5.4: Same as Figure 5.3, except for the (1/2,1/2) beam. 1 8 1 r i 1 1 1 1 1 1 4o IOO iso ENERGY CEV ) Figure 5.5: Same as Figure 5 .3 , except for the (3/2,-1/2) beam. 182 CO OQ CC < >-LO Z LU (3/2r3/2)BEAM PP=0.0236 ZJ=0.0104 PP =0.0497 PZJ=Q0I06 PP=0.0757 R*. =0.0316 . 1 1 i i I I 40 100 160 ENERGY (EV) Figure 5.6: Same as Figure 5.3, except for the (3/2,-3/2) beam. 183 c u r v e s a r e a l m o s t v i s u a l l y i n d i s t i n g u i s h a b l e , w h i c h i s a l s o i n d i c a t e d so by a v e r y low v a l u e o f R z j . P e n d r y ' s R - f a c t o r (Rp) e m p h a s i z e s p o s i t i o n s of p e a k s and t r o u g h s , and i t s v a l u e s t e n d t o be c o n s i d e r a b l y l a r g e r t h a n t h e c o r r e s p o n d i n g v a l u e s o f R z j ? a p p a r e n t l y R p can e x a g g e r a t e t h e e f f e c t o f s m a l l d i f f e r e n c e s i n 1 ( E ) c u r v e s b e c a u s e u n l i k e R „ J i t d o e s n o t put e x t r a w e i g h t s on p r o m i n e n t p e a k s . T h i s i s most e v i d e n t from F i g u r e 5.7. Here R z j i n c r e a s e s a l m o s t 10 t i m e s w h i l e R p i s o n l y d o u b l e d , from a p a i r o f n e a r l y v i s u a l l y i n d i s t i n g u i s h a b l e c u r v e s t o a p a i r o f c u r v e s showing e a s i l y d e t e c t a b l e d i f f e r e n c e s i n s h a p e . T h a t s u g g e s t s R z j i s more u s e f u l f o r c o m p a r i n g 1 ( E ) c u r v e s w h i c h match c o m p a r a t i v e l y w e l l ( e . g . f o r r e f i n e m e n t s t a g e s o f a LEED a n a l y s i s ) whereas R i s b e t t e r f o r t h e i n i t i a l s t a g e s of a s u r f a c e c r y s t a l l o g r a p h i c s t u d y . T h i s work d e v e l o p e d f r o m a LEED c r y s t a l l o g r a p h i c i n v e s t i g a t i o n f o r oxygen a d s o r b e d a t a Z r ( 0 0 0 l ) s u r f a c e ( C h a p t e r 6 ) . I t was f o u n d t h a t w i t h n o r m a l i n c i d e n c e e x p e r i m e n t a l i n t e n s i t y d a t a f o r s e v e n (3 i n t e g r a l and 4 f r a c t i o n a l o r d e r ) d i f f r a c t e d beams m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s were u n a b l e t o d i s t i n g u i s h between t h e (2x1) and (2x2) t r a n s l a t i o n a l s y m m e t r i e s f o r a v a r i e t y o f l o c a l c o o r d i n a t i o n m o d e l s , a l t h o u g h an e a r l y r e p o r t by S h i h et al. [40] d i d s u g g e s t t h a t t h e r e may be s u f f i c i e n t d i f f e r e n c e s i n c a l c u l a t e d 1(E) c u r v e s f o r t h e (2x2) and (2x1) m o d e l s o f d i s s o c i a t e d CO on T i ( 0 0 0 l ) t o d i s t i n g u i s h between s u c h m o d e l s . T h i s i s i n t e r e s t i n g s i n c e C i s not 184 CI,I) BEAM I i i i 1 1 r 40 80 I20 ie ENERGY (EV) Figure 5.7: Comparison of calculated 1(E) curves for the (1,1) beam for the corresponding (2x1) (dotted line) and (2x2) ( s o l i d l i ne) structures for adsorption of (A) 0 and (B) Zr, at the 3h s i t e on the Zr(OOOl) surface. Single beam R and R ^ are indicated for each pair of 1(E) curves. 185 l i k e l y t o be a s t r o n g e r s c a t t e r e r t h a n 0; however s m a l l d i f f e r e n c e s may a c c u m u l a t e w i t h a more e x t e n s i v e s e t of i n t e n s i t y d a t a . T h e r e f o r e t h e o b s e r v a t i o n of S h i h et al . may have been p o s s i b l e b e c a u s e o f t h e r e l a t i v e l y l a r g e amount o f e x p e r i m e n t a l d a t a t h a t t h e y had a v a i l a b l e ( s p e c i f i c a l l y , 10 beams o v e r 2 d i r e c t i o n s o f i n c i d e n c e ) . F u r t h e r s t u d y c o u l d be u s e f u l t o a s s e s s w hether r e s u l t i n g d i f f e r e n c e s i n R - f a c t o r v a l u e s ( f o r c o m p a r i s o n s between e x p e r i m e n t a l and t h e c o r r e s p o n d i n g c a l c u l a t e d 1 ( E ) c u r v e s f r o m e a c h c o v e r a g e ) c o u l d be l a r g e enough t o d i s t i n g u i s h between t h e two c o v e r a g e s . I t i s p o s s i b l e t h a t t h e s i m i l a r i t y o f 1(E) c u r v e s from t h e (2x1) and (2x2) s t r u c t u r e s may be most p r o n o u n c e d a t n o r m a l i n c i d e n c e where t h e m u l t i p l e s c a t t e r i n g w i t h i n t h e o v e r l a y e r s i s s m a l l ; t h i s i s b e c a u s e a t o m i c s c a t t e r i n g f a c t o r s f o r s c a t t e r i n g a n g l e s c l o s e t o 90° a r e g e n e r a l l y s m a l l . T h i s s u g g e s t s a s t r a t e g y f o r LEED a n a l y s e s f o r s u c h s y s t e m s . F i r s t use n o r m a l i n c i d e n c e d a t a t o d e t e r m i n e t h e l o c a l a d s o r p t i o n s i t e f o r t h e ad-atom, t h e n use d a t a f o r s h a l l o w a n g l e s o f i n c i d e n c e t o d e t e r m i n e t h e u n i t mesh o f t h e a d l a y e r . 5.4 APPROXIMATE SCHEMES FOR MULTIPLE SCATTERING CALCULATIONS The s t a b i l i t y o f t h e f r a c t i o n a l o r d e r beam 1( E ) c u r v e s w i t h d i f f e r e n t a d s o r b a t e c o v e r a g e s i m p l i e s t h a t (1) i n t r a l a y e r m u l t i p l e s c a t t e r i n g ( a t l e a s t i n t h e a d l a y e r ) i s weak compared t o i n t e r l a y e r s c a t t e r i n g , and (2) t h e u n i t mesh a r e a o f t h e a d l a y e r c a n be r e d u c e d , b u t so as t o keep 186 an a p p r e c i a b l e number o f beams i n common w i t h t h e s i t u a t i o n f o r t h e o r i g i n a l s t r u c t u r e . T h e s e two a s p e c t s have been u s e d i n v a r i o u s a t t e m p t s t o s i m p l i f y m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s f o r more complex s y s t e m s . 5.4'.1 APPROXIMATIONS IN L-SPACE At o r n e a r n o r m a l i n c i d e n c e , i n t r a l a y e r m u l t i p l e s c a t t e r i n g c a n o f t e n be r e l a t i v e l y weak b e c a u s e s u c h p r o c e s s e s r e q u i r e a t l e a s t two l a r g e - a n g l e (£90°) s c a t t e r i n g e v e n t s [ l 3 9 ] , whereas o n l y one s u c h e v e n t (back s c a t t e r i n g ) n eed o c c u r w i t h i n t e r l a y e r m u l t i p l e s c a t t e r i n g . Thus t h e e f f e c t of i n t r a l a y e r s c a t t e r i n g may, t o a r e a s o n a b l e a p p r o x i m a t i o n , be t r e a t e d k i n e m a t i c a l l y . The l a t t e r t r e a t m e n t y i e l d s a d i a g o n a l i n t r a l a y e r s c a t t e r i n g m a t r i x X ( e q u a t i o n ( 3 . 2 3 ) ) , w h i c h r e s t r i c t s a n g u l a r momentum m i x i n g between n e i g h b o r i n g atoms t o t h e same (/m) p a i r (/ and m a r e t h e a n g u l a r momentum quantum numbers f o r c h a r a c t e r i z i n g s p h e r i c a l w a v e s ) . One o f t h e most t i m e - c o n s u m i n g s t e p s i n t h e c a l c u l a t i o n o f t h e d i f f r a c t i o n m a t r i x M " i n v o l v e s t h e i n v e r s i o n o f t h e m a t r i x [ £ - X ] ( s e e e q u a t i o n s (3.22) and ( 3 . 2 3 ) ) ; b u t t h e k i n e m a t i c a l t r e a t m e n t l e a d s t o j u s t t h e t r i v i a l i n v e r s i o n o f a d i a g o n a l m a t r i x . In t h e quasi-dynami cal method (QDM)[140,141], a s t r o n g e r a s s u m p t i o n i s made t h a t i n t r a l a y e r m u l t i p l e s c a t t e r i n g c a n be n e g l e c t e d i n a l l l a y e r s . I n d e e d f o r m o d e r a t e l y w e a k l y s c a t t e r i n g c r y s t a l s , s u c h a s S i and Ge, t h e q u a s i - d y n a m i c a l and f u l l d y n a m i c a l methods g i v e s i m i l a r i n t e n s i t y c u r v e s [ l 4 7 ] . F o r 187 s t r o n g l y s c a t t e r i n g atoms i n more-compact s t r u c t u r e s , t h i s a p p r o x i m a t i o n i s l e s s s a t i s f a c t o r y [ 1 4 8 ] . The r e s u l t s o f t h i s work s u g g e s t t h a t QDM may be more g e n e r a l l y a p p l i c a b l e i f t h e n e g l e c t o f i n t r a l a y e r m u l t i p l e s c a t t e r i n g i s r e s t r i c t e d t o t h e a d s o r b e d l a y e r s , r e g a r d l e s s o f s c a t t e r i n g s t r e n g t h s . A h i g h e r o r d e r a p p r o x i m a t i o n o f i n t r a l a y e r m u l t i p l e s c a t t e r i n g i s c o n s i d e r e d i n t h e diagonal dominant method (DDM), w h i c h was p r o p o s e d by S h i h and T a m [ l 4 2 ] . I n t h i s method, a n g u l a r momentum m i x i n g between two atoms i s a l l o w e d o n l y when m=m'; i t t h e r e f o r e g i v e s r i s e t o a s p a r s e ( b u t banded) i n t r a l a y e r s c a t t e r i n g m a t r i x X. The i n v e r s i o n of t h e l a t t e r i s c o m p u t a t i o n a l l y much l e s s demanding t h a n t h e c o r r e s p o n d i n g f u l l m a t r i x . S h i h and Tarn have shown t h a t t h e r e s u l t s f r o m DDM f o r s e v e r a l i o n i c c r y s t a l s a g r e e v e r y w e l l w i t h e x a c t c a l c u l a t i o n s , a l t h o u g h t h e c o m p a r i s o n h as been l i m i t e d t o n o r m a l i n c i d e n c e c a l c u l a t i o n s . 5.4.2 APPROXIMATIONS IN K-SPACE The s i m i l a r i t y between t h e c a l c u l a t e d 1 ( E ) c u r v e s f o r t h e c o r r e s p o n d i n g (2x1) and (2x2) s t r u c t u r e s i m m e d i a t e l y shows t h a t t h e a r e a o f t h e u n i t mesh o f t h e a d s o r b e d l a y e r c a n be h a l v e d w i t h o u t s i g n i f i c a n t l y a f f e c t i n g t h e r e s u l t s o f t h e c a l c u l a t i o n s . A v e r y u s e f u l c o n s e q u e n c e o f t h i s f a c t i s t h e r e d u c t i o n i n t h e number o f beams r e q u i r e d i n a m u l t i p l e s c a t t e r i n g c a l c u l a t i o n , b e c a u s e t h e number o f beams r e q u i r e d i s d i r e c t l y p r o p o r t i o n a l t o t h e u n i t mesh a r e a i n a p a r t i c u l a r a d s o r p t i o n s y s t e m ( e q u a t i o n ( 3 . 4 0 ) ) . I n t h i s work 188 n o r m a l i n c i d e n c e i s assumed, and t h e r e f o r e t h e s c a t t e r i n g p r o b l e m f o r t h e (2x2) s t r u c t u r e c an u t i l i z e a h i g h e r s u r f a c e symmetry (namely a 3 - f o l d r o t a t i o n and a m i r r o r p l a n e ) w h i c h l e a d s t o fewer s y m m e t r i z e d beams compared t o t h e (2x1) c a l c u l a t i o n . In g e n e r a l , however, a s m a l l e r u n i t mesh a r e a i s more f a v o r a b l e when o f f - n o r m a l i n c i d e n c e i s c o n s i d e r e d . The u n i t mesh s i z e r e d u c t i o n method has been f u r t h e r t e s t e d i n t h i s l a b o r a t o r y f o r t h e s y s t e m C u ( l 0 0 ) - p ( 2 x 2 ) - S [ 1 4 9 ] . P r e l i m i n a r y r e s u l t s have i n d i c a t e d t h a t even a t o f f - n o r m a l i n c i d e n c e (6=14°) 1 ( E ) c u r v e s f o r f r a c t i o n a l o r d e r beams from p ( 1 x 2 ) , p ( 2 x 1 ) and c ( 2 x 2 ) ( f o r c o n s t a n t l o c a l s t r u c t u r e ) a r e a l m o s t i d e n t i c a l t o t h e c o r r e s p o n d i n g 1 ( E ) c u r v e s f r o m t h e f u l l p ( 2 x 2 ) c a l c u l a t i o n . From a p h y s i c a l s t a n d p o i n t , t h e u n i t mesh s i z e r e d u c t i o n method e s s e n t i a l l y b r e a k s t h e s u r f a c e s t r u c t u r e i n t o s m a l l e r a d s o r p t i o n n e t s ; a n d t h e r e c i p r o c a l l a t t i c e s o f t h e l a t t e r t h e n combine t o mimic t h a t o f t h e o r i g i n a l s u r f a c e s t r u c t u r e . From a c o m p u t a t i o n a l s t a n d p o i n t , t h i s p r o c e d u r e i s e q u i v a l e n t t o i g n o r i n g a l l b u t two s e t s o f d i f f r a c t e d beams ( t h e c o n c e p t of beam s e t s has been d i s c u s s e d i n S e c t i o n 3.5.2) i n t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n f o r beam i n t e n s i t i e s . More b r o a d l y , t h e f u l l beam s e t s a r e r e q u i r e d t o d e f i n e t h e l o n g r a n g e 2 - d i m e n s i o n a l p e r i o d i c i t y of t h e a d s o r b e d l a y e r , b u t t h e y a r e l e s s i m p o r t a n t i n s o f a r a s beam i n t e n s i t i e s a r e c o n c e r n e d . The l a t t e r have been shown[51,139] (and f u r t h e r s u p p o r t e d by t h i s work) t o be 189 d e t e r m i n e d p r i m a r i l y by s h o r t - r a n g e o r d e r ( i . e . l o c a l a t o m i c e n v i r o n m e n t ) . F o r a commensurate (pxq) s u p e r l a t t i c e , pq beam s e t s a r e f o r m a l l y r e q u i r e d f o r t h e c a l c u l a t i o n o f i t s l a y e r d i f f r a c t i o n m a t r i x ( S e c t i o n 4.5.2) i n an e x a c t m u l t i p l e s c a t t e r i n g c a l c u l a t i o n , t h e r e b y t a k i n g i n t o a c c o u n t a l l t h e p o s s i b l e s c a t t e r i n g p a t h s between p a i r s o f d i f f r a c t e d beams. By r e s t r i c t i n g t h e number of n o n - z e r o - a n g l e s c a t t e r i n g e v e n t s t o j u s t one o r two, Van Hove et C / . [ 5 1 ] n o t e d t h a t an emergent i n t e g r a l o r d e r beam i s n o t a f f e c t e d by any f r a c t i o n a l o r d e r beams; whereas an emergent f r a c t i o n a l o r d e r beam i s a f f e c t e d o n l y by beams w h i c h b e l o n g t o i t s own beam s e t o r t o t h e i n t e g r a l o r d e r beam s e t . T h i s i s t h e b a s i s f o r t h e i r p r o p o s e d beam set neglect (BSN) method f o r t h e c a l c u l a t i o n o f t h e d i f f r a c t i o n m a t r i x M~~ f o r t h e s u p e r l a t t i c e . In t h i s method, a f r a c t i o n a l o r d e r beam s e t i s c o u p l e d w i t h t h e i n t e g r a l o r d e r beam s e t t o fo r m a m i n i - d i f f r a c t i o n m a t r i x f o r t h e a d l a y e r . The s u b s e q u e n t s t a c k i n g o f l a y e r s t o o b t a i n d i f f r a c t e d beam i n t e n s i t i e s f o r t h i s p a r t i c u l a r s e t o f f r a c t i o n a l o r d e r beams i s t h e same as t h e f u l l d y n a m i c a l t r e a t m e n t . T h i s p r o c e d u r e c a n be r e p e a t e d f o r a l l t h e o t h e r f r a c t i o n a l o r d e r beam s e t s . The p ( 2 x 1 ) c a l c u l a t i o n s i n t h i s work t h u s r e p r e s e n t a s p e c i a l c a s e of t h i s g e n e r a l i z e d scheme; t h e d i f f r a c t i o n m a t r i x f o r t h e p ( 2 x l ) s t r u c t u r e c a n be t a k e n a s one o f t h e m i n i - d i f f r a c t i o n m a t r i c e s o f t h e p ( 2 x 2 ) s t r u c t u r e , b u t b e c a u s e o f symmetry, o n l y one s u c h m a t r i x i s r e q u i r e d . In g e n e r a l , f o r a (pxq) s u p e r l a t t i c e s t r u c t u r e , i n s t e a d of c a l c u l a t i n g a 190 s u p e r l a t t i c e d i f f r a c t i o n m a t r i x of d i m e n s i o n pqn (where n i s t h e a v e r a g e number o f beams i n e a c h beam s e t ) , t h e beam s e t n e g l e c t method s i m p l y c a l c u l a t e s (pq-1) s m a l l e r m a t r i c e s of d i m e n s i o n 2n. The s a v i n g s i n c o m p u t i n g e f f o r t i s s i g n i f i c a n t , e s p e c i a l l y i n LEED s t u d i e s i n v o l v i n g a v e r y l a r g e s u p e r l a t t i c e , and f o r o f f - n o r m a l i n c i d e n c e where s y m m e t r i z a t i o n of beam s e t s i s m i n i m a l . T h i s method has been a p p l i e d s a t i s f a c t o r i l y t o a LEED a n a l y s i s of benzene a d s o r b e d on t h e Rh(111) s u r f a c e [ 5 l ] , a l t h o u g h more t e s t s a r e needed t o a s s e s s i t s g e n e r a l a p p l i c a b i l i t y . 5.5 CONCLUSION T h i s work has e x t e n d e d an o b s e r v a t i o n of Yang et al. [ 1 3 9 ] , and shown t h a t c o r r e s p o n d i n g 1 ( E ) c u r v e s f o r s u r f a c e s t r u c t u r e s w i t h (2x1) and (2x2) t r a n s l a t i o n a l s y m m e t r i e s on h c p ( 0 0 0 1 ) s u r f a c e s c a n be r e m a r k a b l y s i m i l a r . F o r a (2x1) s t r u c t u r e , atoms i n t h e a d s o r b e d l a y e r a r e s e p a r a t e d by j u s t t h e s u b s t r a t e i n t e r a t o m i c d i s t a n c e , but t h e p r e v i o u s s t a t e m e n t i s f o u n d t o h o l d c l o s e l y even o v e r a r a n g e o f s c a t t e r i n g s t r e n g t h s . O v e r a l l t h i s s t u d y adds f u r t h e r s u p p o r t t o s e v e r a l a p p r o x i m a t e schemes f o r s i m p l i f y i n g m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s , a t l e a s t f o r t h e e x p l o r a t o r y i n v e s t i g a t i o n s of s y s t e m s w i t h l o n g r e p e a t d i s t a n c e s . S i n c e t h e c o m p a r i s o n s h e r e a r e made f o r c u r v e s w h i c h a r e b a s i c a l l y r a t h e r s i m i l a r , t h e Z a n a z z i - J o n a and P e n d r y R - f a c t o r p r o c e d u r e s c o u l d be compared i n ways t h a t 191 r a r e l y have been t r i e d b e f o r e ( f o r example, t h e v a l u e s of R • a r e low compared w i t h t h e more u s u a l c o m p a r i s o n s between e x p e r i m e n t a l and c a l c u l a t e d 1 ( E ) c u r v e s ) . F i n a l l y t h i s work e m p h a s i z e d a g a i n [ l 5 0 ] t h a t an a p p r e c i a b l e number o f LEED 1( E ) c u r v e s a r e s t i l l r e q u i r e d when t h e o b j e c t i v e i s t o d i s t i n g u i s h between f i n e d i f f e r e n c e s i n a s t r u c t u r a l a n a l y s i s (and t h i s i n c l u d e s d i s t i n g u i s h i n g (2x1) and (2x2) s t r u c t u r e s on hcp(OOOl) and f c c ( ! 1 l ) s u r f a c e s ) . C H A P T E R 6 O X Y G E N A D S O R P T I O N ON Z R ( 0 0 0 1 ) 192 193 6.1 INTRODUCTION Zirconium ranks t w e l f t h among the elements i n abundance. I t has found a p p l i c a t i o n s i n c o n s t r u c t i o n f o r nuc l e a r r e a c t o r s because i t has a low a b s o r p t i o n c r o s s s e c t i o n f o r neutrons as w e l l as a hig h r e s i s t a n c e to the c o r r o s i v e environments i n s i d e a r e a c t o r . The outstanding c o r r o s i o n r e s i s t a n c e of zi r c o n i u m and i t s a l l o y s i s due to the formation of a p r o t e c t i v e but very t h i n f i l m of oxide on the metal s u r f a c e . In f a c t the ease of oxide formation makes zir c o n i u m a u s e f u l g e t t e r m a t e r i a l f o r oxygen, which i s widely used i n the e l e c t r o n i c s i n d u s t r y . The g e n e r a l l y high r e a c t i v i t y of z i r c o n i u m toward gaseous s p e c i e s a l s o makes i t an important c o n s t i t u e n t of many c a t a l y t i c a l l o y s [ l 5 l ] . There i s a growing i n t e r e s t i n the a d s o r p t i o n p r o p e r t i e s of zir c o n i u m due to i t s t e c h n o l o g i c a l a p p l i c a t i o n s . S e v e r a l s t u d i e s of the chemisorption of small gaseous molecules on zi r c o n i u m s u r f a c e s under UHV environments have been reported!69,152-158], but most of these are concerned e i t h e r with p o l y c r y s t a l l i n e z i r c o n i u m or with t h i c k f i l m s of oxide. S t r u c t u r a l i n f o r m a t i o n of adsorbates on metal s u r f a c e s r e p r e s e n t s a fundamental requirement f o r developing a t o m i s t i c models both f o r simple c h e m i s o r p t i o n and u l t i m a t e l y f o r s u r f a c e r e a c t i v i t y with the adsorbing s p e c i e s . D e s p i t e the numerous t e c h n o l o g i c a l a p p l i c a t i o n s of zir c o n i u m metal and i t s a l l o y s , no s t r u c t u r a l data are c u r r e n t l y a v a i l a b l e f o r a d s o r p t i o n systems that i n v o l v e s u r f a c e s of zi r c o n i u m . T h i s may be a t t r i b u t e d i n pa r t to the 194 d i f f i c u l t i e s i n o b t a i n i n g a w e l l c h a r a c t e r i z e d z i r c o n i u m s u r f a c e ( S e c t i o n 1.4). The p r e s e n t LEED c r y s t a l l o g r a p h i c s t u d y aims t o r e v e a l t h e a t o m i c g e o m e t r y of oxygen a d s o r p t i o n on t h e (0001) s u r f a c e o f z i r c o n i u m a t r e l a t i v e l y low oxygen c o v e r a g e s . S p e c i f i c a l l y t h e i n t e n s i t i e s o f LEED beams f r o m Z r ( 0 0 0 1 ) - ( 2 x 2 ) - O and f r o m Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 were measured and a n a l y z e d . T h i s s t u d y i s p o t e n t i a l l y c h a l l e n g i n g , not o n l y b e c a u s e of t h e d i f f i c u l t i e s o f w o r k i n g w i t h z i r c o n i u m , b u t a l s o b e c a u s e some uncommon s t r u c t u r a l r e s u l t s may be e x p e c t e d f o l l o w i n g s t u d i e s by S h i h et a l . [41] f o r n i t r o g e n a d s o r p t i o n on t h e (0001) s u r f a c e of a n o t h e r g r o u p 4 hep m e t a l , namely t i t a n i u m . The l a t t e r s t u d y s u g g e s t e d t h a t N atoms f o r m an u n d e r l a y e r s t r u c t u r e by o c c u p y i n g a l l o c t a h e d r a l h o l e s i t e s between t h e f i r s t and s e c o n d T i l a y e r s . Then t h e f i r s t t h r e e l a y e r s o f t h e s u r f a c e r e g i o n c o r r e s p o n d c l o s e l y t o t h r e e s u c c e s s i v e (111) l a y e r s of T i N . S i n c e ZrO has t h e same N a C l t y p e c r y s t a l l a t t i c e a s T i N , i t i s l o g i c a l t o ask w hether oxygen a d s o r p t i o n on Z r ( 0 0 0 l ) s u r f a c e s w o u l d y i e l d t h e same phenomenon o f u n d e r l a y e r f o r m a t i o n , and t h i s i s one o f t h e m o t i v a t i o n s f o r t h e p r e s e n t LEED c r y s t a l l o g r a p h i c s t u d y . The p r e s e n t work was i n i t i a t e d by o b s e r v a t i o n s made by M o o r e [ l 3 0 ] a t t h e end o f h i s Ph.D. t h e s i s , where s e v e r a l 1 ( E ) c u r v e s were r e p o r t e d from t h e (2x2) and (1x1) LEED p a t t e r n s . However, i n i t i a l c a l c u l a t e d 1 ( E ) c u r v e s o b t a i n e d i n t h e e a r l y s t a g e s o f t h i s work y i e l d e d p o o r agreement w i t h Moore';s measured 1 ( E ) c u r v e s . I t h e n u n d e r t o o k a r e - a n a l y s i s 195 of Moore's photographic r e c o r d of the LEED p a t t e r n s . That confirmed the o r i g i n a l 1(E) curves, except f o r the (3/2,-3/2) beam (see F i g u r e 5.2 f o r beam n o t a t i o n ) f o r the (2x2) p a t t e r n , where the e n e r g i e s were 40 eV higher than the c o r r e c t v a l u e s . With the c o r r e c t e d measured 1(E) curves, only s l i g h t l y b e t t e r agreement with c a l c u l a t e d curves was ach i e v e d . S i m i l a r LEED p a t t e r n s to those formed with oxygen a d s o r p t i o n on Zr(0001) have been r e p o r t e d f o r oxygen on T i ( 0 0 0 1 ) [ 1 5 9 ] , but no s t r u c t u r a l r e s u l t s have yet been p u b l i s h e d . Indeed once my work was at an advanced stage, we l e a r n e d that Shih and Jona[159,160] had been unable to complete a LEED c r y s t a l l o g r a p h i c a n a l y s i s f o r 0 on T i ( 0 0 0 l ) . The l a t t e r system has s i n c e been s t u d i e d by a wide v a r i e t y of s u r f a c e techniques[161], yet so f a r there has been no consensus on the q u e s t i o n of oxygen o v e r l a y e r versus underlayer,. l e t alone s t r u c t u r a l d a t a . From the r e s u l t s r e p o r t e d f o r oxygen a d s o r p t i o n on t i t a n i u m and on aluminum[162], i t can be concluded that the i n i t i a l stages of o x i d a t i o n of e l e c t r o p o s i t i v e metal s u r f a c e s may not be s t r a i g h t f o r w a r d , but that only makes the present s t u d i e s more worthwhile. In t h i s work separate LEED experiments and s e v e r a l AES s t u d i e s were c a r r i e d out with an independent z i r c o n i u m sample to e s t a b l i s h that the experimental data are r e p r o d u c i b l e . The 1(E) curves f o r the (1x1) p a t t e r n were a l s o measured with the new TV camera method while t h i s 1 9 6 t h e s i s w a s b e i n g w r i t t e n , a n d t h e y s h o w a h i g h d e g r e e o f r e p r o d u c i b i l i t y w i t h t h e p r e v i o u s l y m e a s u r e d d a t a . T h e l a t e s t v e r s i o n o f t h e m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s c o m p u t e r p r o g r a m s p r o v i d e d b y V a n H o v e [ 9 7 ] i n 1 9 8 3 f a c i l i t a t e d m o d e l c a l c u l a t i o n s f o r m a n y c o m p l i c a t e d s u r f a c e m o d e l s , s o m e o f w h i c h h a v e b e e n f o u n d t o g i v e b e t t e r a g r e e m e n t w i t h e x p e r i m e n t a l i n t e n s i t i e s f r o m t h e Z r ( 0 0 0 - 1 ) - ( 2 x 2 ) - 0 s t r u c t u r e . 6 . 2 E X P E R I M E N T A L 6 . 2 . 1 S A M P L E P R E P A R A T I O N A N D C L E A N I N G I n i t i a l z i r c o n i u m s p e c i m e n s w e r e c u t b y s p a r k e r o s i o n f r o m 9 9 . 9 7 % p u r i t y s i n g l e c r y s t a l r o d s ( g r o w n b y A . A k h t a r , D e p t . o f M e t a l l u r g y , U . B . C . ) o f d i a m e t e r . T h e n o r m a l t o t h e c u t t i n g p l a n e w a s o r i e n t e d t o w i t h i n 1 ° o f t h e ( 0 0 0 1 ) d i r e c t i o n i n t h e c r y s t a l . A f t e r c u t t i n g , t h e s a m p l e s w e r e m e c h a n i c a l l y p o l i s h e d w i t h i n c r e a s i n g l y f i n e r d i a m o n d p a s t e ( 9 - 1 u), a n d f i n a l l y w i t h 0 . 0 2 5 u a l u m i n a p a s t e f o r a b o u t 1 m i n u t e . T h e s a m p l e s w e r e d e g r e a s e d t h o r o u g h l y w i t h t r i c h l o r o e t h y l e n e , a n d t h e n c h e m i c a l l y e t c h e d i n a c i d ( 4 5 % H N 0 3 , 5 0 % H 2 0 2 , 5 % H F , b y v o l u m e f 1 6 3 ] ) f o r 3 0 - 6 0 s e c o n d s . T h i s p r o c e d u r e r e s u l t e d i n a s h i n y s u r f a c e w i t h o u t o b v i o u s s m e a r i n g w h e n v i e w e d w i t h a 1 0 X m a g n i f i e r . T h e c r y s t a l s l i c e w a s m o u n t e d o n a V a r i a n r e s i s t i v e h e a t e r a n d a c h r o m e l - a l u m e l t h e r m o c o u p l e w a s s p o t - w e l d e d t o t h e s a m p l e e d g e . 197 F i g u r e 2.12(a) shows an Auger spectrum measured with the c y l i n d r i c a l m i r r o r a n a l y z e r (16 kHz modulation frequency) from a Zr(0001) s u r f a c e on e n t e r i n g the FC12 chamber. Large q u a n t i t i e s of C and 0 were i n d i c a t e d by the Auger peaks at 272 eV ( C 2 7 2 ) and 510 eV ( 0 5 1 0 ) r e s p e c t i v e l y . The Mo peaks came from the sample holder cup, which i s sometimes s t r u c k by a s m a l l p a r t of the e l e c t r o n beam from the g l a n c i n g i n c i d e n c e gun. Argon ion bombardment (0.8-1.2 keV, 4-5 uh) was then c a r r i e d out at room temperature u n t i l i m p u r i t i e s other than C were reduced to below the d e t e c t a b l e amounts. At t h i s stage, the sample was annealed at about 600°C f o r 30 minutes to d r i v e most of the bulk carbon and a l s o s u l f u r to the s u r f a c e . T h i s was f o l l o w e d by s e v e r a l c y c l e s of A r + bombardment (with d e c r e a s i n g primary e n e r g i e s to minimize damage to the s u r f a c e ) and a n n e a l i n g u n t i l both C and S were at t h e i r minimum coverages. S u l f u r cannot be d e t e c t e d d i r e c t l y by AES with the present r e s o l u t i o n because the s u l f u r Auger peak at 150 eV (or S 1 5 0 ) o v e r l a p s with the z i r c o n i u m Auger peak at 147 eV ( Z r , „ 7 ) . However the presence of s u l f u r can be d e t e c t e d i n d i r e c t l y by monitoring the peak r a t i o of Z r 1 a 7 / Z r 9 2 . The use of the Z r 9 2 peak as a r e f e r e n c e was prompted by my o b s e r v a t i o n that i t was unattenuated when the Zr(000l) s u r f a c e was exposed to l a r g e doses of H 2S. A f t e r a t o t a l of about 50 hours of A r + bombardment, the peak r a t i o Z r 1 j ( 7 / Z r 9 2 reached a l i m i t i n g value of approximately 1.35, which compared w e l l with p u b l i s h e d data f o r a cleaned z i r c o n i u m s u r f a c e [ 1 6 4 ] . Carbon contamination c o u l d not be 198 removed to undetectable l e v e l s , and the minimum C 2 7 2 / Z r i 7 « peak r a t i o obtained was about 0.1, which i s b e l i e v e d to correspond to carbon coverage of l e s s than 3% monolayer ( d i s c u s s e d f u r t h e r i n S e c t i o n 6.5). 6.2.2 MEASUREMENTS OF 1(E) CURVES Exposure of a c l e a n Z r(000l) s u r f a c e to r e s e a r c h grade oxygen (Matheson 99.99% p u r i t y ) at a pr e s s u r e of approximately 5x10" 9 t o r r at temperatures below 100°C gave r i s e f i r s t to the formation of a f a i n t (2x2) LEED p a t t e r n with d i f f u s e d i f f r a c t i o n spots, a f t e r an exposure of about 1 Langmuir (1 L = 10" 6 t o r r s ) . However the spots became b r i g h t e r and sharper when the sample was annealed at about 250°C f o r 2-3 minutes. Annealing at temperatures above 400°C f o r a few minutes r e s u l t e d i n a d e p l e t i o n of oxygen, which i s presumably due to d i f f u s i o n of oxygen i n t o the bulk as has been observed i n a d s o r p t i o n s t u d i e s on p o l y c r y s t a l l i n e zirconium!157]. With i n c r e a s i n g oxygen exposure, the (2x2) p a t t e r n became f a i n t e r and e v e n t u a l l y a (1x1) p a t t e r n appeared, whose d i f f r a c t e d beam 1(E) curves were d i f f e r e n t from those of the c l e a n s u r f a c e . For even g r e a t e r oxygen exposures, the LEED p a t t e r n remained (1x1) but became d i f f u s e with an i n c r e a s i n g background. The l a t t e r stage p o s s i b l y s i g n a l e d m u l t i - l a y e r a d s o r p t i o n , as has been proposed f o r r e a c t i v e adsorbates on s u r f a c e s of e l e c t r o p o s i t i v e t r a n s i t i o n metals such as t i t a n i u m and zirco n i u m [ 1 6 5 ] , 199 The sharpest (2x2) LEED p a t t e r n was o b t a i n e d a f t e r the sample was annealed a t 250°C f o r 5 minutes, and then cooled to room temperature. The Auger peak r a t i o 0 5 1 0 / Z r 1 7 , was then about 0.9, and t h i s corresponds to 0.37 monolayer 0 coverage, a c c o r d i n g t o a c a l i b r a t i o n method with CO ( S e c t i o n 6.5). For the sharp (1x1) p a t t e r n , a f t e r the sample was annealed at 250°C f o r about 3 minutes, the r a t i o 0 5 1 0 / Z r l 7 l ( was 2.0, which corresponds to 0.8 monolayer oxygen coverage. D i f f r a c t e d beam i n t e n s i t i e s f o r both oxygen s t r u c t u r e s were measured at normal i n c i d e n c e by the photographic method at 2 eV i n t e r v a l s . Before the i n t r o d u c t i o n of the TV camera method to t h i s l a b o r a t o r y , normal i n c i d e n c e was e s t a b l i s h e d by a d j u s t i n g the sample manipulator micrometers u n t i l , by v i s u a l i n s p e c t i o n , the appearance and disappearance of the supposedly s y m m e t r i c a l l y e q u i v a l e n t beams were synchronized as the i n c i d e n t e n e r g i e s were v a r i e d . F i g u r e 6.1 shows the 1(E) curves of the s i x ' e q u i v a l e n t ' (1,0) beams, as w e l l as the average of these curves f o r Zr(0001)-(2x2)-0, at normal i n c i d e n c e a d j u s t e d by the above v i s u a l method. With experience, t h i s method works reasonably w e l l , and an assessment i s made i n F i g u r e 6.1 by r e p o r t i n g v a l u e s of R and R_^ f o r the i n d i v i d u a l beams i n comparison with the z 3 averaged beam. In the l a t e r stages of t h i s work, the normal i n c i d e n c e adjustment c o u l d be made more o b j e c t i v e with the use of the video LEED a n a l y z e r ; 1(E) curves of e q u i v a l e n t beams were then compared o n - l i n e t o assess the accuracy of normal i n c i d e n c e . With the photographic method, I 200 40 80 120 160 ENERGY CEV) Figure 6.1: Comparison of experimental 1(E) curves of s i x ' e q u i v a l e n t ' (1,0) beams ( s o l i d l i n e s ) o b t a i n e d at normal i n c i d e n c e f o r Zr(0001)-(2x2)-0 with the averaged curve (do t t e d l i n e ) . Single-beam R p and R z j are a l s o given f o r such comparisons. 201 re-measured 1 ( E ) curves f o r seven d i f f r a c t e d beams design a t e d (1,0), (1,1), (2,0), (1/2,-1/2), (1/2,1/2), (3/2,-1/2) and (3/2,-3/2) f o r the (2x2) s t r u c t u r e , and f o r the two d i f f r a c t e d beams designated (1,0) and (1 ,1) f o r the normal i n c i d e n c e (1x1) s t r u c t u r e (beam n o t a t i o n f o l l o w s that of F i g u r e 5.2). Measured 1 ( E ) curves from d i f f e r e n t beams which are expected to be e q u i v a l e n t from symmetry and f o r equal p o p u l a t i o n s of the p o s s i b l e r o t a t i o n a l l y r e l a t e d domains (Figure 5.1) were averaged to take up minor d e f i c i e n c i e s i n alignment[54], but a l l e s s e n t i a l s t r u c t u r e was apparent i n the i n d i v i d u a l beams. My data agreed c l o s e l y with those of Moore; the hi g h r e l i a b i l i t y of the experimental data i s suggested by the r e p r e s e n t a t i v e comparison i n F i g u r e 6.2 f o r 1 ( E ) curves of the (3/2,-1/2) beam measured i n t h i s work and by Moore f o r the Zr(0001)-(2x2)-0 d i f f r a c t i o n p a t t e r n . During the w r i t i n g of t h i s t h e s i s , 1 ( E ) curves f o r the two d i f f r a c t e d beams of the (1x1) p a t t e r n were again measured i n a c o l l a b o r a t i v e p r o j e c t on a d i f f e r e n t z i r c o n i u m s i n g l e c r y s t a l t . i prepared the new Zr(000l) s u r f a c e , and P. Wong completed the LEED measurements with the VLA; the r e s u l t i n g i n t e n s i t y data compare w e l l with p r e v i o u s measurements. F i g u r e 6.3 shows 1 ( E ) curves f o r the (1,1) beam measured by the two methods. The p o s i t i o n s of major peaks and v a l l e y s compare r a t h e r f a v o r a b l y , which i s t The s i n g l e c r y s t a l was k i n d l y p r o v i d e d by P.R. Norton, AECL, Chalk River Nuclear L a b o r a t o r i e s . 202 C O I— CH < >-t z co Zr(OOOI)-(2x2)-0 C3/2, -1/2) BEAM MOORE THIS WORK I I I r 100 140 ENERGYCEV) 60 180 Figure 6.2: Comparison of expe r imen ta l 1(E) cu r ves measured by Moore [ l30 ] ( s o l i d l i n e ) and by t h i s work (do t ted l i n e ) f o r the (3/2,-1/2) beam fo r Zr (0001)- (2x2)-0 at normal inc i d e n c e . 203 Zr(OOOI)-(lxl)-0 (I.I ) BEAM VLA PHOTO. RP= 0J640 RZJ= 0.2016 1 1 1 140 180 ENERGY (EV) 100 220 F i g u r e 6 .3 : Comparison of e x p e r i m e n t a l 1(E) c u r v e s measured w i t h the VLA ( s o l i d l i n e ) and w i t h the p h o t o g r a p h i c method ( d o t t e d l i n e ) f o r the (1,1) beam f o r Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 at normal i n c i d e n c e . 204 supported by a reasonably low value of Rp. The s l i g h t l y higher value of R 2 j i s a result of differences in peak shape; 1(E) curves measured by the VLA tend to y i e l d broader peaks and v a l l e y s . This may be a consequence of the rather simple background subtraction scheme employed in the VLA method; s p e c i f i c a l l y the intensity of the areas between neighboring spots were used for background estimation. 6.3 STRUCTURE ANALYSIS OF ZR(0001)-(2X2)-Q 6.3.1 MULTIPLE SCATTERING CALCULATIONS 1(E) curves for an extensive set of adsorption models were obtained using the 'combined space' approach to multiple scattering c a l c u l a t i o n s . The s p e c i f i c methods used for d i f f e r e n t categories of surface models are discussed below. The following d e t a i l s the non-structural parameters which were kept constant for a l l surface models considered. The potential was expressed in the usual 'muffin-tin' form. The real part of the constant pot e n t i a l ( V 0 r ) between the atomic spheres was set i n i t i a l l y at -10.0 eV, and the imaginary potential (V0/. ) was fixed at -5.0 eV. For atomic potentials, zirconium was characterized by phase s h i f t s (6^, up to 1=1) derived from a band structure potential[96], and for oxygen the superposition potential obtained by Demuth et al . [98] was used. The Debye temperature (0 D) used for zirconium was 270K (an average value from the compilation by Schneider[166]), and for oxygen 843K (a value suggested by 205 other LEED analyses i n v o l v i n g oxygen a d s o r p t i o n on metal s u r f a c e s [ 5 6 , 9 7 ] ) . The experimental temperature was estimated at 300K. The dramatic i n c r e a s e i n s t i c k i n g p r o b a b i l i t y f o r oxygen molecules on p o l y c r y s t a l l i n e z i r c o n i u m s u r f a c e s from room temperature to temperatures even below 100°C observed by Hoflund et a l . [152], and the a p p r e c i a b l e charge t r a n s f e r from z i r c o n i u m to adsorbed oxygen repo r t e d by Tapping[l56] are i n d i c a t i v e of d i s s o c i a t i v e c h e m i s o r p t i o n [ 1 6 7 ] . The LEED p a t t e r n s observed i n t h i s work are b e l i e v e d to i n v o l v e atomic a d s o r p t i o n , and no molecular models were t e s t e d i n the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . F i g u r e 6.4 shows s e v e r a l p o s s i b l e a d s o r p t i o n s i t e s f o r oxygen atoms on top of or underneath the topmost l a y e r of a Zr(000l) s u r f a c e conforming to the hep s t r u c t u r e or the r e c o n s t r u c t e d fee s t r u c t u r e . The s u r f a c e models c o n s i d e r e d i n the c a l c u l a t i o n s i n c l u d e d oxygen o v e r l a y e r (Zr-0 i n t e r l a y e r spacing from 1.4 to 0.0 A), s i n g l e oxygen t e t r a h e d r a l underlayer (both a t and b f c i n F i g u r e 6.4, with the s h o r t e r Zr-0 i n t e r l a y e r d i s t a n c e being v a r i e d from 0.7 to 0.4 A), as w e l l as s i n g l e , double and i n f i n i t e ( i . e . bulk) oxygen o c t a h e d r a l u n d e r l a y e r s . Due to the l a r g e number of a d s o r p t i o n models t e s t e d , they cannot be d e s c r i b e d .adequately by simple d e s i g n a t i o n s such as those used i n Chapter 5. A g e n e r a l i z e d nomenclature f o r s u r f a c e models i n c l u d e d i n t h i s work i s e x p l a i n e d as f o l l o w s . The symbols A, B, C (with r e f e r e n c e to F i g u r e 6.4) i d e n t i f y close-packed z i r c o n i u m l a y e r s which are l a t e r a l l y 206 b c • A -® b. B a t B AC AC — - © — -(a) hep substrate AC B B A- -A-(b)fee substrate F i g u r e 6.4: Side view of the Zr(000l) surface with (a) hep and (b) reconstructed fee stacking sequences to show some poss ib le oxygen adsorption s i t e s . Upper and lower case l e t t e r s represent the r e g i s t r i e s of Zr and O layers r e s p e c t i v e l y . Except for b f c and occupy octahedral holes . a l l underlayer 0 atoms 207 d i s p l a c e d so that the hep and fee s t r u c t u r e s f o l l o w r e s p e c t i v e l y the f a m i l i a r s t a c k i n g sequences ABABAB.. and ABCABC... Lower case symbols i n parentheses have analogous meaning f o r oxygen, except that the c a l c u l a t i o n s are made f o r O l a y e r s which have (2x2) or (2x1) t r a n s l a t i o n a l symmetry with r e s p e c t to the z i r c o n i u m s u b s t r a t e ; ( c ) , (c') i n d i c a t e two such l a y e r s which are d i s p l a c e d l a t e r a l l y from one another by the v e c t o r sum of the s u b s t r a t e u n i t t r a n s l a t i o n a l v e c t o r s p a r a l l e l to Z r ( 0 0 0 l ) . A sequence of l a y e r s from vacuum to the bulk i s s p e c i f i e d by l i s t i n g the a p p r o p r i a t e symbols from l e f t to r i g h t . The bulk l a y e r p e r i o d i c i t y i s i n d i c a t e d by the number of dots t r a i l i n g the l i s t : i f there are n such dots, the l a s t n symbols are to be repeated i n f i n i t e l y to represent the bulk. Using t h i s system of nomenclature, the s u r f a c e models (3h, 3f and 6u) of Chapter 5 are d e s c r i b e d as (b)AB.., (c)AB.. and A(c)BA.. r e s p e c t i v e l y , and the s u r f a c e model which i n v o l v e s oxygen u n d e r l a y e r s occupying o c t a h e d r a l h o l e s i n the bulk of a fee l a t t i c e i s w r i t t e n as A(c)B(a)C(b) and so on. Unless otherwise s t a t e d , the n e i g h b o r i n g Zr-Zr i n t e r l a y e r spacing i s h e l d at the value O f o r z i r c o n i u m metal (2.57 A). An oxygen underlayer occupying o c t a h e d r a l h o l e s of the l a t t i c e i s assumed to be midway between two s u c c e s s i v e Zr l a y e r s . For a t e t r a h e d r a l - h o l e oxygen un d e r l a y e r , there are two d i f f e r e n t Zr-0 i n t e r l a y e r o spacings; the s h o r t e r one was v a r i e d from 0.4 to 0.6 A, while the longer one takes a value which g i v e s the m e t a l l i c 208 Zr-Zr spacing when combined with the s h o r t e r one. The c l o s e s i m i l a r i t y of corresponding 1(E) curves for the (2x2) and (2x1) models (with constant l o c a l environment f o r the oxygen atoms as d i s c u s s e d i n Chapter 5) was e x p l o i t e d i n the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . In each case, the t r a n s l a t i o n a l symmetry which leads to the more economical c a l c u l a t i o n s was used. The general g u i d e l i n e s f o r choosing between the (2x2) and (2x1) models are as f o l l o w s . 1. For composite l a y e r c a l c u l a t i o n s which i n v o l v e both Zr and 0 i n the same composite l a y e r , the oxygen a d l a y e r s are a s s i g n e d a (2x1) t r a n s l a t i o n a l symmetry. Such a c h o i c e l i m i t s the number of subplanes in the composite l a y e r to two (compared to f i v e i f a (2x2) t r a n s l a t i o n a l symmetry i s assumed f o r oxygen). 2. For c a l c u l a t i o n s which i n v o l v e only B r a v a i s l a t t i c e l a y e r s , the (2x2) t r a n s l a t i o n a l symmetry i s assumed f o r oxygen whenever the r e s u l t i n g s u r f a c e model preserves the 3 - f o l d r o t a t i o n a l a x i s and m i r r o r plane symmetry of the c l e a n s u r f a c e at normal i n c i d e n c e ; otherwise the c a l c u l a t i o n s assume the (2x1) t r a n s l a t i o n a l symmetry f o r oxygen, where the h i g h e s t p o s s i b l e symmetry f o r the s u r f a c e r e g i o n i s j u s t a m i r r o r plane. T h e r e f o r e , c a l c u l a t i o n s f o r s u r f a c e models with two or more l a y e r s of oxygen were done with the (2x1) model except when the s u c c e s s i v e oxygen l a y e r s possess the same r e g i s t r y (e.g. i n A ( c ) B ( c ) . . . . ) . The input beams f o r the (2x1) and (2x2) c a l c u l a t i o n s f o l l o w e d the beam l a b e l s of domain (B) f o r 209 F i g u r e 5.2(a) and F i g u r e 5.2(b) r e s p e c t i v e l y . The s t a c k i n g of l a y e r s was performed by the renormalized forward s c a t t e r i n g method f o r Zr-0 i n t e r l a y e r spacings ( d Z r _ 0 ) g r e a t e r than 1.15 A, and by the l a y e r d o u b l i n g method f o r d Z r _ 0 l e s s than 1.10 A. C a l c u l a t e d 1(E) curves of beams which are r e l a t e d by a 180° r o t a t i o n about the normal were averaged with equal weightings to take i n t o account the e f f e c t s of domains due to the two t e r m i n a t i o n s of the bulk ( F i g u r e 5.1). 6.3.2 RESULTS AND DISCUSSIONS F i g u r e 6.5 compares the experimental 1(E) curye of the (1/2,1/2) beam with the corresponding 1(E) curves from r e p r e s e n t a t i v e model c a l c u l a t i o n s . The l a t t e r are arranged a c c o r d i n g to the extent of oxygen i n c o r p o r a t i o n i n t o the bulk. The o v e r a l l agreement between c a l c u l a t e d and experimental 1(E) curves i s not very good; t h i s i s e s p e c i a l l y t r u e f o r the o v e r l a y e r models. More s t r u c t u r e i n c a l c u l a t e d 1(E) curves i s observed f o r models which i n v o l v e underlayer oxygen, and these s t r u c t u r e s become more pronounced as the number of oxygen u n d e r l a y e r s i s i n c r e a s e d ; t h i s i s presumably due to more m u l t i p l e s c a t t e r i n g of the LEED e l e c t r o n s by oxygen atoms. These statements are g e n e r a l l y t r u e f o r the other beams. Comparisons of the complete set of experimental 1(E) curves (four f r a c t i o n a l and three i n t e g r a l order) with the c o r r e s p o n d i n g set from each s u r f a c e model c a l c u l a t i o n were 210 Zr(OOOI)-(2x2)-0 Figure 6 .5: Comparison of the experimental 1(E) curve of the ( 1 / 2 , 1 / 2 ) beam with the 'best' curves from s e l e c t e d model c a l c u l a t i o n s f o r Z r ( 0 0 0 1 ) - ( 2 x 2 ) - 0 at normal i n c i d e n c e . 21 1 I ^Ll I I ^ 40 80 120 ENERGY CEV) Figure 6.5: (continued) 212 performed using the r e l i a b i l i t y index (Rp) proposed by Pendry[l45] m u l t i p l i e d with a n o r m a l i z a t i o n constant of 1/2 proposed by Van Hove and Koestner[146]. For a p a r t i c u l a r s u r f a c e model, s e v e r a l sets of 1(E) curves corresponding to d i f f e r e n t Zr-0 i n t e r l a y e r spacings (d Z r_Q) were c a l c u l a t e d . Each set of these 1(E) curves was compared with corresponding curves from experiment, with r i g i d s h i f t s of the c a l c u l a t e d curves with energy ( i n steps of 2 eV) f o r f i n e - t u n i n g the value of the r e a l part of the constant p o t e n t i a l V 0 r between the atomic spheres. To o b t a i n the minimum value of Rp f o r a p a r t i c u l a r s u r f a c e model, the values of Rp at v a r i o u s d Z r _ Q and V 0 r combinations were i n t e r p o l a t e d and p l o t t e d as contour l i n e s ; these l i n e s t h e r e f o r e represent d i s c r e t e v a l u e s of Rp as a f u n c t i o n of both d Z r _ 0 and V 0 r . F i g u r e s 6.6-6.8 show such contour p l o t s f o r s u r f a c e models which i n v o l v e one, two and bulk u n d e r l a y e r s . Table 6.1 r e p o r t s minimum valu e s of R obtained from contour p l o t s , and the corresponding d Z r_Q and V 0 r , f o r a s e r i e s of model c a l c u l a t i o n s i n which the i n c i d e n t energy i s v a r i e d from 40 to 180 eV. Surface models f o r very short d Z r_Q ( i n c l u d i n g 0 atoms coplanar with Zr) and the t e t r a h e d r a l underlayer models were t e s t e d f o r a r e s t r i c t e d energy range, and the c a l c u l a t e d 1(E) curves appear very u n f a v o r a b l e , hence these models are not i n c l u d e d i n the R- f a c t o r a n a l y s i s presented i n Table 6.1. The models i n v e s t i g a t e d i n Table 6.1 can be c a t e g o r i z e d as: ( i ) O o v e r l a y e r , ( i i ) s i n g l e 0 un d e r l a y e r , ( i i i ) double 213 Zr(OOOI)-(2x2)-0, C(b)AB.. O. INTERLAYER SPACING CA) F i g u r e 6.6: Contour p l o t of multi-beam R p f o r Zr(0001)-(2x2)-0 versus V 0 r and Zr-0 i n t e r l a y e r spacing f o r the a d s o r p t i o n model des i g n a t e d C(b)AB.. at normal inc idence. 214 Zr(OOOI)-(2x2)-0, A(c)B(a)CAB... ' 1200 1.244 1.287 1.330 1.374 1.418 INTERLAYER SPACING (A) 1.461 F i g u r e 6.7: Contour p l o t of multi-beam Rp f o r Zr(0001)-(2x2)-0 versus V 0 r and Zr-0 i n t e r l a y e r spacing f o r the a d s o r p t i o n model designated A(c)B(a)CAB... at normal inc idence. 215 Zr(OOOI)-(2x2)-0J A(c)B(a)C(b). o INTERLAYER SPACING CA) F i g u r e 6 .8: Contour p l o t of multi-beam R p f o r Zr(0001)-(2x2)-0 versus V 0 r and Zr-0 i n t e r l a y e r spacing f o r the a d s o r p t i o n model desi g n a t e d A(c)B(a)C(b) at normal inc idence. 216 Surface R d z r - 0 ( A ) V 0 r ( e V ) (b)AB..(hep) 0.410 0 .99 -16.0 (c)AB..(hep) 0.380 1 .03 -13.9 (b+c)AB..(hep) 0.365 1 .10 -9.5 (c)A(c)BA..(hep) 0.373 1 .25 -18.6 A(c)BA..(hep) 0.360 1 .35 -14.5 C(b)AB..(hep) 0.332 1 .36 -14.0 A(b)CAB..(hep) 0.360 1 .26 -13.4 A(c)BCA...(fee) 0.363 1 .36 -14.2 A(c)B(c)AB..(hep) 0.350 1 .37 -14.5 A(c)B(c')AB..(hep) 0.352 1 .28 -14.8 C(b)A(b' )CAB'. . (hep) 0.350 1 .27 -16.0 B(c)A(b)CAB..(hep) 0.337 1 .33 -1 3 ..6 C(a)B(c)AB..(hep) 0.332 1 .34 -12.4 A(c)B(a)CAB...(fee) 0.324 1 .34 -16.0 A ( c ) B ( c ) . . . . ( h e p ) 0.350 1 .35 -10.4 A ( c ) B ( c ' ) (hep) 0.328 1 .29 -14.0 A(c)B(a)C(b) (fee) 0.306 1 .33 -14.0 Table 6.1: Minimum values of multi-beam R p with the cor r e s p o n d i n g Zr-0 i n t e r l a y e r spacings ( d z r - 0 ^ a n d V ° r obtained from the comparisons of experimental and c a l c u l a t e d 1(E) curves based on oxygen a d s o r p t i o n models l i s t e d i n the f i r s t column f o r Zr(0001)-(2x2)-0 at normal i n c i d e n c e . 217 0 u n d e r l a y e r , and ( i v ) bulk 0 s t r u c t u r e s . In p r i n c i p l e 0 a d s o r p t i o n i n o c t a h e d r a l h o l e s may occur w i t h i n the un r e c o n s t r u c t e d hep s t r u c t u r e f o r z i r c o n i u m ( i . e . ABAB.. s t a c k i n g sequence), w i t h i n a r e c o n s t r u c t e d hep s t r u c t u r e ( i . e . ACABAB..), or w i t h i n s t r u c t u r e s which i n c l u d e v a r y i n g degrees of fee r e c o n s t r u c t i o n (e.g. from top l a y e r r e c o n s t r u c t i o n CABAB.. through to the complete fee s t r u c t u r e ) . The examples i n Table 6.1 gi v e some p o s s i b i l i t i e s f o r each category, and three i n t e r e s t i n g t rends appear: (a) o v e r l a y e r models y i e l d very high v a l u e s of minimum R p; the s m a l l e s t value i s 0.365, which i s g r e a t e r than the l a r g e s t v a l u e s i n the other groups; (b) the average v a l u e s of minimum R p decrease with i n c r e a s i n g numbers of 0 u n d e r l a y e r s , the values being 0.354 f o r category ( i i ) , 0.340 fo r category ( i i i ) and 0.328 f o r ( i v ) ; (c) w i t h i n c a t e g o r i e s ( i i ) - ( i v ) the lowest value of R p occurs f o r the s i m p l e s t fee type r e c o n s t r u c t i o n which spans the depth of 0 atoms, that i s C(b)AB.. f o r category ( i i ) , A(c)B(a)CAB... f o r ( i i i ) and A(c)B(a)C(b) f o r ( i v ) . The h i g h v a l u e s of R p f o r a l l the o v e r l a y e r models support the view t h a t chemisorption occurs v i a some form of 0 u n d e r l a y e r s ; t h i s i s c o n s i s t e n t with a c o n c l u s i o n f o r oxygen a d s o r p t i o n on p o l y c r y s t a l l i n e t i t a n i u m [ 1 6 1 ] . The lowest value of R found so f a r f o r the tr Z r ( 0 0 0 l ) - ( 2 x 2 ) - 0 s u r f a c e s t r u c t u r e i s 0.306 f o r the model i n which 0 atoms occupy o c t a h e d r a l holes w i t h i n the fee r e c o n s t r u c t e d form of zi r c o n i u m . F i g u r e 6.9 compares the set 218 of 1(E) curves c a l c u l a t e d from the s u r f a c e model A(c)B(a)C(b) (with the optimum d Z r_Q and V 0 r ) with the c o r r e s p o n d i n g curves from experiment. The c a l c u l a t i o n assumes a b u l k - l i k e s t r u c t u r e , although i n p r a c t i c e LEED p r o v i d e s i n f o r m a t i o n on j u s t a r e s t r i c t e d number of Zr l a y e r s (e.g. 3-5); i n any event, from the i n i t i a l exposure g i v e n , s u b s t a n t i a l 0 i n c o r p o r a t i o n seems u n l i k e l y to extend s i g n i f i c a n t l y deeper than the depth probed by LEED. Although F i g u r e 6.9 r e p o r t s the best correspondence found to date between experiment and c a l c u l a t i o n f o r the Zr(0001)-(2x2)-0 s t r u c t u r e , the match-up i s s t i l l l e s s than i d e a l . N e v e r t h e l e s s the favored model does have some reasonable f e a t u r e s . For example, the r e p o r t e d value of d Z r _ 0 equal to o 1.33 A seems q u i t e p l a u s i b l e , s i n c e i t suggests that the i n t e r s t i t i a l 0 atoms expand the Zr-Zr i n t e r l a y e r spacing by j u s t 3.5% from that i n z i r c o n i u m metal. F u r t h e r , the o r e s u l t i n g LEED-determined Zr-0 bond d i s t a n c e of 2.29 A o agrees c l o s e l y with the value 2.31 A given by X-ray d i f f r a c t i o n f o r bulk Z r O [ l 6 8 ] . I n c i d e n t a l l y the l a t t e r compound has the sodium c h l o r i d e l a t t i c e s t r u c t u r e , and t h e r e f o r e ( i ) 0 atoms in ZrO have the same l o c a l c o o r d i n a t i o n as i n d i c a t e d by LEED f o r the i n t e r s t i t i a l Zr(0001)-(2x2)-0 s u r f a c e s t r u c t u r e ; and ( i i ) the Zr atoms in ZrO form the fee arrangement. 219 >-t— I l 1 1 1 1 I 1 1 1 1 1 I 0 0 I 4 0 I 6 0 8 0 I 2 0 I 6 0 E N E R G Y ( E V ) Figure 6 .9: Comparison of experimental 1(E) curves (dotted l i n e s ) f o r three i n t e g r a l and four f r a c t i o n a l order beams from Zr(0001)-(2x2)-0 with the corresponding 1(E) curves c a l c u l a t e d f o r the a d s o r p t i o n model A(c)B(a)C(b) at normal i n c i d e n c e with d 7 _ _ n at e i t h e r 1.33 or 1.37 A. 220 CD cr < Figure 6 . 9 : (continued) 221 6.3.3 STRUCTURAL REFINEMENT Although the LEED a n a l y s i s r e p o r t e d here a p p a r e n t l y has some c o n s i s t e n c y with other s t r u c t u r a l i n f o r m a t i o n , the d i s c r e p a n c i e s i n LEED i n t e n s i t i e s ( p a r t i c u l a r l y f o r the (1/2,-1/2) beam where so f a r no a d s o r p t i o n models t e s t e d y i e l d a s a t i s f a c t o r y match-up with the experimental curve) s t i l l make t h i s a c h a l l e n g i n g s u r f a c e s t r u c t u r e . Some i n g r e d i e n t s r e q u i r e d f o r s t r u c t u r a l refinement are d i s c u s s e d i n the f o l l o w i n g . The t r e n d of b e t t e r correspondence with i n c r e a s i n g number of oxygen l a y e r s opens the p o s s i b i l i t y t h a t the o r i g i n a l atomic s c a t t e r i n g f a c t o r used f o r oxygen may not be adequate, and that t h i s d e f i c i e n c y i s a r t i f i c i a l l y compensated f o r by the presence of more oxygen l a y e r s i n the model c a l c u l a t i o n s . Since the LEED a n a l y s i s i n d i c a t e s a b u l k - l i k e oxide s t r u c t u r e , that may suggest i t would be more a p p r o p r i a t e to use an atomic p o t e n t i a l f o r n e g a t i v e l y charged 0, d e r i v e d from s e l f - c o n s i s t e n t c r y s t a l l a t t i c e or c l u s t e r s t r u c t u r e . A p r e l i m i n a r y i n v e s t i g a t i o n of t h i s was undertaken in the a n a l y s i s of the Zr(0001)-(1x1)-0 s t r u c t u r e ( S e c t i o n 6.4.2). Even i f the s u r f a c e s t r u c t u r e used f o r F i g u r e 6.9 i s b a s i c a l l y c o r r e c t , some warping of Zr l a y e r s would be l i k e l y i n the presence of (2x2) 0 l a y e r s . That would introduce both an e x t r a parameter, and a c o m p l i c a t i o n i n the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s . N e v e r t h e l e s s i t i s a refinement that should be c o n s i d e r e d in f u t u r e work. With i n c o r p o r a t i o n of 0 i n t o the bulk i t seems i n e v i t a b l e that there would be some degree of random 222 o c c u p a t i o n of 0 i n the o c t a h e d r a l h o l e s . A f i n a l refinement i s to c o n s i d e r the p o s s i b i l i t y of v a r i a t i o n of s t r u c t u r e in d i f f e r e n t s u r f a c e domains, perhaps with d i f f e r e n t l e v e l s of fee r e c o n s t r u c t i o n . Although the f i r s t two i n g r e d i e n t s f o r refinement l i s t e d above ( i . e . n e g a t i v e l y charged 0 and Zr l a y e r warping) can be i n v e s t i g a t e d with f u r t h e r m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s , assessments of the other two i n g r e d i e n t s ( d i s o r d e r , d i f f e r e n t domains) w i l l a p p a r e n t l y r e q u i r e a n a l y s i s with the LEED fine-beam angular spot p r o f i l e technique!65]. That the f o u r t h i n g r e d i e n t may be r e l e v a n t i s perhaps t e n t a t i v e l y suggested by the c l o s e correspondence (as judged by optimal values of R ) f o r the Zr d i f f e r e n t models c o n s i d e r e d i n Table 6.1. I n c i d e n t a l l y a l l the models optimize Rp with Zr-0 i n t e r l a y e r spacings i n the o range 1.28 to 1.37 A; the corresponding set of Zr-0 bond l e n g t h s match the i n t e r a t o m i c d i s t a n c e i n bulk ZrO to w i t h i n about 2%. 6.4 STRUCTURE ANALYSIS OF ZR(0001)-(1X1)-Q 6.4.1 MULTIPLE SCATTERING CALCULATIONS The nomenclature of s u r f a c e models f o l l o w s the convention f o r the (2x2) a n a l y s i s , except that i t i s now understood that the oxygen a d l a y e r s have (1x1) t r a n s l a t i o n a l symmetry. The non-geometrical parameters used i n the (1x1) c a l c u l a t i o n s were e x a c t l y the same as those used f o r the 223 (2x2) a n a l y s i s . o The bulk Zr-Zr i n t e r l a y e r spacing used was 2.57 A. For oxygen u n d e r l a y e r s occupying o c t a h e d r a l h o l e s i n the bulk, each 0 l a y e r was assumed to be midway between two s u c c e s s i v e Zr l a y e r s , u n l e s s otherwise s t a t e d . For s u r f a c e models with both 0 o v e r l a y e r and 0 underlayer (e.g. (b)A(c)BA.. and (c ) A ( c ) B A . . ) , d Z R _ Q f o r the 0 underlayer was f i x e d at 1.35 A (an average value from the (2x2) a n a l y s i s ) , while that f o r o the o v e r l a y e r was v a r i e d from 0.7 to 1.4 A. For the t e t r a h e d r a l underlayer c a l c u l a t i o n s , the s h o r t e r Zr-0 spacing was v a r i e d from 0.6 to 1.1 A, while the longer Zr-0 spacing was v a r i e d from 2.1 to 1.6 A i n such a way that the maximum spacing between the two topmost Zr l a y e r s d i d not o exceed 2.7 A. The same c r i t e r i a f o r s e l e c t i o n of e i t h e r the renormalized forward s c a t t e r i n g or the l a y e r d o u b l i n g method d e s c r i b e d i n S e c t i o n 6.3.1 were a p p l i c a b l e . The n o t a t i o n f o r input beams f o l l o w s that of the i n t e g r a l beams of the (2x2) r e c i p r o c a l l a t t i c e d e p i c t e d i n F i g u r e 5.2(b). 1(E) curves were c a l c u l a t e d at normal i n c i d e n c e , and were domain-averaged before comparing to experimental c u r v e s . 6.4.2 RESULTS AND DISCUSSION F i g u r e 6.10 compares the experimental 1(E) curve f o r the (1,0) beam with the corresponding curves c a l c u l a t e d from s u r f a c e models ranging from 0 o v e r l a y e r to bulk 0 u n d e r l a y e r s . The f i r s t o b s e r v a t i o n i s that 1(E) curves 224 Zr(OOOI)-(lxl)-0 ,0 ) BEAM (b) ABAB.. (c) ABAB.. (c)A(c)BAB.. A(b t)BAB.. A(c)BAB.. A(b)CABAB. C(b)A(c)BAB. A(c)B(a)C(b). , '\ EXPT. 100 160 ENERGY CEV) 220 F i g u r e 6.10: Comparison of the experimental 1(E) curve of the ( 1 , 0 ) beam with the 'best' curves from s e l e c t e d model c a l c u l a t i o n s f o r Z r ( 0 0 0 1 ) - ( 1 x 1 ) - 0 at normal i n c i d e n c e . 225 c a l c u l a t e d from models in which an 0 o v e r l a y e r i s present show l i t t l e s t r u c t u r e i n the energy range of comparison. The same o b s e r v a t i o n has been r e p o r t e d f o r f r a c t i o n a l beams in the (2x2) a n a l y s i s . The second o b s e r v a t i o n i s that 1(E) curves c a l c u l a t e d from models with s i n g l e , double, or i n f i n i t e ( i . e . bulk) 0 l a y e r s occupying o c t a h e d r a l holes appear to be v i s u a l l y s i m i l a r almost r e g a r d l e s s of the number of 0 l a y e r s . T h i s suggests that the c a l c u l a t e d beam i n t e n s i t i e s f o r i n t e g r a l beams are mostly c o n t r i b u t e d by the s u b s t r a t e Zr atoms, which i s presumably the r e s u l t of the gr e a t e r s c a t t e r i n g s t r e n g t h of Zr compared with 0. The l a t t e r group of 1(E) curves a l s o i n d i c a t e b e t t e r correspondence with the experimental curve, although the 'match' i s l e s s than i d e a l . Table 6.2 r e p o r t s the minimum values of Rp and t h e i r c o r r e s p o n d i n g d Z r_Q and V 0 r obtained from the comparisons of experimental 1(E) curves ( f o r (1,0) and (1,1) beams) with those from model c a l c u l a t i o n s . These values were d e r i v e d from contour p l o t s , which are d e s c r i b e d i n S e c t i o n 6.3.2. The models i n v e s t i g a t e d i n Table 6.2 can again be c a t e g o r i z e d as: ( i ) o v e r l a y e r , or combinations of o v e r l a y e r and a s i n g l e u n d e r l a y e r , ( i i ) s i n g l e u n d e r l a y e r , and ( i i i ) m u l t i - u n d e r l a y e r s . Category ( i ) has high average Rp values, which tend to support the c o n c l u s i o n that 0 o v e r l a y e r s do not form on Zr(OOOl) at low exposures, thereby showing some c o n s i s t e n c y with o b s e r v a t i o n s from d i f f e r e n t techniques f o r s t u d i e s of oxygen chemisorption on the Ti(OOOl) 226 Surface R p d z r - 0 ( A ) V 0 r ( e V ) (b)AB..(hep) 0.388 1 .30 -14.0 (c)AB..(hep) 0.332 1 .47 -10.0 A(b t)BA..(hep) 0.365 0.70 1,2. OO2 -8.0 (b)A(c)BA..(hep) 0.330 1 .40 3 , 1 . 35" -8.0 (c)A(c)BA..(hep) 0.294 1 .OO3,1. 35 s -8.2 A(c)BA..(hep) 0.264 1 .38 -7.4 C(b)AB..(hep) 0.265 1 .39 -7.0 A(b)CAB..(hep) 0.240 1 .37 -7.4 A(c)B(c)AB..(hep) 0.280 1 .37 -6.2 A(c)B(c)AB..(hep) 0.283 1 .37 -6.2 C(b)A(c)BA..(hep) 0.300 1 .37 -8.3 A(b)C(a)BA..(hep) 0.300 1 .37 -8.6 A(c)B(a)C(b) (fee) 0.306 1 .33 -14.0 'Top Zr l a y e r to O l a y e r ; 20 l a y e r to second Zr l a y e r ; 3 F o r O o v e r l a y e r ; "For O u n d e r l a y e r . Table 6.2: Minimum v a l u e s of multi-beam R p with the corr e s p o n d i n g Zr-0 i n t e r l a y e r spacings (&zr-0^ a n d V ° r obt a i n e d from the comparisons of experimental and c a l c u l a t e d 1(E) curves based on oxygen a d s o r p t i o n models l i s t e d i n the f i r s t column f o r Zr(0001)-(1x1)-0 at normal i n c i d e n c e . 227 s u r f a c e [ 1 6 2 ] . U n l i k e the (2x2) a n a l y s i s , the s m a l l e s t values of Rp f a l l i n the group of s i n g l e 0 underlayer i n which 0 atoms occupy the o c t a h e d r a l h o l e s between the f i r s t two Zr l a y e r s . However not much weight should be given to t h i s c o n c l u s i o n c o n s i d e r i n g the f a c t t h a t the experimental 1(E) data are only a v a i l a b l e f o r two beams. Neve r t h e l e s s i n f o r m a t i o n at t h i s stage i s s t i l l u s e f u l s i n c e i t se t s l i m i t a t i o n s on p o s s i b l e models f o r f u r t h e r a n a l y s e s . The lowest value of Rp found so f a r f o r the Zr(0001)-(1x1)-0 s u r f a c e s t r u c t u r e i s 0.240 f o r the s u r f a c e model A(b)CAB.., i n which the 0 atoms occupy o c t a h e d r a l h o l e s between the f i r s t two Zr l a y e r s , while the r e g i s t r y of the second l a y e r i s s h i f t e d so that the second, t h i r d and f o u r t h Zr l a y e r s resemble three f e e d 11) l a y e r s . The l a t t e r s h i f t of r e g i s t r y again i n d i c a t e s t h a t some degree of fee r e c o n s t r u c t i o n i s favored f o r the i n c o r p o r a t i o n of oxygen i n t o the bulk of zi r c o n i u m . F i g u r e 6.11 compares the two experimental 1(E) curves with the corresponding curves c a l c u l a t e d from A(b)CAB.., while the R - f a c t o r contour p l o t f o r t h i s model i s d e p i c t e d i n F i g u r e 6.12. Although the value of Rp f o r t h i s model c a l c u l a t i o n i s q u i t e s a t i s f a c t o r y , t h i s study only y i e l d s p r e l i m i n a r y r e s u l t s f o r the Zr(0001)-(1x1)-0 system s i n c e a very l i m i t e d beam set was a v a i l a b l e . For a complete a n a l y s i s , more experimental data (e.g. off-normal i n c i d e n c e 1(E) curves) are r e q u i r e d . N e v e r t h e l e s s f o r the favored model, the f i r s t three atomic l a y e r s correspond to the (111) l a y e r s of ZrO, which i s ZKOOOD-OxO-O, A(b)CAB.. I 1 1 1 1 1 1 1 1 1 I 1 r~—i ~ J T i 40 80 120 160 200 100 140 180 200 ENERGY (EV) F i g u r e 6.11: Comparison of experimental 1(E) curves (dotted l i n e s ) f o r two d i f f r a c t e d beams from Zr(0001)-(1x1)-0 with the corresponding 1(E) curves c a l c u l a t e d f o r the a d s o r p t i o n model A(c)CAB.. at normal inci d e n c e with d Z r _ 0 at e i t h e r 1.37 or 1.41 A. 229 ZKOOOD-OxD-O, A(b)CAB.. Q INTERLAYER SPACING (A) F i g u r e 6.12: Contour p l o t of multi-beam R p f o r Zr ( 0 0 0 l ) - ( 1 x 1 ) - 0 versus V 0 r and Z r - 0 i n t e r l a y e r spacing f o r the a d s o r p t i o n model designated A(b)CAB.. at normal inc idence. 230 c o n s i s t e n t with the c o n c l u s i o n f o r Ti(0001)-(1x1)N r e p o r t e d by Shih et al. [41], Moreover, the r e p o r t e d value of o equal to 1.37 A seems q u i t e p l a u s i b l e . T h i s value suggests that the i n t e r s t i t i a l 0 atoms expand the Zr-Zr i n t e r l a y e r s pacing by 6.6% from that i n z i r c o n i u m metal. The r e s u l t i n g o LEED-determined Zr-0 bond d i s t a n c e of 2.31 A agrees e x a c t l y with the value given by X-ray d i f f r a c t i o n f o r bulk ZrO[!68]. 6.4.3 VARIATION OF PHASE SHIFTS With the p r e l i m i n a r y knowledge of the p o s s i b l e formation of ZrO (at l e a s t i n the top three l a y e r s ) , i t seems a p p r o p r i a t e to a s s i g n some negative charge to oxygen and to repeat the m u l t i p l e s c a t t e r i n g c a l c u l a t i o n s f o r the s u r f a c e models A(c)BA.., C(b)AB.. and A(b)CAB.. to see i f any improvement can be obtained. For t h i s purpose, f i v e a d d i t i o n a l s e t s of phase s h i f t s were c o n s i d e r e d f o r 0 and Zr c o r r e s p o n d i n g to i o n i c charges of 0.00, 0.50, 1.00, 1.50 and 2.00 (here i t i s understood that the charge i s p o s i t i v e f o r Zr and n e g a t i v e f o r O). The ion core p o t e n t i a l s f o r each of these s p e c i e s were d e r i v e d f o l l o w i n g Matheiss[93] f o r a sodium c h l o r i d e type ZrO l a t t i c e ( F i gure 3.4), i n which O has the same l o c a l environment as i n the a d s o r p t i o n model; the input wavefunctions f o r atomic Zr and 0 were obtained from Clementi and R o e t t i [ l 6 9 ] , and the- exchange terms (a) used were 0.70 and 0.74 f o r Zr and 0 r e s p e c t i v e l y . The ' m u f f i n - t i n ' r a d i i f o r Zr (rgj.) and O (TQ) were estimated by matching the r a d i a l d i s t r i b u t i o n s of the p o t e n t i a l s f o r Zr 231 Charge r o ^ A ^ r Z r ^ A ^ P o t e n t i a l (eV) 0.00 1.01 1.30 -13.12 0.50 1.05 1.26 -11.12 1.00 1.09 1.22 -9.06 1.50 1.11 1.20 -6.14 2.00 1.13 1.18 -4.78 Table 6 .3: V a r i a t i o n of ' m u f f i n - t i n ' r a d i i f o r 0 ( r Q ) and Zr ( r Z r ) with i o n i c charge on O (negative) and Zr ( p o s i t i v e ) r e s p e c t i v e l y from c a l c u l a t i o n s of ion core p o t e n t i a l s f o r the two s p e c i e s i n a ZrO c r y s t a l l a t t i c e . The v a l u e s of the p o t e n t i a l at hte ' m u f f i n - t i n ' r a d i i f o r each ZrO l a t t i c e with the s p e c i f i e d i o n i c c h a r a c t e r f o r the two s p e c i e s are a l s o g i v e n . 232 Surface R p d Z r _ 0 ( A ) V 0 r ( e V ) ( i ) Of. A(c)BA..(hep) 0.330 1.41 -6.0 C(b)AB..(hep) 0.360 1.39 -4.5 A(b)CAB..(hep) 0.340 1.39 -6.0 ( i i ) CrJ_ A(c)BA..(hep) 0.316 1.20 -5.0 C(b)AB..(hep) 0.326 1.38 -6.0 A(b)CAB..(hep) 0.307 1.38 -5.5 ( i i i ) o-± A(c)BA..(hep) 0.330 1.15 -5.0 C(b)AB..(hep) 0.300 1.20 -8.0 A(b)CAB..(hep) 0.360 1.39 -5.0 Table 6 .4: Minimum valu e s of multi-beam R f o r Z r ( 0 0 0 l ) -(1x1)-0 a t normal i n c i d e n c e . The corresponding v a l u e s of d Z r_Q and V 0 r were obtained from the comparisons of experimental and c a l c u l a t e d 1(E) curves, f o r a d s o r p t i o n models i n which the three topmost l a y e r s correspond to the three (111) l a y e r s of ZrO l a t t i c e . 0 phase s h i f t s used were d e r i v e d from ion core p o t e n t i a l s of ( i ) 0 ° , ( i i ) 0" 1 and ( i i i ) O"2 i n a ZrO l a t t i c e . 233 o and 0. The n u c l e i of these s p e c i e s were f i x e d at 2.31 A a p a r t , and the d i s t r i b u t i o n curves of p o t e n t i a l s were drawn in o p p o s i t e d i r e c t i o n s u n t i l they cro s s e d each other. The l a t t e r c r o s s i n g p o i n t to the r e s p e c t i v e c e n t e r s of the two s p e c i e s then d e f i n e d the ' m u f f i n - t i n ' r a d i i . Table 6.3 summarizes values so o b t a i n e d f o r the ' m u f f i n - t i n ' r a d i i and the a s s o c i a t e d p o t e n t i a l ( i n each case the l a t t e r may be seen as a f i r s t approximation f o r V 0 r ) . Multi-beam minimum R r e s u l t i n g from m u l t i p l e Cr s c a t t e r i n g c a l c u l a t i o n s u t i l i z i n g three s e t s of these new phase s h i f t s f o r 0 are given i n Table 6.4. Although no improvement on minimum R p i s i n d i c a t e d i n t h i s study, the t r e n d i n the average value of minimum R f o r the three kr s u r f a c e models suggests that some p a r t i a l n e g ative charge on 0 may be more f a v o r a b l e . The ' m u f f i n - t i n ' r a d i i f o r 0 given i n Table 6.3 appear u n u s u a l l y l a r g e . In f u t u r e s t u d i e s i t may be p r e f e r a b l e to c a l c u l a t e the ion core p o t e n t i a l s i n c l u s t e r s with some s e l f - c o n s i s t e n t charge adjustment. 6.5 INTERPRETATION OF ADSORBATE COVERAGES In order to determine the oxygen uptake c h a r a c t e r i s t i c s of the Zr(OOOl) s u r f a c e , r e s e a r c h grade oxygen (Matheson 99.99% p u r i t y ) was leaked i n t o the FC12 chamber through a leak v a l v e (with both the t i t a n i u m s u b l i m a t i o n pump and the main ion pump operat i n g ) at room temperature, at a pressure of 5x10" 9 t o r r i n d i c a t e d by an i o n i z a t i o n gauge i n the chamber. No d e t e c t a b l e changes were observed in the mass 234 1.2 Hi.0 0.8 > < _ l o 0.6 o 0.4 y >-X o -10.2 0.0 1 2 3 4 EXPOSURE (L) F i g u r e 6.13: A p l o t of Auger peak r a t i o 0 5 l 0 / Z r 1 7 « and est i m a t e d oxygen monolayer as a f u n c t i o n of 0 2 exposure ( i n Langmuir, 1 L = 10" 6 t o r r s) to the (0001) s u r f a c e of z i r c o n i u m . The oxygen coverages f o r the appearance of the (2x2) and (1x1) p a t t e r n s are a l s o g i v e n . 235 s p e c t r a , or i n s u r f a c e contamination l e v e l s as i n d i c a t e d by AES. The Auger peak height r a t i o 0 5 1 0 / Z r 1 7 f t was measured as a f u n c t i o n of oxygen exposure, as shown in F i g u r e 6.13. No breaks were apparent i n t h i s curve around the coverages that correspond to the (2x2) and (1x1) LEED p a t t e r n s . The other o b s e r v a t i o n (not shown i n the f i g u r e ) was that the curve d i d not l e v e l o f f even at the high oxygen exposure (=20 L) c o n s i d e r e d i n the experiment, although the slope d i d decrease. T h i s behavior may be a t t r i b u t e d to the ease of formation of oxygen m u l t i - o v e r l a y e r s on group 4 metal s u r f a c e s [ 1 6 4 ] , or to the f a c t t h a t the Z r 1 7 a peak (which corresponds to an MW Auger t r a n s i t i o n ) i s a t t e n u a t e d with high oxygen coveragesf157 ]. The l a c k of apparent breaks i n the oxygen uptake curve means that there are no r e f e r e n c e coverages f o r c a l i b r a t i o n of the 0 5 1 0 / Z r 1 7 < , r a t i o . The f r a c t i o n a l monolayer oxygen coverages given i n F i g u r e 6.13 were a c t u a l l y c a l i b r a t e d u s i n g CO uptake data of Moore[l30], which are shown in F i g u r e 6.14. Here C 2 7 2 / Z r , 7 , and 0 5 1 0 / Z r 1 7 f l Auger peak height r a t i o s are p l o t t e d a g a i n s t CO exposure. In c o n t r a s t to the oxygen exposure experiment, the uptake curve of CO shows a sharp break at s a t u r a t i o n coverage, where a (1x1) LEED p a t t e r n a p p e a r s t . The same uptake behavior was a l s o observed f o r h e t e r o n u c l e a r molecules (e.g. CO, NO and N 20) on p o l y c r y s t a l l i n e z i r c o n i u m s u r f a c e s by Foord et al. [157]. 1*This p a t t e r n was a l s o observed in t h i s work, and some p r e l i m i n a r y 1(E) data were c o l l e c t e d at normal i n c i d e n c e . I t i s hoped that a n a l y s i s can be done i n the f u t u r e when the oxygen problem i s f u l l y s o l v e d . 236 o 24 LU §0.8 Or UJ o < C z 7 2 / Z r l 7 A 1 S — _ i . - - c - - - a — ° / o To' __o _ _ o o 05I0/Zr|74 V i i 4 8 12 IS 32 EXPOSURE (ARB. UNITS) F i g u r e 6.14: A p l o t of Auger peak r a t i o s 0 5 1 0 / Z r 1 7 A and C 2 7 2 / Z r i 7 « a s a f u n c t i o n of CO exposure to the (0001) s u r f a c e of z i r c o n i u m . The v a l u e s of 0 5 1 0 / Z r 1 7 « and C 2 7 2 / Z r i 7 « a t the i n t e r s e c t i o n s of the tangents of the steep p a r t and of the f l a t p a r t of the curves are used as r e f e r e n c e f o r h a l f monolayer each of 0 and C r e s p e c t i v e l y ( a f t e r Moore[130]). 237 These authors suggested that these molecules chemisorb d i s s o c i a t i v e l y on the metal s u r f a c e , probably with an i n i t i a l 'head-on' p h y s i s o r p t i o n at s p e c i f i c s i t e s ; once these s i t e s are occupied (e.g. by a monolayer of CO) subsequent chemisorption i s slow. I f an a d d i t i o n a l assumption i s made that C and 0 atoms are i n the same plane (which i s suggested by the p r e l i m i n a r y , but incomplete, LEED a n a l y s i s of Ti(0001)-(2x2)-CO[40]), then the s a t u r a t i o n coverage corresponds to h a l f monolayer each of C and 0 atomst. By e x t r a p o l a t i o n of the tangent of the r a p i d l y r i s i n g p a r t and of the r e l a t i v e l y l e v e l e d p a r t of the uptake curve ( F i g u r e 6.14), s a t u r a t i o n Auger peak r a t i o s f o r C 2 7 2 / Z r i 7 « and 0 5 1 o / Z r 1 7 4 are found to be approximately 1.9 and 1.2 r e s p e c t i v e l y . The l e v e l of carbon contamination f o r a cleaned Zr(000l) s u r f a c e ( S e c t i o n 6.2.1) was estimated by d i v i d i n g the minimum recorded value of C 2 7 2 / Z r 1 7 , (about 0.1) by 3.8, which g i v e s the approximately 3% of a monolayer r e f e r r e d to e a r l i e r . Using 0 5!o/Zr, 7„=1.2 to i n d i c a t e a h a l f monolayer of 0, the peak height r a t i o s f o r oxygen uptake curve i n F i g u r e 6.13 were a l l d i v i d e d by 2.4 to y i e l d the f r a c t i o n a l monolayer coverage on the r i g h t hand y - a x i s . The (2x2)-0 p a t t e r n was observed f o r oxygen coverages between 0.3 and 0.65 monolayer, while the (1x1) p a t t e r n was observed from t i n p r i n c i p l e C c o u l d form an o v e r l a y e r while 0 forms an u n d e r l a y e r , i n which case the s a t u r a t i o n coverage corresponds to one monolayer each of C and 0. 238 0.65 to gr e a t e r than monolayer oxygen coverages. The sharpest (1x1) LEED p a t t e r n was obtained at about 0.8 monolayer coverage, and t h e r e f o r e that i s ap p a r e n t l y c o n s i s t e n t with a s i n g l e underlayer of 0 atoms conforming to the (1x1) t r a n s l a t i o n a l symmetry. The Auger data f o r the (2x2) p a t t e r n are not as c l e a r cut as those f o r (1x1). Here the sharpest p a t t e r n was observed f o r oxygen coverage around 0.37 monolayer. With t h i s value alone, i t i s d i f f i c u l t to decide whether 0 atoms conform t o the (2x2) or (2x1) t r a n s l a t i o n a l symmetries, which correspond to 0.25 and 0.5 monolayer of O atoms r e s p e c t i v e l y . Perhaps the . (2x2) t r a n s l a t i o n a l symmetry i s favored, however, s i n c e then i t i s p o s s i b l e to form almost two unde r l a y e r s of O atoms i n r e l a t i v e l y f u l l y occupied domains. 6.6 CONCLUSIONS AND FUTURE WORK T h i s study has y i e l d e d p r e l i m i n a r y s t r u c t u r a l data f o r the Zr(0001)-(2x2)-0 and Zr(0001)-(1x1)-0 s u r f a c e s . T h i s r e p r e s e n t s the f i r s t s u r f a c e s t r u c t u r a l i n f o r m a t i o n a v a i l a b l e f o r oxygen chemisorption on a hep metal. In both s t r u c t u r e s c o n s i d e r e d here O atoms are i n d i c a t e d to occupy o c t a h e d r a l holes between Zr l a y e r s which i n turn r e c o n s t r u c t to show fee packing. Although the favored models have some c o n s i s t e n c y with the bond l e n g t h i n bulk ZrO, and with the p r e v i o u s l y r e p o r t e d underlayer formation i n Ti(0001)-(1x1)-N, the s u r f a c e s t r u c t u r e s f o r 0 on Zr(000l) s t i l l appear c h a l l e n g i n g . 239 Now that the g e o m e t r i c a l s t r u c t u r e s f o r these two oxygen a d s o r p t i o n s t r u c t u r e s have become c l e a r e r , i t seems a p p r o p r i a t e to have f u r t h e r experimental 1(E) curves (e.g. measurements c a r r i e d out at off-normal i n c i d e n c e ) to enable more d e t a i l e d comparisons with c a l c u l a t e d 1(E) curves to r e f i n e both g e o m e t r i c a l and p h y s i c a l parameters. To r e s o l v e the problem of (2x2) versus (2x1) t r a n s l a t i o n a l symmetry for adsorbed oxygen, experimental 1(E) curves c o l l e c t e d at shallow angles of i n c i d e n c e should be e s p e c i a l l y u s e f u l . In t h i s regard, i t w i l l a l s o be h e l p f u l to measure the angular spot p r o f i l e s of f r a c t i o n a l order beams as a f u n c t i o n of oxygen exposure. The f u l l width at h a l f maximum (FWHM) of the p r o f i l e s i s expected to i n c r e a s e as the degree of order decreases; c o r r e s p o n d i n g l y a (2x2) to (2x1) t r a n s i t i o n should be i n d i c a t e d by a r i s e i n the FWHM. A p r e l i m i n a r y study of t h i s i s c u r r e n t l y being undertaken with our VLA[170], but a fine-beam spot p r o f i l e a n a l y z e r (SPA) would be very h e l p f u l t o probe as p e c t s of d i s o r d e r and domain s t r u c t u r e over g r e a t e r dimensions than we can do c u r r e n t l y . As to the extent of 0 i n c o r p o r a t i o n i n t o the bulk, the oxygen s i g n a l from XPS can be monitored as a f u n c t i o n of i n c i d e n c e angles of the X-ray beam. The number of l a y e r s probed by the primary beam w i l l i n c r e a s e as the i n c i d e n c e d i r e c t i o n moves towards the normal. Such s t u d i e s should give a s e m i - q u a n t i t a t i v e depth p r o f i l e f o r oxygen. The c a l i b r a t i o n of oxygen coverage using the CO uptake data ( S e c t i o n 6.5) can s t i l l be improved. To answer whether 240 the 0 and C atoms are i n the same plane, the C 2 7 2 / 0 5 1 o Auger peak r a t i o c o u l d be measured as a f u n c t i o n of A r + bombardment of the Zr(000l) s u r f a c e t h a t g i v e s a (1x1)-CO p a t t e r n . A decrease of the r a t i o would presumably i n d i c a t e C o v e r l a y e r and 0 underlayer (assuming C and 0 have s i m i l a r s p u t t e r i n g y i e l d s ) . 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