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Volume 188—Reflection H i g h - 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 n d R e f l e c t i o n E l e c t r o n I m a g i n g o f S u r f a c e s e d i t e d by P. K. L a r s e n a n d P. J . D o b s o n

c£inü Senes S; Physics

Reflection High-Energy Electron Diffraction and Reflection Electron Imaging of Surfaces E d i t e d by

P. K. Larsen Ph i l i p s R e s e a r c h L a b o r a t o r i e s E i n d h o v e n , T h e N e t h e r l a n d s

a n d

P. J. Dobson Ph i l i p s R e s e a r c h L a b o r a t o r i e s Redh i l l , U n i t e d K i n g d o m

Plenum Press New York and London Published in Cooperation with NATO Scientific Affairs Division

P r o c e e d i n g s o f a N A T O A d v a n c e d R e s e a r c h W o r k s h o p o n R e f l e c t i o n H igh -Ene rgy E l e c t r o n D i f f r a c t i o n a n d R e f l e c t i o n E l e c t r o n I m a g i n g o f S u r f a c e s , he ld J u n e 1 5 - 1 9 , 1987, in V e l d h o v e n , T h e N e t h e r l a n d s

L i b r a r y o f C o n g r e s s C a t a l o g i n g in P u b l i c a t i o n D a t a

N A T O A d v a n c e d R e s e a r c h W o r k s h o p o n R e f l e c t i o n H i g h - 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 n d R e f l e c t i o n E l e c t r o n I m a g i n g o f S u r f a c e s (1987: V e l d h o v e n , N e t h e r l a n d s )

R e f l e c t i o n h i g h - 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 n d r e f l e c t i o n e l e c t r o n i m a g i n g o f s u r f a c e s / e d i t e d b y P. K. L a r s e n a n d P. J . D o b s o n .

p. c m . — ( N A T O A S I s e r i e s . S e r i e s B, P h y s i c s ; v. 188) " P r o c e e d i n g s o f a N A T O A d v a n c e d R e s e r a c h W o r k s h o p o n R e f l e c t i o n H i g h -

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 n d R e f l e c t i o n E l e c t r o n I m a g i n g o f S u r f a c e s , h e l d J u n e 1 5 - 1 9 , 1 9 8 7 , in V e l d h o v e n , T h e N e t h e r l a n d s " — T . p . v e r s o .

" P u b l i s h e d in C o o p e r a t i o n w i t h N A T O S c i e n t i f i c A f f a i r s D i v i s i o n . " I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s a n d i n d e x . I S B N 0-306-43035-5 1. S u r f a c e s ( P h y s i c s ) — T e c h n i q u e — C o n g r e s s e s . 2. R e f l e c t i o n h i g h 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 — C o n g r e s s e s . 3. R e f l e c t i o n e l e c t r o n m i c r o s c o p y — C o n ­g r e s s e s . I. L a r s e n , P. K. I I . D o b s o n , P. J . I I I . N o r t h A t l a n t i c T r e a t y O r g a n i z a t i o n . S c i e n t i f i c A f f a i r s D i v i s i o n . IV. T i t l e . V. S e r i e s . Q C 1 7 3 . 4 . S 9 4 N 3 7 1987 88 -28843 5 3 0 . 4 ' 1 - d c 1 9 C I P

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CONTENTS

SURFACE STRUCTURAL DETERMINATION

EXPERIMENTAL

Experimental Overview of Surface S t r u c t u r e Determination by RHEED

S. Ino 3

THEORY

Surface S t r u c t u r a l Determination Using RHEED J.L. Beeby 29

Theory of RHEED by Reconstructed Surfaces M.G. Knibb and P.A. Maksym 43

Accurate Dynamical Theory f o r RHEED Rocking-curve I n t e n s i t y Spectra

S.Y. Tong, T.C. Zhao and H.C. Poon 63

INELASTIC EFFECTS

I n e l a s t i c S c a t t e r i n g E f f e c t s i n RHEED and R e f l e c t i o n Imaging A.L. B l e l o c h , A. Howie, R.H. M i l n e and M.G. Wal l s 77

E x c i t a t i o n of D i e l e c t r i c Spheres by E l e c t r o n Beams P.M. Echenique 91

RESONANCE AND CHANNELING EFFECTS

Resonance E f f e c t s i n RHEED G. Meyer-Ehmsen 99

I n e l a s t i c S c a t t e r i n g and Secondary E l e c t r o n Emission Under Resonance C o n d i t i o n s i n RHEED from P t ( l l l )

H. Marten 109

Adatom S i t e Determination Using Channeling E f f e c t s i n RHEED on X-ray and Auger E l e c t r o n P r o d u c t i o n

J.C.H. Spence and Y. Kim 117

A Note on the Bloch Wave and I n t e g r a l Formulations of RHEED Theory

Jon Gj^nnes

DISORDERS AND STEPS IN ELECTRON DIFFRACTION

D i f f r a c t i o n from Disordered Surfaces: An Overview M.G. L a g a l l y , D.E. Savage and M.C. T r i n g i d e s

Theory of E l e c t r o n S c a t t e r i n g from Defect: Steps on Surfaces w i t h Non-equivalent Terraces

W. M o r i t z

D i f f r a c t i o n from Stepped Surfaces M. Henzler

RHEED and Disordered Surfaces B. Böiger, P.K. Larsen and G. Meyer Ehmsen

Temperature D i f f u s e S c a t t e r i n g i n RHEED M. A l b r e c h t and G. Meyer-Ehmsen

Temperature +Dependence of the Surface D i s o r d e r on Ge(OOl) Due to Ar Ion Bombardment

A.J. Hoeven, J.S.C. K o o l s , J . A a r t s and P.C. Zalm

Two-dimensional F i r s t - o r d e r Phase S e p a r a t i o n i n an E p i t a x i a l Layer

T.-M. Lu and S.-N. Yang

CONVERGENT BEAM DIFFRACTION

Surface Convergent-Beam D i f f r a c t i o n f o r C h a r a c t e r i z a t i o n and Symmetry Determination

J.A. Eades and M.D. Shannon

Convergent Beam RHEED C a l c u l a t i o n s Using the Surface P a r a l l e l M u l t i s l i c e Approach

A.E. Smith

REFLECTION ELECTRON MICROSCOPY

USING CONVENTIONAL INSTRUMENTS

R e f l e c t i o n E l e c t r o n Microscopy i n TEM and STEM Instruments J.M. Cowley

R e f l e c t i o n E l e c t r o n Microscopy w i t h Use of CTEM: Studies of Au Growth on P t ( l l l )

K. Y a g i , S. Ogawa and Y. T a n i s h i r o

A p p l i c a t i o n of R e f l e c t i o n E l e c t r o n Microscopy f o r Surface Science (Observation of cleaned c r y s t a l s urfaces of S i , P t , Au and Ag)

Y. Uchida

R e f l e c t i o n Microscopy i n a Scanning Transmission E l e c t r o n Microscope

R.H. Milne

xii

C o n t r a s t of Surface Steps and D i s l o c a t i o n s Under Resonance, Non-resonance, Bragg, and Non-Bragg C o n d i t i o n s

Tung Hsu and L.-M. Peng 329

MICROPROBE RHEED

Microprobe R e f l e c t i o n High-energy E l e c t r o n D i f f r a c t i o n M. Ichikawa and T. D o i 343

Scanning RHEED S t u d i e s of S i l i c i d e Formation i n a UHV-SEM P.A. Bennett and A.P. Johnson 371

LOW ENERGY INSTRUMENTS

Low Energy E l e c t r o n R e f l e c t i o n Microscopy (LEEM) and I t s A p p l i c a t i o n t o the Study of S i Surfaces

E. Bauer and W. T e l i e p s 381

Low Energy Scanning E l e c t r o n Microscope T. Ichinokawa 385

ELECTRON DIFFRACTION STUDIES OF GROWTH

SEMICONDUCTORS

RHEED I n t e n s i t y O s c i l l a t i o n s Düring MBE Growth of I I I - V Compounds -An Overview

B.A. Joyce, J.H. Neave, J . Zhang and P.J. Dobson 397

RHEED O s c i l l a t i o n s C o n t r o l of GaAs and AlAs MBE Growth Using Phase­l o c k Modulated Beams

F. B r i o n e s , L. Gonzalez and J.A. V e l a 419

The C o n t r i b u t i o n of Atomic Steps to R e f l e c t i o n High energy E l e c t r o n D i f f r a c t i o n from Semiconductor Surfaces

P.R. P u k i t e , P.I. Cohen and S. B a t r a 427

RHEED S t u d i e s of Growing Ge and S i Surfaces J . A a r t s and P.K. Larsen 449

LEED I n v e s t i g a t i o n s of S i MBE onto Si(100) M. Horn, U. G o t t e r and M. Henzler 463

METALS

Q u a n t i t a t i v e S t u d i e s of the Growth of Metals on GaAs(llO) Using RHEED

D.E. Savage and M.G. L a g a l l y 475

RHEED I n t e n s i t y O s c i l l a t i o n s i n M e t a l E p i t a x y

G. L i l i e n k a m p , C. K o z i o l and E. Bauer 489

THEORY

C a l c u l a t i o n of RHEED I n t e n s i t y from Growing Surfaces T. Kawamura 501

Studies of Growth K i n e t i c s on Surfaces w i t h D i f f r a c t i o n M.C. T r i n g i d e s and M.G. L a g a l l y 523

Index 539

xiii

THEORY OF ELECTRON SCATTERING FROM DEFECT: STEPS

ON SURFACES WITH NON-EQUIVALENT TERRACES

W. M o r i t z

I n s t i t u t f . K r i s t a l l o g r a p h i e und M i n e r a l o g i e U n i v e r s i t a e t Muenchen T h e r e s i e n s t r . 41, 8000 Muenchen 2, FRG

INTRODUCTION

One important parameter f o r the c h a r a c t e r i z a t i o n of surfaces i s the surfac e roughness. The d e n s i t y and d i s t r i b u t i o n of steps i n f l u e n c e s most of the p h y s i c a l and chemical p r o p e r t i e s of s u r f a c e s . I t i s found, f o r example, that a number of c a t a l y t i c r e a c t i o n s at sur f a c e s a c t u a l l y take p l a c e at edge or k i n k s i t e s . In the p r o d u c t i o n of semiconductor d e v i c e s the f l a t n e s s of surfaces or i n t e r f a c e s p l a y s an e s s e n t i a l r o l e . T h e r e f o r e a number of techniques have been developed to i n v e s t i g a t e the topography of s u r f a c e s e i t h e r by d i r e c t imaging or by d i f f r a c t i o n u s i n g both, X-rays and e l e c t r o n s . The d i f f r a c t i o n method i s the most convenient method t o o b t a i n I n f o r m a t i o n about the su r f a c e roughness on an atomic s c a l e .

The q u a n t i t i e s which are e a s i l y obtained form a d i f f r a c t i o n experiment are average q u a n t i t i e s l i k e t e r r a c e w i d t h s , i s l a n d s i z e d i s t r i b u t i o n f u n c t i o n s , mean roughness depths e t c . , because u s u a l l y a l a r g e area of the su r f a c e i s probed. The d e t e r m i n a t i o n of the d e t a i l s of the atomic geometry at and around a defect or a Step u s u a l l y r e q u i r e s a l a r g e experimental and t h e o r e t i c a l e f f o r t because of lack of s e n s i t i v i t y to such d e t a i l s , and i n the case of e l e c t r o n d i f f r a c t i o n , because m u l t i p l e s c a t t e r i n g e f f e c t s become important.

Surface roughness, domain or i s l a n d d i s t r i b u t i o n s cause broadening of the d i f f r a c t e d beams, i n general depending on the d i f f r a c t i o n c o n d i t i o n s and the beam i n d i c e s . I n many cases only the width of the beam (füll wi d t h at h a l f maximum) i s measured as an estimate of the average i s l a n d or t e r r a c e s i z e . However, from an a n a l y s i s of the angular beam p r o f i l e s the s i z e d i s t r i b u t i o n can be o b t a i n e d . This i s u s u a l l y done i n the k i n e m a t i c approximation ( s i n g l e s c a t t e r i n g ) where the c a l c u l a t i o n of beam p r o f i l e s becomes e s p e c i a l l y easy. The k i n e m a t i c a l theory i s used f o r low energy e l e c t r o n d i f f r a c t i o n (LEED) as w e l l as f o r r e f l e c t i o n h i g h energy e l e c t r o n d i f f r a c t i o n (RHEED). Although m u l t i p l e s c a t t e r i n g e f f e c t s are i n g e n e r a l s t r o n g f o r e l e c t r o n d i f f r a c t i o n , f o r the c a l c u l a t i o n of angular beam p r o f i l e s the k i n e m a t i c a l theory i s i n many cases s u f f i c i e n t . The q u e s t i o n a r i s e s whether m u l t i p l e s c a t t e r i n g can always be neglected i n e v a l u a t i n g beam p r o f i l e s and what e f f e c t s occur. In the case of LEED w i t h i t s back s c a t t e r i n g geometry, m u l t i p l e s c a t t e r i n g e f f e c t s are short ranged and the k i n e m a t i c e v a l u a t i o n of angular beam p r o f i l e s i s u s u a l l y based on t h a t argument. For RHEED w i t h i t s h i g h e l e c t r o n energies and the occurrence of

1 7 5

forward s c a t t e r i n g t h i s argument should be l e s s a p p l i c a b l e . On the other hand, l a r g e t e r r a c e s i z e s can be measured w i t h a RHEED instrument u s i n g the low index beams where the r e s o l u t i o n i s h i g h . In such cases m u l t i p l e s c a t t e r i n g e f f e c t s should again be n e g l i g i b l e . In g e n e r a l , f o r RHEED m u l t i p l e s c a t t e r i n g e f f e c t s are more d i s t u r b i n g i n the beam p r o f i l e a n a l y s i s than f o r LEED and i n a d d i t i o n , shadowing e f f e c t s occur.

M u l t i p l e s c a t t e r i n g e f f e c t s can be d i v i d e d i n t o two p a r t s , one where the m u l t i p l e s c a t t e r i n g path crosses a domain boundary or a step and another p a r t where t h i s i s not the case and m u l t i p l e s c a t t e r i n g occurs o n l y w i t h i n one domain and w i t h i n the bul k . The l a t t e r e f f e c t i s i n ge n e r a l s t r o n g and cannot be ne g l e c t e d , n e i t h e r f o r LEED nor f o r RHEED, but i s not r e l e v a n t f o r the angular beam p r o f i l e s i f only antiphase domains e x i s t at the s u r f a c e . Therefore i t i s u s u a l l y not considered i n the a n a l y s i s of beam p r o f i l e s from stepped s u r f a c e s where a l l t e r r a c e s are e q u i v a l e n t . At surf a c e s w i t h non-equivalent t e r r a c e s i t can be t r e a t e d i n a si m p l e q u a s i - k i n e m a t i c a l approximation as w i l l be d i s c u s s e d below. The o t h e r c o n t r i b u t i o n to the m u l t i p l e s c a t t e r i n g e f f e c t , where subsequent s c a t t e r i n g i n d i f f e r e n t t e r r a c e s i s i n v o l v e d , i s c e r t a i n l y important f o r RHEED but r e q u i r e s a more s o p h i s t i c a t e d treatment than that given here and i s not considered i n t h i s a r t i c l e .

D i f f r a c t i o n from stepped su r f a c e s has been d e s c r i b e d i n a number of s t u d i e s d e a l i n g w i t h one dimensional d i s t r i b u t i o n s [ 1 - 8 ] . The reference l i s t i s not intended to be complete, an e x t e n s i v e l i s t of references can be found i n the review a r t i c l e by L a g a l l y et a l . [ 9 ] . I t i s u s u a l l y assumed that the surface c o n s i s t of t e r r a c e s of i d e n t i c a l s c a t t e r i n g p r o p e r t i e s but separated by a s h i f t v e c t o r w i t h v e r t i c a l and l a t e r a l components. This i s i n general the case f o r a l l u n r e c o n s t r u c t e d su r f a c e s of the monoatomic l a t t i c e s i n c l u d i n g the f c c and bcc metals. The (111) s u r f a c e of the diamond l a t t i c e a l s o e x h i b i t s only one type of t e r r a c e , whereas the (100) su r f a c e s i m i l a r to the basal plane of the hcp l a t t i c e , c o n s i s t s of two d i f f e r e n t t e r m i n a t i o n s . Non-equivalent t e r r a c e s may be the i n h e r e n t property of the l a t t i c e or the r e s u l t of r e c o n s t r u c t i o n . The S i ( 1 0 0 ) s u r f a c e , f o r example, has two r o t a t i o n a l domains due to the f o u r -f o l d screw a x i s of the bulk l a t t i c e which reduces t o a t w o - f o l d a x i s at the s u r f a c e . These two d i f f e r e n t domains e x i s t even without a r e c o n s t r u c t i o n . Another example i s the W(100) s u r f a c e where the r e d u c t i o n of the symmetry i n a s i n g l e domain i s due to the r e c o n s t r u c t i o n . The f o u r - f o l d symmetry of the unr e c o n s t r u c t e d s u r f a c e i s destroyed by atomic displacements and two r o t a t i o n a l domains occur i n a d d i t i o n to the an t i p h a s e domains. I t i s i n such cases that the e f f e c t of m u l t i p l e s c a t t e r i n g w i t h i n one t e r r a c e cannot be neglected f o r the c a l c u l a t i o n of beam p r o f i l e s even f o r the s p e c u l a r beam. In a kinematic c a l c u l a t i o n , where only s i n g l e s c a t t e r i n g i s co n s i d e r e d , the s t r u c t u r e f a c t o r of the s p e c u l a r beam would be the same f o r a l l domains. Therefore a d i s t r i b u t i o n of r o t a t i o n a l domains would not cause a broadening of the spe c u l a r beam. Th i s i s not so i n case of m u l t i p l e s c a t t e r i n g . Here the s t r u c t u r e f a c t o r s f o r r o t a t i o n a l domains can be very d i f f e r e n t . N e g l e c t i n g d i f f e r e n t s t r u c t u r e f a c t o r s at the domain boundaries (and shadowing e f f e c t s e t c . ) the s u r f a c e can be approximately d e s c r i b e d by domains w i t h d i f f e r e n t s t r u c t u r e f a c t o r s . That means the su r f a c e c o n s i s t s of domains, or i n the case of Si( 1 0 0 ) of t e r r a c e s , which f o r e l e c t r o n s d i f f e r i n the e f f e c t i v e s c a t t e r i n g p r o p e r t i e s . These t e r r a c e s are t h e r e f o r e denoted i n the f o l l o w i n g as non-equivalent t e r r a c e s , though the a c t u a l s t r u c t u r e s may be s y m m e t r i c a l l y e q u i v a l e n t . That the d i f f r a c t i o n from r o t a t i o n a l domains or t e r r a c e s has to be de s c r i b e d by d i f f e r e n t s t r u c t u r e f a c t o r s i s c l e a r l y a m u l t i p l e s c a t t e r i n g e f f e c t and would not occur, f o r example, i n the s p e c u l a r beam w i t h X-ray d i f f r a c t i o n . This e f f e c t i s larg e and depends s t r o n g l y on the d i f f r a c t i o n c o n d i t i o n s as can be e a s i l y seen from the experimental r o t a t i o n a l diagrams where lar g e i n t e n s i t y v a r i a t i o n s occur.

1 7 6

In the case of LEED, r o t a t i o n a l diagrams could even be used f o r s t r u c t u r e a n a l y s i s . Azimuthai dependencies of i n t e n s i t i e s and beam p r o f i l e s have a l s o been r e p o r t e d i n RHEED s t u d i e s [10-13].

The d i f f r a c t i o n from stepped s u r f a c e s w i t h non-equivalent t e r r a c e s can be e a s i l y d e s c r i b e d by a q u a s i - k i n e m a t i c approximation by assuming d i f f e r e n t s t r u c t u r e f a c t o r s f o r each k i n d of t e r r a c e and u s i n g the kinem a t i c formalism o t h e r w i s e . M u l t i p l e s c a t t e r i n g e f f e c t s a r i s i n g from s c a t t e r i n g paths w i t h i n one t e r r a c e and w i t h i n the bulk are i n c l u d e d i n t h i s way. Those m u l t i p l e s c a t t e r i n g paths c r o s s i n g a step are ne g l e c t e d here. T h i s i s of course an approximation but i t i s w e l l a p p l i c a b l e f o r LEED at s u r f a c e w i t h l a r g e t e r r a c e s and e l u c i d a t e s a l s o some of the e f f e c t s observed w i t h RHEED.

In the next chapter the d i f f r a c t i o n from stepped s u r f a c e s w i l l be dis c u s s e d from the very g e n e r a l c o n s i d e r a t i o n t h a t the d i f f r a c t e d i n t e n s i t y can be d i v i d e d i n t o two p a r t s , one a r i s i n g from long ränge order and the other from s h o r t ränge ord e r . The i n f l u e n c e of the e x i s t e n c e of two d i f f e r e n t t e r r a c e s on the i n t e n s i t y o s c i l l a t i o n s d u r i n g l a y e r by l a y e r growth i s d i s c u s s e d b r i e f l y and a d e t a i l e d c a l c u l a t i o n of beam p r o f i l e s which would be expected from a S i ( 1 0 0 ) s u r f a c e w i t h s i n g l e and double steps and a geometric d i s t r i b u t i o n of t e r r a c e widths i s g i v e n .

GENERAL ASPECTS OF DIFFUSE SCATTERING

The i n t e n s i t y s c a t t e r e d from a d i s o r d e r e d s t r u c t u r e can be d i v i d e d i n t o two terms, a sharp r e f l e c t i o n and a d i f f u s e i n t e n s i t y . The sharp r e f l e c t i o n i s due to the ordered l a t t i c e and e x i s t s as long ränge order e x i s t s . The ränge i s gi v e n by the r e s o l u t i o n l i m i t of the instrument. The d i f f u s e i n t e n s i t y a r i s e s from the f l u c t u a t i o n of the s c a t t e r i n g amplitudes along the s u r f a c e ,

N i ( k - k * ) R I ( k , k ' ) = | Z F ( k , k ' ) e ~^| 2 +

n=l

N N _ * _* Z Z C(F (k,k') - F ( k , k

?

) ) ( F ,(k,k') - F ' ( k , k ' ) ) J x ( i ) n=l n' = l n ~ ~~~ n ~ ~ i ( k - k ' ) ( R -R ,)

x e ~*

where R n i s a two dimensional l a t t i c e v e c t o r , N i s the number of u n i t c e l l s , k and k' are the wave v e c t o r s of the incoming and outgoing waves and F n ( k , k ' ) are s t r u c t u r e amplitudes to u n i t c e l l column v a l u e s . F ( k , k ' ) i s the s p a t i a l average of s t r u c t u r e amplitudes. To s i m p l i f y the above e x p r e s s i o n we assume that the s t r u c t u r e amplitudes are s l o w l y v a r y i n g f u n c t i o n s of the d i f f r a c t i o n angle. I t may be f u r t h e r assumed that the d e n s i t y of d e f e c t s or steps i s low so that m u l t i p l e s c a t t e r i n g e f f e c t s at the edges can be n e g l e c t e d . The d i f f r a c t e d i n t e n s i t y can then be approximately w r i t t e n i n the convenient form

I ( k , k ' ) = |F| 2G(h)G(k) + N ( ] F T 2 - | F | 2 ) P ( k - k ' )

s i n N TTX

G(x) = ^ sin 2 -rrx

where P(k - k') i s the F o u r i e r t r a n s f o r m of the p a i r - c o r r e l a t i o n f u n c t i o n s and d e s c r i b e s the angular beam p r o f i l e , h and k are the beam i n d i c e s . N

1 7 7

i s the t o t a l number of u n i t c e l l s . Eqs. 1 or 2 d e s c r i b e t h a t the i n t e n s i t y i n the sharp r e f l e c t i o n s i s due to the mean va l u e of the s c a t t e r i n g amplitudes and that the i n t e g r a t e d d i f f u s e i n t e n s i t y i s p r o p o r t i o n a l to the mean square d e v i a t i o n of s c a t t e r i n g amplitudes.

A comment has to be made on m u l t i p l e s c a t t e r i n g e f f e c t s . I t should be noted that Eq. 1 i s i n general v a l i d , a l s o f o r the m u l t i p l e s c a t t e r i n g case, when f o r each u n i t c e l l at the l a t t i c e p o i n t an i n d i v i d u a l s t r u c t u r e f a c t o r i s taken a c c o r d i n g t o the i n d i v i d u a l surroundings of the u n i t c e l l . Eq. 2 i s i n general not v a l i d because i t n e g l e c t s a l l m u l t i p l e s c a t t e r i n g e f f e c t s between d i f f e r e n t t e r r a c e s or i s l a n d s i f P(k - k') i s the F o u r i e r - t r a n s f o r m of the p a i r - c o r r e l a t i o n f u n c t i o n s . I t corresponds to the column approximation used i n high energy e l e c t r o n d i f f r a c t i o n [14]. A gen e r a l m u l t i p l e s c a t t e r i n g f o r m u l a t i o n has to i n c l u d e m u l t i - s i t e c o r r e l a t i o n s . The a p p l i c a b i l i t y of Eq. 2 i s j u s t i f i e d by the f a c t that f o r an e x p o n e n t i a l decay of p a i r c o r r e l a t i o n f u n c t i o n s , a l l m u l t i - s i t e c o r r e l a t i o n f u n c t i o n s w i l l a l s o e x h i b i t the same e x p o n e n t i a l decay. As long as no sharp i n t e r f e r e n c e s i n the s c a t t e r i n g f a c t o r s occur, the beam p r o f i l e s w i l l t h e r e f o r e keep the L o r e n t z i a n form as i n the ki n e m a t i c case. This has a l s o been shown a n a l y t i c a l l y f o r one dimension [3,15]. I t has t o be kept i n mind f u r t h e r that i n the ki n e m a t i c l i m i t the t o t a l i n t e g r a t e d i n t e n s i t y remains constant independent of the State of order w h i l e i n the m u l t i p l e s c a t t e r i n g theory t h i s i s i n general not the case. I n t e r f e r e n c e i n the m u l t i p l e s c a t t e r i n g paths can d r a s t i c a l l y enhance the i n t e n s i t y of a c e r t a i n beam. The i n t e n s i t y , of course, i s then m i s s i n g i n other beams. Co n s e r v a t i o n of the t o t a l d i f f r a c t e d i n t e n s i t y cannot be observed, however, because only the b a c k s c a t t e r i n g p a r t can be measured. F u r t h e r -more, s t r o n g a b s o r p t i v e e f f e c t s depending on the d i f f r a c t i o n c o n d i t i o n s d e s t r o y the c o n s e r v a t i o n of the i n t e g r a t e d i n t e n s i t y .

I t i s o f t e n convenient to S t a r t w i t h Eq. 2 f o r a q u a l i t a t i v e d i s c u s s i o n of the c h a r a c t e r i s t i c f e a t u r e s of the d i f f r a c t i o n p a t t e r n . I f u n c o r r e l a t e d d e f e c t s e x i s t at the s u r f a c e the d i f f u s e i n t e n s i t y w i l l be u n i f o r m l y d i s t r i b u t e d over the r e c i p r o c a l u n i t c e l l , the angular d i s t r i b u t i o n being determined onl y by the angular dependence of the atomic c r o s s s e c t i o n . I f the defect atoms form C l u s t e r s or t e r r a c e s , the d i f f u s e i n t e n s i t y w i l l be concentrated around c e r t a i n p o i n t s i n the r e c i p r o c a l u n i t c e l l . The i n t e n s i t y of the sharp r e f l e c t i o n as w e l l as the d i f f u s e i n t e n s i t y may van i s h at c e r t a i n p o i n t s , i . e . , the mean value of the s c a t t e r i n g amplitudes vanishes f o r a s u r f a c e w i t h an equal d i s t r i b u t i o n of t e r r a c e s at the "out of phase" c o n d i t i o n . The mean Square d e v i a t i o n may a l s o v a n i s h , e.g., at i n phase s c a t t e r i n g from two d i f f e r e n t t e r r a c e s , and u s u a l l y has i t s maximum when the mean value has i t s minimum. The q u a l i t a t i v e f e a t u r e s of the d i f f r a c t i o n p a t t e r n s f o r the two simple cases as shown i n the Fi g u r e s 1-4 f o l l o w s d i r e c t l y from the i n t e r p r e t a t i o n of Eq. 2.

The i n t e n s i t y and angular p r o f i l e of the s p e c u l a r beam f o r a two l e v e l System as a f u n c t i o n of coverage and momentum t r a n s f e r p e r p e n d i c u l a r to the surface i s d i s p l a y e d i n F i g u r e s 1 and 2. I t i s assumed here t h a t the two t e r r a c e s are e q u i v a l e n t . This case has been d e s c r i b e d i n d e t a i l and has been f r e q u e n t l y v e r i f i e d e x p e r i m e n t a l l y [4-9] but w i l l be s h o r t l y repeated here f o r the sake of completeness. Thus the two l e v e l s have s c a t t e r i n g f a c t o r s ^ ( k ^ k ' ) and F 2 ( k , k ' ) d i f f e r i n g by a phase f a c t o r

F . ( k , k ' ) = F . C k . l O e 1 ^ " ^ (3)

| F 2 ( k , k ' ) | = i F ^ k . k ' ) !

where d i s the s h i f t v e c t o r between t e r r a c e s . The mean value of the s c a t t e r i n g f a c t o r at coverage 0 i s d e f i n e d as

1 7 8

F i g . 1. Model of a s u r f a c e w i t h i d e n t i c a l t e r r a c e s i n two l e v e l s and the corresponding angular p r o f i l e of the sp e c u l a r beam. The beam p r o f i l e c o n s i s t s i n general of two components, a c e n t r a l peak and a d i f f u s e p a r t . The w idth of the d i f f u s e i n t e n s i t y corresponds roughly to the mean t e r r a c e s i z e .

F(k,k') = (1 - 0 ) F 1 ( k , k ' ) +0F 2(k,k') (4)

and the i n t e n s i t y of the c e n t r a l peak i s

I = |F. | 2 C l - 20(1 - 0) (1 - C O S2T T S)] (5) P 1

where s = (k - k')d/2TT i s the momentum t r a n s f e r i n r e c i p r o c a l l a t t i c e u n i t s . The arguments k and k 1 have been dropped f o r convenience. The i n t e g r a l d i f f u s e i n t e n s i t y i s gi v e n by

I D = 20(1 - 0) \ T 1 | 2 ( 1 - COS2TTS) . (6)

The peak i n t e n s i t y becomes an o s c i l l a t i n g f u n c t i o n i n r e c i p r o c a l space as w e l l as being a f u n c t i o n of coverage as i l l u s t r a t e d i n F i g u r e 2a-c. The minimum of the peak i n t e n s i t y of the sp e c u l a r beam corresponds t o h a l f coverage. Because a t t h i s p o i n t the d i f f r a c t i o n from a l l l a y e r s should be out of phase, the i n t e n s i t y of the sharp r e f l e c t i o n has t o v a n i s h . The remaining i n t e n s i t y at s = 1/2 i s t h e r e f o r e e n t i r e l y due t o the maximum of the d i f f u s e term and corresponds roughly to the Square of the average i s l a n d diameter. The width of the d i f f u s e i n t e n s i t y i n r e c i p r o c a l u n i t s i s approximately the i n v e r s e of the i s l a n d diameter, the exact value depends on the i s l a n d s i z e d i s t r i b u t i o n f u n c t i o n . How the maximum of the d i f f u s e i n t e n s i t y behaves as a f u n c t i o n of coverage cannot be g e n e r a l l y g i v e n i n t h i s q u a l i t a t i v e d i s c u s s i o n s i n c e t h i s depends on the s p e c i f i c

1 7 9

d

0 0.5 1

F i g . 2. I n t e n s i t y o s c i l l a t i o n s as a f u n c t i o n of coverage and momentum t r a n s f e r . I n t e n s i t y of the c e n t r a l peak (a) and d i f f u s e p a r t (b) at d i f f e r e n t coverages. c: I n t e n s i t y of the c e n t r a l peak as a f u n c t i o n of coverage at the "out of phase" c o n d i t i o n , s = 1/2, where (k - k')d = 2TTS.

growth model. At each maximum of the c e n t r a l peak one l a y e r i s completed. The l a y e r growth mechanism i s o b v i o u s l y best s t u d i e d at the "out of phase" c o n d i t i o n where the maximum e f f e c t on the peak i n t e n s i t y and d i f f u s e i n t e n s i t y i s observed. From the slope of the peak i n t e n s i t y at füll coverage (F i g u r e 2c) the number of l a y e r s i n v o l v e d i n the growth process can be determined [16].

We c o n s i d e r next a su r f a c e w i t h non-equivalent t e r r a c e s and assume a l s o l a y e r by l a y e r growth. We assume f u r t h e r that the su r f a c e w i t h a complete l a y e r c o n s i s t s of one t e r r a c e o n l y , see F i g u r e 3a. The s t r u c t u r e amplitudes f o r both t e r r a c e s d i f f e r now i n modulus and phase i n a d d i t i o n t o the geometric phase

i4> (k,k') i<J> (k,k') F l " ! F J e 1 • F 2 - l F 2 | e 2

( 7 )

I V * |F 2|.

1 8 0

F i g . 3. a. Model of a s u r f a c e w i t h two non-equivalent t e r r a c e s i n two l e v e l s . b. I n t e n s i t y o s c i l ­l a t i o n s of the c e n t r a l peak as a f u n c t i o n of coverage. S o l i d l i n e : "out of phase" c o n d i t i o n , broken l i n e : " i n phase" c o n d i t i o n . A r e l a t i o n of J F j l 2 = 0 . 5 | F 2 I 2 n a s been assumed. A phase f a c t o r between the complex s t r u c t u r e amplitudes F^ and F 2 i n f l u e n c e s o n l y the " i n phase" and "out of phase" c o n d i t i o n s but not the p o s i t i o n s of maxima and minima on the coverage s c a l e .

The mean value of s c a t t e r i n g amplitudes does not v a n i s h any more at the out of phase c o n d i t i o n (s = 1 / 2 ) and at h a l f coverage. A l s o the maxima and minima may be s h i f t e d due to the phase d i f f e r e n c e . The peak i n t e n s i t y i s now given by

I = (1 - 0 ) 2 | F 1 | 2 + 0 2 | F 2 | 2 + 2 0 ( 1 - 0 ) I F J F J C O S ^ T T S + A<j>) ( 8 )

and shows now a double p e r i o d i c i t y as a f u n c t i o n of coverage, F i g u r e 3b. The i n t e g r a l d i f f u s e i n t e n s i t y i s

I ß = 0 ( 1 - 0 ) C | F 1 | 2 + | F 2 | 2 - 2 | F 1 F 2 | C O S ( 2 T T S + A<J>)3 ( 9 )

and cannot show a double p e r i o d i c i t y as a f u n c t i o n of coverage as i t r e s u l t s from the mean Square d e v i a t i o n i n s t r u c t u r e amplitudes. T h i s case i s i l l u s t r a t e d i n F i g u r e 4a and b. The phase d i f f e r e n c e A4»(k,k') between the s t r u c t u r e f a c t o r s s h i f t s the maxima and minima of the i n t e n s i t y of the c e n t r a l peak away from the p o i n t s s = 0 and s - 1 / 2 . The same occurs f o r the maxima and minima of the d i f f u s e i n t e n s i t y i n r e c i p r o c a l space a c c o r d i n g to Eqs. 8 and 9 . I t should be noted that a double p e r i o d i c i t y i n r e c i p r o c a l space f o r the peak i n t e n s i t y or angular w i d t h , which has been observed f o r S i ( 1 0 0 ) at i n c i d e n c e i n [011] d i r e c t i o n [17], i s due t o the e x i s t e n c e of f o u r l e v e l s w i t h two r o t a t i o n a l domains and two a n t i p h a s e domains. The s i m p l e r System which i s i l l u s t r a t e d i n F i g u r e s 3 and 4 c o n s i s t s of o n l y two l e v e l s .

The i n t e n s i t y o s c i l l a t i o n s as observed w i t h RHEED or LEED d u r i n g l a y e r by l a y e r growth (see F i g u r e 3b) e x h i b i t a double p e r i o d i c i t y but s t i l l have a maximum and minimum a t füll coverage because the d i f f u s e term

1 8 1

a

F i g . 4. I n t e n s i t y o s c i l l a t i o n s of the peak i n t e n s i t y and the d i f f u s e i n t e n s i t y f o r two d i f f e r e n t t e r r a c e s and h a l f coverage as a f u n c t i o n of momentum t r a n s f e r . | F i | * = 0.5|F 2| 2. The phase d i f f e r e n c e A<j>(k,k*) s h i f t s the p o s i t i o n of maxima and minima i n r e c i p r o c a l space.

always vanishes at füll coverage. The i n t e n s i t y o s c i l l a t i o n s as a f u n c t i o n of momentum t r a n s f e r s and coverage 0 are now d i f f e r e n t . I t i s i n t e r e s t i n g t o note that double p e r i o d i c i t i e s and azimuthal dependencies of i n t e n s i t y o s c i l l a t i o n s d u r i n g MBE growth of S i ( 1 0 0 ) [11,12,18] and GaAs(lOO) [10,13] have been observed. From the above d i s c u s s i o n f o l l o w s that the s h i f t of maxima and minima i n the i n t e n s i t y o s c i l l a t i o n s as observed d u r i n g MBE growth of S i ( 1 0 0 ) and GaAs(lOO) cannot be e x p l a i n e d only w i t h a sequence of two d i f f e r e n t l a y e r s . The occurrence of two maxima d u r i n g the completion of only one l a y e r has o b v i o u s l y to be e x p l a i n e d by a s t r o n g l y peaked d i f f u s e s c a t t e r i n g at h a l f coverage [10]. On the other hand, the f a c t that at c e r t a i n d i f f r a c t i o n c o n d i t i o n s , the i n t e n s i t y i n i t i a l l y i n c r e a s e s w i t h i n c r e a s i n g coverage, as has been observed f o r GaAs(lOO) [10], f o l l o w s d i r e c t l y from the assumption of a phase d i f f e r e n c e between the two s c a t t e r i n g f a c t o r s . For the S i ( 1 0 0 ) s u r f a c e the azimuthal dependence of the i n t e n s i t y o s c i l l a t i o n s has been e x p l a i n e d by m u l t i p l e s c a t t e r i n g e f f e c t s i n v o l v i n g the form of the growing t e r r a c e s [19].

At s p e c i a l d i f f r a c t i o n c o n d i t i o n s the s t r u c t u r e f a c t o r s of the two t e r r a c e s can become equal, that i s when the plane of i n c i d e n c e c o i n c i d e s w i t h a symmetry plane t r a n s f o r m i n g one t e r r a c e to the o t h e r . For the S i ( 1 0 0 ) s u r f a c e t h i s occurs f o r i n c i d e n c e i n the [010] d i r e c t i o n . The two t e r r a c e s have then i d e n t i c a l s c a t t e r i n g f a c t o r s f o r the s p e c u l a r beam. This i s not so at i n c i d e n c e i n the [011] d i r e c t i o n , f o r which s c a t t e r i n g f a c t o r s are d i f f e r e n t . The dependence of the s t r u c t u r e f a c t o r on the azimuthal angle i s s p e c i f i c f o r m u l t i p l e s c a t t e r i n g . From the aboye d i s c u s s i o n , i t f o l l o w s that the S i ( 1 0 0 ) s u r f a c e looks l i k e a two l e v e l

182

surface w i t h e q u i v a l e n t t e r r a c e s at Observation i n [010] and w i t h non-equi v a l e n t t e r r a c e s i n the [011] d i r e c t i o n . The double and s i n g l e p e r i o d i c i t i e s i n r e c i p r o c a l space at d i f f e r e n t azimuths have indeed been observed e x p e r i m e n t a l l y [17,20]. Double p e r i o d i c i t y i n the o s c i l l a t i o n of beam widths has a l s o been observed f o r Os(0001) [21] and has been i n t e r p r e t e d by the e x i s t e n c e of double s t e p s .

I t i s h i g h l y u n r e a l i s t i c to assume that the area of the su r f a c e i l l u m i n a t e d by the primary beam c o n s i s t s of only one or two t e r r a c e s , unless the beam i s very w e l l focused. U s u a l l y a s u p e r p o s i t i o n of a l l p o s s i b l e domains corresponding t o the po i n t symmetry of the c r y s t a l i s observed i n the d i f f r a c t i o n p a t t e r n . For GaAs(lOO) or S i ( 1 0 0 ) , f o r example, the two o r i e n t a t i o n s of the (4 x 2) and ( 2 x 1 ) s t r u c t u r e s r e s p e c t i v e l y , are always observed un l e s s one o r i e n t a t i o n i s suppressed at v i c i n a l f a c e s . For the above mentioned case of a two l a y e r sequence t h i s means that the s c a t t e r i n g amplitudes from two regions of the su r f a c e have to be summed up w i t h a l t e r n a t e l a y e r sequences. This s u p e r p o s i t i o n can be done c o h e r e n t l y or i n c o h e r e n t l y . The l a t t e r i m p l i e s that l a r g e areas e x i s t where one l a y e r i s completed before the next begins to grow. The doubling of the p e r i o d of the peak i n t e n s i t y w i t h i n c r e a s i n g coverage (F i g u r e 3b) i s then removed. I f a double p e r i o d i c i t y i s observed, however, i t can be concluded t h a t only one domain i s present w i t h i n the area of the i n c i d e n t beam. E x p e r i m e n t a l l y t h i s has been observed f o r S i ( 1 0 0 ) , see the a r t i c l e by Horn et a l . i n t h i s volume, F i g u r e 1.

The other case, coherent s u p e r p o s i t i o n , corresponds to the simultaneous growth of fo u r l a y e r s which may not be r e a l i z e d at steady State c o n d i t i o n s but may occur f o r the i n i t i a l growth c o n d i t i o n s . T h i s w i l l not be d i s c u s s e d here because i t depends too much on the s p e c i f i c model f o r the i n i t i a l t e r r a c e width d i s t r i b u t i o n and the s p e c i f i c growth model that general c o n c l u s i o n s can be drawn from the q u a l i t a t i v e p i c t u r e used above. The coherent s u p e r p o s i t i o n of the s c a t t e r i n g amplitudes from four l e v e l s , however, and the consequences f o r the beam widths i n r e c i p r o c a l space w i l l be d i s c u s s e d f o r the S i ( 1 0 0 ) sur f a c e i n some d e t a i l i n the next s e c t i o n .

I t has been assumed up to now that the s u r f a c e i s f l a t and c o n s i s t s of only two l e v e l s at l e a s t w i t h i n the coherence len g t h of the e l e c t r o n s which may be as la r g e as s e v e r a l u. I t has been shown e x p e r i m e n t a l l y that these assumptions are j u s t i f i e d f o r many cases of molecular beam e p i t a x y i n c l u d i n g the case of S i ( 1 0 0 ) [ 1 8 ] . However, the S i t u a t i o n where the surfac e i s rough on an atomic s c a l e , which means that many l e v e l s e x i s t w i t h i n the coherence len g t h of the beam, i s a l s o q u i t e f r e q u e n t l y r e a l i z e d , and the consequences f o r the beam p r o f i l e s can be s h o r t l y mentioned. For a two l e v e l system, see F i g u r e 1, the beam p r o f i l e c o n s i s t s of two components, a c e n t r a l peak and a d i f f u s e p a r t . The i n t e n s i t y of the c e n t r a l peak vanishes only at c e r t a i n p o i n t s i n r e c i p r o c a l space and at c e r t a i n coverages. The width of the d i f f u s e p a r t remains independent of the d i f f r a c t i o n c o n d i t i o n s and depends only on the i s l a n d d i s t r i b u t i o n . As the number of d i f f e r e n t l e v e l s i n c r e a s e s , the i n t e n s i t y of the c e n t r a l peak decreases r a p i d l y as a f u n c t i o n of momentum t r a n s f e r and a d d i t i o n a l narrow d i f f u s e f e a t u r e s occur [ 4 ] . I n the l i m i t i n g case of an i n f i n i t e number of l e v e l s , a su r f a c e which i s rough on an atomic s c a l e , the c e n t r a l peak vanishes at a l l d i f f r a c t i o n c o n d i t i o n s except at the B r a g g - c o n d i t i o n where a l l l e v e l s s c a t t e r " i n phase". The width of the peak then broadens c o n t i n u o u s l y . This has been shown p r e v i o u s l y i n an a n a l y s i s of the beam p r o f i l e s of the d i s o r d e r e d Au(110) surface [ 3 ] . The i n f l u e n c e of the number of l e v e l s on the beam p r o f i l e s has a l s o been i n v e s t i g a t e d i n d e t a i l r e c e n t l y [ 6 - 8 ] .

1 8 3

ONE DIMENSIONAL MODEL WITH SINGLE AND DOUBLE STEPS

The c a l c u l a t i o n of beam p r o f i l e s from one-dimensional d i s o r d e r e d surfaces w i t h a geometric d i s t r i b u t i o n of steps f o l l o w s the methods developed f o r the a n a l y s i s of s t a c k i n g f a u l t s i n c r y s t a l s [22-24] and the a p p l i c a t i o n to steps on sur f a c e s has been d e s c r i b e d i n d e t a i l p r e v i o u s l y [ 3 - 8 ] . Therefore the formalism w i l l be only s h o r t l y repeated here as f a r as i t i s necessary to i n c l u d e double s t e p s . So f a r most c a l c u l a t i o n s of beam p r o f i l e s from stepped s u r f a c e s have assumed s i n g l e steps o c c u r r i n g at the surface w i t h a given p r o b a b i l i t y a. That model i n c l u d e s the occurrence of double steps w i t h the p r o b a b i l i t y a 2 and assumes that there i s no c o r r e l a t i o n between s t e p s . A s l i g h t l y m o d i f i e d model where the occurrence of a double step i s gi v e n a d i f f e r e n t p r o b a b i l i t y 3 and how t h i s w i l l become v i s i b l e i n the beam p r o f i l e s w i l l be i n v e s t i g a t e d here. To assume a d i f f e r e n t p r o b a b i l i t y f o r the occurrence of double steps w i t h i n the formal frame of a geometric d i s t r i b u t i o n of t e r r a c e s i z e s i s p o s s i b l e because f o u r d i f f e r e n t t e r r a c e s , or f o u r l e v e l s , are assumed.

The S i ( 1 0 0 ) s u r f a c e has been found to show a preference f o r double s t e p s . V i c i n a l s u r f a c e s form double steps as has been concluded from the f a c t that only one o r i e n t a t i o n of the r e c o n s t r u c t e d ( 2 x 1 ) u n i t c e l l occurs [25,26]. The e x i s t e n c e of double steps has been observed r e c e n t l y i n the d i r e c t image w i t h R e f l e c t i o n E l e c t r o n Microscopy (REM) [27] and Transmission E l e c t r o n Microscopy (TEM) [28]. The formation of double steps has a l s o been p r e d i c t e d t h e o r e t i c a l l y [24]. I t i s t h e r e f o r e i n t e r e s t i n g t o i n v e s t i g a t e how double steps can be observed i n the beam p r o f i l e s . The 1-D model w i t h a geometric d i s t r i b u t i o n of steps w i l l be a p p l i e d here because of i t s s i m p l i c i t y . The L o r e n t z i a n shape of the beam p r o f i l e r e s u l t i n g from the e x p o n e n t i a l decay of c o r r e l a t i o n s has been found o f t e n to f i t the experimental p r o f i l e s s u f f i c i e n t l y w e l l [ 4 ] . The ex p o n e n t i a l decay of c o r r e l a t i o n s corresponds i n 1-D to the geometric d i s t r i b u t i o n of t e r r a c e w i d t h s . The 1-D model has the advantage t h a t the beam p r o f i l e s can be c a l c u l a t e d a n a l y t i c a l l y and the i n f l u e n c e of d i f f e r e n t parameters can be e a s i l y seen. The 1-D model i s a l s o u s e f u l f o r the d e s c r i p t i o n of v i c i n a l f a c e s . I t i s not expected that the gene r a l f e a t u r e s of the d i f f r a c t i o n p a t t e r n s and beam p r o f i l e s d i f f e r s u b s t a n t i a l l y from the exact c a l c u l a t i o n f o r the 2-D model. For a q u a n t i t a t i v e e v a l u a t i o n of t e r r a c e s i z e d i s t r i b u t i o n s , however, a two-dimensional c a l c u l a t i o n may be necessary.

Two l i m i t i n g cases can be d i s t i n g u i s h e d . F i r s t the case of a f o u r l e v e l s u r f a c e , corresponding to the f o u r l a y e r sequence i n the bulk of Si( 1 0 0 ) w i t h two d i f f e r e n t p r o b a b i l i t i e s f o r s i n g l e and double s t e p s . T h i s i s the l i m i t i n g case of a f l a t s u r f a c e showing a l l f o u r types of domains. The other l i m i t i n g case i s a rough s u r f a c e w i t h an i n f i n i t e number of l a y e r s (at a l a t e r a l l y i n f i n i t e s u r f a c e ) and c o n t i n u o u s l y broadened beams. At s p e c i a l d i f f r a c t i o n c o n d i t i o n s , that i s at " i n phase" and "out of phase" s c a t t e r i n g from s i n g l e step height t e r r a c e s , both cases r e s u l t i n the same beam p r o f i l e s . In the model di s c u s s e d below a rough surf a c e i s assumed by a l l o w i n g the occurrence of up and down steps i n each l e v e l .

A model of the Si ( 1 0 0 ) s u r f a c e w i t h the f o u r d i f f e r e n t t e r r a c e s i s shown i n F i g u r e 5. The d i r e c t i o n of the edges i s assumed to be al o n g the densely packed atomic rows i n [011]. There are two antiphase domains and two r o t a t i o n a l domains i n the f o u r t e r r a c e s having s t r u c t u r e f a c t o r s

i ( k - k ' ) ( ^ + 2 d ) F ^ k . k ' ) , F (kfk») - F ^ k . k ^ e

i ( k - k ' ) d i ( k - k » ) ( ^ + 3 d ) F 2 ( k , k ' ) e Z , F u ( k , k ' ) = F 2 ( k , k ' ) e

1 8 4

[100]

F 1 ! F 2 I F 3 I F 4

F i g . 5. Model of the S i ( 1 0 0 ) s u r f a c e w i t h f o u r d i f f e r e n t t e r r a c e s . The r e c o n s t r u c t i o n of the s u r f a c e i s neg l e c t e d here. The assignment of d i f f e r e n t s t r u c t u r e amplitudes to the d i f f e r e n t atomic columns as used i n the t e x t i s i n d i c a t e d i n the F i g u r e .

a and b are the l a t t i c e v e c t o r s of the p r i m i t i v e p l a n a r u n i t c e l l and d z

i s the l a y e r s p a c i n g . The r e c o n s t r u c t i o n of the surfa c e i s neglected here because i t i s not r e l e v a n t f o r the beam p r o f i l e s of the sp e c u l a r beam. M u l t i p l e s c a t t e r i n g events i n v o l v i n g the edge atoms as w e l l d e t a i l s of the atomic geometry near an edge are a l s o not considered here. As mentioned above, at i n c i d e n c e i n [011] d i r e c t i o n the s t r u c t u r e f a c t o r s F} and F 2 are d i f f e r e n t i n modulus and phase even f o r the s p e c u l a r beam as the r e s u l t of m u l t i p l e s c a t t e r i n g , w h i l e at i n c i d e n c e i n [010] t h i s i s not the case.

The phase f a c t o r s due to the antiphase r e l a t i o n between t e r r a c e 1 and 3, and 2 and 4 r e s p e c t i v e l y , as w e l l as the phase f a c t o r s due to the height d i f f e r e n c e s are a t t r i b u t e d here to the s t r u c t u r e f a c t o r s F i to F i * . This i s i n general p o s s i b l e f o r a s u r f a c e having only these f o u r l e v e l s . For a rough s u r f a c e w i t h more l e v e l s t h i s d e f i n i t i o n of the s t r u c t u r e f a c t o r s can be a l s o used as long as only s p e c i a l d i f f r a c t i o n c o n d i t i o n s are c o n s i d e r e d . I t i s i n general convenient t o i n c l u d e the phase f a c t o r s i n the c o r r e l a t i o n m a t r i x f o r the d e s c r i p t i o n of a rough s u r f a c e [ 3 ] . However, the a n a l y t i c a l e v a l u a t i o n of the eigenvalues f o r the general case becomes inconvenient here and the beam p r o f i l e s may be c a l c u l a t e d only f o r the s p e c i a l d i f f r a c t i o n c o n d i t i o n s of , f i n phase" and "out of phase" s c a t t e r i n g from s i n g l e and double s t e p s .

There e x i s t s two d i f f e r e n t monoatomic steps at t h i s s u r f a c e . The atomic geometry at the edge from t e r r a c e 1 to t e r r a c e 2 d i f f e r s from the step from t e r r a c e 2 t o 3. Both steps are assumed here t o occur w i t h the same p r o b a b i l i t y , f o r the sake of s i m p l i c i t y . To a l l o w d i f f e r e n t

1 8 5

coverages f o r the t e r r a c e s 1 and 2, and 3 and 4, r e s p e c t i v e l y , the st e p down from t e r r a c e land 2 may occur w i t h p r o b a b i l i t y and the step up from t e r r a c e 2 to 1 w i t h p r o b a b i l i t y a 2 . The assumption of d i f f e r e n t p r o b a b i l i t i e s f o r these steps up and down enables the d e s c r i p t i o n of a surface w i t h a preference f o r one k i n d of t e r r a c e s . This i s not u n r e a l i s t i c , because i t has been e x p e r i m e n t a l l y observed that at v i c i n a l faces one k i n d of t e r r a c e s dominates [25-28]. The occurrence of double steps i s g i v e n by the p r o b a b i l i t y ß. The edge of double steps i s normal to the dimer rows i n the (2 x 1) r e c o n s t r u c t e d s u r f a c e , as i t becomes v i s i b l e i n the S p l i t t i n g of s u p e r s t r u c t u r e spots at v i c i n a l s u r f a c e s [25,26]. Therefore i t may be assumed f o r s i m p l i c i t y that double steps occur o n l y between the t e r r a c e s 1 and 3. Double steps between t e r r a c e s 2 and 4 would occur i n the other d i r e c t i o n and cannot be considered i n the one-dimensional model. I t i s f u r t h e r assumed that double steps up and down occur w i t h equal p r o b a b i l i t y which ensures that the average o r i e n t a t i o n of the s u r f a c e i s i n [100]. Because the p r o b a b i l i t y f o r the occurrence of a s i n g l e step has to be d i v i d e d i n t o two c o n t r i b u t i o n s , the p r o b a b i l i t y f o r a s i n g l e step downward from t e r r a c e 1 t o 2, or upward from t e r r a c e 1 t o 4, i s a/2. For double steps t h i s d i s t i n c t i o n needs not to be made because the step from t e r r a c e 1 to 3 may be upward or downward. To use the same d e f i n i t i o n f o r the p r o b a b i l i t i e s f o r s i n g l e and double steps the double step gets the p r o b a b i l i t y ß and not ß/2. More s o p h i s t i c a t e d models of the topography of the surf a c e are p o s s i b l e , but then the e v a l u a t i o n of the c o r r e l a t i o n m a t r i x has to be done n u m e r i c a l l y . I t i s f u r t h e r d o u b t f u l t h a t a more d e t a i l e d p i c t u r e i s u s e f u l w i t h i n the l i m i t s of a model of a geometric d i s t r i b u t i o n of t e r r a c e s i z e s .

The c o r r e l a t i o n m a t r i x f o r the above d e s c r i b e d model i s given by

1 -

P ( D =

* l - ß

a l 2 ß

a i 2

2 1 - a 2 2 0

Ö a 1 1 - a i - ß

a i P 2 1 - a i - ß 2

a 2

2 0 a 2

2 1 - c

(10)

With

? P i P i k ( i ) l

= P 'k' (11)

the t o t a l area of d i f f e r e n t t e r r a c e s i s obtained from the l e f t e i g e n v e c t o r of the c o r r e l a t i o n m a t r i x t o the eigenvalue 1. The p r o b a b i l i t i e s p being the coverages of the corresponding t e r r a c e s , are

P l = P 3 = 2 ( a } + a 2) P 2 - P, = 2 ( a 1 + a 2) (12)

I n t h i s model the d i f f e r e n c e i n the coverage between the two r o t a t i o n a l domains depends s o l e l y on the r e l a t i o n between the p r o b a b i l i t i e s a\ and ct 2

f o r the steps up and down. The double step p r o b a b i l i t y only i n f l u e n c e s the s i z e d i s t r i b u t i o n f o r the d i f f e r e n t t e r r a c e s .

I n the 1-D case the p a i r - c o r r e l a t i o n f o r a l l d i s t a n c e s are e a s i l y c a l c u l a t e d by powers of the m a t r i x P ( l ) . (See a l s o a recent review a r t i c l e by H. J a g o d z i n s k i [30].) The beam p r o f i l e as the F o u r i e r transform of the p a i r c o r r e l a t i o n s i s given by

1 8 6

Ä - i ( k - k ' ) j a I ( k , k ' ) = G ( k ) E P i p i k ( j ) F . F k e . (13)

j With

P i k - p ( D J k <>*)

and

p j = i T ^ U , (15)

the d i f f r a c t e d i n t e n s i t y can be w r i t t e n i n the form

4 4 4 -N,N Ä _ . - i ( k - k ' ) j a T.F.TJ^U . X Je

I k i r r k r I ( k , k ' ) = G(k) Z Z Z Z F . F ^ Ü . X i , x i e ' 0 6 )

i = l k=l r=l j

D e f i n i n g q u a n t i t i e s B r ( k , k ' ) and D r(k,k')

Z Z l l T 1 ! ! ^.(k.k^Ffck.k 1) = B (k,k') + iD ( k , k f ) (17) . , . . i r r k l — — k r r i= 1 k=1 the beam p r o f i l e i s f i n a l l y g i ven by

l 4 1 - |X | 2

I ( k , k ' ) = Z B (k,k') ; -r r 1 - 21 X | c o s ( ( k - k ' ) a + <j> )+|X | 2

|X | s i n ( ( k - k f ) a + <\> ) - 2D (k,k') r r

r > ' 1 - 2 | X r | c o s ( ( k - k , ) a +* r) + |X r| 2

(18)

The eigenvalues of the m a t r i x P ( l ) are i n general complex i<j>

X = |X |e r . (19) r 1 r 1

The phase <J>r of an eigenvalue d e s c r i b e s the p o s i t i o n of the maximum of the beam p r o f i l e and the modulus |X r| the wi d t h .

The f i r s t term i n Eq. 18 d e s c r i b e s a Symmetrie p r o f i l e ( i n general the main c o n t r i b u t i o n ) and the second term d e s c r i b e s an asymmetric c o r r e c t i o n . The o r i g i n of the asymmetric c o r r e c t i o n may be t w o - f o l d . For a Symmetrie d i s t r i b u t i o n of up and down steps the kine m a t i c beam p r o f i l e i s always Symmetrie and the q u a n t i t i e s D r(k,k') v a n i s h . This has been denoted a l s o by the term " r e v e r s i b l e " s u r f a c e [ 7 , 8 ] . For a v i c i n a l plane the p r o b a b i l i t i e s of up and down steps are not equal i n both d i r e c t i o n s ( " i r r e v e r s i b l e s u r f a c e " ) and the beam p r o f i l e s become asymmetric. The second o r i g i n f o r an asymmetry of beam p r o f i l e s i s m u l t i p l e s c a t t e r i n g at the edges. Asymmetric p r o f i l e s w i l l not be d i s c u s s e d here.

The q u a n t i t i e s B r ( k , k ' ) and D r(k,k') are l i n e a r combinations of the s c a t t e r i n g amplitudes and may be a l s o denoted as s t r u c t u r e f a c t o r s i n the f o l l o w i n g . I t i s i n general convenient to c a l c u l a t e the s t r u c t u r e f a c t o r s n u m e r i c a l l y by the ei g e n v e c t o r s of the m a t r i x P ( l ) aecording to Eq. 17. In simple cases, such as the t w o - l e v e l System d i s c u s s e d above, an a n a l y t i c e x p r e s s i o n showing the r e l a t i o n t o the coverage 0 i s u s e f u l [4-8] but f o r the s p e c i a l case d i s c u s s e d here w i t h f o u r eigenvalues and f o u r d i f f e r e n t

187

s t r u c t u r e f a c t o r s the e x p r e s s i o n of the s t r u c t u r e f a c t o r s i n terms of the coverages becomes unhandy. However, the approximate a n a l y t i c e x p r e s s i o n of the s t r u c t u r e f a c t o r s and the eigenvalues of the m a t r i x P ( l ) can be found f o r s p e c i a l d i f f r a c t i o n c o n d i t i o n s . I n s p e c t i o n of the eigenvalues A r and s t r u c t u r e f a c t o r s B r ( k , k ' ) at the " i n phase" (cos(k - k' )d_ = 1) and "out of phase" (cos(k - k')d = -1) d i f f r a c t i o n c o n d i t i o n f o r s i n g l e steps shows the V a r i a t i o n of beam i n t e n s i t i e s and p r o f i l e s i n r e c i p r o c a l space. The f o u r eigenvalues of the (4 x 4) m a t r i x are:

S • 1

X 3 - 1 - o, - 26 ( 2 0>

\ ~ ' " ° 2

The s t r u c t u r e f a c t o r s to the f i r s t two eigenvalues at i n phase d i f f r a c t i o n from a l l l e v e l s are

B j C k . k 1 ) = |Zp.F.(k,k')| 2

i

B 2(k,k') * P 1 P 2 | F 1 ( k , k ' ) ~ F 2 ( k , k ' ) | 2 .

The other two s t r u c t u r e f a c t o r s t u r n out to be very s m a l l , because J F i | = |F 3 j and |F 2| = | j - Therefore the c o n t r i b u t i o n s from A 3 and A 4 become very weak and t'he beam p r o f i l e c o n s i s t s of two p a r t s , a c e n t r a l s p i k e and a broad p r o f i l e . The i n t e n s i t y of the c e n t r a l s p i k e i s p r o p o r t i o n a l to the average s t r u c t u r e f a c t o r and the d i f f u s e p a r t i s due to the mean Square d e v i a t i o n s , as d i s c u s s e d above. At the out of phase c o n d i t i o n f o r s i n g l e steps the beam p r o f i l e c o n s i s t s a l s o of two p a r t s w i t h the r e v e r s e r e l a t i o n between the i n t e n s i t i e s . At the out of phase c o n d i t i o n f o r double s t e p s , cos(k - k 1 ) d = 0, the c e n t r a l s p i k e and" the broad p r o f i l e f o r A 2 v a n i s h and the two broadened p r o f i l e s from the two eigenvalues A 3 and A 4 become i n t e n s i v e . The r e l a t i v e i n t e n s i t i e s can be c a l c u l a t e d from Eq. 17, the a n a l y t i c e x p r e s s i o n being not very i n s t r u c t i v e i s not g i v e n here

The q u a l i t a t i v e p i c t u r e f o r i n c i d e n c e along [011] i s given i n F i g u r e 6. The beam p r o f i l e s show a double p e r i o d i c i t y i n r e c i p r o c a l space corresponding to the double step h e i g h t . At the " i n phase" c o n d i t i o n f o r a s i n g l e step a l l t e r r a c e s s c a t t e r i n phase and the width of the d i f f u s e i n t e n s i t y shows the s i n g l e t e r r a c e w i d t h . The d i f f u s e i n t e n s i t y a r i s e s from the d i f f e r e n t s c a t t e r i n g f a c t o r s of the two r o t a t i o n a l domains. The same occurs f o r the "out of phase" c o n d i t i o n f o r a s i n g l e s t e p . The t e r r a c e s 1 and 3, or 2 and 4 r e s p e c t i v e l y , are then i n phase and the d i f f u s e i n t e n s i t y again shows the average t e r r a c e w i d t h . In between the d i f f u s e i n t e n s i t y c o n s i s t s of a s u p e r p o s i t i o n of two L o r e n t z i a n p r o f i l e s , the narrower one showing the average width of the two t e r r a c e s , and the other one showing an a d d i t i o n a l broadening from which the p r o b a b i l i t y ß f o r double steps can be o b t a i n e d .

The beam p r o f i l e s f o r a s u r f a c e having o n l y s i n g l e steps i s shown f o r comparison i n F i g u r e 7. Assuming only the one p r o b a b i l i t y a f o r a s i n g l e s t e p , a l l t e r r a c e s occur w i t h equal p r o b a b i l i t y . The width of the d i f f u s e i n t e n s i t y shows a l s o a double p e r i o d i c i t y i n r e c i p r o c a l space, that i s because there are f o u r d i f f e r e n t t e r r a c e s at the s u r f a c e . The double p e r i o d i c i t y of the beam widths has been e x p e r i m e n t a l l y observed at a s u r f a c e where s i n g l e steps were o b v i o u s l y predominant [17]. The c e n t r a l s p i k e does not v a n i s h at the "out of phase" c o n d i t i o n f o r s i n g l e s t e p s because of the d i f f e r e n t s t r u c t u r e f a c t o r s f o r the two r o t a t i o n a l domains. At (001), (003) e t c . (s = 1/4, 3/4 e t c . ) the d i f f u s e i n t e n s i t y becomes narrower because then the average repeat d i s t a n c e i s twice the average t e r r a c e s i z e . The c e n t r a l spike vanishes at a l l d i f f r a c t i o n c o n d i t i o n s

1 8 8

F i g . 6. Schematic drawing of the V a r i a t i o n of the angular p r o f i l e s of the s p e c u l a r beam f o r a Si( 1 0 0 ) s u r f a c e e x h i b i t i n g s i n g l e and double s t e p s . A geometrical d i s t r i b u t i o n of t e r r a c e s i z e s has been assumed w i t h equal p r o b a b i l i t y a f o r s i n g l e steps up or down, and fB f o r double s t e p s . The " i n phase" c o n d i t i o n f o r double steps i s at (002), (004), w h i l e (002) i s the "out of phase" c o n d i t i o n f o r s i n g l e s t e p s . The eigenvalue X of the c o r r e l a t i o n m a t r i x i n d i c a t e d i n the F i g u r e determines the widths of the d i f f u s e i n t e n s i t i e s . The i n c i d e n t beam has been assumed along the [011] d i r e c t i o n .

except at the " i n phase" and "out of phase" c o n d i t i o n f o r s i n g l e steps because a raulti-level System has been assumed by a l l o w i n g up and down steps at a l l l e v e l s .

CONCLUSIONS

I t has been shown that at su r f a c e s w i t h non-equivalent t e r r a c e s the i n t e n s i t y o s c i l l a t i o n s i n r e c i p r o c a l Space and as a f u n c t i o n of coverage may d i f f e r s u b s t a n t i a l l y from the simple case where only antiphase domains e x i s t . For a q u a n t i t a t i v e e v a l u a t i o n of beam p r o f i l e s m u l t i p l e s c a t t e r i n g e f f e c t s cannot be neg l e c t e d and a l s o the o r i e n t a t i o n of the i n c i d e n t beam has to be co n s i d e r e d . For a S i ( 1 0 0 ) s u r f a c e w i t h f o u r d i f f e r e n t t e r r a c e s and a c e r t a i n model f o r double steps the beam p r o f i l e s have been t h e o r e t i c a l l y d e s c r i b e d . The beam p r o f i l e e x h i b i t s at the " i n phase" and "out of phase" c o n d i t i o n f o r s i n g l e steps a d i f f u s e i n t e n s i t y w i t h a width a c c o r d i n g t o the s i n g l e step p r o b a b i l i t y . I f double steps are present at the surface w i t h a c o n s i d e r a b l e p r o b a b i l i t y the corresponding broadening of the beam becomes v i s i b l e at a momentum t r a n s f e r normal t o the surfa c e of s = 1/4, 3/4 i n r e c i p r o c a l u n i t s , which the "out of phase" c o n d i t i o n f o r double s t e p s . The p r o f i l e c o n s i s t s then of a s u p e r p o s i t i o n of two L o r e n t z i a n p r o f i l e s w i t h d i f f e r e n t w i d t h s . The p r o b a b i l i t i e s f o r double

1 8 9

Ak„ F i g . 7. Same as F i g u r e 6 f o r a s u r f a c e having only

s i n g l e s t e p s .

steps can be determined i f both components can be separated. The e x i s t e n c e of two d i f f u s e components should be i n d i c a t e d by d e v i a t i o n s from the L o r e n t z i a n shape of the p r o f i l e . This should be a l s o observable at i n c i d e n c e i n C0103 d i r e c t i o n where the s t r u c t u r e f a c t o r s f o r the two r o t a t i o n a l domains become e q u a l . The beam p r o f i l e would then a l s o d e v i a t e from the L o r e n t z i a n shape.

Acknowledgement

The author would l i k e to thank P r o f . M. G. L a g a l l y , Dr. D. Saloner, J . M a r t i n and D. Savage f o r s t i m u l a t i n g d i s c u s s i o n s . Part of t h i s work was completed at the Dept. of M e t a l l u r g i c a l E n g i n e e r i n g , Univ. Wisconsin-Madison w i t h support from ONR, Chemistry Program. F i n a n c i a l support of the Deutsche Forschungsgemeinschaft (SFB 128) i s a l s o g r a t e f u l l y acknowledged.

REFERENCES

1. M. He n z l e r , Surface S e i . , 22:12 (1979). 2. M. Henzler, E l e c t r o n d i f f r a c t i o n and su r f a c e d e f e c t s t r u c t u r e , i n :

" E l e c t r o n Spectroscopy f o r Surface A n a l y s i s " , H. Ibach, ed., Spr i n g e r V e r l a g , Berlin-Heidelberg-New York (1977).

3. H. J a g o d z i n s k i , W. M o r i t z and D. Wolf, Surface S e i . , 77:233 (1978). W. M o r i t z , H. J a g o d z i n s k i and D. Wolf, Surface S e i . , 77:249 (1979). D. Wolf, H. J a g o d z i n s k i and W. M o r i t z , Surface S e i . , 77:265 (1979). D. Wolf, H. J a g o d z i n s k i and W. M o r i t z , Surface S e i . , 77:283 (1979).

4. T. M. Lu and M. G. L a g a l l y , Surface S e i . , 120:47 (1982). 5. T. M. Lu, G.-C. Wang and M. G. L a g a l l y , Surface S e i . , 108:494 (1981). 6. J . M. Pimbley and T. M. Lu, J . A p p l . Phys., 55:182 (1984).

J . M. Pimbley and T. M. Lu, J . A p p l . Phys., 57:1121 (1985). J . M. Pimbley and T. M. Lu, J . A p p l . Phys., 58:2184 (1985).

1 9 0

7. C. S. Lent and P. I . Cohen, Surface S e i . , 139:121 (1984). 8. P. R. P u k i t e , C. S. Lent and P. I . Cohen, Surface S e i . , 161:39

(1985) . 9. M. G. L a g a l l y , D. E. Savage and M. C. T r i n g i d e s , t h i s volume, p. 139.

10. J . Zhang, J . H. Neave, P. J . Dobson and B. A. Joyce, A p p l . Phys., A42:317-326 (1987).

11. T. Sakamoto, N. J . Kawai, N. J . Kagakawa, K. Ohta and T. Kojima, A p p l . Phys. L e t t . , 47:617-619 (1985).

T. Sakamoto, T. Kawamura and G. H a s h i g u c h i , A p p l . Phys. L e t t . , 48 (1986) .

12. J . A a r t s , W. M. G e r i t s and P. K. Larsen, A p p l . Phys. L e t t . , 48:931 (1986).

13. B. A. Joyce, J . H. Neave, J . Zhang and P. J . Dobson, t h i s volume, p. 397.

14. J . M. Cowley, " D i f f r a c t i o n P h y s i c s " , E l s e v i e r Science P u b l . , Amsterdam-New York (1981).

15. W. M o r i t z , I n s t . Phys. Conf. Ser., 41:261 (1978). 16. M. H e n z l e r , t h i s volume, p. 193. 17. J . A. M a r t i n , C. E. Aumann, D. E. Savage, M. C. T r i n g i d e s , M. G.

L a g a l l y , W. M o r i t z and F. Kretschmar, J . Vac. S e i . Technol. (1986).

18. M. Horn, U. G o t t e r and M. H e n z l e r , t h i s volume, p. 463. 19. T. Kawamura, T. Sakamoto and K. Ohto, Surface S e i . , 171:L409-L414

(1986). 20. D. S a l o n e r , J . A. M a r t i n , M. C. T r i n g i d e s , D. E. Savage, C. A. Aumann

and M. G. L a g a l l y , J . A p p l . Phys., 61:2884 (1987). 21. H. S a a l f e l d , S. Tougaard, K. Bolwin and M. Neumann, Surface S e i .

(1986) . 22. T. Sakamoto, N. J . Nagakawa, K. Ohta and T. Kojima, A p p l . Phys.

L e t t . , 47:617-619 (1985). 23. S. B. Hendricks and E. T e l l e r , J . Chem. Phys., 10:147 (1942). 24. H. J a g o d z i n s k i , Acta C r y s t . , 2:201, 208 and 298 (1949). 25. J . K a k i n o k i and Y. Kamura, J . Phys. Soc. Japan, 9:169, 176 (1954). 26. M. Henzler and J . Clabes, Proc. 2nd I n t e r n . Conf. on S o l i d Surfaces

1974, Japan. J . A p p l . Phys., Suppl. 2, P t . 2:389-396 (1974). 27. R. Kaplan, Surface S e i . , 93:145-158 (1980). 28. N. Inoue, Y. T a n i s h i r o and K. Y a g i , Japan. J . A p p l . Phys., 26:L293

(1987) . 29. T. Nakayama, Y. T a n i s h i r o and K. Takanayagi, Japan. J . A p p l . Phys.,

26:L280 (1987). 30. D. E. Aspnes and J . Ihm, Phys. Rev. L e t t . , 57:3054-3057 (1986). 31. H. J a g o d z i n s k i , Prog. C r y s t a l Growth and Charact., 14:47-102 (1987).

191

INDEX

Adatom m i g r a t i o n d u r i n g growth, 409 Aluminium ar s e n i d e

growth by MBE u s i n g phase l o c k modulated beams, 419

Antimony grown on G a A s ( l l O ) , 482

Atomic steps e f f e c t on r o c k i n g curves, 511, 516 ki n e m a t i c d i f f r a c t i o n , 428 RHEED, 427, 504

Auger e l e c t r o n e m i s s i o n , 109, 117

Beam - Specimen i n t e r a c t i o n , 325

C h a n n e l l i n g e f f e c t s , 117 i n R e f l e c t i o n E l e c t r o n Microscopy,

274 Column approximation and REM, 268 Convergent Beam D i f f r a c t i o n

s u r f a c e c h a r a c t e r i s a t i o n , 237 s u r f a c e m u l t i s l i c e approach, 251

Copper o x i d a t i o n , 320

De f e c t s and D i s o r d e r , 139, 201, 217 m u l t i l e v e l Systems, 157 s t e p s , 175, 193, 217 su r f a c e roughness and 3-D

s t r u c t u r e , 160 two l e v e l Systems, 146 v i c i n a l s u r f a c e s , 154

Depth of focus i n REM, 269 D i f f u s e s c a t t e r i n g , 177

thermal d i f f u s e s c a t t e r i n g , 211 D i s l o c a t i o n s

c o n t r a s t i n REM, 329, 332 resonance c o n d i t i o n , 338

D i s o r d e r , 33 i n S i ( 1 0 0 ) ( 2 x n ) , 533

Dynamical Theory i n RHEED, 29, 43, 31

resonances, 100

E l e c t r o n Microscopy of Surfaces w i t h Low Energy E l e c t r o n s , 381, 385 r e f l e c t i o n , 77, 261, 285, 303, 317,

329, 343

G a l l i u m A r s e n i d e growth by MBE, 397 growth us i n g phase-lock modulated

beams, 419 growth of Antimony on, 482 (O O l ) s u r f a c e , 49, 204, 318, 508 ( H O ) s u r f a c e , 72, 276, 390, 530 r e f l e c t i o n microscopy, 84, 318 thermal a n n e a l i n g , 530

Germanium (O O l ) s u r f a c e , 217 RHEED from growing s u r f a c e , 449

Gold s u r f a c e s s t u d i e d by REM, 303, 315

Growth k i n e t i c s , 163, 523 annealed G a A s ( l l O ) , 530 d i s o r d e r of S i ( 1 0 0 ) ( 2 x n ) , 533 oxygen p ( 2 x l ) o v e r l a y e r s on

W(110), 527 Growth of metals on G a A s ( l l O ) , 475 Growth modes, 362, 493 Growth Processes, 407. 456, 476

e f f e c t of I n t e r r u p t i o n , 414

Image formation i n REM, 268, 285, 296

Indium on G a A s ( l l O ) , 482 I n e l a s t i c s c a t t e r i n g i n RHEED

phonons, 38 Plasmons, 39 resonance c o n d i t i o n s , 277

I n t e r r u p t e d Growth, 414 Ion bombardment damage, 217

K i k u c h i e f f e c t s , 201, 403, 437

Low Energy E l e c t r o n D i f f r a c t i o n , i n t e n s i t y o s c i l l a t i o n s , 466 S i MBE on S i ( 0 0 1 ) , 463 spot p r o f i l e , 467

Low Energy E l e c t r o n R e f l e c t i o n Microscope, 381

Low Energy E l e c t r o n Scanning Microscope, 385

5 3 9

M e t a l E p i t a x y , O v e r l a y e r s e t c . . phase t r a n s f o r m a t i o n s , 11 s u p e r s t r u c t u r e s , 11 A g / S i , 15, 17 A l / S i , 351 A u / S i , 9, 11, 16, 18, 349 Cu/Nb, 489 Cu/Ni, 489 Cu/W, 489 In/Ge, 16, 17 In/GaAs, 482 Mo/W, 489 Nb/W, 489 N i / S i , 17 Ni/Mo, 489 Ni/W, 489 Sb/GaAs, 482 Sn/Ge, 19, 17

Microprobe RHEED, 343 M o l e c u l a r Beam E p i t a x y

AlGaAs, 412, 443 GaAs, 201, 397, 419, 435 Ge, 443, 449 m e t a l s , 489 S i , 355, 449, 463 s u r f a c e s t r u c t u r e s d u r i n g growth,

367 Monolayer s c a t t e r i n g , 101, 103 M u l t i l e v e l Systems, 157

N i c k e l d e p o s i t e d onto S i l i c o n , 373

O x i d a t i o n s t u d i e s of copper, 320

P e n e t r a t i o n depth, 80, 81 Phase-lock modulated beams i n MBE,

419 Photoemission

angle r e s o l v e d , of s u r f a c e d u r i n g growth, 452

P l a t i n u m ( l l l ) s u r f a c e , 104, 109, 213, 281 g o l d d e p o s i t e d on P t ( l l l ) , 285 r e f l e c t i o n e l e c t r o n microscopy,

281, 303, 312, 330

Reconstructed s u r f a c e s s t u d i e d w i t h RHEED, 3, 43

R e f l e c t i o n E l e c t r o n M i c r o s c o p y , 77, 261, 285, 303, 317, 329, 343

column appro x i m a t i o n , 268 c o n t r a s t of s t e p s , 329 f a c e t t i n g , 307 image computations, 86 image f o r m a t i o n , 267 i n e l a s t i c s c a t t e r i n g , 277, 287 resonance e f f e c t s , 274

specimen p r e p a r a t i o n , 279 i n STEM, 261, 317 i n TEM, 261 su r f a c e d e f e c t s , 85

R e f l e c t i o n High Energy E l e c t r o n D i f f r a c t i o n (RHEED)

apparatus, 4 atomic s t e p s , e f f e c t o f , 427 Bloch wave f o r m u l a t i o n , 131, 270 d i s o r d e r , 32, 33, 139, 201 dynamical theory, 29, 34, 43, 63,

131 e l a s t i c s c a t t e r i n g , 30 i n e l a s t i c s c a t t e r i n g , 38, 109 i n t e n s i t y o s c i l l a t i o n s , 165, 208,

356, 397 apparent phase s h i f t , 401, 492 damping, 495 d i f f r a c t i o n c o n d i t i o n s 399, 453,

510 dynamical theory, 492 low growth temperature, 450 metal e p i t a x y , 489 phase-lock modulated beams, 419 r e s i d u a l gas e f f e c t s , 496 t h e o r e t i c a l m o d e l l i n g , 501

i n t e g r a l f o r m u l a t i o n , 131 i n v a r i a n t imbedding method, 67 kin e m a t i c theory, TDS, 212 microprobe, 343 m u l t i p l e s c a t t e r i n g , 506 m u l t i - s l i c e method, 271 novel techniques, 6 o s c i l l a t i o n / r o t a t i o n p a t t e r n s , 8 p a t t e r n s , I n t e r p r e t a t i o n , 4 r e c o n s t r u c t e d s u r f a c e s , 3, 43 resonance e f f e c t s , 99 r o c k i n g curves, 52, 63, 104 s t r e a k s , 241 s t r e a k shape, 208 su r f a c e s t r u c t u r e d e t e r m i n a t i o n ,

29 symmetry, 242 thermal d i f f u s e s c a t t e r i n g , 211 three l e v e l System, 517 two l e v e l system, 429 v i c i n a l s u r f a c e s , 432 x-ray spectroscopy, 20

Resonance c o n d i t i o n s i n RHEED, 83, 99

c o n t r a s t of steps i n REM, 329, 331 c o n t r a s t of d i s l o c a t i o n s i n REM,

339 d e f e c t images, 279 dynamical theory, 100, 109, 122 r e f l e c t i o n e l e c t r o n microsopy, 274

Rough su r f a c e s by REM, 319

5 4 0

Scanning microprobe f o r REM, 385 Scanning t r a n s m i s s i o n e l e c t r o n microscope f o r REM, 317

Scanning t u n n e l l i n g microscope, 391 Secondary e l e c t r o n e m i s s i o n , 109, 118 S i l i c i d e f o r m a t i o n observed by UHV

SEM, 371 S i l i c o n ,

growth s t u d i e s by RHEED, 449, 513 by low energy e l e c t r o n microscopy,

381, 388 by r e f l e c t i o n e l e c t r o n microscopy,

303 S u b s t r a t e c l e a n i n g , 354

S i ( 0 0 1 ) , 185 growth, 463, 513 (2xn) phase, 533

S i ( 1 1 0 ) , 389 S i ( l l l ) , 5, 12, 195, 347, 352, 374 S i ( l l l ) + Au ( 5 x 1 ) , 9, 11 S i l v e r ,

(001) RHEED c a l c u l a t i o n , 71 s u r f a c e s by REM, 303, 315

Specimen p r e p a r a t i o n f o r REM, 279, 292, 303, 334

Steps, c o n t r a s t i n REM, 329, 335 e l e c t r o n d i f f r a c t i o n , 193 e l e c t r o n s c a t t e r i n g , 175

S u p e r l a t t i c e s , RHEED o s c i l l a t i o n s d u r i n g growth

o f , 496

Surf a c e d e f e c t s , 85 Surf a c e phase t r a n s f o r m a t i o n s , 11,

225 S u r f a c e r e c o n s t r u c t i o n , 3

REM of the ( l l l ) A u s u r f a c e , 297 Surface roughness, 160 Su r f a c e s t r u c t u r e d e t e r m i n a t i o n , 3,

29, 43 S u r f a c e s u p e r s t r u c t u r e s

metal induced on S i and Ge, 13, 18 Su r f a c e wave resonance, 99, 109, 122

Time dependent changes of RHEED, 163, 208 (see a l s o RHEED I n t e n s i t y O s c i l l a t i o n s )

T o t a l R e f l e c t i o n Angle X-ray Spectroscopy (TRAXS), 22

Two L e v e l Systems, 146

U l t r a h i g h vacuum e l e c t r o n microscope, 289, 371

V i c i n a l S u r f a c e s , 154 d i f f r a c t i o n from, 432 GaAs and Ge, 440

X-ray s p e c t r o s c o p y and RHEED, 20

Bayerische StaatsbibiJothoi

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