S. Y. Tong M. A. Van Hove K. Takayanagi X. D. Xie (Eds.)

The Structure of Surfaces III Proceedings of the 3rd International Conference on the Structure of Surfaces (ICSOS III) Milwaukee, Wisconsin, USA, July 9-12,1990

With 374 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

Contents

Introduction 1

Part I Techniques

LI Theory Microscopic Origins of Stress on Semiconductor Surfaces By R.D. Meade and D. Vanderbilt (With 2 Figures) 4 Energy Density Calculations Applied to GaAs Polar Surfaces By N. Chetty and R.M. Martin (With 1 Figure) 9 First Principles Calculation of Lattice Relaxation at Low Index Surfaces of Ag and Cu By K.P. Bohnen, Th. Rodach, and K.M. Ho 16 Surface Diffusion of Al(l 10) By P. Stoltze and J.K. N0rskov (With 2 Figures) 19 Theory of Submonolayer Alkali-Metal Chemisorption By Dingsheng Wang, Kailai Chen, Shiwu Gao, Ruqian Wu, and Ning Wang (With 9 Figures) 24 Downward Funneling Model of Low-Temperature Epitaxial Growth: A Hybrid Molecular Dynamics-Monte Carlo Study By D.E. Sanders and J.W. Evans (With 4 Figures) 38 Mapping of Crystal Growth onto the 6-Vertex Model By C. Garrod, M. Kotria, A.C. Levi, and M. Touzani (With 4 Figures) . . 44 Characterizing the Evolution of Non-equilibrium Structure Düring Adsorption By J.W. Evans (With 1 Figure) 49 Dependence of Cluster Diffusivity upon Cluster Structure By H.C. Kang, P.A. Thiel, and J.W. Evans (With 3 Figures) 55

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Interaction of Heavy Rare-Gas Atoms with Metal Surfaces: A Model Based on the Effective Medium Theory By M. Karimi, D. IIa, I. Dahns, and G. Vidali 60

1.2 Angle-Resolved Electron Diffraction Angular Diffraction Patterns of Photoelectrons and Auger Electrons from Single-Crystal Cu(lll) By X.-D. Wang, Y. Chen, Z.-L. Han, S.Y. Tong, and B.P. Tonner (With 3 Figures) 65 Surface Termination of Binary Epitaxial Compounds by High Angular Resolution X-Ray Photoelectron Diffraction By S.A. Chambers (With 3 Figures) 72 Multiple-Scattering Effects in Auger Electron Diffraction and Photoelectron Diffraction: Theory and Applications By A.P. Kaduwela, DJ. Friedman, YJ. Kim, T.T. Tran, G.S. Herman, CS. Fadley, JJ . Rehr, J. Osterwalder, H.A. Aebischer, and A. Stuck (With 4 Figures) 77 Inelastic Photoelectron Diffraction By G.S. Herman, A.P. Kaduwela, T.T. Tran, YJ. Kim, S. Lewis, and CS. Fadley (With 2 Figures) 85 Electron Diffraction from Al(001) By J. Osterwalder, T. Greber, S. Hüfner, H.A. Aebischer, and L. Schlapbach (With 2 Figures) 91 Photoelectron Diffraction Patterns by Display-Type Spherical Mirror Analyzer By H. Daimon, Y. Tezuka, N. Kanada, A. Otaka, S.K. Lee, S. Ino, H. Namba, and H. Kuroda (With 5 Figures) 96 Energy-Angle Multidetection-Type Electron Spectrometer for X-Ray Photoelectron Diffraction Studies By S. Kanayama, S. Teramoto, M. Owari, and Y. Nihei (With 4 Figures) 102 3D Images of Surface Structure from Photoelectron Holography By JJ . Barton and LJ. Terminello (With 2 Figures) 107 1.3 Low-Energy Electron Diffraction New Developments in the Theory of LEED By PJ. Rous (With 2 Figures) 118 Progress in Automatic Structure Refinement with LEED By W. Moritz, H. Over, G. Kleinle, and G. Ertl (With 3 Figures) 128 x

Application of Direct Methods in LEED: Multilayer Relaxation, Adsorption Sites and Adsorbate Induced Reconstruction By K. Heinz, W. Oed, and J.B. Pendry (With 1 Figure) 139 Real-Space Multiple Scattering Theory Calculations of LEED Intensities for Stepped Surfaces By X.-G. Zhang, P.J. Rous, J.M. MacLaren, A. Gonis, M.A. Van Hove, and G.A. Somorjai (With 3 Figures) 144 A Simple Scheme for LEED Intensity Calculations By R.F. Lin, P.M. Marcus, and F. Jona (With 2 Figures) 150 A Video-LEED System Based on a Personal Computer and Frame-Grabber By D.L. Adams, S.P. Andersen, and J. Buchhardt (With 3 Figures) . . . . 156 1.4 Other Electron-Based Methods One-Beam RHEED for Surface Structure Analysis By A. Ichimiya (With 3 Figures) 162 Direct Observation of Surface Structures by a High Resolution UHV-SEM By S. Ino, A. Endo, and H. Daimon (With 3 Figures) 168 High Resolution Electron Energy Loss Spectroscopy as a New Tool to Determine the Structure of a Surface Reconstructed System By J. Szeftel, J. Colin de Verdifere, G.S. Dong, and D. Muller (With 3 Figures) 174 Auger Photoelectron Coincidence Spectroscopy of Cobalt By S.M. Thurgate (With 5 Figures) 179 1.5 Atomic and Molecular Scattering Exact Theory of One-Phonon Amplitudes in Diffractive Atom-Surface Scattering By M.E. Flatt6 185 Metastable Spin-Polarized He as a Probe of Surface Magnetism on NiO(lOO) By A. Swan, W. Franzen, M. El-Batanouny, and K.M. Martini (With 3 Figures) 190 Scattering of O2 from the Ag(110) Surface By X.Y. Shen, K.Q. Zhang, and Y.L. Tan (With 2 Figures) 197 1.6 STM and AFM hnaging Adsorbates on Metals by Scanning Tunneling Microscopy By S. Chiang, D.D. Chambliss, V.M. Hallmark, R.J. Wilson, and Ch. Wöll (With 9 Figures) 204

XI

Scanning Tunneling Microscopy of Organic Conductors By A. Kawazu, N. Ära, and M. Yoshimura (With 12 Figures) 214 Restructuring and Symmetry of Surfaces: \/3x\/3-Ag on Si(l 11)7x7 By K. Takayanagi (With 9 Figures) 227 Atomic Resolution Imaging of Glycine Molecules on Graphite in Water By W.S. Yang, Wenjun Sun, Jianxun Mou, and Junjue Yan (With 4 Figures) 237 Theory of Atomic Force Microscopy on Elastic Surfaces By W. Zhong, G. Overney and D. Tomänek (With 3 Figures) 243 Part II Clean Metals H l Elemental Metals Microscopic Kinetics of the (1 x2) Missing Row Reconstruction of the Au(llO) Surface By L.D. Roelofs and E.I. Martir (With 2 Figures) 248 Surface Structural Analysis of the (1 x2) and (1 x3) Phases of Pt{110} by Time-of-Flight Scattering and Recoiling Spectrometry By F. Masson and J.W. Rabalais (With 5 Figures) 253 Analysis of the Reconstructed Ir{l 10} Surface from Time-of-Flight Scattering and Recoiling Spectrometry By M. Shi, H. Bu, and J.W. Rabalais (With 3 Figures) 259 Thermally Activated Defects on Crystal Surfaces By B. Salanon, HJ. Ernst, F. Fabre, L. Barbier, and J. Lapujoulade (With 3 Figures) 264 The Surface Barrier Structure of Metals from Barrier-Induced Surface States and Resonances By A.S. Christopoulos and M.N. Read (With 1 Figure) 271 Ü.2 Alloys Anomalous Surface Phase Formation on Pt3Sn(110) By A.N. Haner, P.N. Ross, and U. Bardi (With 3 Figures) 276 Surface Relaxation of Fe72Cr2s(110) By DJ. O'Connor and C. Xu (With 2 Figures) 282 The Structure of CuPd(110)(2xl): A Case of Underlayer Ordering By M. Lindroos, CJ. Barnes, M. Bowker, and D.A. King (With 3 Figures) 287 Effect of Segregation on Angular Distributions of Cu and Pt from a CuPt Alloy Sputtered by Low-Energy Ar Ions By L.-P. Zheng, R.-S. Li, M.-Y. Li, and W.-Z. Shen (With 1 Figure) . . . 293 XII

Studies of Surface and Interface Structure of Ni3Al Alloy by AP-FIM By D.G. Ren (With 7 Figures) 298 Simulation of the FesoB2o Metallic Glass Surface By W. Kowbel, P. Tlomak, and W.E. Brower, Jr. (With 2 Figures) 303 Part HI Adsorption on Metals ni.l Metallic Adsorption and Growth Energetics in Structure Transformation of Ir Clusters and Atomic Reconstruction of Ir Surfaces By Chong-lin Chen and Tien T. Tsong (With 5 Figures) 312 Observations of Cluster Structures on fcc (111) By S.C. Wang and G. Ehrlich (With 7 Figures) 317 Structure of Metal Overlayers by Low Energy Alkali Ion Scattering: Cu/Ru(0001) and Sn/Pt(lll) By S.H. Overbury, D.R. Mullins, M.T. Paffett, and B. Koel (With 4 Figures) 323 Growth and Structure of Ordered Thin Films of Cu on Pd{001} By H. Li, S.C. Wu, D. Tian, J. Quinn, Y.S. Li, F. Jona, and P.M. Marcus (With 3 Figures) 328 Electric and Magnetic Hyperfine Fields at Ni(lll) Surfaces and Ni/In Interface Compound Formation By R. Platzer, X.L. Ding, R. Fink, G. Krausch, B. Luckscheiter, J. Voigt, U. Wöhrmann, and G. Schatz (With 4 Figures) 333 Structure and Surface Alloy Formation of the Fe(100)/Pt Overlayer System Studied with HEIS By G.W.R. Leibbrandt, R. van Wijk, and F.H.P.M. Habraken (With 4 Figures) 339 Initial Growth and Structure of Pt on Ni(100) By S. Deckers, W.F. van der Weg, and F.H.P.M. Habraken (With 3 Figures) 345 Structural Studies of Mercury Overlayers on Cu(001) by Atom Beam Scattering and LEED By Wei Li, J.-S. Lin, M. Karimi, C. Moses, P.A. Dowben, and G. Vidali (With 4 Figures) 350 Yb on Al(001): A Two-Dimensional Disordered System By T. Greber, J. Osterwalder, and L. Schlapbach (With 3 Figures) 355 Order-Disorder Transition of 6 ML Fe Films on the Cu(100) Surface By C. Stuhlmann, U. Beckers, J. Thomassen, M. Wuttig, H. Ibach, and G. Schmidt (With 3 Figures) 360

XIII

The Growth of Au on Ag(l 10): From Bilayer to Multilayer By P. Fenter and T. Gustafsson (With 4 Figures) 366 Structure of Au and Pt Films on Pd(110): Deposition Temperature and Coverage Dependent (1 x2) and (1 x3) Reconstructions By PJ. Schmitz, H.C Kang, W.-Y. Leung, and P.A. Thiel (With 3 Figures) 372 Faceting Induced by Ultrathin Metal Films: Pt and Au on W(ll 1) By K.J. Song, C.Z. Dong, and T.E. Madey (With 3 Figures) . 378 Core Level Photoemission for the Study of Metallic Interfaces and Two-Dimensional Compounds By J.N. Andersen, A. Nilsson, O. Björneholm, and N. Märtensson (With 1 Figure) 383 Temperature Dependence of the Work Function of Alkali-Metal Atoms Adsorbed on a Metal Surface By T. Kato and M. Nakayama (With 6 Figures) 388 m.2 Non-metallic Adsorption A LEED Study of the Clean and Hydrogen-Covered W(310)-(lxl) Surfaces By D.L. Adams and S.P. Andersen (With 3 Figures) 395 Coverage Dependent Hydrogen Induced Restructuring of Rh(110) By K. Heinz, W. Nichtl-Pecher, W. Oed, H. Landskron, M. Michl, and K. Müller (With 2 Figures) 401 Atomic Beam Diffraction Studies of the Ordered Hydrogen Chemisorption Phases on Rh(110) By G. Parschau, E. Kirsten, and K.H. Rieder (With 3 Figures) 406 Structural Studies of Highly Corrugated Systems with Atomic Beam Diffraction: Hydrogen-Chemisorption on Missing-Row Reconstructed Pt(110) and Ir(110) By E. Kirsten and K.H. Rieder (With 3 Figures) 411 Temperature and Coverage Dependence of the Structure and Dynamics of Hydrogen on Pd(l 11) By C.-H. Hsu, B.E. Larson, M. El-Batanouny, CR. Willis, and K.M. Martini (With 3 Figures) 416 Hydrogen Titration Studies of fcc Cobalt Layers on Cu(001) By M.T. Kief, G.J. Mankey, and R.F. Willis (With 3 Figures) 422 Disordered Adsorption of O and S on Ni(100) Studied by Diffuse LEED By U. Starke, W. Oed, P. Bayer, F. Bothe, G. Fürst, PL. de Andres, K. Heinz, and J.B. Pendry (With 3 Figures) 427 XIV

A Directed LEED Search for Many Structural Parameters: Substrate Relaxations and Bückling in Mo(100)-c(2x2)-C and -S By PJ. Rous, D. Jentz, D.G. Kelly, R.Q. Hwang, M.A. Van Hove, and G.A. Somorjai (With 1 Figure) 432 Lattice Gas Models for Order-Disorder Behaviour of Oxygen on Ruthenium(OOl) By K. De'Bell, H. Pfnür, and P. Piercy (With 2 Figures) 437 Phase Separation of the O/W(110)p(2xl)+p(2x2) System By M.C. Tringides (With 2 Figures) 442 STM Studies of the Initial Stage of Oxygen Interaction with Ni(100) By G. Wilhelmi, A. Brodde, D. Badt, H. Wengelnik, and H. Neddermeyer (With 3 Figures) 448 0 2 Induced (1 x3)->(l x l ) Phase Change of an Ir{110} Surface from TOF-SARS By H. Bu, M. Shi, and J.W. Rabalais (With 4 Figures) 456 Oxygen Induced Reconstruction of Cu(llO) and Cu(100) Studied by STM By F. Jensen, F. Besenbacher, E. Laegsgaard, and I. Stensgaard (With 4 Figures) 462 Geometrical Structure of Molecular Adsorbates from Core-Level Binding Energies By A. Nilsson, H. Antonsson, A. Sandell, and N. Märtensson (With 3 Figures) 467 Lateral Surface Stark Effect in Chemisorbed Molecules: CO on Metal Surfaces By B. Gumhalter, K. Hermann, and K. Wandelt (With 2 Figures) 473

Part IV Clean Semiconductors IV.l Elemental Semiconductors Deposition and Annealing of Silicon on Cleaved Silicon Surfaces Studied by Scanning Tunneling Microscopy By R.M. Feenstra and M.A. Lutz (With 3 Figures) 480 Simulation and STM Studies of Equilibrium Properties of Vicinal Surfaces By T.L. Einstein, N.C. Bartelt, J.L. Goldberg, X.-S. Wang, E.D. Williams, and B. Joos (With 3 Figures) 486 The Precipitation of Kinks on Stepped Si(l 11) Surfaces By Jian Wei, N.C. Bartelt, and E.D. Williams (With 4 Figures) 492

xv

Disordering of the (3x1) Reconstruction of Si(l 13): Realization of the Chiral Three-State Potts Model By Y.-N. Yang, N.C. Bartelt, T.L. Einstein, R.L. Park, and E.D. Williams (With 3 Figures) 497 Current Effects on Clean Si(ll l) and (001) Surfaces Studied by Reflection Electron Microscopy By A. Yamanaka, N. Ohse, H. Kahata, and K. Yagi (With 6 Figures) . . . 502 Morphology and Electron States Along Clean Si(100) Vicinal Surfaces By M. Khial, J.-P. Lacharme, and CA. S<Sbenne (With 2 Figures) 507 2x1 Reconstructions of Si(001) Interfaces: Dimer and Non-dimer Models Derived from Grazing X-Ray Diffraction Data By N. Jedrecy, N. Greiser, M. Sauvage-Simkin, R. Pinchaux, J. Massies, and V.H. Etgens (With 3 Figures) 512 Kinematic Low-Energy Electron Diffraction Study of the Atomic Structure of the Si(001)2xl Surface By R.G. Zhao, Jinfeng Jia, Yanfang Li, and W.S. Yang (With 3 Figures) 517 X-Ray Photoelectron Diffraction Study of a High-Temperature Surface Phase Transition on Ge(ll l) By T.T. Tran, DJ. Friedman, YJ. Kim, G.A. Rizzi, and CS. Fadley (With 3 Figures) 522 Surface Mosaic of Multi-Component Systems By V.L. Avgustimov, D.I. Bidnyk, L.A. Bystraya, and S.P. Kostenko (With 4 Figures) 527 IV.2 Compound Semiconductors Electron-Hole Counting Rule at DI-V Surfaces: Applications to Surface Structure and Passivation By DJ. Chadi (With 4 Figures) 532 The Dimerization of the Reconstructed Clean and Chemisorbed (100) Surfaces of Si and SiC By B.I. Craig and P.V. Smith (With 1 Figure) 545 Towards an Understanding of the Composition of the Reconstructed a SiC(0001) Surface By P. Badziag, M.A. Van Hove, and G.A. Somorjai (With 2 Figures) . . . 550 Structure of the GaAs(-l-l-l) 2x2 and GaAs (-1-1-1) VT9 x >/l9 Surfaces By R.D. Bringans, D.K. Biegelsen, L.-E. Swartz, and J.E. Northrup (With 5 Figures) 555

xvi

A New Reconstruction of the GaP(lll) Surface By X.Y. Hou, X.K. Lu, P.H. Hao, X.M. Ding, P. Chen, and X. Wang (With 4 Figures) 560 Atomic Geometry of the CdS(l 120) Surface By A. Kahn and C.B. Duke (With 2 Figures) 566 Strain Relaxation in Metastable Cubic CdS Epilayers By D.W. Niles and H. Höchst (With 3 Figures) 570 Part V Adsorbates on Semiconductors V.l Metallic Adsorption Overview of Metal/Semiconductor Interfaces By Xie Xide (With 5 Figures) 576 Scanning Tunneling Microscopy Investigation of the K/Si(100)2xl Stepped (4°) Surface By P. Soukiassian and J.A. Kubby (With 5 Figures) 584 Lead Adsorption on Ge(001) and Si(001) Studied by Core-Level Spectroscopy By K. Hricovini, G. Le Lay, A. Kahn, A. Taleb-Ibrahimi, and J.E. Bonnet (With 2 Figures) 589 Location of Ag in Si(lllMV^x V̂ )R30°-Ag from X-Ray Standing Waves By E. Vlieg, E. Fontes, and J.R. Patel (With 2 Figures) 595 Study of (\/3x \/3)R30° Ag on Si(lll) by Photoelectron Diffraction By G.S. Herman, A.P. Kaduwela, DJ. Friedman, M. Yamada, E.L Bullock, CS. Fadley, Th. Lindner, D. Ricken, A.W. Robinson, and A.M. Bradshaw (With 4 Figures) 600 Electronic and Atomic Structure of the Cu/Si(lll)-Quasi-5x5 Overlayer By T.N. Rhodin and D.D. Chambliss (With 3 Figures) 606 Structures Formed by Co-deposition of Two Metals on Si(l 11)7x7 Surfaces Studied by Reflection Electron Microscopy By I. Homma, Y. Tanishiro, and K. Yagi (With 6 Figures) 610 Catalytic Effect of Metals (Sn, Ag, and Pb) on Homoepitaxial Growth of Ge and Si By K. Fukutani, H. Daimon, and S. Ino (With 5 Figures) 615 REM and RHEED Studies of Lead Adsorption on Silicon (111) Surfaces By Y. Tanishiro, M. Fukuyama, and K. Yagi (With 5 Figures) 623

XVII

Electron Accumulation Near the a-Sn/CdTe(lll) Interface Region? By I. Hemändez-Calderön and H. Höchst (With 3 Figures) 628 V.2 Non-metallic Adsorption Successive Oxidation Stages of Adatoms on the Si(l 11)7x7 Surface Observed with Scanning Tunneling Microscopy and Spectroscopy By J.P. Pelz and R.H. Koch (With 3 Figures) 633 Füll Ab Initio Determination of the Structural Parameters of HonGaAs(llO) By CM. Bertoni, F. Finocchi, M. Buongiorno Nardelli, F. Bernardini, and E. Molinari (With 2 Figures) 639 Part VI Oxides Observation of a Corrugated Structure for the TiO2(100) 1 x3 Surface Reconstruction By P. Zschack (With 2 Figures) 646 Simultaneous Epitaxial Growth of Two Differently Oriented Ni Island Structures upon TiO2(110) By Ming-Cheng Wu and P.J. M0ller (With 5 Figures) 652 Characterization of (100) and (111) Surfaces of MgO by Reflection Electron Microscopy By M. Gajdardziska-Josifovska, P.A. Crozier, and J.M. Cowley (With 3 Figures) 660 MgO(lOO) Structural Investigation Using EELFS and EXFAS Techniques By M. De Crescenzi, N. Motta, F. Patella, A. Sgarlata, F. Arciprete, A. Balzarotti, M. Benfatto, and CR. Natoli (With 6 Figures) 665 LEED Studies of Krypton and Nitrogen Monolayers on MgO Single Crystals By T. Angot and J. Suzanne (With 4 Figures) 671 Structural Analysis of a Thin CaO Layer Formed by Electron Bombardment Hearing on CaF2 ( l l l ) by Means of Chemical-State Discriminated XPED By C. Akita, T. Tomioka, M. Owari, A. Mizuike, and Y. Nihei (With 3 Figures) 676 Index of Contributors 681

XVIII

Progress in Automatic Structure Refinement with LEED W. Moritz1, H. Over2, G. Kleinle2, and G. Ertl2 1 Institut für Kristallographie und Mineralogie, Universität München, Theresienstr. 41, W-8000 München 2, Fed. Rep. of Germany

2Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, W-1000 Berlin 33, Fed. Rep. of Germany

I t i s demonstrated t h a t c o n v e n t i o n a l l e a s t Squares o p t i -m i s a t i o n techniques can be s u c c e s s f u l l y u s e d f o r automa-t i c s t r u c t u r e r e f i n e m e n t w i t h LEED. Examples a r e g i v e n f o r two a d s o r b a t e Sys tems , H / N i ( 1 1 0 ) - ( 1 x 2 ) and O / N i ( 1 1 0 ) - ( 2 x 1 ) where r a p i d conve rgence i s r e a c h e d i n a s i m u l t a n e o u s o p t i m i s a t i o n of a l l s t r u c t u r a l p a r a m e t e r s w i t h i n t h e top t h r e e l a y e r s . 1 . I n t r o d u c t i o n The a p p l i c a t i o n of LEED as a Standard t e c h n i q u e f o r s u 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 r e l i e s c r i t i c a l l y on i t s c o n v e n i e n t u s e and i t s c a p a b i l i t y t o s o l v e complex s t r u c -t u r e s . An a u t o m a t i c s t r u c t u r e r e f i n e m e n t t e c h n i q u e , a c l e a r r e c i p e t o l o c a l i s e t h e b e s t f i t model and a s t a n d a -r d i s e d c r i t e r i o n t o j u d g e t h e q u a l i t y of t h e r e s u l t would be most u s e f u l t o make t h e method a p p l i c a b l e by t h e n o n - s p e c i a l i s t . C l e a r l y , t h i s S i t u a t i o n h a s n o t y e t been r e a c h e d and, compared t o X- ray d i f f r a c t i o n , t h e method i s s t i l l l i m i t e d . Recen t d e v e l o p m e n t s , however , show t h a t c o n s i d e r a b l e improvements a r e p o s s i b l e . The c o m p u t a t i o n a l e f f o r t , which i s s t i l l a l i m i t i n g f a c t o r , can be r e d u c e d t o a l a r g e e x t e n t [ 1 - 3 ] . F u r t h e r improvem­e n t s can be made i n t r o d u c i n g o p t i m i s a t i o n t e c h n i q u e s i n t o t h e s t r u c t u r e a n a l y s i s .

S e v e r a l p r o c e d u r e s have been p r o p o s e d f o r an automa­t i c s t r u c t u r e r e f i n e m e n t i n which a m i n i m i s a t i o n of t h e c o n v e n t i o n a l l y u s e d R - f a c t o r s i s o b t a i n e d by g r a d i e n t methods o r s e a r c h p r o c e d u r e s [ 4 - 6 ] . The f i r s t a p p r o a c h had been p r o p o s e d by Powel l and de Carva lho [4] , who a p p l i e d a s e a r c h p r o c e d u r e w i t h an i n d e p e n d e n t o p t i m i s a ­t i o n of each p a r a m e t e r . The method i s g e n e r a l l y a p p l i c a ­b l e and does n o t r e q u i r e t h e c a l c u l a t i o n of d e r i v a t i v e s . A more s o p h i s t i c a t e d method which a l l o w s t h e s i m u l t a -128 Springer Series in Surface Sciences, Vol. 24 The Structure of Surfaces III

Editors: S.Y. Tong • M.A. Van Hove • K. Takayanagi • X.D. Xie © Springer-Verlag Berlin, Heidelberg 1991

neous r e f i n e m e n t of a l l p a r a m e t e r s h a s been r e c e n t l y p roposed by Rous, Van Hove and Somorja i [6] . Th i s method i s a g r a d i e n t method which , i n t h e f o r m u l a t i o n u s e d t h e r e , a l s o r e q u i r e s R - f a c t o r c a l c u l a t i o n s o n l y and no d e r i v a t i v e s . A d i f f e r e n t a p p r o a c h , somewhat r e l a t e d t o o p t i m i s a t i o n methods i s t h e d i r e c t method p r o p o s e d by Pendry , Heinz and Oed [5] i n which t h e d e v i a t i o n from a r e f e r e n c e s t r u c t u r e i s d e t e r m i n e d i n one s t e p . The p r o -cedure can be i t e r a t e d i f t h e l i n e a r e x p a n s i o n from t h e r e f e r e n c e s t r u c t u r e i s n o t s u f f i c i e n t . The c o m b i n a t i o n w i t h Tensor LEED t e c h n i q u e s [1] makes t h i s method v e r y e f f i c i e n t .

We p r o p o s e h e r e an a l t e r n a t i v e approach which c l o s e l y r e s e m b l e s t h e methods c o n v e n t i o n a l l y a p p l i e d i n x - r a y c r y s t a l l o g r a p h y [ 7 , 8 ] . Both methods a r e d i f f r a c t i o n t e c h n i q u e s and t h e s t r u c t u r e i s u s u a l l y d e t e r m i n e d by f i t t i n g model c a l c u l a t i o n s t o e x p e r i m e n t a l d a t a . The only d i f f e r e n c e i n t h e two t e c h n i q u e s i s g i v e n by t h e way 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 a r e compared, and, of c o u r s e , i n X- ray d i f f r a c t i o n , f u r t h e r methods a r e a p p l i c a b l e which canno t be a p p l i e d w i t h LEED. Us ihg X- rays a s e t of d i f f r a c t e d i n t e n s i t i e s i s measured a t c o n s t a n t w a v e l e n g t h , w h i l e w i t h LEED füll s p e c t r a a r e measured and f i t t e d t o t h e o r e t i c a l c u r v e s . The f i t i s u s u a l l y done by a d j u s t i n g t h e p o s i t i o n of maxima and minima i n t h e s p e c t r a .

The re i s a - p r i o r i no obv ious r e a s o n why t h e s i m p l e compar i son of r e l a t i v e i n t e n s i t i e s , which works w e l l i n t h e c a s e of X - r a y s , s h o u l d n o t work i n t h e c a s e of LEED. I t s h o u l d be n o t e d t h a t t h e p o s i t i o n of maxima and m i n i ­ma i n t h e s p e c t r a i s n o t d i r e c t l y u s e d . Measur ing t h e d i s t a n c e be tween e x p e r i m e n t a l and c a l c u l a t e d s p e c t r a by t h e l i n e a r o r q u a d r a t i c d e v i a t i o n h a s t h e a d v a n t a g e t h a t w e l l d e v e l o p e d o p t i m i s a t i o n t e c h n i q u e s can be u s e d . With t h e c o n v e n t i o n a l R - f a c t o r s , such a s d e f i n e d by Zanazz i and J o n a [9] and Pendry [10] t h e f i t - f u n c t i o n becomes f a i r l y c o m p l i c a t e d . A s i m p l e r R - f a c t o r t h e r e f o r e seems t o be a d v a n t a g e o u s and i t h a s i n d e e d been shown t h a t an R - f a c t o r b a s e d on t h e mean Square d e v i a t i o n i s w e l l a p ­p l i c a b l e and l e a d s t o r e l i a b l e r e s u l t s . A füll d e s c r i p -t i o n of t h i s method and t h e o p t i m i s a t i o n p r o c e d u r e h a s b e e n p u b l i s h e d r e c e n t l y [ 7 , 8 ] . We t h e r e f o r e g i v e h e r e a s h o r t r e v i e w of t h e n o v e l s t r u c t u r e r e f i n e m e n t t e c h n i q u e and d i s c u s s some c a l c u l a t i o n a l improvements i n d e t a i l .

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2. O p t i m i s a t i o n p r o c e d u r e The R - f a c t o r s u sed i n X- ray c r y s t a l l o g r a p h y a r e e i t h e r t h e l i n e a r o r t h e mean Square d e v i a t i o n be tween o b s e r v e d and c a l c u l a t e d i n t e n s i t i e s a t a f i x e d e n e r g y . The a n a l o -gous approach i n LEED h a s been p r o p o s e d p r e v i o u s l y a s t h e I ( g ) method [11] . The s h o r t c o m i n g s of t h a t method a r e t h a t a s u p e r s t r u c t u r e p r o d u c i n g o n l y weak e x t r a s p o t s , such as a hydrogen s u p e r s t r u c t u r e , c a n n o t be w e l l d e t e r m i n e d b e c a u s e t h e I ( E ) spec t rum of t h e s u p e r s t r u c t u -r e s p o t i s n o t p r o p e r l y w e i g h t e d . T h i s p rob l em c a n be overcome by e v a l u a t i n g each beam w i t h a s e p a r a t e w e i g h t f a c t o r . The R - f a c t o r i s d e f i n e d f o r d i s c r e t e e n e r g i e s and i s g i v e n by

S | J 6 X - c J t h | RnT7 = S W 1 1 , 8 S - 1 , 8 ; (1)

DE g g s j ex i i , g

c - 2 J ? / S J . ; W - n / En . (2) g l i , g i , g g g g g

ng i s t h e number of p o i n t s p e r beam g. T h i s R - f a c t o r can be compared w i t h t h e u s u a l R - f a c t o r i n x - r a y d i f f r a c ­t i o n . The q u a n t i t y which i s a c t u a l l y m i n i m i s e d i n t h e o p t i m i s a t i o n p r o c e d u r e i s , however , t h e mean s q u a r e d e v i a t i o n R2. RßE i s u s e d f o r compar i son w i t h x - r a y d i f f r a c t i o n where t h e unweigh ted l i n e a r d e v i a t i o n i s a S t a n d a r d R - f a c t o r .

To save Computing t ime i t i s a d v a n t a g e o u s t o choose a s t e p w i d t h on t h e ene rgy s c a l e which i s a s l a r g e a s p o s s i b l e . We have e x t e n s i v e l y t e s t e d which s t e p w i d t h can be chosen w i t h o u t l o s i n g p r e c i s i o n of t h e r e s u l t . A s t e p w i d t h up t o 15 - 20 eV c o r r e s p o n d i n g t o a b o u t 10 -15 d a t a p o i n t s p e r spec t rum seems t o b e c o m p l e t e l y s u f f i c i e n t [ 9 , 1 0 ] . The upper l i m i t of t h e s t e p w i d t h a p p e a r s t o be s e t by t h e r e q u i r e m e n t t h a t enough p o i n t s p e r beam r e m a i n .

Having d e f i n e d t h e f i t - f u n c t i o n an a u t o m a t i c min imi -s a t i o n p r o c e d u r e can be i n t r o d u c e d . A v e r y e f f i c i e n t met ­hod h a s been d e v e l o p e d by Marquard t [ 1 2 ] . His method combines t h e a d v a n t a g e s of t h e g r a d i e n t method and of t h e e x p a n s i o n method. I n t h e g r a d i e n t method t h e s t e e -p e s t d e c e n t of t h e R - f a c t o r i n t h e p a r a m e t e r s p a c e i s d e t e r m i n e d from i t s p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o 130

a l l v a r i a b l e p a r a m e t e r s . The method works w e l l f a r away from t h e mimimum b u t c o n v e r g e s on ly s l o w l y n e a r t h e minimum where t h e d e r i v a t i v e s become v e r y s m a l l . The e x p a n s i o n method , on t h e o t h e r hand , works w e l l n e a r t h e minimum and may l e a d t o s e r i o u s e r r o r s f a r away from t h e minimum.

I n t h e e x p a n s i o n method a l i n e a r a p p r o x i m a t i o n of t h e i n t e n s i t y f u n c t i o n i s u s e d

k t h I t h (£o^ £ ) - I t h ( £ o ) + 2 C ^ - ^ } ] Ap + . . . (3)

j = l j where j>0 = (p]_, . . . Pk) d e n o t e s t h e s e t of s t r u c t u r a l , p a r a ­m e t e r s and I e x , I " 1 a r e t h e n o r m a l i z e d i n t e n s i t i e s . Eq. (3) i s i n s e r t e d i n t o t h e minimum c o n d i t i o n

dR/3pj - 0 , ( j = l , . . . k ) (4) leading to a set of l i n e a r equations

_, r Tex Tth/ N „ i o A i 1 0 _ / c .

. V i ' h ~W.— A p j > — ä F ~ " = ( )

1=1 j - i j m

which i s solved by matrix inversion

Ap = ß • a " 1 , (6)

1=1 rm

«. _ £ C a i ! h ( P o ) • 3 l t! (Po)3 • (8) jm . = 1 3 p . 3pm

The method of Marqua rd t r e p l a c e s ocjm by a ! - a . ( 1+5 . - A) (9)

jm jm jm

5 j m i s t h e Kronecker symbol . I f A i s l a r g e t h e n t h e d i a ­g o n a l t e r m s domina te and t h e r e s u l t i s s i m i l i a r t o t h e g r a d i e n t method , i f A i s s m a l l t h e e x p a n s i o n method i s r e c o v e r e d . A i s d y n a m i c a l l y a d j u s t e d by

131

k ß? . A = c 2 ( - i - ) (10)

i-1 a i i R2 The speed of t h e o p t i m i s a t i o n can be i n f l u e n c e d by t h e p a r a m e t e r c .

3. C a l c u l a t i o n of d e r i v a t i v e s The c a l c u l a t i o n a l e f f o r t i n t h e p r o c e d u r e d e s c r i b e d above i n c r e a s e s l i n e a r l y w i t h t h e number of f r e e P a r a ­m e t e r s b e c a u s e t h e n u m e r i c a l c a l c u l a t i o n of each d e r i v a ­t i v e r e q u i r e s an a d d i t i o n a l füll dynamica l c a l c u l a t i o n p e r p a r a m e t e r a t a l l e n e r g i e s . Th i s c a l c u l a t i o n c o u l d be done v e r y e f f i c i e n t l y by a p p l y i n g t h e Tensor LEED t e c h n i ­que [1] . However, an app rox ima te c a l c u l a t i o n of d e r i v a t i ­ve s seemed t o be e a s i e r t o implement i n t h e e x i s t i n g program and t u r n s o u t t o be q u i t e e f f i c i e n t . A l i n e a r a p -p r o x i m a t i o n i n c a l c u l a t i n g d e r i v a t i v e s can be u s e d .

The l a y e r s c a t t e r i n g m a t r i c e s a r e app rox ima ted b y : 3M ( 2 o)

M , - M , (Eo) + 2 — |s Ap. + . . . (11)

gg ' gg' *° J 3p *J j

where 3Mgg'/<9pj can be o b t a i n e d from a l i n e a r e x p a n s i o n of t h e i n v e r s e of t h e p r o p a g a t o r m a t r i x . The d e f i n i t i o n of t h e p r o p a g a t o r m a t r i x and t h e l a y e r s c a t t e r i n g m a t r i c e s i s g i v e n i n r e f . 13. I n a l i n e a r e x p a n s i o n

(l-Xteo+AE))" 1- [ l-X( E o)]" 1+ ( l-Xteo)]" 1.

[ X ( £ 0 + A 2 ) - X ( 2 o ) ] • [ l - X ( 2 o ) ] ~ l (12)

and o n l y m a t r i x m u l t i p l i c a t i o n s a r e r e q u i r e d . The i n v e r ­se of [ 1 - X ( e 0 ) ] h a s been a l r e a d y c a l c u l a t e d f o r t h e r e f e r e n c e s t r u c t u r e and can be r e u s e d a g a i n . Next an app rox ima te c a l c u l a t i o n of t h e l a t t i c e sum i s d e s i r a b l e . The sum over s c a t t e r e d waves from a l l atoms i n t h e l a y e r w i t h i n a l i m i t i n g r a d i u s of abou t 10 - 15 i n t e r a t o m i c d i s t a n c e s r e q u i r e s a good p a r t of t h e Computing t i m e and i t i s wor th w h i l e c o n s i d e r i n g s i m p l i f i c a t i o n s . The l a t t i ­ce sum of s c a t t e r i n g p a t h s be tween p l a n e s v and v' i s 132

g i v e n by [13]

im P+p - p ' (13) The s p e r i c a l ha rmon ics Y^m(Op+pv-pv#) do n o t change much by i n c r e a s i n g p^, by A p v and f o r t h e Hankel f u n c t i o n s t h e a s y m p t o t i c b e h a v i o u r a t l a r g e P can be used t o c a l o u l a t e t h e d e r i v a t i v e s f o r v a l u e s P > P m i n . Pmin c a n ke s e t t o 3-4 i n t e r a t o m i c d i s t a n c e s w i t h o u t l o s i n g p r e c i s i o n .

i h ^ ( z ) + e 2 4 (14) The l a t t i c e sum i s t h e r e f o r e s p l i t up i n t o two p a r t s where on ly one p a r t c o n t a i n i n g t h e n e a r ne ighborhood of an atom needs t o be r e c a l c u l a t e d . The minimum d i s t a n c e ^min c a n ^ e chosen t o abou t t h r e e i n t e r a t o m i c d i s t a n c e s . The c a l c u l a t i o n a l e f f o r t f o r t h e l a t t i c e sum i s r e d u c e d by a f a c t o r of 15 by t h i s a p p r o x i m a t i o n . The compar i son of an app rox ima te c a l c u l a t i o n of d e r i v a t i v e s w i t h t h e füll dynamic c a l c u l a t i o n i s shown i n f i g . 1 .

The app rox ima te c a l c u l a t i o n of d e r i v a t i v e s t u r n s o u t t o be c o m p l e t e l y s u f f i c i e n t f o r a l l t y p e s of p a r a m e t e r s .

-Ar—^~^\/'^

(10) beam Ni (110) (2x1)0

}• BU

D34

100 200 300

ENERGY (eV)

F i g . l . Comparison of app rox ima te (dashed l i n e ) and füll dynamica l ( s o l i d l i n e ) c a l c u l a t i o n of d e r i v a t i v e s . The s t r u c t u r e p a r a m e t e r s a r e d i s p l a y e d i n f i g . 2 .

133

The i n c r e a s e of t h e speed of t h e c a l c u l a t i o n d e p e n d s , however , on t h e number of v a r i a b l e p a r a m e t e r s and t h e number of p h a s e s h i f t s u s e d . I n t h e examples shown be low w i t h 6 and 8 v a r i a b l e p a r a m e t e r s a f a c t o r of 2 . 5 f o r t h e whole c a l c u l a t i o n i s g a i n e d . F u r t h e r improvements a r e p o s s i b l e . A l i n e a r e x p a n s i o n s i m i l a r t o t h a t d e s c r i b e d i n eq . (12) can be u s e d f o r t h e m a t r i c e s t o be i n v e r t e d i n t h e l a y e r d o u b l i n g method.

4 . A p p l i c a t i o n t o H / N i ( 1 1 0 ) - ( 1 x 2 ) and O/Ni(110) - (2x1) To i l l u s t r a t e t h e c a p a b i l i t y of t h e method t h e r e s u l t s of two a d s o r b a t e Systems w i l l be p r e s e n t e d . Füll s t r u c t u ­r a l r e s u l t s have b e e n p u b l i s h e d r e c e n t l y [ 7 , 1 4 , 1 5 ] , we p r e s e n t h e r e o n l y t h e r e s u l t s of t h e f i t - p r o c e d u r e u s i n g t h e b u l k v a l u e s of Ni w i t h a s l i g h t b u c k l i n g i n t h e t h i r d l a y e r as s t a r t p a r a m e t e r s .

The s t r u c t u r e of t h e H/Ni(110) - (1x2) i s shown i n f i g . 2; t h e hydrogen atoms a r e i g n o r e d i n t h e c a l c u l a ­t i o n . The f i n a l s t r u c t u r a l p a r a m e t e r s a s w e l l a s t h e r e s u l t of two c a l c u l a t i o n s w i t h d i f f e r e n t s t a r t p a r a ­m e t e r s a r e g i v e n i n t a b l e 1 . The c h o i c e of t h e b u l k s t r u c t u r e a s s t a r t p a r a m e t e r i s n o t p o s s i b l e i n t h i s c a s e . The b u l k s t r u c t u r e does n o t p roduce s u p e r s t r u c t u r e beams and t h e d e r i v a t i v e s w i t h r e s p e c t t o t h e s u p e r s t r u c -t u r e p a r a m e t e r s LS and BU v a n i s h . The d e r i v a t i v e v a n i s -h e s b e c a u s e a t a h i g h l y symmetry p o i n t two c h o i c e s of t h e d e r i v a t i v e a r e s y m m e t r i c a l l y e q u i v a l e n t . I t i s t h e r e ­f o r e n e c e s s a r y t o s h i f t a t l e a s t one atom o f f i t s b u l k l a t t i c e p o s i t i o n .

A second example i s O / N i ( 1 1 0 ) - ( 2 x 1 ) . The s t r u c t u r a l model i s shown i n f i g . 3 . Here 6 i n d e p e n d e n t p a r a m e t e r s w i t h i n t h e t op t h r e e l a y e r s had t o be r e f i n e d , a s suming

LS, LS,

LS3 LS3

F i g . 2 . Model of t h e (1x2) s t r u c t u r e of H / N i ( 1 1 0 ) . Hydrogen atoms a r e n o t shown.

134

Table 1. R e s u l t of a s i m u l t a n e o u s f i t of 8 s t r u c t u r a l p a r a m e t e r s . The f i n a l s t r u c t u r e was r e a c h e d a f t e r 10 i t e r a t i o n s , Rj)g = 0 . 3

Pa rame te r S t a r t Value F i n a l Value [Ä] D 1 2 1 246 1 223 D23 1 246 1 331 D 3 4 1 246 1 272 D 4 5 1 246 1 220 LS! 0 0 30 LS 3 0 0 12 BU2 0 1 0 25 BU4 0 0 02

Tab le 2 . R e s u l t of a s i m u l t a n e o u s f i t of 6 p a r a m e t e r s , t h e minimum R - f a c t o r was r e a c h e d a f t e r 7 i t e r a t i o n s , Rj)g = 0 .24 P a r a m e t e r S t a r t Value F i n a l Value [Ä]

7 0 3 0 .224 D 1 2 1 246 1 293 D23 1 246 1 246. LS 2 0 0 011 BU4 0 0 052

t h a t oxygen s i t s i n t h e Symmetrie p o s i t i o n . R e s u l t s a r e shown i n t a b l e 2 . 15 eV s t e p s were u s e d i n t h e e n e r g y ränge b e t w e e n 40 and 340 eV, c o r r e s p o n d i n g t o 106 d a t a p o i n t s i n 8 I /V s p e c t r a .

I n t h e above c a l c u l a t i o n s oxygen was f i x e d i n t h e Sym­m e t r i e s i t e . A d e t a i l e d s t u d y showed t h a t a s l i g h t p r e f e -r e n c e was found f o r an a sysmmet r i c s i t e [14] . The s e a r c h

135

fo r an asymmetr ic s i t e was s t i m u l a t e d by a HREELS s t u d y [16] which showed an a d d i t i o n a l oxygen mode n o t c o m p a t i -b l e w i t h t h e a s s u m p t i o n of a Symmetrie oxygen p o s i t i o n . Th i s asymmetry h a s been n e g l e c t e d h e r e , b e c a u s e i t i s f e i t t h a t from t h e LEED d a t a a t normal i n c i d e n c e a l o n e t h i s asymmetry c a n n o t be d e f i n i t e l y c o n c l u d e d [ 1 4 ] . The i n f l u e n c e of a l a t e r a l s h i f t of t h e oxygen atom on t h e o t h e r s t r u c t u r a l p a r a m e t e r s can be n e g l e c t e d .

5 . D i s c u s s i o n A c e n t r a l p o i n t i n any o p t i m i s a t i o n scheme i s of c o u r s e t h e r a d i u s of conve rgence w i t h i n which a minimum w i l l be l o c a l i s e d . Tha t t h e minimum may be a l o c a l mimimum h a s been a l r e a d y p o i n t e d o u t . Loca l minima can be a v o i d e d on ly by c h o o s i n g s t a r t s t r u e t u r e s on a wide g r i d i n t h e p a r a m e t e r s p a c e . The r a d i u s of conve rgence t h e r e f o r e d e t e r m i n e s t h e g r i d s i z e which must be a p p l i e d t o e x c l u -de l o c a l minima. T h i s same prob lem o c c u r s , by t h e way, i n t h e c o n v e n t i o n a l R - f a c t o r a n a l y s i s . The a v e r a g e d i s -t a n c e be tween l o c a l minima i n t h e R - f a c t o r h y p e r f a c e may be e s t i m a t e d t o be a b o u t 0 . 5 Ä. Th i s r e s u l t s from t h e s imp le c o n s i d e r a t i o n t h a t w i t h an a v e r a g e w a v e l e n g t h of 1.0 Ä an i n t e r f e r e n c e maximum of backward and fo rward s c a t t e r i n g be tween two atoms o c c u r s a g a i n a f t e r a s h i f t of 0 .5 Ä. I t f o l l o w s t h a t r o u g h l y a d e v i a t i o n of 0 . 2 -0 . 3 Ä can be t o l e r a t e d i n t h e s t a r t p a r a m e t e r s . T h i s i s a rough e s t i m a t e , of c o u r s e , and a p p l i e s t o p a r a m e t e r s p a r a l l e l t o Ak. Sma l l e r d i s t a n c e s be tween l o c a l minima may a l s o occu r due t o domain a v e r a g i n g [ 6 ] .

To check t h e r a d i u s of conve rgence we pe r fo rmed s e v e r -a l r u n s w i t h d i f f e r e n t s t a r t v a l u e s f o r two p a r a m e t e r s i n t h e 0 /N i (110) - (2x1) s t r u c t u r e , Z o x and D ; l 2 , k e e p i n g a l l o t h e r p a r a m e t e r s a t t h e i r optimum v a l u e . The r e s u l t s a r e shown i n t a b l e 3 .

I t may be c o n c l u d e d t h a t t h e r a d i u s of c o n v e r g e n c e i s abou t 0 . 2 Ä i n agreement w i t h t h e e s t i m a t e c o n s i d e r e d above . I n t e s t r u n s w i t h a s i m u l t a n e o u s f i t of a l l p a r a ­m e t e r s t h e same r a d i u s of conve rgence was found which i n d i c a t e s t h a t t h e p a r a m e t e r s a r e o n l y weakly c o r r e l a -t e d .

The convergence depends a l s o on t h e p a r a m e t e r c i n eq . 10 , t h i s h a s been chosen a s 0 .2 i n t h e above exam-p l e s . A l a r g e r v a l u e d e c r e a s e s t h e speed of t h e c a l c u l a -136

Table 3 Check of t h e r a d i u s of convergence f o r two p a r a m e t e r s .

s t a r t f i n a l R^E No. of i t e r a t i o n s ^ox 0. .5 0, .83 0. .43 6 7

^ox 0. .4 0. .22 0. .243 4 D 1 2 1. .5 1. .58 0. .73 1 D 1 2 1. .4 1. .31 0. .242 3

tion and has b e e n found sometimes t o increase t h e radius of conve rgence because t h e large Steps in t h e beginning of t h e i t e r a t i o n p r o c e s s are damped. Acknowledgement P a r t of t h i s work was f i n a n c i a l l y s u p p o r t e d by t h e Deut­sche F o r s c h u n g s g e m e i n s c h a f t , SFB 338.

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