II
Editor: Gerhard Ertl
Springer Series in Surface Sciences Editors: Gerhard Ertl and Robert Gomer
Volume 1: Physisorption Kinetics By H.J. Kreuzer, Z. W. Gortel
Volume 2: The Structure of Surfaces Editors: M. A. Van Hove, S. Y. Tong
Volume 3: Dynamical Phenomena at Surfaces, Interfaces and Superiattices Editors: F. Nizzoli, K.-H. Rieder, R.F. Willis
Volume 4: Desorption Induced by Electronic Transitions, DIET II Editors: W. Brenig, D. Menzel
Volume 5: Chemistry and Physics of SoHd Surfaces VI Editors: R. Vanselow, R. Howe
Volume 6: Low-Energy Electron Diffraction Experiment, Theory and Surface Structure Determination By M. A. Van Hove, W. H. Weinberg, C.-M. Chan
Volume 7: Electronic Phenomena in Adsorption and Catalysis By V. F. Kiselev, o. V. Krylov
Volume 8: Kinetics of Interface Reactions Editors: M. Grunze, H. J. Kreuzer
Volume 9: Adsorption and Catalysis in Transition Metals and their Oxides III By o. V. Krylov, V. F. Kiselev
Volume 10: Chemistry and Physics of Solid Surfaces VII Editors: R. Vanselow, R. Howe
Volume 11: The Structure of Surfaces II Editors: J. F. van der Veen, M. A. Van Hove
J. F. van der Veen M. A. Van Hove (Eds.)
The Structure of Surfaces II Proceedings of the 2nd International Conference on the Structure of Surfaces (I CSOS II), Amsterdam, The Netherlands, June 22-25,1987
With 343 Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Professor Dr. Johannes Friso van der Veen FOM-Institute for Atomic and Molecular Physics, Kruislaan 407, NL-I098 SJ Amsterdam, The Netherlands
Dr. Michel A. Van Hove Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA
Series Editors
Professor Robert Gomer
The James Franck Institute, The University of Chicago, 5640 Ellis Avenue, Chicago, IL 60637, USA
ISBN-13:978-3-642-73345-1 e-ISBN-13:978-3-642-73343-7 DOl: 10.1007/978-3-642-73343-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Softcover reprint of the hardcover 1st edition 1988
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and thereof free for general use.
2153/3150-543210
Preface
This book collects together selected papers presented at the Second Interna­ tional Conference on the Structure of Surfaces (ICSOS-II). The conference was held at the Royal Thopical Institute in Amsterdam, The Netherlands, June 22-25, 1987. It was held in part to celebrate the 25th anniversary of the NEVAC (Netherlands Vacuum Society). The International Organizing Committee members were:
M.A. Van Hove (Chairman) W.F. van der Weg (Treasurer) A.M. Bradshaw D.J. Chadi J. Eckert S. Ino B.I. Lundqvist Y. Petroff G.A. Somorjai S.Y. Tong
J.F. van der Veen (Vice-Chairman) D.L. Adams M.J. Cardillo J.E. Demuth G. Ertl D.A. King J.B. Pendry J.R. Smith J. Stohr X.D. Xie
The ICSOS meetings serve to assess the status of surface structure determination and the relationship between surface or interface structures and physical or chemical properties of interest. The papers in this book cover: theoretical and experimental structural techniques; structural aspects of metal and semiconductor surfaces, including relaxations and reconstruc­ tions, as well as adsorbates and epitaxial layers; phase transitions in two dimensions, roughening and surface melting; defects, disorder and surface morphology.
Amsterdam, Berkeley October 1987
v
Acknow ledgements
We wish to acknowledge the many organizations and individuals whose contributions made possible the Second International Conference on the Structure of Surfaces and these Proceedings. We express our gratitude to our host institution: FOM-Institute for Atomic and Molecular Physics; and our many sponsors: Balzers, EOARD, EPS (European Physical Society), Foundation FOM (Fundamental Research on Matter), Foundation Physica, IBM (Nederland) NV, IUPAP (International Union of Pure and Applied Physics), IUVSTA (International Union for Vacuum Science, Technique and Applications), KLM (Royal Dutch Airlines), KNAW (Royal Netherlands Academy of Sciences), Leybold Hereaus, Ministry of Education and Sciences, NEVAC (Nether­ lands Vacuum Society), NNV (Netherlands Physical Society), Philips, Shell Research, The City of Amsterdam, US-Army, and VG Instruments B.V. We also thank our ex­ hibitors: Balzers, De Jong TH, Hositrad, Leybold Hereaus, North-Holland Publishing, Positronica, Intechmij, VG Instruments B.V. and D. Reidel Publ. Co.
Particular thanks go to all the individuals who contributed much to the well­ being of both the conference and the proceedings, especially Louise Roos, Jan Verhoe­ ven, Dorine Heynert and the members of the Local Organizing Committee: F.H.P.M. Habraken, A.G.J. van Oostrom, G.A. Sawatzky, and W.F. van der Weg. An important element was of course the contribution from the International Advisory Committee members: D. Aberdam, J.C. Bertolini, M. Cardona, G. Comsa, L.C. Feldman, F. Garcia Moliner, D.R. Hamann, D. Haneman, A.A. Lucas, T.E. Madey, K. Miiller, S. Nakamura, A.G. Naumovetz, P.R. Norton, G. Rovida, W.E. Spicer, A.G.J. van Oostrom, and R.F. Willis.
VI
Contents
Resolution in Scanning Tunneling Microscopy By J. Tersoff ...................................... 4
Tunneling Current Between Two Nonplanar Surfaces By W. Sacks, S. Gauthier, S. Rousset, and J. Klein (With 2 Figures) ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Tensor LEED; New Prospects for Surface Structure Determination by LEED. By P.J. Rous and J.B. Pendry .................. 14
Comparison of the Quasidynamical and Tensor LEED Approximation for LEED Intensity Spectra from a Reconstructed Surface. By N. Bickel, K. Heinz, H. Landskron, P.J. Rous, J.B. Pendry, and D.K. Saldin (With 3 Figures) .... . . . . . . . . . . 19
Surface Barrier Bound State Energies from Elastic Electron Scattering By M.N. Read and A.S. Christopoulos (With 3 Figures) ....... 26
The Theory of SEELFS from Adsorbates By D.K. Saldin (With 2 Figures) ........................ 32
Multiple-Scattering Studies of Normal and Off-Normal Photoelectron Diffraction of C(2x2)S-Ni(001) By Jing Chang Tang (With 4 Figures) .................... 38
I. 2 Experiment
High Resolution Profile Imaging of Reconstructed Gold Surfaces By T. Hasegawa, N. Ikarashi, K. Kobayashi, K. Takayanagi, and K. Yagi (With 13 Figures) ......................... 43
VII
Low Energy Electron Microscopy (LEEM) By W. Telieps and E. Bauer (With 3 Figures)
Surface Structure Analysis by Scanning LEED Microscopy By T. Ichinokawa, Y. Ishikawa, Y. Hosokawa, 1. Hamaguchi,
53
Structural Information from Stimulated Desorption: A Critical Assessment. By D. Menzel ............................ 65
Comparative Study of Graphite and Intercalated Graphite by Tunneling Microscopy. By S. Gauthier, S. Rousset, J. Klein, W. Sacks, and M. Belin (With 4 Figures) .................. 71
Auger Neutralization Lifetimes for Low-Energy Ne+ Ions Scattered from Pt(111) Surfaces. By E.A. Eklund, R.S. Daley, J.H. Huang, and R.S. Williams (With 4 Figures) ...................... 75
Determination of Surface Structure from the Observation of Catastrophes. By T.C.M. Horn and A.W. Kleyn (With 3 Figures) 83
Part II Clean Metals
II. 1 Relaxation and Reconstruction
Surface Structures from LEED: Metal Surfaces and Metastable Phases. By F. Jona and P.M. Marcus (With 2 Figures) ....... 90
Electrostatic Models of Relaxation on Metal Surfaces By P.M. Marcus, P. Jiang, and F. Jona (With 1 Figure)
Asymptotic Behavior of Relaxation and Reconstruction Near Crystalline Surfaces: Application to V(100) and Al(331) Surfaces
100
By G. Allan and M. Lannoo (With 1 Figure) ............... 105
Ion Channeling and Blocking Investigations of the Structure of Ideal and Reconstructed Metal Surfaces By T. Gustafsson, M. Copel, and P. Fenter (With 6 Figures) 110
Reconstruction of fcc(110) Surfaces By K.W. Jacobsen and J.K. N~rskov (With 1 Figure) ......... 118
Calculations of Structural Phases of Transition Metal Surfaces Using the Embedded Atom Method By M.S. Daw and S.M. Foiles (With 3 Figures) ............. 125
The (111) Surface Reconstruction of Gold in the Glue Model By A. Bartolini, F. Ercolessi, and E. Tosatti (With 1 Figure) 132
VIII
Second Layer Displacements in the Clean Reconstructed W (100) Surface. By I.K. Robinson, M.S. Altman, and P.J. Estrup (With 2 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
Calculation of Ni(100) Vibrational Properties Using the Matching Procedure. By J. Szeftel, A. Khater, and F. Mila (With 4 Figures) 142
II. 2 Alloys
LEED Study of the Structure of the Pt3 Ti(510) Stepped Single Crystal Surface. By U. Bardi, A. Santucci, G. Rovida, and P.N. Ross (With 3 Figures) .. . . . . . . . . . . . . . . . . . . . . . .. 147
Atomic Structure of Three Low-Index Surfaces of the Ordered Binary Alloy NiAI By H.L. Davis and J.R. Noonan (With 5 Figures) . . . . . . . . . . .. 152
Structure, Electronic Properties and Dynamics of the NiAI(110) Surface. By M.H. Kang and E.J. Mele (With 4 Figures) ....... 160
Multilayer Segregation on Pt-Ni(l11), (100) and (110): Influence of the Variation of Pair Interactions at the Surface By B. Legrand and G. Treglia (With 3 Figures) ............. 167
Part III Adsorbates on Metals
Surface EXAFS on Low-Z Elements By K. Baberschke (With 5 Figures) ...................... 174
An Application of SEXAFS to Sub-Monolayer Complexes on Polycrystalline Surfaces. By D. Norman, R.A. Tuck, H.B. Skinner, P.J. Wadsworth, T.M. Gardiner, 1.W. Owen, C.H. Richardson, and G. Thornton (With 4 Figures) ......... 183
X-Ray Absorption Fine Structure Study of Mercaptide on Cu(lll) By D.L. Seymour, C.F. McConville, M.D. Crapper, D.P. Woodruff, and R.G. Jones (With 3 Figures) ........................ 189
Adsorption Position of Deuterium on the Pd(100) and Ni(111) Surface Determined by Transmission Channeling. By F. Besenbacher, 1. Stensgaard, and K. Mortensen (Wi th 2 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195
Adsorption of Hydrogen on Rhodium (110) By W. Nichtl, L. Hammer, K. Miiller, N. Bickel, K. Heinz, K. Christmann, and M. Ehsasi (With 5 Figures) ......... . . .. 201
Relaxation and Reconstruction on Ni(110) and Pd(110) Induced by Adsorbed Hydrogen. By W. Moritz, R.J. Behm, G. Ertl, G. Kleinle, V. Penka, W. Reimer, and M. Skottke (With 1 Figure) 207
IX
EELFS Determination of Interatomic Distances in Adsorbed Monolayers. By A. Atrei, U. Bardi, G. Rovida, M. Torrini, E. Zanazzi, and M. Maglietta (With 4 Figures) . . . . . . . . . . . . .. 214
The Structures of CO, NO and Benzene on Various Transition Metal Surfaces: Overview of LEED and HREELS Results By H. Ohtani, M.A. Van Hove, and G.A. Somorjai (With 1 Figure) .................................... 219
Formation and Stability of a Metastable c(2x4)O Structure on an Unreconstructed Ni(llO) Surface By J. Wintterlin and R.J. Behm (With 4 Figures) ........... 225
Surface Structures Determined by Kinetic Processes: Adsorption and Diffusion of Oxygen on Pd(100). By S.-L. Chang, D.E. Sanders, J.W. Evans, and P.A. Thiel (With 2 Figures) 231
Mercury Adsorption on Ni(l11) By N.K. Singh and R.G. Jones (With 3 Figures) ............ 238
Ion Scattering Study of the W(OOl )-( 5x 1 )-C Surface By S.H. Over bury and D.R. Mullins (With 5 Figures) ......... 244
Structural Determination of Oxygen Chemisorption-Site Geometry on W(211) by Low-Energy He+ ISS By W.P. Ellis and R. Bastasz (With 5 Figures) ............. 250
Structure of Oxygen on Ni3AI(1l0). By D.J. O'Connor, C.M. Loxton, and R.J. MacDonald (With 3 Figures) . . . . . . . . .. 256
Early Stages of Ni(llO) Oxidation - An STM Study By E. Ritter and R.J. Behm (With 3 Figures) .............. 261
Symmetry Rules in Chemisorption By R.A. van Santen (With 2 Figures)
The Electronic Structure of Adsorbed Oxygen on Ag(llO) By W. Segeth, J.H. Wijngaard, and G.A. Sawatzky
267
(With 4 Figures) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 271
Part IV Clean Semiconductors
Understanding the Si 7x7: Energetics, Topology, and Stress By D. Vanderbilt (With 3 Figures) ...................... 276
Scanning Tunneling Microscopy of Semiconductor Surfaces and Interfaces. By R.M. Tromp, E.J. van Loenen, R.J. Hamers, and J .E. Demuth (With 6 Figures) ...................... 282
x
Surface X-Ray Diffraction: the Ge(001)2x1 Reconstruction and Surface Relaxation. By F. Grey, R.L. Johnson, J. Skov Pedersen, R. Feidenhans'l, and M. Nielsen (With 4 Figures) ............ 292
RHEED Intensity Analysis on a Single Domain Si(100)-2x 1 By T. Kawamura, T. Sakamoto, K. Sakamoto, G. Hashiguchi, and N. Takahashi (With 3 Figures) ...................... 298
Surface Electronic Structure of Si(100)2x 1 Studied with Angle- Resolved Photoemission. By R.I.G. Uhrberg, L.S.O. Johansson, and G.V. Hansson (With 5 Figures) ...................... 303
Screened Coulomb Interaction at Semiconductor Surfaces: The Contribution of Surface States By R. Del Sole and L. Reining (With 3 Figures) ............. 309
On the Reconstruction of the Diamond (111) Surface By P. Badziag (With 2 Figures) .. . . . . . . . . . . . . . . . . . . . . . .. 316
Charge Self-Consistent Empirical Tight Binding Cluster Method for Semiconductor Surface Structures By V.M. Dwyer, J.N. Carter, and B.W. Holland (With 2 Figures) 320
Two New Models for the As-Stabilized GaAs (111)-(2x2) Surface By Huizhou Liu, Geng Xu, and Zheyin Li (With 2 Figures) 327
Part V Adsorbates on Semiconductors
High Sensitivity Detection of a Few Atomic Layers of Adsorbate by RHEED-TRAXS (Total Reflection Angle X-Ray Spectroscopy) By S. Ino, S. Hasegawa, H. Matsumoto, and H. Daimon (With 6 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 334
Submonolayers of Lead on Silicon (111) Surfaces: An X-Ray Standing Wave Analysis. By B.N. Dev, G. Mat erlik , F. Grey, and R.L. Johnson (With 3 Figures) ...................... 340
Atomic Geometry of the Si(111)v'3 x v'3-Sn Surface by X-ray Photoelectron and Auger Electron Diffraction By K. Higashiyama, C.Y. Park, and S. Kono (With 3 Figures) 346
Surface X-Ray Diffraction: The Atomic Geometry of the Ge(111)7x7-Sn and Ge(111)5x5-Sn Reconstructions By J. Skov Pedersen, R. Feidenhans'l, M. Nielsen, K. Kjrer, F. Grey, R.L. Johnson, and C. Reiss ..................... 352
Chemisorption Geometry of Molybdenum on Silicon Surfaces By Tang Shaoping, Zhang Kaiming, and Xie Xide (With 4 Figures) ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 357
XI
Si(lOO) Surface Reordering upon Ga Adsorption By 1. Andriamanantenasoa, J.P. La,charme, and C.A. Sebenne (With 3 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 363
Synchrotron Radiation Study of the Au-Si(lOO) Interface By B. Carriere, J.P. Deville, M. Hanbiicken, and G. Le Lay (With 5 Figures) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 368
Studies of the High Temperature Nitridation Structures of the Si(111) Surface by LEED, AES and EELFS By Hongchuan Wang, Rongfu Lin, and Xun Wang (With 3 Figures) ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375
Hydrogen Bonding onto Microcrystalline Surfaces within Anodized Porous Silicon Crystals Studied by Infrared Spectroscopy By T. Ito, Y. Kato, and A. Hiraki (With 6 Figures) .......... 378
The Oxygen Coverage on Diamond Surfaces By T.E. Derry, J.O. Hansen, P.E. Harris, R.G. Copperthwaite, and J.P.F. Sellschop (With 4 Figures) .................... 384
H-Induced Reconstruction at the (110) Faces of GaAs and InP By F. Proix, O. M'hamedi, and C.A. Sebenne (With 3 Figures) .. 393
Indiffusion and Chemisorption of B, C, and N on GaAs and InP By M. Menon and R.E. Allen (With 8 Figures) ............. 399
Structure of Platinum Metal Clusters Deposited on the Ti02 Surface by X-Ray Photoelectron Diffraction (XPED) By K. Tamura, U. Bardi, M. Owari, and Y. Nihei (With 4 Figures) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 404
Part VI Epitaxy
Strained Layer Epitaxy.By L.C. Feldman, M. Zinke-Allmang, J. Bevk, and H.-J. Gossmann (With 3 Figures) ............. 412
Critical Misfits for Lattice-Matched Strained Monolayers By T. Hibma (With 5 Figures) ......................... 419
Interface Pseudomorphism Detected by Mossbauer Spectroscopy By M. Przybylski and U. Gradmann (With 2 Figures) ........ 426
The Study of Epitaxy with Spot Profile Analysis of LEED (SPA-LEED). By M. Henzler (With 5 Figures) .............. 431
Epitaxial Growth Studied by Surface X-Ray Diffraction By J.E. Macdonald, C. Norris, E. Vlieg, A. Denier van der Gon, and J.F. van der Veen (With 4 Figures) ................... 438
XII
The Epitaxial Growth of Nickel on Cu(lOO) Studied by Ion Channeling. By P.F.A. Alkemade, H. Fortuin, R. Balkenende, F.H.P.M. Habraken, and W.F. van der Weg (With 2 Figures) 443
Structure and Ferromagnetism of Thin Magnetic Layers By R.F. Willis (With 4 Figures) ........................ 450
Part VII Phase Transitions
Temperature-Dependent Dynamics of a Displacively Reconstructed Surface: W(OOl) By C.Z. Wang, A. Fasolino, and E. Tosatti (With 3 Figures) .. .. 458
Surface Core Level Shifts for the Clean-Surface and Hydrogen- Induced Phase Transitions on W(lOO) By J. Jupille, K.G. Purcell, G. Derby, J. Wendelken, and D.A. King (With 3 Figures) ........................ 463
Theory of Phase Transitions on H/W(llO) and H/Mo(llO) Systems By D. Sahu, S.C. Ying, and J.M. Kosterlitz (With 3 Figures) 470
Critical Phenomena of Surface Phase Transitions: Theoretical Studies of the Structure Factor By T.L. Einstein, N.C. Bartelt, and L.D. Roelofs ............ 475
Order-Disorder Critical Behaviour in the System Oxygen on Ru(OOl). By P. Piercy, M. Maier, and H. Pfnur (With 4 Figures) 480
Structure and Phase Transitions of Incommensurate Xe Layers on Pt(lll). By K. Kern, P. Zeppenfeld, R. David, and G. Comsa (With 3 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 488
The Temperature Dependence of the Near Order Structure of Au(llO) Studied by Ion Scattering Spectrometry (ISS) By H. Derks, J. Moller, and W. Heiland (With 4 Figures) ...... 496
High Resolution He-Scattering Studies of Physisorbed Films By K. Kern, R. David, P. Zeppenfeld, and G. Comsa (With 4 Figures) .................................... 502
VII. 2 Roughening
The Step Roughening of the CU(1l3) Surface: A Grazing Incidence X-Ray Scattering Study By K.S. Liang, E.B. Sirota, K.L. D'Amico, G.J. Hughes, S.K. Sinha, and W.N. Unertl (With 3 Figures) .............. 509
XIII
Roughening on (11m) Metal Surfaces By E.H. Conrad, L.R Allen, D.L. Blanchard, and T. Engel (With 4 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 514
Determination of the Kink Formation Energy and Step-Step Interaction Energy for Vicinal Copper Faces by Thermal Roughening Studies By F. Fabre, B. Salanon, and J. Lapujoulade (With 3 Figures) 520
The Phase Diagram of Vicinal Si(111) Surfaces Misoriented Toward the [110] Direction By RJ. Phaneuf and E.D. Williams (With 4 Figures) ......... 525
Atom Scattering from a Markovian bcc(001) Surface By A.C. Levi, R Spadacini, and G.E. Tommei (With 3 Figures) 530
VII. 3 Surface Melting
Theory of Surface Melting and Non-Melting By E. Tosatti (With 7 Figures) ......................... 535
Experimental Investigations of Surface Melting By J.W.M. Frenken, J.P. Toennies, Ch. Woll, B. Pluis, A.W. Denier van der Gon, and J.F. van der Veen (With 6 Figures) 545
Mean-Field Theory of Surface Melting By A. Trayanov and E. Tosatti (With 5 Figures) 554
Mobility of the Surface Melted Layer of CH4 Thin Films By M. Bienfait and J.P. Palmari (With 2 Figures) ........... 559
Diffraction Studies of Langmuir Films. By J .B. Peng, B. Lin, J.B. Ketterson, and P. Dutta (With 3 Figures) .............. 564
Part VIII Defects, Disorder and Morphology
Simulation of Substitution Disorder Within Chemisorbed Monolayers By V. Maurice, J. Oudar, and M. Huber (With 2 Figures) 570
Ordered Dimer Structures and Defects on Si(001) Studied by High Resolution Helium Atom Scattering. By D.M. Rohlfing, J. Ellis, B.J. Hinch, W. Allison, and RF. Willis (With 4 Figures) ...... 575
Structure of the CaF2 (111) Surface and Its Change with Electron Bombardment Studied by Impact Collision Ion Scattering Spectroscopy (ICISS) By R. Souda and M. Aono (With 7 Figures) . . . . . . . . . . . . . . .. 581
XIV
Electron-Beam-Induced Surface Reduction in Transition-Metal Oxides. By D.J. Smith, L.A. Bursill, and M.R. McCartney (With 5 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 588
Surface Structure of Metallic Glasses Studied by Scanning Tunneling Microscopy By R. Wiesendanger, L. Eng, H.R. Hidber, L. Rosenthaler, L. Scandella, U. Staufer, H.-J. Giintherodt, N. Koch, and M. von Allmen (With 4 Figures) ..................... 595
The Surface and Near Surface Structure of Metal-Metalloid Glasses By W.E. Brower Jr., P. Tlomak, and S.J. Pierz (With 4 Figures) 601
Calculation of Diffracted Laser Beam Intensities from Non-Sinusoidal Periodic Surface Profiles Extending in the [OOl]-Direction on Pt(llO) By E. Preuss and N. Freyer (With 3 Figures) ............... 606
Structural Changes on Ni Surfaces Induced by Catalytic CO Hydrogenation By D.A. Wesner, F.P. Coenen, and H.P. Bonzel (With 3 Figures) 612
Surface Generation of Rayleigh Waves by Picosecond Laser Pulses By D. Jost, H.P. Weber, and G. Benedek (With 2 Figures) ..... 618
Index of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . .. 625
xv
Introduction
In the past years, our knowledge of the structure of solid surfaces has advanced considerably. It now proves possible to make atoms on a surface directly visible and to have their positions theoretically explained in terms of their electronic bonding configuration. These breakthroughs have been realized on the exper­ imental side through the introduction of a variety of novel surface analytical tools and on the theoretical side through the use of new computational tech­ niques. As a result, the data base of solved structures has expanded to the point that common trends among different crystal faces and overlayer systems can be identified. These proceedings cover the latest developments in the field, grouped according to the following scheme.
Part I treats various theoretical and experimental aspects of new analyt­ ical techniques. In particular, it focuses on the microscopy of surfaces, based either on electron diffraction or on scanning tunneling microscopy.
Relaxation and reconstruction phenomena in metal surfaces are the sub­ ject of Part II. The analyses have been refined to a level where even sub-surface atom displacements can be determined with reasonable accuracy. Not only the static displacements but also the vibrational properties can be measured and calculated.
An important topic in surface crystallography is the characterization of adsorbate overlayers on metals. This subject is extensively covered in Part III. The results are relevant for understanding heterogeneous catalysis and corrosion phenomena. Noteworthy is the progress that is being made with the detection and location of hydrogen atoms at surfaces.
Parts IV and V present recent results that have been obtained on clean and adsorbate-covered semiconductor surfaces. In particular, the atomic ge­ ometries of various metal overlayers on silicon are investigated as well as the structural rearrangements induced by the adsorbate.
Part VI provides new information on the structural aspects of epitaxial growth of thin films. A number of articles deal with the influence of lattice strain on various physical properties of the film; other papers focus on the growth process itself.
Phase transitions form the subject of Part VII. For two-dimensional over­ layer systems both theory and experimental methods are well developed. In­ triguing are the atomic-scale observations of surface melting. In this chapter,
various phenomenological descriptions of surface melting are discussed and a first microscopic theory is presented.
Part VIII discusses various types of surface defects and disorder, which are induced either by the preparation technique or by the surface probe. At­ tention is also given to near-macroscopic changes in surface morphology that may occur upon heating or exposure to a reactive gas. Results in these ar­ eas are significant for understanding the behaviour of surfaces under practical circumstances.
2
J. Tersoff
IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA
In any microscopy, it is extremely useful to know the resolution (or more precisely the resolution function) of the instrument. Here the present under­ standing of the resolution of Scanning Tunneling Microscopy (STM) is reviewed.
The principles of STM have been described in detail by the inventors [1], and are not repeated here. Understanding, or even defining, the resolution of STM raises tricky issues for two reasons. First, STM is inherently nonlinear, so the usual definition of resolution in terms of convolution with an instrumental func­ tion cannot be applied directly. Second, STM is actually a spectroscopy, in that it is sensitive to the electronic structure of the sample. It is therefore hard to say what is the "ideal" image which one would expect for an arbitrarily sharp resol­ ution.
The resolution of STM has been previously addressed by Tersoff and Hamann [2] and by Stoll [3] in the case of simple metals. More recently, it was pointed out [4] that the resolution may be quite different for semiconducting or semimetallic surfaces. In fact, the resolution of STM is inherently sample-dependent in princi­ ple, and often in practice.
Here, a specific convention is described for defining the "ideal" STM image. Then, using this convention, a formal expression is derived for the resolution function. This expression is evaluated for metals, giving a very reasonable in­ strumental resolution function which is relatively independent of the sample. Then the case of a semiconductor or semimetal surface is briefly discussed. In that case, the resolution function becomes very sample-dependent, with a rather pe­ culiar lineshape.
In general, one defines resolution by assuming that there exists some ideal image Io(x) , which would be seen in the case of perfect instrumental resolution.
4
The actual measured image 1(x) is then related to 10 via the resolution function F,
1(x) = j1o(X - y)F(y)dy . (1)
Equation (1) should be viewed as the definition of the resolution function F(x). It is often convenient to Fourier transform (1) to obtain
1(q) = 1o(q)F(q) . (2)
Here I(q) and F(q) are the Fourier transforms of I(x) and F(x). (Arguments r, x, or y indicate real-space quantities here, while arguments k, G, or q indicate reciprocal-space or Fourier transformed quantities.) The obvious advantage of (2) is that F(q) can be determined directly as l(q)/10(q).
It is important to recognize that F(x) is a well-defined function, independent of the specific image, only if the measured image 1 (the "output") is a linear function of the "true" image 10 (the "input"). While the meaning of 10 may be obvious in the case of an optical microscope, it is not so for STM.
We now consider the general form of the STM image, in order to motivate a choice of 10 and to set the groundwork for evaluating the resolution function F. Within the model of Tersoff and Hamann [21 the image corresponds to a contour of constant surface local density of states p(r , EF), where
(3)
Here E~ k i is the energy of the eigenstate 1/;, k II is the surface wavevector, and the index II runs over the remaining quantum numbers.
The approximations involved in this model are expected to be rather accurate in most cases [2]. Moreover, the model has proven adequate for the quantitative interpretation of STM images [2,5], and has been tested by comparison with more exact calculations in simple cases [6]. We therefore accept without further dis­ cussion that the STM image does in fact correspond to a contour of p(r , EF), to sufficient accuracy for t~e present discussion. (For finite voltage, it is merely necessary to integrate p(r , E) over the appropriate range of energy, with the added difficulty that 1/; should in principle be calculated in the presence of the electric field.)
Let us rewrite p(r , EF) as p(x, z), where we separate lateral and vertical po­ sition as r = (x, z), an~ we suppress the energy argument for notational simplic­ ity. The STM image z(x) is implicitly defined by
5
p(x, z) = PT' (4)
where PT is proportional to the tunneling current at which the microscope is op­ erated [2]. In the limit of weak corrugation, it is convenient to write
z = Zo + r(x, zo), (5)
where Zo is some average tip height, which depends on PT but is not in general experimentally accessible, and r is the small corrugation which const~utes the image, and which depends on Zo or equivalently on PT. Expanding p(x, z) about z = zo, (4) and (5) give
- - d-r(x, zO)~[PT - p(x, zo)] / dz p(x, zo)· (6)
Because of the eXJ!onential decay 01 the wavefunction, for weak corrugation one can write [2] dp(x, zo)/dz ~ - p(x, zo)/"A. The decay length "A is discussed below. Also, the ch~acteristic tip height Zo is defined by the condition that the lateral average of p(x, zo) is
J p(x, zo)dx == Po(zo) = PT·
To lowest order in the small quantity [p(x, zo) - PT] / PT one can write
(7)
The important point is that, in the limit of weak corru~ation, the image rex , zo) is simply proportional to the fractional variation of p(x, zo) about its mean value.
In analogy_with (7), we therefore propose to identify 10 with the fractional variation of p(x, zo) about its mean value, evaluated in a plane at or very near the surface. Taking the origin in the surface plane,
IO(x) == "A[p(x, 0) - Po(O)] / PoCO) • (8)
There remains a minor ambiguity in the choice of the surface plane z=O. Rea­ sonable choices might include the average classical turning point for electrons at the Fermi level, or more simply, the plane of the outermost layer of nuclei.
It is now convenient to assume that the surface is periodic, and to work with the Fourier transformed quantities. The periodicity can later be taken to be ar­ bitrarily large, to include nonperiodic surfaces. Then
p(x, z) = ~>(G, z) exp(iGx), (9) G
6
where G are the surface reciprocal lattice vectors, and Po was defined above to be just the G=O term of (9). Now (7) and (8) may be rewritten as
- - r(G, ZO)~Ap(G, zo)1 Po(zo), (10)
and
10(G) = Ap(G, 0)1 Po(O) . (11)
Combining (10) and (11) with (2) gives the desired expression for the resolution function, valid for weak corrugation:
(12)
Thl.s formula may of course be Fourier transformed to give an explicit lineshape F(x). Note that the resolution depends explicitly on Zo ' which is determined by the choice of tunneling current and by the tip radius [2].
The approach used here for evaluating the resolution function in specific cases clo~ely follows Ref. [2]. Specifically, by neglecting the variation in the potential V(r) over the region of interest, the wavefunctions can be expanded in general­ ized (complex) planewaves, i.e.
"'kU = ~:a<i~II' G) exp [i(k ll + G)i] exp( - KGZ) , G
(13)
where KC -= 1 k ~ +G 12 + ;;. , and fj2K2/2m = V - Ek II. V is the potential in the region of interest, and the index JJ of (3) has been suppressed for brevity. It is then
a simple exercise to expand p(G, z) in terms of products of such complex planewaves, using (3) and (13) as in Ref. [2].
In principle, a precise evaluation of the resolution from (12) requires a de­ tailed knowledge of the surface electronic structure. However, an ansalz based on the superposition of atomic-like densities [2,7] yields
(14)
where A is a constant independent of G.
Two other approaches appropriate for metal surfaces, based respectively on an asymptotic analysis and on a "most typical" wavefunction [2,7], give results
7
virtually identical to (14) for the first couple of Fourier components of (9). These lowest components are usually the only ones of interest, because for large G, KG
is rather large compared to K. Then for reasonable values of z, p(G, z) is negligible compared with Po(z) , and so rG is unobservably small.
From (14), one may immediately obtain an explicit form for the resolution function. Using (12),
(15)
However, because the higher G components are somewhat model dependent, the most useful part of (15) in the case of a periodic surface is its value for the smallest non-zero G.
Often the observable G components obey G«2K, and (15) may be expanded to give
(16)
This model resolution function is simply a Gaussian with rms width (zo /2/C) 1/2. According to Ref. [2], the relevant value for Zo is the sum R+d of the effective tip radius of curvature R and the tunneling gap distance d.
Equations (14)-(16) are based on the assumption that a range of k I contrib­ utes to the tunneling. This assumption is particularly appropriate for metals, al­ though it may sometimes apply well to semiconductors, especially at moderately large tunneling voltage. In contrast, in STM of semiconducting surfaces at the lowest possible tunneling voltage, tunneling takes place into or out of states at the band edge. The analysis is then actually simpler than for a metal surface, since only one or a few states (or pockets of states) contribute to (3).
Consider in particular tunneling to states which are quasi-two-dimensional. These could be either surface states, as for Si(111) 2xl [8], or states of a quasi­ two-dimensional material such as IT-TaS2 or graphite. Then for semiconductors (and often for semimetals), the states at the band edge (or Fermi level) generally fall at either the center or edge of the surface Brillouin zone. The case of tunneling to zone-edge states is particularly interesting, and has been discussed briefly in Ref. [4]. Here the implications of that work for STM resolution are merely sum­ marized.
For tunneling to states at the edge of the surface Brillouin zone, k I = g/2 , where g is the smallest G. Substituting this into (13) and then (3), and assuming
8
reflection symmetry, one finds that asymptotically p(g, z) = po(z)/2 for all z, al-
though for G>g, p(G, z) decays faster with increasing G, as expected.
Substituting this result into (12) gives F(g)=F(O)=l, independent of z, while for G>g, F(G) decreases with increasing z as for a metal. This is a very peculiar result. It implies that, for large z (i.e. large tunneling distance or tip radius), the ability to resolve structure within the unit cell decreases and is lost, just as for a metal surface; but because F(g) = 1, the unit cell itself is well resolved even if it is very small (large g), and even if the tip is relatively blunt, as long as the model of Ref. [2] is applicable.
This enhanced resolution of the unit cell was noted in Ref. [4]. The effect is particularly striking for graphite, where the 2 A unit cell is easily resolved, even though such small structures have never been successfully resolved on metal sur­ faces. In fact, it may well be the case for most semiconductor surfaces, that the resolution is enhanced over that expected for metals by electronic structure ef­ fects.
References
1. G. Binnig and H. Rohrer, Helv. Phys. Acta 55, 726 (1982), and Surf. Sci. 152/153,17 (1985).
2. J. Tersoff and D.R. Hamann, Phys. Rev. Lett. 50, 1998 (1983), and Phys. Rev. B 31, 805 (1985).
3. E. Stoll, Surf. Sci. 143, L411 (1984).
4. J. Tersoff, Phys. Rev. Lett. 57, 440 (1986).
5. R. M. Feenstra, J. A. Stroscio, J. Tersoff, and A. P. Fein, Phys. Rev. Lett. 58, 1192 (1987).
6. N. D. Lang, Phys. Rev. Lett. 56,1164 (1986).
7. J. Tersoff, M. J. Cardillo and D. R. Hamann, Phys. Rev. B32, 5044 (1985).
8. J. A. Stroscio, R. M. Feenstra, and A. P. Fein, Phys. Rev. Lett. 57, 2579 (1986).
9
w. Sacks, S. Gauthier, S. Rousset, and J. Klein
Groupe de Physique des Solides de l'Ecole Normale Superieure, Universite Paris VII, Tour 23, 2 place Jussieu, F-75251 Paris Cedex OS, France
The problem of determining the tunneling current between two electrodes that are nonplanar is essential to the theory of Scanning Tunneling Microscopy (S.T.M.) [1-10]. J. Tersoff and D.R. Hamann have shown that for a spherical tip as one electrode, and including only an evanescent s wave function, the current is proportional to the local density of states (LDOS) of the second electrode, at the Fermi level, evaluated at the center of the tip [1,2].
We have calculated the wave functions for a free-electron metal with a corrugated soft-wall potential in a perturbation expansion in terms of the surface profile function h S (x), as well as the Fermi level LDOS. This is applied to the problem of S. T. M. imaging, (surface topography and image resolution) within the spherical tip model, as in this case the image consists of contours of constant current or constant LDOS.
In the free-electron case, the isodensity contours (equivalently the probe path) take the simple asymptotic form:
(1)
where z R+d (tip radius plus average tip-surface distance) The corrugation dD is a simple convolution over hS(x):
dD(x,Z) (2)
f (x, z) = (K/1t2) exp (-x2 K/2) (3)
and K = (2m$/fi2)1/2, $ the work function.
The f.w.h.m. of f(x,2) determines the lateral resolution as Ll/2 = 2 (ln2 2/K) 1/2, a result in agreement with many authors [2-5] .
As an example of the use of (2), consider a simple stepped surface defined by hS(x) h S El(x), then the S.T.M. image follows as:
z (x,y) z + h S (1 + <I> [x (K/2) 1/2]) /2 (4)
10
where ~(y) is the error function. These contours are illus­ trated in Fig.1 for different values of Z. While the step height is the same in each successive trace, i.e. equal to h S ,
the broadening of the step image is significant and roughly equal to (8 Z/K) 1/2 = (8 (R+d)/K)1/2. A more complete discussion of these results will be given elsewhere [11).
18
16
14
12
10
X (a.u.)
~: Contours of constant current along the x­ direction for a step (hS = 4 a.u., K = 1/2 a.u.-1).
It is instructive to question whether or not the tip follows the LDOS of the sample surface, if the tip is not spherical (6). We therefore propose an alternative model specifically to address this question.
As depicted in Fig. 2, we consider two electrodes bounded by surfaces that are nonplanar: the right surface has a finite protuberance on an otherwise perfect plane, which will represent the tip, while the left corrugated surface represents the sample. The tunneling current for this system, subtracting the contribution due to the plane, is explicitly dependent on both the surface and the tip profiles, and is generally not proportional to the LDOS of the left isolated system [6,10). The main purpose of this work is thus to compare the corrugation of the isocurrent, ~c(x,z), with that of the LDOS, or ~D (x, z) .
In this new geometry, ~c(x,z) is found to be:
(5)
11
with
Fig. 2: Geometry of the two electrode system.
where f is approximately the same as in (3) and <ht> is the average tip height [12]. In general, therefore, the corrugation 8 C is different than that of the LDOS. Due to the additional
convolution over ht (x), the tip significantly influences the image, and indeed also the resolution, due to the width of g. If the tip is asymmetric about the z-axis, even the form of the image may not resemble the surface. Such examples will be considered elsewhere [12].
In the simple case of a symmetric gaussian tip of the form ht(x) = h t exp(-br2), with f.w.h.m. r = 2(ln2/b)1/2, we have the
result that: g (y, z) = f (y, z + K/b). The isocurrent contours follow the isodensity contours however translated away from the left surface by z ~ z + K/b. From this result, the resolution
for the gaussian tip is Ll/2 = 2(ln2(z + K/b)/K)1/2, which for large tip widths becomes Ll/2 = 2(ln2/b)1/2 = r. This is to be
contrasted with the spherical tip, in which Ll/2 ~ Rl/2, in the same limit.
In conclusion, the methods outlined in this report enable one to obtain simple, although approximate, analytic expressions for the S.T.M. image in the free-electron case. It also permits a comparison between the spherical tip, in which the LDOS of the second electrode is the measured quantity, and an arbitrary tip, which only approximately follows the LDOS. The S.T.M. resolution in both cases is discussed.
12
References
1. J.Tersoff and D.R.Hamann: Phys. Rev. Lett.50, 1998 (1983) 2. J.Tersoff and D.R.Hamann: Phys. Rev. B 31, 805 (1985) 3. E. Stoll: Surf. Sci. 143, L411 (1984) 4. E.Sto11, A.Baratoff, A.Selloni, and P.Carnevali: J.Phys.
17,3073 (1984) 5. N. Garcia: I.B.M. J. Res. Develop.30 5, 533 (1986) 6. Feuchtwang and Cutler, Phys. Scr.35, 132 (1987) 7. N.D. Lang: Phys. Rev. Lett.56, 1164 (1986) 8. N.D. Lang: Phys. Rev. Lett.58, 45 (1987) 9. N.D. Lang: Phys. Rev. B 34, 5947 (1986) 10. A.Baratoff: Physica 127 B, 143 (1984) 11. W.Sacks, S.Gauthier, S.Rousset, J.Klein, and M.A.Esrick:
Phys. Rev. B 35, (1987) 12. W.Sacks, S.Gauthier, S.Rousset, and J.Klein, (unpublished)
13
Tensor LEED; New Prospects for Surface Structure Determination by LEED
P.J. Rous1 and J.B. Pendry2
IThe Cavendish Laboratory, University of Cambridge, Madingley Rd, Cambridge CB30HE, United Kingdom
2The Blackett Laboratory, Imperial College, London SW72BZ, United Kingdom
Recently we have proposed a new calculational scheme for the evaluation of LEED I!V spectra from complex surfaces called Tensor LEED (TLEED) [1-3]. This is primarily a perturbative scheme which allows the rapid evaluation of I!V spectra from a complex trial structure which is considered to be a distortion of a (simpler) reference surface. This technique is, in many cases, two to three orders of magnitude faster than conventional methods when applied to a standard trial and error structure determination.
Tensor LEED is ideally suited to calculations involving complex reconstructed surfaces, such as the III-V compound semic~nductors, for which conventional approaches suffer from the N scaling of computational effort with the number of atoms in the surface unit cell. If the undistorted structure is chosen to be the unreconstructed surface then the calculation of I!V spectra from each reconstructed, trial, surface scales as N the optimum scaling for LEED calculations [4]. However, in attempting to determine the structure of such complex surfaces one encounters additional obstacles. Often the atomic displacements are too large for the direct application of TLEED for which the reference structure is the unreconstructed surface. It is necessary to consider a number of reference surfaces as starting points for a series of Tensor LEED calculations. More importantly, however as the number of inequivalent atoms in the surface increases so does the volume of parameter space of possible trial structures which must be explored for a reliable structure determination.
Whilst the comparative efficiency of TLEED can go some way to alleviating this difficulty, the conventional trial and error approach becomes extremely cumbersome in such cases and often one is forced to confine the structure search to only a part of the possible parameter space. The solution to this problem lies in the use of a systematic search strategy to efficiently locate the best fit structure. Such a scheme, based upon a standard method of optimisation, has recently been proposed and tested by COWELL et al [5,6,7]. The purpose of this paper is to demonstrate that Tensor LEED is ideally suited for incorporation into such an optimised structure search and, in particular, can be used to determine directly the derivatives on the R-factor hypersurface with respect to variations of the structural parameters. First, however, we review the pertintent aspects of the theory of Tensor LEED.
14
The Theory of Tensor LEED
The concept which is fundamental to the theory of Tensor LEED is that of a pair of surface structures; the reference surface and the trial structure. The reference surface is a particular surface structure from which we treat the scattering from a fully dynamical standpoint by performing a LEED calculation including full multiple scattering corrections. The trial surface is regarded as a distortion of the reference structure generated by displacing atoms from their positions in the undistorted surface. If these displacements are small then we can consider this distortion as a weak perturbation of the crystal potential which can be treated by a simple perturbation theory.
Tensor LEED comes in two levels of sophistication. At the simplest level, which treats the smallest distortions, we write the change in potential generated by displacing the ith atom of the reference structure through or i as
OVi = YV(E-Ei).ori . (1)
The induced change in the amplitude of the LEED beam with momentum transfer parallel to the surface of ~//-~// is
OA(k/'/,k//) = E. E. T .. (k/'/,k//)or .. , (2) - - ~ J ~J - - ~J
where the sums are taken over j=x,y,z and the N displaced defined as
where ik > is the LEED state having p&tallel momentum ~L/ vicinity of each displacea calculation.
the three cartesian coordinates atoms i=l .. N. The tensor T is
excited by an incident LEED which can be evaluated in
atom by a conventional
LEED
The linear relationship between the atomic displacements or. and the corresponding change in the amplitude of each LEED beam implies that once T is known the generation of I/V spectra from a large number of -trial structures for which (1) holds can be achieved with virtually negligible computational effort. This linearity can also be exploited to evaluate R-factor derivatives as we shall show in the next section.
Whilst this first version of Tensor LEED has been shown to be capable of treating a~omic displacements, even of entire atomic planes, of up to O.15~, for larger distortions we require a more sophisticated theory. In this case we employ a renormalised perturbation scheme to rewrite (2) in an angular momentum basis as
(4 )
ot(or.) is the change in the single site atomic t matrix generated by the displacement or. and referred to an origin at the position of the ith atom in~the reference surface. The sum is taken over the pair of angular momentum components L=(l,m).
15
In a plane wave basis ot expressed as a simple phase the undisplaced atom t
has a kinematic form and can be factor multiplying the t matrix of
<~' lot(or) I~> = (exp{i(~I-~).or}-l)<~' Itl~>·
Whilst it is now necessary to evaluate the ots for each structure, this is a comparatively efficient procedure so sUbstantial time savings over conventional methods remain.
Tensor LEED and R-factor Optimisation
(5 )
trial that
For a number of years there has been some interest in the use of optimisation methods within the conventional trial and error structure search in which calculated I/V spectra are compared to the experimental spectra [8,10). Only recently, however, has such a scheme been put into practice by COWELL et al who have succesfully applied Hooke-Jeeves optimisation to the determination of the structure of a compound semiconductor surface [5,6,7). These techniques use standard methods of optimisation as the basis for a systematic structure search reducing the number of distinct LEED calculations which must be performed to locate the R-factor minima and thus the best-fit structure.
Before starting an optimised search it is necessary, in the terminology of COWELL et al, to define a number of "base points". A base point is a position in parameter space defining a particular surface structure. The optimisation scheme is then used to explore efficiently the region of parameter space in the vicinity of each base point, locating any nearby R-factor minima. Clearly, the original set of base points must be carefully chosen so that all physically reasonable local R-factor minima are found from which the global R-factor minimum corresponding to the actual surface structure can be selected.
It is clear that if these base points are considered as reference surfaces for Tensor LEED then, since the structure search is confined to the vicinity of each base point, the second version of Tensor LEED can be directly applied, in its present form, to any existing optimisation scheme. Apart from its comparative speed, Tensor LEED also has an additional advantage over more conventional methods. As has been pointed out by ADAMS [7), LEED calculations are not well suited to optimisation since it is usually most efficient to calculate LEED intensities for many different trial structures at a each energy point (i.e. the loop over structures is inside the energy loop). This is because it is possible to store and reuse energy dependent quantities such as layer reflection matrices which are identical for several different surface structures. This is not the case in the calculational scheme which we employ for Tensor LEED [8). Once the investment in the time consuming reference calculation has been made then it is, in fact, just as efficient to place the loop over energies inside the loop over strucutures. This transposition of the conventional order of calculation simply follows from the method used to construct the ots of (4) further details of which can be found elsewhere [9). Thus by using Tensor LEED it is possible to employ a systematic
16
procedure in which the theory/experiment comparison is performed as the theoretical I/V spectra for each trial structure are calculated. The structural parameters for the next trial surface can then be determined "a posteriori" by the application of a suitable optimisation algorithm.
However, of more importance is the potential of the first version of Tensor LEED for the calculation of R-factor derivatives with respect to variations of the structural parameters. In any optimisation scheme it is necessary to choose the direction in parameter space in which the structure search must proceed from the base point. To do this one must calculate the R-factor derivatives which define the local curvature of the R-factor hypersurface. This is usually done by performing a number of full dynamical LEED calculations around the base point and then selecting the direction of steepest descent. In the next section we shall show that, by using the first version of Tensor LEED, it is possible to determine directly the R-factor derivatives by performing a single LEED calculation at the base point.
R-factor Derivatives From Tensor LEED
Let us use the first version of Tensor LEED and consider an infinitesimal set of displacement vectors or.. denoting the change in the jth coordinate of the ith atom ofJthe base point structure. By performing a full dynamical LEED calculation for this reference structure we generate the tensor ~ and find the infinitesimal change in amplitude of each reflected LEED beam (2). The new intensity at each energy point is now
IO + oIo =: Ao + Ei Ej
where I and A denote the beam re~lectedOby the base order we have
Io + OIo :AO:2 + 2Ei
* Re(AoTij)orij'
(6 )
(7 )
(8 )
Thus we have obtained an exact expression for the derivatives of the intensity at each energy point with respect to variations in each of the 3N coordinates of the multidimensional parameter space. What we actually require is the R-factor derivatives which we evaluate by sUbstitution of (7) into the formula for the particular R-factor we wish to use. Since many popular R-factors have a complex functional dependence upon the calculated intensity this may have to done numerically. However, a more direct approach is to attempt to preserve the linearity of (8) and adopt a R-factor which is linear in the calculated intensities or their derivatives with respect to energy.
For example, substituting (7) into the formula for the X-ray R-factor R2 [10]
17
we obtain a direct expression for its derivatives
dR2/dr ij = 4J{ItheOry(E)-CIexpt(E)}Re{A:Tij(E)} dE. (10)
The constant c norma1ises the two spectra to each other at the base point. Similar relationships hold for the R-factor R1, and R-factors such as RPP1 and RPP2 which depend upon the derivatives of the intensity with respect to incident electron energy [10]. Clearly (10) allows us to examine the sensitivity of the R-factor to variations in each of the structural parameters- information which can be used to selectively restrict the number of parameters which are varied simultaneously in the initial stages of the subsequent structure search. Of course, it is well known that such simple R-factors are often inadequate for determining the final best-fit structure in a conventional structure search. However the X-ray R-factor has been shown to be adequate for determining the direction of steepest descent through its derivatives, as one of a hierachy of R-factors used within the optimisation scheme [7].
Thus we have shown that Tensor LEED is ideally suited for incorporation into an R-factor optimisation algorithm. This can be achieved by the direct application of the second version of Tensor LEED, the calculation of the curvature of the R-factor hypersurface by the first version, or even a combination of both methods. Clearly, the next step is to put the theory presented here into practice- work upon which is already underway.
We would like to acknowledge lengthy discussions with Dr. P.G. Cowell, Dr S.P. Tear and Professor M. Prutton upon the subject of R-factor optimisation. PJR thanks the U.K. SERC for continued financial support through the award of a Postdoctoral Research Fellowship.
References
1. P.J. ROus, J.B. Pendry, D.K. Saldin, K. Heinz, K. Mueller, N. Bickel: Phys. Rev. Lett. 57 2951 (1986).
2. P.J. Rous, J.B. Pendry: Submitted to Surface Sci. 3. P.J. Rous: PhD thesis, University of London (1987) 4. J.B. Pendry: In Determination of Surface Structure by
LEED, eds. P.M. Marcus, F. Jona (Plenum, New York, London 1984).
5. P.G. Cowell, M. Prutton, S.P. Tear: Surface. Sci. 177 L915 (1986).
6. V.E. De Carvalho, M. Prutton, S.P. Tear: Surface. Sci. 184 198 (1987).
7. P.G. Cowell and V.E. de Carvalho: This conference. 8. D.L. Adams: presented at the International Seminar on
Surface Structure Determination by LEED, Erlangen 1985. 9. P.J. Rous and J.B. Pendry: Submitted to Comput. Phys.
Commun 1987. 10. M.A. Van Hove, W.H. Weinberg, C.-M. Chan: Low Energy
Electron Diffraction, Springer Ser. Surface Sci., Vol 6 (Springer, Berlin, Heidelberg 1986) p. 237-244. and references therein.
18
Comparison of the Quasidynamical and Tensor LEED Approximation for LEED Intensity Spectra from a Reconstructed Surface
N. Bickell, K. Heinz l , H. Landskron l , P.J. Rous2, J.B. Pendry2, and D.K. Saldin 2
1 Lehrstuhl fiir Festkorperphysik, University of Erlangen-Niirnberg, Erwin-Rommel-Str. 1, D-8520 Erlangen, Fed. Rep. of Germany
2The Blackett Laboratory, Imperial College, London SW72BZ, United Kingdom
The quasi dynamical and Tensor LEED approximations of intensity spectra are compared to each other and to full dynamical data. Both approximations save considerable computer time. Tensor LEED approaches exact data more closely when starting from a close enough reference structure. On the other hand the quasi dynamical method puts no restrictions for an inspection of a larger parameter space. A combination of both methods is therefore suggested for optimization of computational efforts. Finally full dynamical and approxi­ mative results are compared to experimental data from reconstructed c(2x2) W(100) .
1. I ntroducti on
The development of surface structure determination by LEED towards more complex structures is mainly inhibited by the existence of multiple elec­ tron diffraction. A full dynamical calculation is in general necessary to reproduce experimental spectra. Especially intralayer multiple scattering requires both large computer time and memory. As the necessary computing efforts scale with N3 (N = number of atoms within the unit cell) practical limits enforced even by fast computers are usually reached with values not much higher than N = 5. Therefore, much efforts were undertaken in the past in order to avoid the full dynamical treatment by using approximations, which neglect certain parts of the various multiple scattering processes. Among them are quasidynamical approach /1/ and the recently developed Tensor LEED method /2/. It is the purpose of this contribution to compare both approximations with respect to reliability and computational efforts.
2. The Quasidynamical and Tensor LEED Approach
The quasidynamical method (QD) relies on the assumption that electrons im­ pinging on an atomic layer are only negligibly scattered into the layer, where they undergo multiple scattering. Then, of course, the layer dif­ fraction is approximately kinematic and easy to be calculated. Originally the method was applied at normal LEED energies (20-200 eV) and sometimes oblique incidence of the primary beam /3-6/. However, it was found that it works much better when applied for normal incidence and higher energies (> 150 eV) /1, 7, 8/. In this case forward scattering is highly dominating and so layer diffraction matrices computed kinematically are a good approxi-
19
mation for forward diffraction. This makes layer interference peaks appear at about the correct position though not necessarily with the correct height. Applying a mainly peak position sensitive R-factor for the theory-experiment fit, e.g. the Pendry R-factor /9/, the correct structure could be retrieved in a number of cases.
The approach through Tensor LEED (TL) is much more sophisticated and basically different. The main idea appears from the observation that com­ puted I(E)-spectra change only gradually when structural model parameters are changed by amounts of the order of 0.1 A. Therefore, starting from a full dynamical calculation of a certain initial reference structure modi­ fications of the latter should be computable using a proper perturbation scheme. The corresponding theory was developed only very recently and it was shown that reliable data can be obtained for structure parameter changes of up to 0.4 A /2/. As the change of the total diffraction amplitude can be formally written as the product of a tensor by the vector of atomic dis­ placements or, more sophistically, of modified t-matrices, the method is called Tensor LEED.
3. Application to zig-zag c(2x2) Model Reconstruction of a bcc(lOO) Surface
In this section QD and TL are applied to a bcc(lOO) surface whose top layer is reconstructed according to the zig-zag model in fig. 1. This model was proposed for the c(2x2) reconstruction of W(lOO) /10, 11/ and we adjusted all parameters of the calculations for this surface. However, previous LEED structure determinations of this phase /12, 13/ resulted in an only re­ stricted quality of the theory-experiment fit. So, though the model is be­ lieved to display the important features of the reconstruction, some space is left open for further structural refinements. Therefore, in order to get rid of the structural uncertainty, in this section we compare the approxi­ ative results of QD and TL with the full dynamical calculation (FD) only, and leave the comparison to the experiment for the next section.
The calculations were performed in the energy range 20-250 eV using 9 phase shifts up to 104 eV and 10 of them above. The same phase shifts as in a previous investigation of the non reconstructed phase were used /14/. For FD matrix inversion was used to calculate the layer diffraction matrices, both for the reconstruction models and the non reconstructed surface used as a reference model for TL. The stacking of layers was realized by RFS in both FD and QD. The imaginary part of the inner potential was fixed to be V . = -5 eV for FD as well as TL and -7 eV for QD in order to guarantee RFS c~~vergence. All data correspond to normal incidence of the primary beam and a total of up to 89 non equivalent beams was considered.
Figures 1a and 1b display as an example the 1/2 1/2 beam spectra for TL and QD, respectively, each of them in comparison to FD data. The reference structure for TL is the unreconstructed surface with a first to second layer distance of d = 1.53 A. QD data are computed beginning only at 100 eV because QD works better at higher energies /1/. The lateral shift s as a parameter describes the zig-zag amplitude of the model (fig. 1 inset) while interlayer distances are kept constant. It is obvtous that TL approaches FD up to s = 0.44 A very closely and much better than QD. Most peaks are well reproduced by TL both with respect to position and relative height, while with QD shoulders can develop as peaks and vice versa and sometimes also peak shifts appear. As usual for normal incidence data all spectra react with only poor
20
(a) (b) -00 ••• FD
o 100 200 300 E (eV) o 100 200 300 E (eV)
Fig. 1: Results of the TL (a) and QD method (b) in comparison to full dy­ namical data for the inset model (1/2 1/2 beam)
sensitivity to parallel atomic shifts within a layer, i ,e. the zig-zag ampli­ tude.
The crucial question for any approximative scheme is whether a R-factor comparison produces the best fit minimum at the right position of the para­ meter space. Therefore we present R-factor maps and cuts throught it in figs. 2a and 2b again for both the TL and QD approximation, respectively, whereby the Pendry R-factor /9/ is used. The true structure is represented by FD spectra calculated for a zig-zag amplitude of s = 0.28 A and d = 1.49 ~, the latter corresponding to a 0.04 ~ contraction with respect to the reference structure. It appears that both TL as well as QD R-factor maps have their minimum near the right structure (s, d) = (0.28 ~, 1.49 ~). Both of them extract a zig-zag amplitude of s = 0.21 ~, i.e. smaller than the correct one. TL reproduces the correct d value, while QD predicts a slightly smaller one of 1.45 ~. In both cases the contour lines have nearly horizon­ tal orientation displaying the much higher sensitivity on d compared to that on s as appearing from the horizontal and vertical cuts through the maps, too.
Non zero R-factors and deviations from the correct structure raise the question about the reliability of the structure determination, i.e. the re­ liability of reliability factors. As pointed out earlier /9/, the variance of the R-factor can be taken to define an R-factor width 6R = R • /8 Voi /6E whereby R is the minimum R-factor and 6E the total energy range of the data used. In turn 6R defines error widths for the structural parameters which are given by bars in fig. 2. According to this procedure the TL R-factor re­ sults to be R = 0,2 (6R = 0.04) with structure parameters s = 0.21 ± 0.09 ~
21
1.50
1.45
RF ·55
OR= .005
. 1 .15 . 2 .25 . J .)5 . 1 S (a) . I . I 5 . 2 .25 . ) . )5 . 1 S (~)
R 06
: ~ --L6R : I. ._,. , ,
R 0.6V 0.4
(a) 1.431.1.7 1.51 155 l59 d (AI
~I~ ___ .,...______ __ '-061- . , AR -.:::- ' : --I' Iii , .. i , • i " • ,)r
0.12 Q20 028 036 Q4/, 5 (AI
~r.-d.-r.. _.J. U6 " AR
U3 1.1.7 1.51 1.551.59 d \Al ( b)
Fig. 2: R-factor comparison of TL (a) and QD (b) data with "experimental" data simulated by a full dynamical calculation for s = 0.28 ~ and d = 1.49 ~
and d = 1.49 ± 0.02 ~, which means that the correct structure is within the error width. For QD the R-factor levels are much higher, R = 0.55 (6R = 0. 13), as expected. For the inter1ayer distance this gives d = 1.45 ± 0.05 ~ which again includes the correct value. For the parallel atomic shift, however, the high R-factor level gives s = 0.21 ± 0.21 ~ so that no information is yielded. This is the price to be paid for the neglection of intra1ayer scat­ tering, which makes both the quality of the fit as well as the sensitivity on parallel atomic displacements decrease compared to the TL results .
Both the programs for TL and QD are much less complex than that for FD. Consequently, also much less computer memory is necessary, i.e. only about 30% for TL and even less for QD. However, this figure is true for the pre­ sent structure and might not generally hold. Moreover, the structural sim­ plicity of a program is more important than the necessary memory space. In this respect QD is the most favourable method, as layer matrices are com­ puted kinematically and coupled by e.g . RFS. For TL a full dynamical refer­ ence calculation is necessary in a time reversed mode for each beam wanted /2/ " Once this is done, the subsequent structure variations are computation­ ally simple as well. With respect to run times the full dynamical reference calculation is negligible when many trial structures are calculated. So, leaving that part out, the most striking feature of TL is that the calcula­ tion of I(E) spectra does not increase with energy, i.e. the calculation
22
of each intensity-energy point takes the same time. This is completely dif­ ferent for the FD and QD calculations where the increasing number of phase shifts and beams makes run times considerably increase with energy. Conse­ quently, it is very hard to give general figures by which one approach is faster than the other. Moreover, the saving of computer time compared to FD is strongly dependent on the number N of atoms in the unit cell. So, for a given energy, tro - N3 whilst tTL - tao - N. In the present c(2x2) recon­ struction, N = 2 applies. For this case we observed that TL is faster than FD by a factor of 2.5 at 20 eV and of 10 at 230 eV. This compares to a nearly energy independent saving for QD by a factor of about 10, so that QD and TL computer times meet near 200 eV. It should be emphasized, however, that the time saving increases with N2 for both TL and QD.
4. Application to Reconstructed c(2x2)W(100)
In this section we want to present a comparison of FD, TL and QD results to experimental data for c(2x2)W(100) taken at T ~ 100 K and normal incidence. So far, however, we could not yet inspect the complete parameter space which should also include vertical atomic displacements as well as such in the second layer. Therefore, the following comparison to the experiment is not meant as careful structure determination though the resulting R-factors are fairly low for a reconstructed surface. The procedure of measurement will be published elsewhere.
d ( J!) . 37, OR . 005 (a)
d (J!) (b)
1.15
. 1 . 15 . 2 .25 . 3 .35 . 1 s <Ill . 1 . I 5 . 2 .25 . 3 . 35 .1 s (J!)
Fig. 3: R-factor maps comparing FD (a) and TL (b) model calculations to ex­ perimental intensities from c(2x2)W(100).
Because of lack of space we cannot display spectra but only the resulting R-factor maps for 7 non equivalent beams (fig. 3a, b). Only sand d were varied and so we leave out QD results because of their very low sensitivity with respect to s. The FD minimum R-factor is R = 0.37 (~R = 0.07) and ap­ pears at s = 0.22 ± 0.13 a and d = 1.47 ± 0.04 a. Though the TL spectra are close to the FD ones, the minimum R-factor for TL is only R = 0.43 (~R = 0.08) at parameters s = 0.19 ± 0.14 a and d = 1.47 ± 0.04 a. This is in fair agreement to the FD results.
23
5. Discussion and Conclusion
It is obvious from the results above that TL approaches FD results much more closely than QD. This seems to be clear from the basics of the two methods. Whilst QD starts with nothing than the positions and phase shifts of atoms, the start for TL are exact spectra for a reference crystal whose structure is already verx near the true one, So, the Pendry R-factor between TL and FD for a 0.28 A in plane and a 0.04 A vertical shift is R = 0.24 as can be taken from fig. 1. Comparison of QD and FD results for the same structure results in a much poorer level. Whilst the sensitivities of TL and QD with respect to vertical atomic displacements are comparable, these for in plane displacements are much in favour for TL. This is easily understood by the fact that in plane atomic movements affect layer diffraction mainly by mul­ tiple intralayer scattering which is neglected in the QD approximation. So, TL appears to be more flexible than QD. However, the latter requires the less complex program which can also be used by the non specialist. Moreover, at least for the present case of two atoms in the unit cell, QD is even fas­ ter than TL and saves an order of magnitude computer time compared to FD calculations. This might change in favour of TL for an increasing size of the unit cell, because then the number of beams and so the computational efforts for QD but not for TL increase. The saving of TL over FD might then grow towards several orders of magnitude. However, the success of a TL cal­ culation heavily depends on the quality of the reference structure, which should be as close as possible to the structure wanted. From this point of view it might be a reasonable strategy to use the QD approximation for the search of a good reference structure which subsequently is refined through TL. Different from the above examples there is no need for the reference structure to be unreconstructed. So, the QD and TL approaches could be com­ bined to minimize computational efforts with simultaneous saving of relia­ bility. This might be especially true for cases where reconstruction cannot be modelled by small atomic displacements.
Concerning the comparison to experimental data of reconstructed W(100) the R-factor for FD and TL calculations resulted to give the same structural parameters for an assumed zig-zag model /10, 11/, i.e. a diagonal in plane displacement of 0.22 ± 0.13 A. The large error of ±0.13 A reflects the poor sensitivity of normal incidence data with respect to lateral displacements. Though the minimum R-factor for the full dynamical calculations is fairly low for a reconstructed surface, R = 0.37, a final structural conclusion can only be drawn after inspection of the whole paramenter space including also vertical and second layer atoms displacements. This as well as measurements at oblique incidence in order to overcome to insensitiveness with respect to lateral displacements are in progress.
Acknowledgements: The Erlangen authors would like to thank Professor K. MUller for his steady encouragement. We are indebted to Miss G. Schmidtlein for making the measurements available prior to publication. Financial support through Deutsche Forschungsgemeinschaft is gratefully acknowledged as well.
References
1. K. Heinz and G. Besold: Surf, Sci. 125, 515 (1983) 2. P. J. Rous, J. B. Pendry, D. K. Saldin, K. Heinz, K. MUller and N. Bickel:
Phys. Rev. Lett. 57, 2951 (1986) 3. D. Aberdam, R. Baudoing and C. Gaubert: Surf. Sci. ~, 125 (1975)
24
4. D. Aberdam: In Electron Diffraction 1927-1977, ed, by P. J. Dobson, J. B. Pendry and C. J. Humphreys, intern. Phys. Conf. Ser. 41 (Institute of Physics, London), 239 (1978) .
5. S. Y. Tong, M. A. Van Hove and B. J. Mrstik: In Proc. 7th Intern. Vacuum Con r. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 2407 (1977)
6. G. Cisneros: J. Vac. Sci. Technol. 16, 584 1979) 7. K. Heinz, N. Bickel, G. Beso1d and ~ MUller: J. Phys. C 18, 933 (1985) 8. N. Bickel and K. Heinz: Surf. Sci. 163,435 (1985) -- 9. J. B. Pendry: J. Phys. C 13, 937 (1980) 10. M. K. Debe and D. A. King;-Phys. Rev. Lett. 39, 708 (1977) II. M. K. Debe and D. A. King: Surf. Sci. 81, 19~(1979) 12. R. A. Barker, P. J. Estrup, F. Jona an~P. M. Marcus: Sol. St. Comm. 25,
375 (1978) 13. J. A. Walker, M. K. Debe and D. A. King: Surf. Sci. 104, 405 (1981) 14. P. Heilmann, K. Heinz and K. MUller: Surf. Sci. 89, 84 (1979)
25
M.N. Read and A.S. Christopoulos
School of Physics, University of New South Wales, P.O. Box 1, Kensington, Sydney NSW 2033, Australia
Recently there has been interest in unfilled electron surface states at clean metal surfaces where the electron is bound between the surface barrier potential and the outer layer of atoms. Information has been obtained from photoemission and inverse photoemission experiments /1,2/ and calculations from model surfaces /3,4/. States whose wave functions are localised far from the surface are more dependent on the barrier potential and are termed "barrier-induced" or "image-induced" states if the barrier approaches the image form. Those states localised closer to the crystal substrate are more dependent on the potential in that region and are termed "crystal-induced" surface states.
In the elastic scattering of electrons from metal surfaces the backscattered electrons which have insufficient energy normal to the surface to escape into the vacuum may undergo sustained multiple scattering between the barrier potential and the metal substrate. In this case the electron is temporarily trapped in this region and enters unfilled surface states of the types mentior.~d above /5/. It has been found that in many cases this multiple scattering does not actually occur because the electron loses energy due to inelastic collisions and only one internal scat~ering between the barrier and the substrate takes place /6,7/. In either case the interference between the electron directly scattered from the substrate and that indirectly scattered, once or a number of times by the barrier and substrate, gives rise to a fluctuation in the reflectivity data. These fine structure effects are termed barrier scattering features in general. If they are due to sustained multiple scattering they are called barrier resonance features and if due only to a single internal scattering at the barrier they are called interference features. Recently McRae et al. have suggested that the resonance mechanism may occur on the W(OOl) surface when the incident electrons have energies less than 10 eV and are within 260 of normal incidence /8/. Read and Christopoulos have examined in detail the scattering processes occurring under these conditions on W(OOl) for a realistic substrate and barrier model /9/. They found that the mechanism responsible for some of the features does indeed involve higher order scattering between substrate and barrier. Therefore some of the structure in the reflectivity data can be identified as barrier resonance features.
Thus the reflectivity data from W(OOl) contain information about the surface states discussed above. In order to show how this information can be extracted, we have calculated and analysed over the required range of energies and angles the reflectivity from W(OOl) using a realistic model of the substrate and barrier. We show how bound state energies can be found for comparison with theoretical calculations and results from inverse photoemission experiments. The present calculations show the wealth of features present on W(OOl) in this range and the same principles of analysis could be used for experimental data from the real surface when an extensive set of high resolution reflectivity data for low energies and small polar angles of incidence becomes available. In
26
0 0
0·4
0·2
0·6
0·4
0·2
0·6
0·4
0·2
0·0 5·0 10·0 Energy (eV )
Fig. 1. Plot of the 00 beam reflectivity for W(OOl) at various angles of incidence and for the [10] azimuth. Downward full arrows indicate the energies of emergences of the io beam into the vacuum in each case. Vertical lines indicate the energy positions of the resonance features which are plotted in Fig. 3
Fig. 1, we show calculated reflectivity data for the 00 beam for angles of incidence from 50 to 200 which includes all orders of internal scattering between the surface barrier and the substrate. The details of the potential models used are described in Refs. 9 and 10. The calculations were performed for a muffin-tin substrate scattering model based on the Mattheiss potential and a barrier model which was of the linear-saturated image form. This was an image barrier with an origin shift Zo= -6 a.u. measured with the negative direction outwards from the centre of the first row of atoms and matched onto a linear form. The value of the potential Us at the join-point between the saturated image barrier and the muffin-tin potential was taken to be zero. The choice of barrier potential was made somewhat arbitrarily at this stage but it does reproduce the gross features of the experimental data near 150 /8,11/. It is expected that refinements to this potential will be made in the future but the present computations serve to illustrate the method by which such data can be analysed.
The effects of different orders of scattering for the 100 profile are shown in Fig. 2. The peak occuring at 4.5 eV is a combination of a Bragg peak which comes entirely from the substrate scattering and a barrier scattering feature. All other features at higher energies are due to scattering between the barrier and substrate. The full curve in the upper frame is the same as that in Fig. 1
27
TO 01,OT
0·2 ,..
Fig. 2. Plot of 00 beam reflectivity for W(OOl) at 6=100
(10 azimuth). Downward full arrows indicate emergences of beams into the vacuum. Upward dashed arrows indicate emergences of beams from the substrate. In the upper frame, the full curve is the exact calculation for all orders of barrier-substrate multiple scattering with all beams included; the dashed curve is for single barrier-substrate scattering for all beams. In the lower frame the full curve is the exact calculation with the 10 beam excluded; the dashed curve is the exact calculation with the 10 and 01,01 beams excluded
~ ! O~~-rTT.,-r",,-rTT,,-r~-rrr"-rrT"~ ~ 0·6
~ CD 0-4
Energy (eV)
and is the result of the exact calculation including all orders of barrier-substrate multiple scattering; the dashed curve in the upper frame is the profile obtained with only single scattering taking place. There are significant differences between these two profiles. Peaks occurring below 9 eV require up to five internal scattering events to reproduce the exact result. Therefore the features occurring at 6.5 and 8.5 eV can be identified as due to the resonance mechanism. The very fine structure features occurring near the rO and 01,01 beam emergences cannot be identified at this stage as due to an interference or a resonance mechanism because of the limit of energy resolution that has been used in the present case.
The beams responsible for the barrier features are those beams which have insufficient energy normal to the surface to emerge into the vacuum. We now examine which of these beams contribute to each of the features. The full curve in the lower frame is the exact profile excluding the To pre-emergent beam and it exactly reproduces the 6.5 eV feature in the all-beam exact case in the upper frame. At this energy the only pre-emergent beams are the 10 beam and the 01,01 degenerate set. Therefore the feature at 6.5 eV is entirely due to resonant scattering of the 01,01 beam set. On the other hand the dashed curve in the lower frame shows the exact result with both the 10 beam and the ol,oT beam set excluded and it is seen that the 8.5 eV feature is exactly reproduced as in the all-beam exact case in the upper frame. Therefore this feature is due exclusively to the 10 beam. Thus not only have barrier resonance features been identified in this reflectivity profile but it has also been shown that each feature is due to sustained multiple scattering of a single, decoupled pre-emergent beam and is well separated in energy from other features in this case.
These resonance features in the reflectivity data occur when the electron associated with the single pre-emergent beam occupies a surface state. Because there is 2D periodicity parallel to the surface, it is the 2D plane wave part of the pre-emergent beam which labels the state. The 2D plane wave has wave vector Ell = !fl + X where :i is the reciprocal mesh vector corresp~nd~ng to the beam and kn is the parallel component of the wave vector of the lncldent beam. Thus for these features the part of the wave function of the surface state for
28
the lateral potential is a single 2D plane wave with energy l~ul2 a.u. In other words the electron is free in a constant potential parallel to the surface. The binding energy of the surface state in the varying potential perpendicular to the surface must be added to give the total surface state energy. The barrier resonance features are centred at energies corresponding to the surface state energies. Thus the binding energy can be obtained by subtracting I Eul 2 from the energy of the centre of the barrier resonance feature. Different, single identifiable beams have been shown in Fig. 2 to be responsible for two isolated features in one of these profiles. From these the binding energies of two surface states for this surface potential model are found to be -3.0 eV and -9.0 eV. These states should give rise to resonances with the same binding energy at all angles. But at other angles it has been found that most resonance features overlap and the positions of each resonance in the composite profile may be shifted from its true position.
The preceding method of analysis could not be used directly to interpret experimental data because the origin of the observed barrier features could not be obtained from the data alone. Thus it would not be known whether a single beam was responsible for the resonance feature and that therefore the 2D free-electron description was appropriate. Even if the free-electron picture were assumed, there is no basis for selecting the correct pre-emergent beam to assign to the feature. Fortunately it is possible to analyse data in this form in a different way by firstly plotting the dispersion of the resonance features against kn. It is legitimate to do this because even if the free-electron picture does not apply and several beams contribute to the resonance, the reduced 2D wave vectors of all the beams is the same, namely k~, and so the energies of the resonances and the corresponding surface states can be unambiguously specified by the reduced 2D wave vector of the state.
The energies of the maxima of the barrier scattering features from the profiles shown in Fig. 1 are plotted versus reduced wave vector kfr in Fig. 3. The kfl values are those of the incident beam which was directed along the [10] reciprocal mesh direction. Reflectivity data for the same polar angles but with the incident beam directed along the [11] reciprocal mesh direction were also calculated and the energies of the maxima of the barrier scattering features are included in Fig. 3. Also plotted is the band structure for an electron which is free in the lateral surface region. McRae analysed barrier features in this way in order to fit a nearly-free electron scheme to the dispersion and thereby obtain the lateral variation of the surface potential of chemisorbed surfaces /5/. Those features which have been identified as resonances using the method described above have been plotted as circles and those not yet identified as due to either resonance or interference effects have been plotted as triangles. The surface barrier resonance features in the [10] direction fall into two groups: one is fitted by the same free-electron dispersion as that labelled 01,01 rigidly displaced downwards in energy while the other group follows the same dispersion as the band labelled 10 rigidly displaced downwards by a different energy. A similar pattern applies in the [11] direction. We conclude from the dispersion that in this scattering situation the electron is free in the direction parallel to the surface. This interpretation requires that the 01,01 beams produce the feature at 6.5 eV in the 100 profile and that the 10 beam produces the 8.5 eV feature. This is exactly what was found from the theoretical analysis of the 100 profile shown in Fig. 2. The displacement downwards in energy from the free-electron band is the binding energy of the surface state in the one-dimensional potential perpendicular to the surface. Therefore from this analysis we find the binding energies of two surface states to be -3.0 eV and -9.0 eV for all k~ values as before. The surface state with binding energy -9.0 eV can be interpreted as a crystal-induced state and the other corresponds to a barrier-induced state. As we are using a realistic but
29
F
15
---- ------------..=..-~ Evac
-5 --------------
-10 IL..J'--'---'---'--'--L-l--L....L--'---'--'--'----L....L--'---'--L....l--'--'---'----.L....J 1·4 1·0 0·5 0 0·5 1·0
k~a/Tt
Fig. 3. Plot of positions of the barrier scattering features as a function of t~~ reduced wave.vector_~ in units of a/~ where a=5.9811 a.u. for [10] azimuth (rX) and [11] aZlmuth irK). Dashed lines are the 2D free-electron band structure. The circles represent features which are identified as resonance effects and the triangles represent features not as yet identified as either resonance or interference effects
arbitrary surface model here, no detailed comparison is made at this stage with other calculations of the crystal-induced surface states /12/.
In the electron scattering technique the surface states which are probed involve large kU values and are therefore at energies above the vacuum level. In the inverse ~hotoemission technique the surface states are probed at smaller kll values near r where the surface states have energies below the vacuum level. When the dispersion is free-electron like as in this case the binding energy is independent of kll and can therefore be obtained by both methods. If the barrier features plotted as triangles in Fig. 3 do correspond to resonance features then the binding energies of these more shallow surface states could also be determined. These shallow surface states would be called image-induced states because the electron is scattering in the outer region of the surface potential where the barrier is of the image form. In the present case these energies have a free-electron dispersion and the binding energies would be -0.2 eV and -0.6 eV. The binding energies of these states below the vacuum level would be the same because once again the dispersion is free-electron like. Recently an image-induced surface state with binding energy of about -0.7 eV was found from inverse photoemission results from W(001) /13/. But this result cannot be compared with the model calculation performed here until the barrier features are identified as being due to the resonance mechanis