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
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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