Springer physics of surfaces and interfaces 2006 (ISBN 3540347097)(652s)

Harald Ibach
Physics of Surfaces and Interfaces


Harald Ibach

Physics of Surfaces
and Interfaces
With 350 Figures

123


Professor Dr. Harald Ibach
Forschungszentrum Jülich GmbH
Institut für Bio- und Nanosysteme (IBN3)
Wilhelm-Johnen-Straße
52425 Jülich
Germany
e-mail:

Library of Congress Control Number: 2006927805

ISBN-10 3-540-34709-7 Springer Berlin Heidelberg New York
ISBN-13 978-3-540- 34709-5 Springer Berlin Heidelberg New York
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Preface

Writing a textbook is an undertaking that requires strong motivation, strong
enough to carry out almost two years of solid work in this case. My motivation
arose from three sources. The first was the ever-increasing pressure of our German
administration on research institutions and individuals to divert time and attention
from the pursuit of research into achieving politically determined five-year plans
and milestones. The challenge of writing a textbook helped me to maintain my
integrity as a scientist and served as an escape.
A second source of motivation lay in my attempt to understand transport processes at the solid/electrolyte interface within the framework of concepts developed
for solid surfaces in vacuum. These concepts provide logical connections between
the properties of single atoms and large ensembles of atoms by describing the
physics on an ever-coarser mesh. The transfer to the solid/electrolyte interface
proved nontrivial, the greatest obstacle being that terms such as surface tension

denote different quantities in surface physics and electrochemistry. Furthermore, I
came to realize that not infrequently identical quantities and concepts carry different names in the two disciplines. I felt challenged by the task of bringing the two
worlds together. Thus a distinct feature of this volume is that, wherever appropriate, it treats surfaces in vacuum and in an electrolyte side-by-side.
The final motivation unfolded during the course of the work itself. After 40
years of research, I found it relaxing and intellectually rewarding to sit back, think
thoroughly about the basics and cast those thoughts into the form of a tutorial text.
In keeping with my own likings, this volume covers everything from experimental methods and technical tricks of the trade to what, at times, are rather
sophisticated theoretical considerations. Thus, while some parts make for easy
reading, others may require a more in-depth study, depending on the reader. I have
tried to be as tutorial as possible even in the theoretical parts and have sacrificed
rigorousness for clarity by introducing illustrative shortcuts.
The experimental examples, for convenience, are drawn largely from the store
of knowledge available in our group in Jülich. Compiling these entailed some
nostalgia as well as the satisfaction of preserving expertise that has been acquired
over three decades of research.
I pondered long and hard about the order of the presentation. The necessarily
linear arrangement of the material in a textbook is intrinsically unsuitable for describing a field in which everything seems to be connected to everything else. I
finally settled for a fairly conventional sequence. To draw attention to relationships between different topics the linear style of presentation is supplemented by
cross-references to earlier and later sections.


Preface
VI
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Despite the length of the text and the many topics covered, it is alarming to
note what had to be left out: the important and fashionable field of adhesion and
friction; catalytic and electrochemical reactions at surfaces; liquid interfaces;
much about solid/solid interfaces; alloy, polymer, oxide and other insulator surfaces; and the new world of switchable organic molecules at solid surfaces, to
name just a few of a seemingly endless list.

This volume could not have been written without the help of many colleagues.
Above all, I would like to thank Margret Giesen for introducing me to the field of
surface transport and growth, both at the solid/vacuum and the solid/electrolyte
interface. This book would not exist without the inspiration I received from the
beautiful experiments of hers and her group and the almost daily discussions with
her. I should also be grateful for the patience she exercised as my wife during the
two years I spent writing this book.
Jorge Müller went through the ordeal of scrutinizing the text for misprints, the
equations for errors, and the text for misconceptions or misleading phrases. I also
express my appreciation for the many enlightening discussions of physics during
the long years of our collaboration.
I greatly enjoyed the hospitality of my colleagues at the University of California Irvine during my sabbatical in Spring 2005 where four chapters of this volume
were written. On that occasion I also enjoyed many discussions with Douglas L.
Mills on thin film magnetism and magnetic excitation, the fruits of which went
into the chapter on magnetism. In addition, the chapter on surface vibrations benefited immensely from our earlier collaboration on that topic.
Of the many other colleagues who helped me to understand the physics of interfaces, I would like to single out Ted L. Einstein and Wolfgang Schmickler. Ted
Einstein initiated me in the statistical thermodynamics of surfaces. Several parts of
this volume draw directly on experience acquired during our collaboration. Wolfgang Schmickler wrote the only textbook on electrochemistry that I was ever able
to understand. The thermodynamics of the solid/electrolyte interface as outlined in
chapter 4 of this volume evolved from our collaboration on this topic.
With Georgi Staikov I had fruitful discussions on nucleation theory and various
aspects of electrochemical phase formation which helped to formulate the chapter
on nucleation and growth. Guillermo Beltramo contributed helpful discussions as
well as several graphs on electrochemistry. Hans-Peter Oepen and Michaela
Hartmann read and commented the chapters on magnetism and electronic properties. Rudolf David contributed to the section on He-scattering. Claudia Steufmehl
made some sophisticated drawings. In drawing the structures of surface, I made
good use of the NIST database 42 [1.1] and the various features of the package.
Last but not least I thank the many nameless students who attended my lectures
on surface physics over the years. Their attentive listening and the awkward questions it led to were indispensable for formulating the concepts described in this
book. Finally, I beg forgiveness from my colleagues in Jülich for having been a

negligent institute director lately.
Jülich, May 2006

Harald Ibach


Contents

1

Structure of Surfaces….….……..……………………………………..
1.1 Surface Crystallography ….……………………………………….
1.1.1 Diffraction at Surfaces ……………………………………
1.1.2 Surface Superlattices ….………….………………………
1.2 Structure of Surfaces ………….…………………………………..
1.2.1 Face Centered Cubic (fcc) Structures ……….……………
1.2.2 Body Centered Cubic (bcc) Structures .…..………………
1.2.3 Diamond, Zincblende and Wurtzite …..…………………..
1.2.4 Surfaces with Adsorbates………………………………….
1.3 Defects at Surfaces…………………………………………………
1.3.1 Line Defects ………….……………….…………………..
1.3.2 Point Defects………………………………………………
1.4 Observation of Defects…………………………………………….
1.4.1 Diffraction Techniques ………………….….…………….
1.4.2 Scanning Microprobes ……………………..…………….
1.5 The Structure of the Solid/Electrolyte Interface ……………………

1
2
2

7
12
12
17
19
30
32
33
46
51
51
55
58

2

Basic Techniques……………………………………………………….
2.1 Ex-Situ Preparation ……………………………………………….
2.1.1 The Making of Crystals……………………………………
2.1.2 Preparing Single Crystal Surfaces …………….………….
2.2 Surfaces in Ultra-High Vacuum ………………………….……….
2.2.1 UHV-Technology …………………………………..…….
2.2.2 Surface Analysis ………………………………………….
2.2.3 Sample Preparation in UHV ………….………………….
2.3 Surfaces in an Electrochemical Cell ………………….…………..
2.3.1 The Three-Electrode Arrangement ……………………….
2.3.2 Voltammograms……………….………….………………
2.3.3 Preparation of Single Crystal Electrodes………………….

63

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63
65
71
71
81
88
95
95
97
101

3

Basic Concepts ………………………………………..………………..
3.1 Electronic States and Chemical Bonding in Bulk Solids ….……..
3.1.1 Metals ……………………………………..………………
3.1.2 Semiconductors ……………………………………….…
3.1.3 From Covalent Bonding to Ions in Solutions …………….

103
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103
106
109


Contents
VIII
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3.2

Charge Distribution at Surfaces and Interfaces ……….…………..
3.2.1 Metal Surfaces in the Jellium Approximation…….………
3.2.2 Space Charge Layers at Semiconductor Interfaces……….
3.2.3 Charge at the Solid/Electrolyte Interface …………………
Elasticity Theory ………………………………………………….
3.3.1 Strain, Stress and Elasticity……………………………….
3.3.2 Elastic Energy in Strained Layers…………………………
3.3.3 Thin Film Stress and Bending of a Substrate……..………
Elastic Interactions Between Defects ……………………………..
3.4.1 Outline of the Problem ……………………..……………..
3.4.2 Interaction Between Point and Line Defects ……………..
3.4.3 Pattern Formation via Elastic Interactions …….………….

112
112
116
121
125
125
129
132
139
139
142
144

4

Equilibrium Thermodynamics ……………………….……………….
4.1 The Hierarchy of Equilibria……………………….………………
4.2 Thermodynamics of Flat Surfaces and Interfaces…………………
4.2.1 The Interface Free Energy……………….………………..
4.2.2 Surface Excesses …………………….………………..…..
4.2.3 Charged Surfaces at Constant Potential …….…………….
4.2.4 Maxwell Relations and Their Applications…………….....
4.2.5 Solid and Solid-Liquid Interfaces …………………………
4.3 Curved Surfaces and Surface Defects……………………………..
4.3.1 Equilibrium Shape of a Three-Dimensional Crystal ……..
4.3.2 Rough Surfaces ………………………………..………….
4.3.3 Step Line Tension and Stiffness …………….……………
4.3.4 Point Defects …………………………..………………….
4.3.5 Steps on Charged Surfaces………………………………..
4.3.6 Point Defect on Charged Surfaces …….…………………
4.3.7 Equilibrium Fluctuations of Line Defects and Surfaces…...
4.3.8. Islands Shape Fluctuations………..………………………

149
149
152
152
158
161
164
168
172
172
180

184
187
188
194
196
201

5

Statistical Thermodynamics of Surfaces ……….……………………
5.1 General Concepts…………………………………………………..
5.1.1 Internal Energy and Free Energy…….……………………
5.1.2 Application to the Ideal Gas ……….……………………..
5.1.3 The Vapor Pressure of Solids …………………………….
5.2 The Terrace-Step-Kink Model ……………………………………
5.2.1 Basic Assumptions and Properties ……………………….
5.2.2 Step-Step Interactions on Vicinal Surfaces ………………
5.2.3 Simple Solutions for the Problem of Interacting Steps …..
5.2.4 Models for Thermal Roughening …………………………
5.2.5 Phonon Entropy of Steps …………………………………
5.3 The Ising-Model …………………………………………………..
5.3.1 Application to the Equilibrium Shape of Islands …..….….
5.3.2 Further Properties of the Model…………………….……..

207
207
207
208
210
211

211
215
218
221
223
225
225
228

3.3

3.4


Contents
IX
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6

7

5.4. Lattice Gas Models ………………………..………………………
5.4.1 Lattice Gas with No Interactions………………………….
5.4.2 Lattice Gas or Real 2D-Gas? …….…………….………..
5.4.3 Segregation ……………………………………………….
5.4.4 Phase Transitions in the Lattice Gas Model ……..……….

233
233

235
238
240

Adsorption ………………………………………………..………….
6.1 Physisorption and Chemisorption  General Issues ……………....
6.2 Isotherms, Isosters, and Isobars ………………….……………….
6.2.1 The Langmuir Isotherm ………….……………………..
6.2.2 Lattice Gas with Mean Field Interaction
the Fowler-Frumkin Isotherm ….………………………
6.2.3 Experimental Determination of the Heat of Adsorption …
6.2.4 Underpotential Deposition …..……………………………
6.2.5 Specific Adsorption of Ions……..………………………...
6.3 Desorption …………………………..….…………………………
6.3.1 Desorption Spectroscopy ………..……………………….
6.3.2 Theory of Desorption Rates ………………………………
6.4 The Chemical Bond of Adsorbates ……………………………….
6.4.1 Carbon Monoxide (CO).…………………………………..
6.4.2 Nitric Oxide……………………………………………….
6.4.3 The Oxygen Molecule………..…………………………...
6.4.4 Water………………………………………………………
6.4.5 Hydrocarbons …………….……………………………….
6.4.6 Alkali Metals ……………………………………………...
6.4.7 Hydrogen…………………….……………………………
6.4.8 Group IV-VII Atoms ………..……………………………

245
245
254
254

Vibrational Excitations at Surfaces .…………………………………
7.1 Surface Phonons of Solids…………………………………………
7.1.1 General Aspects …………………………………………..
7.1.2 Surface Lattice Dynamics …………….…………………
7.1.3 Surface Stress and the Nearest Neighbor
Central Force Model ……….………..……………………
7.1.4 Surface Phonons in the Acoustic Limit …………………..
7.1.5 Surface Phonons and Ab-Initio Theory …………………..
7.1.6 Kohn Anomalies…..……………….…..………………….
7.1.7 Dielectric Surface Waves………………………………….
7.2 Adsorbate Modes ………………………..………………………..
7.2.1 Dispersion of Adsorbate Modes ….………………………
7.2.2 Localized Modes ………………………………………….
7.2.3 Selection Rules……………………………………………
7.3 Inelastic Scattering of Helium Atoms…..…………………………
7.3.1 Experiment ……………………………………………….
7.3.2 Theoretical Background …………………………………..

309
309
309
312

255
260
264
269
273
273

276
284
284
287
288
289
291
295
300
303

315
317
319
321
323
327
327
330
333
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Contents
X
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7.4

Inelastic Scattering of Electrons …………………………………..
7.4.1 Experiment …………………….………………………….
7.4.2 Theory of Inelastic Electron Scattering ……..……………
Optical Techniques ……………………….…..…………………..
7.5.1 Reflection Absorption Infrared Spectroscopy ….………...
7.5.2 Beyond the Surface Selection Rule…….…………………
7.5.3 Special Optical Techniques………….……………………
Tunneling Spectroscopy ……..….….……………………………..

347
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351
362
362
366
369
373

Electronic Properties …………………………………………………..
8.1 Surface Plasmons………………………………………………….
8.1.1 Surface Plasmons in the Continuum Limit………………..
8.1.2 Surface Plasmon Dispersion and Multipole Excitations …
8.2 Electron States at Surfaces …..…..………………………………..
8.2.1 General Issues……………………………………………..
8.2.2 Probing Occupied States  Photoemission Spectroscopy...
8.2.3 Probing Unoccupied States ..……………………………..
8.2.4 Surface States on Semiconductors …..……………………
8.2.5 Surface States on Metals ………….………………………
8.2.6 Band Structure of Adsorbates….…….……………………

8.2.7 Core Level Spectroscopy ….…..….………………………
8.3 Quantum Size Effects ……………………………………………..
8.3.1 Thin Films ……………….……………………………….
8.3.2 Oscillations in the Total Energy of Thin Films …………..
8.3.3 Confinement of Surface States by Defects ……………….
8.3.4 Oscillatory Interactions between Adatoms …..…………...
8.4 Electronic Transport ……………..……………………………….
8.4.1 Conduction in Thin Films  the Effect of Adsorbates …..
8.4.2 Conduction in Thin Films 
the Solution of the Boltzmann Equation …………………
8.4.3 Conduction in Space Charge Layers …………………….
8.4.4 From Nanowires to Quantum Conduction ………………

379
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379
381
383
383
386
391
394
401
407
410
413
413
417
420
425

427
427

Magnetism …………………….……………………………………….
9.1 Magnetism of Bulk Solids …..…………………………………….
9.1.1 General Issues ………..…..……………………………….
9.1.2 Magnetic Anisotropy of Various Crystal Structures ….…..
9.2 Magnetism of Surfaces and Thin Film Systems …………………..
9.2.1 Experimental Methods …………..…………..……………
9.2.2 Magnetic Anisotropy in Thin Film Systems………………
9.2.3 Curie Temperature of Low Dimensional Systems ……….
9.2.4 Temperature Dependence of the Magnetization ………….
9.3 Domain Walls ………………….……..….………………………….
9.3.1 Bloch and Néel Walls ……..….…………………………..
9.3.2 Domain Walls in Thin Films ….………………………….
9.3.3 The Internal Structure of Domain Walls in Thin Films …..

445
445
445
447
451
451
455
459
463
467
467
468
470

7.5

7.6
8

9

431
435
437


Contents
XI
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9.4

Magnetic Coupling in Thin Film Systems ……………………….
9.4.1 Exchange Bias …………….……..……………………….
9.4.2 The GMR Effect ………….….…………………………...
9.4.3 Magnetic Coupling Across Nonmagnetic Interlayers ……
9.5 Magnetic Excitations ………………..……………………………..
9.5.1 Stoner Excitations and Spin Waves ..…………………….
9.5.2 Magnetostatic Spin Waves at Surfaces and in Thin Films..
9.5.3 Exchange-Coupled Surface Spin Waves …………………

473
473

476
479
482
482
485
486

10 Diffusion at Surfaces …………………………….…………………….
10.1. Stochastic Motion ………………….…..…………………………
10.1.1 Observation of Single Atom Diffusion Events .…………..
10.1.2 Statistics of Random Walk………..………………………
10.1.3 Absolute Rate Theory …....………………………………
10.1.4 Calculation of the Prefactor……….………………………
10.1.5 Cluster and Island Diffusion ….…..………………………
10.2 Continuum Theory of Diffusion …….…….………………………
10.2.1 Transition from Stochastic Motion to Continuum Theory..
10.2.2 Smoothening of a Rough Surface…..…..…………………
10.2.3 Decay of Protrusions in Steps and Equilibration of Islands
after Coalescence………..…………..…………………….
10.2.4 Asaro-Tiller-Grinfeld Instability and Crack Propagation…
10.3 The Ehrlich-Schwoebel Barrier …..…………………….………..
10.3.1 The Concept of the Ehrlich-Schwoebel Barrier ….………
10.3.2 Mass Transport on Stepped Surfaces ….…………………
10.3.3 The Kink Ehrlich-Schwoebel Barrier .…………………..
10.3.4 The Atomistic Picture of the Ehrlich-Schwoebel Barrier ..
10.4 Ripening Processes in Well-Defined Geometries …..……………
10.4.1 Ostwald Ripening in Two-Dimensions …………………..
10.4.2 Attachment/Detachment Limited Decay …………………
10.4.3 Diffusion Limited Decay …………….………….………..
10.4.4 Extension to Noncircular Geometries .………….………..

10.4.5 Interlayer Transport in Stacks of Islands ….……………...
10.4.6 Atomic Landslides……………….………………………..
10.4.7 Ripening at the Solid/Electrolyte Interface .……...……….
10.5 The Time Dependence of Step Fluctuations ..……………………
10.5.1 The Basic Phenomenon …………………………………..
10.5.2 Scaling Laws for Step Fluctuations …..…………………..
10.5.3 Experiments on Step Fluctuations ..……………………...

491
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491
495
498
500
503
505
505
508

11 Nucleation and Growth ………………………………………………..
11.1. Nucleation under Controlled Flux ….…………………………….
11.1.1 Nucleation ………………………………………………..
11.1.2 Growth Without Diffusion…………………….…………..
11.1.3 Growth with Hindered Interlayer Transport………………
11.1.4 Growth with Facile Interlayer Transport …………………

555
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561

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Contents
XII
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11.2 Nucleation and Growth under Chemical Potential Control …..….
11.2.1 Two-Dimensional Nucleation ………………………..…..

11.2.2 Two-Dimensional Nucleation in Heteroepitaxy …….…...
11.2.3 Three-Dimensional Nucleation ……………………….…..
11.2.4 Theory of Nucleation Rates ………………………………
11.2.5 Rates for 2D- and 3D-Nucleation …………………….…..
11.2.6 Nucleation Experiments at Solid Electrodes ……….…….
11.3 Nucleation and Growth in Strained Systems ……………………..
11.3.1 2D-Nucleation on Strained Layers………………………..
11.3.2 3D-nucleation on Strained Layers ………………………..
11.4 Nucleation-Free Growth……………………………………………
11.4.1 The Steady State Concentration Profile ………………….
11.4.2 Step Flow Growth ………………………………………..
11.4.3 Meander Instability of Steps .…………….……………….

572
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574
576
580
587
593
597
597
600
603
603
605
607

Appendix: Surface Brillouin Zones……….………………………………. 613
References …….…………………………………….……………………… 615

Subject Index …...………..………………………………………………... 635
List of Common Acronyms ………………………………………………. 645


1. Structure of Surfaces

Surface Physics and Chemistry flourished long before anything was known about
the atomic structure of surfaces. Chemical, optical, electrical and even magnetic
properties were investigated systematically, sometimes in great detail and not
without lasting success. The concept of an ideally terminated bulk structure with
its assumed physical properties frequently served as a base for the rationalization
of the experimental results. Examples are the postulation of specific electric properties that would arise from the broken bonds at surfaces of semiconductors and
the high chemical activity that might be associated with defects on the surface.
Quantitative understanding on an atomic level could not be achieved however
without knowledge the crystallographic structure of surfaces. Vice versa, a tutorial
presentation of our present understanding of the physics of surfaces and interfaces
requires the fundament of facts, concepts and the nomenclature that has evolved
from the analysis of surface structures. The first chapter of this treatise is therefore
devoted to the structure of clean and adsorbate covered surfaces, the important
defects at surfaces and the structural elements of the solid/electrolyte interface.
As for Solid State Physics in general, the quantitative understanding on an
atomic level greatly benefits from the periodic structure of crystalline matter since
the periodicity reduces the electronic and nuclear degrees of freedom from 1023
per cm3 to the degrees of freedom in a single unit cell. However, at surfaces the
reduction in the degrees of freedom by periodicity is less, as the three-dimensional
symmetry is broken. Near surfaces, material properties may differ from the bulk in
several monolayers below the surface. The surface unit cell of periodicity therefore necessarily contains more atoms than the corresponding unit cell of the bulk
structure. Not infrequently, the unit cell of a real surface is substantially larger
than the surface unit cell of a terminated bulk, which increases the number of atoms in the surface unit cell further. For example, the surface cell of the clean
(111) surface of silicon contains 49 atoms in one atom layer and the restructuring

involves 4-5 atom layers! Solving a bulk structure with that many atoms per unit
cell is not an easy, but nowadays tractable problem, but structure analysis at surfaces has to be performed in the presence of the entire bulk below the surface. It is
still one of the greatest successes of surface science that after decades of research
and literally thousands of papers the structure of the Si(111) surface was eventually solved.
Substantial advances in surface crystallography are owed to the experimental
and theoretical achievements in Low Energy Electron Diffraction (LEED) and
Surface X-Ray Diffraction (SXRD). Scanning Tunneling Microscopy (STM) and
other scanning microprobes contributed by providing qualitative images of sur-


1 Structure of Surfaces
2__________________________________________________________________________

faces, which reduced the number of possibilities for surface structure models.
Presently, the structures of more than 1000 surface systems are documented, and
the number keeps growing [1.1].

1.1 Surface Crystallography
1.1.1 Diffraction at Surfaces
The first section of this volume is devoted to the essential elements of surface
crystallography: Laue-equations, Ewald-construction, and symmetry elements.
Elastic scattering of X-rays or particle waves from infinitely extended threedimensional periodic structures undergoes destructive interference, which leaves
scattered intensity only in particular directions. The conditions under which diffracted intensity can be observed are described by the three Laue-equations, which
can be expressed in terms of a single vector equation

k  k0

G,

(1.1)

in which k and k0 are the wave vector of the scattered and incident wave, respectively, and G is an arbitrary vector of the reciprocal space. At the surface, the bulk
periodicity is truncated and the three Laue-equations reduce to two equations concerning the components of the incident and scattered wave vectors parallel to the
surface.

k||  k0||

G||

(1.2)

G|| is a vector of the reciprocal lattice of the two-dimensional unit cell at the surface. Diffracted beams are therefore indexed by two Miller-indices (h,k). The
reduction to two Laue-equations has the consequence that scattering from a surface lattice leads to diffracted beams for all incident k0, unlike for bulk scattering
where diffracted beams occur only for particular wave vectors of the incident
beam. As for the bulk, the Laue-condition is best illustrated with the Ewaldconstruction. Figure 1.1 shows the Ewald-construction as it is typical for LEED: A
beam of low energy electrons (energy E0 between 20 and 500 eV, corresponding
to a wave vector k 0

5.12 nm -1 E0 / eV ) with normal incidence is diffracted

from the surface lattice. Depending on the energy, the {01}, {11}, {02}... beams
are observed in the backscattering direction, providing direct information on the
surface reciprocal lattice.
Early experiments used a Faraday cup for probing the diffracted beams [1.2].
More convenient is the experimental set-up introduced by Lander et al. [1.3],
which is displayed in Fig. 1.2. The equipment was primarily designed for a qualitative quick overview on the diffraction pattern.


1.1 Surface Crystallography
3

__________________________________________________________________________

k

G
k0

( 40) ( 3 0) ( 20) ( 1 0) (00)

(10)

(20)

(30)

(40)

Fig. 1.1. Ewald-construction for surface scattering. The magnitude and orientation of k0
(normal incidence) is representative of a LEED-experiment. Diffracted beams occur if the
wave vector of the scattered electron ends on one of the vertical rods (crystal truncation
rods) representing the reciprocal lattice of the surface. Diffracted electrons are therefore
observed for all energies of the incident beam: The scattering intensity is particular large if
the third Laue-condition concerning the perpendicular component of the scattering vector
(indicated by ellipsoids) is approximately met.

Later the same equipment has been used also for the quantitative analysis of diffracted intensities by monitoring the spots on the screen with the help of a video
camera and specially developed image processing software (Video-LEED).
Like all other experiments using low energy electrons, LEED gains its surface
sensitivity from the relative large cross section for inelastic scattering. The prime
source of inelastic scattering is the interaction with collective excitations of the

valence electrons electron, the plasmons (Sect. 2.2.2, 8.1). The mean free path of
electrons in the relevant range is of the order of 1 nm. All elastically backscattered
electrons therefore stem from the first few monolayers of the crystal. This is the
reason that intensity is observed even for energies for which the third Laue equation for the vertical component of the scattering vector K = k0k is not fulfilled.
The few monolayers, from which the diffraction originates, however, suffice to
impose a weak Laue-condition on the vertical component of the scattering vector
K. In Fig. 1.1 this weak Laue condition is indicated by the ellipsoids. Figure 1.3
displays the measured diffracted intensity of the (10) beam from a Cu(100) surface
[1.4] together with the position of the expected intensity maxima according to the
third Laue-condition. The experimental intensity curve indeed displays pronounced maxima, but only very roughly where they are expected from single
scattering (kinematic scattering) theory. Surely, the complexity of the various
features in the intensity curve cannot be explained based on single scattering


1 Structure of Surfaces
4__________________________________________________________________________
Fluorescence screen

Sample
Filament

U

5 kV

U+'U

1. Grid
2. Grid
3. Grid

Fig. 1.2. Instrument for low electron energy diffraction. Diffracted electrons are observed
on a fluorescent screen. The grids serve for various purposes. Grid 1 establishes a field free
region around the sample, grid 2 repels inelastically scattered electrons so that they cannot
reach the screen, grid 3 prevents the punch-through of the high voltage applied to the screen
to the field at grid 2.

events. Multiple elastic scattering of the electron has to be taken into account (dynamic scattering theory). The difficulty to describe multiple elastic scattering of
electrons theoretically has been a major impediment in the development of surface
crystallography. As Fig. 1.3 demonstrates [1.4-6], theory is now able to describe
the observed intensities quite well. A quantitative structure analysis is performed
by proposing a model for the structure and by comparing experimental and theoretical LEED intensities as a function of the atom position parameters (trial and
error method). Comparison of theory and experiment is quantified in the Pendry
R-factor Rp which is defined on the basis of the logarithmic derivative of the intensities I with respect to the electron energy E0.

Rp
Y

¦ (Ytheory  Yexp ) 2
¦ (Ytheory  Yexp ) 2
I log

(1.3)

1  ( I logV0i ) 2

with

I log

wI
I wE0

(1.4)


Intensity (arb. units)

1.1 Surface Crystallography
5
__________________________________________________________________________

Cu(100)
Intensity of (10) beam
Experiment
Theory

0

100

200

300

400

500

600

Energy / eV
Fig. 1.3. Intensity of the (10) beam diffracted from a Cu(100) surface vs. beam energy for
normal incidence. Experiment and theory are plotted as solid and dashed curves, respectively. The positions of the maxima according to the simple single scattering theory are
indicated as vertical bars.

Here, V0i is the imaginary part of the inner potential (approximately the width of
the intensity peaks on the energy scale) and the sum is over all energies and diffracted beams. The agreement between theory and experiment in Fig. 1.3
corresponds to an Rp-factor of 0.08. In general, Rp-factors below 0.20 are considered as good.
Compared to LEED, X-ray scattering has the definite advantage that X-rays are
scattered only once. The scattering amplitude is therefore the Fourier-transform of
the scattering density [1.7] and intensities are easily calculated for any given structure. Schemes for direct structure determination via the Patterson function can be
employed. Surface sensitivity is achieved by working under condition of grazing
incidence. Since the photon energy is well above all electronic excitations the
complex refraction index n~ for X-rays is described by the dielectric properties of
the free electron gas in the high frequency limit. The real part of n~ is therefore
smaller than one. Total reflection of the X-ray beam occurs at grazing incidence if
the angle between the beam and the surface plane Di is smaller than a critical angle
Dc. Typical values for Dc are between 0.2° and 0.6° for an X-ray wavelength of
0.15 nm [1.8]. Under condition of total reflection the X-ray intensity inside the
solid drops exponentially with a decay length / of about 10 nm. All diffraction
information therefore concerns no more than about 50 atom layers. Information of
just the surface layer is contained in diffracted beams of a surface superlattice. The
intensity of such beams is sufficiently large for detection and stands out from the
diffuse background. The technique is called Grazing Incidence X-Ray Diffraction
(GIXRD). Figure 1.4 shows the structure factor (the modulus of the scattering
amplitude as due to the structure) as a function of the perpendicular component of
the scattering vector [1.9] (a) for a bare Cu(110) surface and (b) for a Cu(110)

1 Structure of Surfaces
6__________________________________________________________________________

(a) bare Cu(110)

(0 1 L)

100

10

0

0.5

L

1.0

1.5

Structure factor (arb. units)

Structure factor (arb. units)

surface covered with oxygen. The parallel component of the scattering vector is
chosen to fulfill the (01) surface diffraction condition. The full line is calculated
using the structural parameters, which gave the best fit to all measured structure
factors (about 150). Note that comparison between experiment and theory is made
for the intensity outside the L=1 peak that results from the third Laue condition.

The intensity in that peak contains mostly information about the structure of the
bulk inside the decay length /.
(b) Cu(110) + O

(0 1 L)

10

1

0

0.5

1.0

L

1.5

Fig. 1.4. Structure factor along the (01) crystal truncation rod as a function of the vertical
component of the scattering vector L expressed in units of the reciprocal lattice vector (a)
for a bare Cu(110) surface and (b) for a Cu(110) surface covered with oxygen [1.9]. The
insets display a top view on the first two layers of surface atoms (see also Sect. 3.4.3). The
structure with oxygen is the so-called added row structure where every second row is
formed by a chain of oxygen atoms (dark circles) and Cu-atoms. Experimental data and
theory for the optimized geometry data are shown as circles and solid lines, respectively.

The applicability of single scattering theory also provides the possibility to use the
elastic diffuse X-ray intensity for an analysis of non-periodic features on surfaces,

such as defects or strain fields associated with domains of adsorbates [1.9, 10].
Furthermore, vacuum is not required, which makes X-ray scattering a technique
suitable also for studies on the solid/liquid interface [1.11] if the liquid layer is
thin enough.
The question which of the two methods LEED or SXRD is the method of
choice depends on circumstances. In principle, both methods can provide equally
precise atom positions for a large number of atoms per unit cell. The scattering
cross section for X-rays scales with the square of the atom number Z. Light elements contribute little to X-ray scattering and data are not sensitive to the position
of light element. LEED does not suffer from that to the same extent. Because of
the larger momentum transfer in the direction of the surface normal, LEED has a
better sensitivity to the vertical atom coordinates, while SXRD is more sensitive to
the lateral position. X-ray scattering experiments require extremely flat surfaces
because of the grazing incidence condition while LEED is more forgiving with
respect to sample quality. At present, most of the surface structure determinations


1.1 Surface Crystallography
7
__________________________________________________________________________

are based on the quantitative analysis of LEED-intensities. However, the balance
may tip as improved synchrotron sources become more available.
1.1.2 Surface Superlattices

Notation
The positions of surface atoms differ from the bulk because of the broken symmetry and the broken bonds. The modifications are referred to as relaxations if the
surface unit cell remains that of the truncated bulk. If the surface unit cell is different, then the corresponding changes in the structures are addressed as
reconstructions. The lattice of an adsorbed phase with a unit cell larger than the
surface cell of the truncated bulk is called a superlattice, the associated structure a
superstructure. Adsorbate superstructures frequently go along with a reconstruction of the substrate. The nomenclature therefore is not unambiguous.

Base vectors of the unit cell of superstructures and surface reconstructions are
expressed in terms of the base vectors of the unit cell of the truncated bulk. With
s1 and s2 as vectors spanning the surface unit cell of a truncated bulk lattice, the
lattice vectors of the actual unit cell on the surface, a1 and a2, are described by the
matrix t

§ a1 ·
¨¨ ¸¸
© a2 ¹

2s 2

§ t11 t12 ·§ s1 ·
¨¨
¸¸¨¨ ¸¸
© t 21 t 22 ¹© s 2 ¹

c2 u 2


(1.5)

s

s2

s2

T=45°

2 s2

s1






2 u 2 R45q

2 s1

s1

2s1

Fig. 1.5. Illustration of the notation of the c(2u2) unit cell of the surface lattice and its

alternative notation as

2 u 2 R45°.

In most cases, this unambiguous notation introduced by E. A. Wood in 1964
[1.12] is unnecessarily complicated and inconvenient. If the surface lattice vectors
are just multiples of the lattice vectors s1 and s2 unit cells are denoted as (2u1),
(2u2), (3u1), etc. Centered and primitive unit cells are denoted by adding a "c"
and a "p" to the notation, e.g. p(2u2) and c(2u2). This type of notation is not al-

1 Structure of Surfaces
8__________________________________________________________________________

ways unique: The c(2u2) lattice on a (100) surface of a cubic crystal can also be
noted as 2 u 2 R45° in which the R45° stand for a rotation by 45° (Fig. 1.5).
§1 1 ·
¸¸
©1 1¹

The unambiguous matrix notation ¨¨

is rarely used in that case, as it is more

difficult to quote.
A few common adsorbate superlattices are displayed in Fig. 1.6 together with
their notation.
p(2x2)

(2x2)

c(2x2)

( 3 x 3 )R30q

Fig. 1.6. Typical adsorbate superlattices on surfaces together with their trivial notation.
Substrate and adsorbate atoms are displayed as black and grey, respectively.

Diffraction pattern of superlattices
The existence of a superlattice on a surface is most easily discovered in a diffraction experiment because the larger unit cell produces extra, fractional order spots
in the diffraction pattern between the normal (hk) spots of the truncated bulk lattice. The determination of the base vectors of the unit cell frequently requires the

consideration of domains. For example, the diffraction pattern of a (1u2) unit cell
on a (111) or (100) surface of a cubic material has half order spots in terms of the
Miller-indices of the substrate at (h r1/2), (h r3/2), etc.. The equivalent second
(2u1) domain, which is rotated by 90°, has spots at (1/2 k), (3/2 k), etc. (Fig. 1.7).
The pattern is distinct from the pattern of a (2u2) lattice since the latter would
produce reflexes also at (r1/2, r1/2), (r3/2, r3/2), etc., which are absent in the
diffraction pattern of the (1u2), (2u1) superlattice (Fig. 1.7).


1.1 Surface Crystallography
9
__________________________________________________________________________

(2u1) + (1u2)

c(2u2)

Fig.1.7. Diffraction pattern of two domains of a (1u2) superlattice and a c(2u2) superlattice
on a (100) surface of a cubic material.

Centered unit cells and unit cell containing glide planes can be identified because
they give rise to systematic extinctions. The extinctions of reflexes (h, k) are calculated from the surface structure factor Shk.

S hk

¦ exp> iʌ (hc u Į  k c vĮ )@

(1.6)

Į

Here, h' and k' are the Miller-indices of the superlattice, uD and vD are the components of the vector rD pointing to the atom D in the unit cells in terms of the base
vectors a1 and a2.

rĮ

uĮ a1  vĮ a 2

(1.7)

We consider the c(2u2) superlattice as an example. Because of the (2u2) lattice,
the Miller-indices of the superlattice h' and k' in terms of the Miller-indices of the
substrate lattice h and k are h' = 2h and k' = 2k. The components uD and vD are
u1 = v1 = 0 and u2 = v2 = 1/2. The structure factor is therefore

S hk

1  (1) 2( h  k )

­° 0 if 2(h  k ) uneven
.
®
°¯ 2 if 2(h  k ) even

(1.8)

The c(2u2) structure is therefore identified by characteristic extinctions in the
half-order spot of the (2u2) lattice. In particular, these extinctions occur for all
half-order spots along the ¢h 0² and ¢0 k²-directions (Fig. 1.7).

Point group symmetry of sites
A very important element of the surface structure is the symmetry of various sites
on surfaces, important, because the local symmetry of an atom or molecular complex determines the classification of the eigenvalues of the electronic quantum
states as well as the selection rules in spectroscopy. The fact that the surface plane


1 Structure of Surfaces
10
__________________________________________________________________________

is never a mirror plane reduces the number of possible point groups on surface to
those, which have rotation axes and mirror planes perpendicular to the surface.
These point groups are Cs, C2v, C3v, C4v, C6v, C3, C4, C6. Figure 1.8 illustrates the
most important point groups Cs, C2v, C3v, C4v together with the point groups C3 and
C4. For the purpose of analyzing and classifying spectroscopic data, it is useful to
have the character tables of the point groups at hand. Characters tables for Cs, C2v,
C3v, C4v, and C6v are listed in Table 1.1.
Cs

C2v

Vx
Vy
C3v

Vv
C3

C4v

Vv

Vd

C4

Fig. 1.8. The point groups Cs, C2v, C3v, C4v, C3, and C4. The upper four point groups are
illustrated with a diatomic molecule like CO or NO (black and gray circle). The species
representing C3 and C4 are hypothetical. The point groups Cs, C2v, C3v, and C4v are frequently encountered with molecules like CO, NO, or NH3.


1.1 Surface Crystallography
11
__________________________________________________________________________

Table 1.1. Character tables of surface point groups. The upper left corner notes the point
group. The first column are the irreducible representations, the following columns are the
characters of the classes of the group. The last column describes to which irreducible representation the translations along the x, y and z-axes and the rotations around these axes
belong. This is important since the translations and rotations of a molecule turn to vibrations when the molecule is adsorbed. (see Sect. 7.2.2).

Cs
'

A

''

A

I

ıxz

+1

+1

+1

-1

C3v

I

C3

ı

A1
A2
E

+1
+1
+2

+1
+1
-1

+1
-1
0

C2

C2v

I

C2

ıxz

ıyz

z, x, Ry

A1

+1

+1

+1

+1

z

y, Rx, Rz

A2

+1

+1

-1

-1

Rz

B1

+1

-1

+1

-1

x, Ry

B2

+1

-1

-1

+1

y, Rx

C4v

I

C4

C4

ıv

ıd

A1

+1

+1

+1

+1

+1

A2
B1
B2
E

+1
+1
+1
+2

+1
-1
-1
0

+1
+1
+1
-2

-1
+1
-1
0

-1
-1

+1
0

z
Rz
x,y,Rx,Ry

2

C6v

I

C6

C6

A1
A2
B1
B2

+1
+1
+1
+1

+1
+1
-1

-1

+1
+1
+1
+1

E1
E2

+2
+2

+1
-1

-1
-1

I

3

ıv

ıd

+1
+1
-1

-1

+1
-1
+1
-1

+1
-1
-1
+1

z
Rz

-2
+2

0
0

0
0

x, y, Rx, Ry

C6

C2

A

+1

+1

B

+1

-1

z, Rz
x, y, Rx, Ry

C4

I

2

C4

C4

C3

I

C3

A
E

+1
+1

+1
-1

2

A

+1

+1

+1

z, Rz

B
E

+1
+2

-1
0

+1
-2

x, y, Rx, Ry

z, Rz
x, y, Rx, Ry

z
Rz

x,y,Rx,Ry


1 Structure of Surfaces
12
__________________________________________________________________________

Space groups
Space groups combine translations with point symmetry operations. In three dimensions, the combination of the 14 Bravais-lattices with the 32-crystallographic
point groups yields the 230 crystallographic space groups. In two dimensions,
only 17 space groups exist. Three important ones are illustrated in Fig. 1.9.

p1g1

p2mg

p4g

Fig. 1.9. Illustration of common space groups at surfaces. All structures contain a combination of translation and mirror symmetry, a glide plane. The p2mg structure contains an
additional mirror plane perpendicular to the glide plane.

1.2 Surface Structures
Many materials, notably metals, have a surface lattice, which corresponds to the
bulk crystallographic (hkl) plane. Merely the atomic distances vertical to the surface plane are changed to a larger or lesser degree, depending on the material, the
surface orientation, and the type of bonding. The surfaces of some 5d-transition
metals, however, reconstruct to form large, sometimes even incommensurate surface cells. Reconstructions are also typical for covalently bonded semiconductors.
This section presents the surface structures of common materials [1.1].
1.2.1 Face Centered Cubic (fcc) Structure

Many metal elements crystallize in the face-centered cubic (fcc) structure. Among
them are the coinage metals copper (Cu), silver (Ag), gold (Au), as well as the
catalytic important metals nickel (Ni), rhodium (Rh), palladium (Pd), iridium (Ir)
and platinum (Pt). Surfaces of these metals have been studied intensively since the
early days of Surface Science. We therefore begin the presentation of surface
structures with the low index surfaces of fcc-crystals. Following the convention in
crystallography, we denote a set of equivalent faces by braced indices, e.g. {100},
and particular faces like (100), (010), or (001) by indices in parenthesis. The three