Handbook of Surface Science Volume I
Handbook of Surface Science Volume I
Handbook of Surface Science Series editors
S. HOLLOWAY Surface Science Research Centre Liverpool, UK N.V. RICHARDSON
Director, Surface Science Research Centre Liverpool, UK
ELSEVIER AMSTERDAM
9 LAUSANNE
9 NEW YORK
9 OXFORD
9 SHANNON
9 TOKYO
Volume I
Physical Structure
Volume editor W.N.
UNERTL
Laboratory for Surface Science and Technology Sawyer Research Center University of Maine Orono, ME 04469 USA
1996 ELSEVIER AMSTERDAM
9 LAUSANNE
9 NEW YORK * OXFORD
9 SHANNON
9 TOKYO
ELSEVIER SCIENCE B.V. Sara B u r g e r h a r t s t r a a t 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
L l b r a r y oF Congress C a t a l o g i n g - I n - P u b l i c a t i o n
Data
Physical structure / volume editor, W.N. Unertl. p. cm. -- (Handbook of surface science ; v . 1) Includes bibliographical references and indexes. ISBN 0-444-89036-X 1. Solids--Surfaces. 2. Surfaces (Physics) 3. Surface chemistry. 4. Energy-band theory of solids. I. Unertl, W. N. (William N.) II. Series QC176.8.S8P49 1996 530.4'17--dc21 96-44174 CIP
ISBN 0-444-89036-X (Volume 1) ISBN 0-444-82526-6 (Series) 9 1996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U . S . A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.
General Preface
How many times has it been said that surface science has come of age? Rather than being a fledgling area of study, it is now patently clear that the investigation of solid surfaces and related interfacial problems is a unique field with profound implications for basic (dare one say academic!) scientific study and the understanding of materials. Surface science provides major support to many technologically ambitious industries. It is widely recognised that it underpins the fabrication of electronic devices but this also extends to any industry working in nanotechnology. Surface science makes significant contributions to product development and problem solving in many materials-based industries where surface finish, cleanliness, adhesion, wear or friction are important. An understanding of surface processes is vital in chemical industries because of the importance of heterogeneous catalysis and is likely to make major contributions to the growing optoelectronics and molecular sensing industries. In social terms, an improved understanding of surfaces will facilitate the development of better catalysts and sensors for improvement of our environment. In this series, we have brought together some of the key players that have made seminal contributions to the study of solid surfaces and their interactions with foreign species. Because of the broad scope of surface science, even when restricted to the solid surfaces which we hope to cover in this series, we have been coerced into 'packaging' the subject in what is, it must be said, a rather arbitrary way. No doubt different editors would have chosen different themes around which to base individual volumes. This first volume is devoted to determining, by measurement and calculation the geometry of the surface. This is probably the oldest field of surface science investigation, dating back to the original experiments of Davisson and Germer employing electrons, and Estermann and Stern using atomic He beams to determine the crystal lattice constants in the surface region. In Volume 1, a wide variety of contemporary techniques are presented that now enable one to determine the location of atoms in the substrate surface as well as those in adsorbed molecules. This has enabled investigators to shed light on such issues as, how molecules can deform in the adsorbed state, and, the role of genuine surface relaxations both for a clean surface as well as for one covered by foreign species. The contributions deal with a wide variety of surface types, metals, insulators, ceramics etc. all of which play a role in real-world technological applications.
Volume 2 deals with the determination of the electronic properties of surfaces. As previously acknowledged, the separation is arbitrary and it is accepted that once it is known where the electrons are, then by calculating the total energy of the system, the geometry is uniquely determined simply by minimization. Well, these are early days and, while we believe that within the next decade this procedure will (hopefully) be routine, the electronic spectroscopies applied to surfaces and interfaces form a section of their own and are reviewed in depth in Volume 2. Volume 3 will address the dynamical aspects of surfaces and surface processes, reflecting in detail current understanding of energy exchange at surfaces and the part this plays in the adsorption and reaction of atoms and molecules at surfaces. Surface dynamics is, of course, intimately related to the geometric and electronic properties of the surface and while Volumes 1 and 2 address the equilibrium geometry and ground state electronic properties, Volume 3 pays particular attention to the response of surfaces to external stimulation. Taken as a set, the three volumes of this series aim to provide an in-depth introduction to the world of surface and interfacial science and would be most suited for scientists having obtained a first degree in the natural sciences.
N. V. Richardson and S. Holloway Surface Science Research Centre Liverpool, UK
Preface to V o l u m e 1
The primary goal of this book is to summarize the current level of accumulated knowledge about the physical structure of solid surfaces with emphasis on well-defined surfaces at the gas-solid and vacuum-solid interfaces. The intention is not only to provide a standard reference for practitioners, but also to provide a good starting point for scientists who are just entering the field. Thus, the presentation in most of the chapters assumes that the typical reader will have a good undergraduate background in chemistry, physics, or materials science but little, or no, knowledge of surface science. At the same time, coverage is comprehensive and at a high technical level with emphasis on fundamental physical principles. It is appropriate that this first volume of the Handbook of Surface Science is devoted to the physical structure of surfaces since knowledge of the atomic positions is often essential for complete understanding of electronic properties or dynamical processes that are the topics of the following two volumes. This volume is divided into four parts. Part I describes the equilibrium properties of surfaces with emphasis on clean surfaces of bulk materials. Chapter 1 defines the terminology and notations used to describe the crystallography of surfaces of crystals. Chapter 2 then outlines the relationships between the atomic structures and the macroscopic thermodynamic properties of surfaces. The major theoretical approaches that are used to calculate the structure of surfaces are presented and critiqued in Chapters 3 and 4. Chapter 3 emphasizes metals and elemental semiconductors and Chapter 4 emphasizes insulators. Chapters 5 and 6 are devoted to crystalline ceramics and to elemental and compound semiconductors, respectively. They demonstrate the complexity that can occur in multicomponent materials. Part II provides an introduction to some of the primary experimental methods that are used to determine surface crystal structures. Chapter 7 covers the three main diffraction methods. Chapter 8 describes techniques that provide more direct images. Spectroscopic techniques will be covered in a future volume of this Handbook. Part III provides an overview of the vast topic of the structure of adsorbed layers. A systematic treatment of the rich variety of structures formed by chemically adsorbed species is given in Chapter 9. Chapter 10 is an in-depth treatment of the structures that occur in physically adsorbed layers. Chapter 11 covers the complex topic of theoretical prediction of the interactions between adsorbed atoms.
vii
Part IV concludes the book. Chapter 12 covers the important topic of defects in surface structures. Chapter 13 is a comprehensive treatment of phase transitions that covers aspects of structure that go beyond local interactions. Surface science is a broad and interdisciplinary subject. This volume should be viewed as a general introduction rather than a comprehensive treatment. Nearly every chapter could easily be expanded into a full volume. Thus, some specific topics that are closely related to surface structure, such as surface magnetism and surface lattice dynamics, have not been included. Look for these in future volumes of the Handbook of Surface Science. I want to express my thanks to the Handbook editors and the publisher for offering me the opportunity to organize this volume. I am especially grateful to Patricia Paul and Barbara Deshane for their assistance in bringing this project to completion.
William N. Unertl Orono, ME, USA
viii
Contents of Volume 1 General Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents o f Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributors to Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List o f Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv vii ix xi xiii
Part L Basic aspects of the structure of crystalline surfaces 1. W.N. Unertl Surface crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. E.D. Williams and N.C. Bartelt Thermodynamics and statistical mechanics of surfaces
3 ............
51
3. C.T. Chan, K.M. Ho and K.P. Bohnen Surface reconstruction: metal surfaces and metal on semiconductor surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
4. J.P. LaFemina Theory o f insulator surface structures
137
....................
5. R.J. Lad Surface structure of crystalline ceramics . . . . . . . . . . . . . . . . . . .
185
6. C.B. Duke Surface structures of elemental and compound semiconductors . . . . . . .
229
Part IL E x p e r i m e n t a l methods to study surface structure 7. E.H. Conrad Diffraction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
271
8. W.N. Unertl and M.E. Kordesch Direct imaging and geometrical methods
361
..................
Part III. Structure of adsorbed layers 9. H. Over and S.Y. Tong Chemically adsorbed layers on metal and semiconductor surfaces . . . . .
ix
425
10. J. Suzanne and J.M. Gay The structure o f physically adsorbed phases . . . . . . . . . . . . . . . . .
503
11. T.L. Einstein Interactions between adsorbate particles . . . . . . . . . . . . . . . . . . .
577
Part IV. Defects a n d p h a s e transitions at surfaces 12. M.C. Tringides Atomic scale defects on surfaces
.......................
653
13. L.D. Roelofs Phase transitions and kinetics o f ordering . . . . . . . . . . . . . . . . . .
713
Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
809 855
Contributors to V o l u m e 1
Norman C. Bartelt, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA K.P. Bohnen, Forschungszentrum Karlsruhe, Institut fur Nukleare Festk6rperphysik, Karlsruhe, Germany Che Ting Chan, Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Edward H. Conrad, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA Charles B. Duke, Xerox Webster Research Center, 800 Phillips Road 0114-38D, Webster, NY 14580, USA Theodore L. Einstein, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA J.M. Gay, Facult6 des Sciences de Luminy, D6partement de Physique, Case 901, 13288 Marseille Cedex 09, France Kai Ming Ho, Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Martin E. Kordesch, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA Robert J. Lad, Laboratory for Surface Science and Technology, 5764 Sawyer Research Center, University of Maine, Orono, ME 04469-5764, USA John P. LaFemina, Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratories, P.O. Box 999, Richland, WA 99352, USA Herbert Over, Fritz Haber Institut der Max Planck Gesellschaft, Faradayweg 4-6, D- 14195 Berlin Dahlem, Germany Lyle D. Roelofs, Physics Department, Haverford College, Haverford, PA 19041, USA J. Suzanne, Facult6 des Sciences de Luminy, D6partement de Physique, Case 901, 13288 Marseille Cedex 09, France David S. Y. Tong, Laboratory for Surface Studies, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA
Michael C. Tringides, Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA William N. Unertl, Laboratory for Surface Science and Technology, Sawyer Research Center, University of Maine, Orono, ME 04469, USA Ellen D. Williams, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA
xii
List of Symbols and Acronyms lengths of unit mesh vectors in real space unit cell and unit mesh vectors in real space reciprocal lattice unit mesh vectors area, unit mesh area in real space, Hamaker constant, aperture A function unit mesh area in reciprocal space A.>g scattered amplitude A(Q) Auger electron spectroscopy AES atomic force microscopy AFM angle resolved photoelectron spectroscopy ARPES bulk modulus B Burger's vector, step height b body centered cubic bcc centered lattice c specific heat, contact function C commensurate-incommensurate transition CIT commensurate site lattice CSL crystal truncation reciprocal lattice rod CTR interrow spacing of rows (hk) dhk distance, diameter, diffusion coefficient D dimer adatom stacking DAS electron charge e kinetic energy, electric field strength, identity transformation, edge E dislocation, effective modulus interaction energy between neighbors separated by i Ei Fermi energy EF embedded atom method EAM electron energy loss spectroscopy EELS extended X-ray absorption fine structure EXAFS face centered cubic fcc f(ko,ki)f(Q,E ) form factor, scattering length F Helmholtz free energy, embedding energy, force F(Q,E), F(Q) crystal structure factor ai, bi ai,bi,ci 9 !, * a i ,I) i ,C i
xiii
F{...} FIM FLAPW FWHM
Ghkl g
g G G(r) H h (hk) {hk} HREELS HWHM I
l(O) i
[i} ISS
J(Q)
Jk k k k,kB l L LCAO LDA LEED LEEM LEIS LEPD LMTO LRO M m mij
MBE MEIS MSHD N
Fourier transformation of {... } field ion microscopy full potential linear augmented plane wave full width at half maximum reciprocal lattice vector two-dimensional reciprocal lattice vector glide line Gibbs' free energy mean-square fluctuation enthalpy, Hamiltonian Planck's constant, external magnetic field, Miller index coordinates of a reciprocal lattice rod, indices of a diffraction beam, indices of a set of parallel rows indices of a form of sets of rows, indices of a form of diffraction beams from such sets of rows high resolution electron energy loss spectroscopy halfwidth at half maximum tunneling current scattered intensity imaginary unit ~/-1 sites on a lattice ion scattering spectroscopy measured diffraction signal exchange constant between neighbors separated by k wavenumber (27t/L), Miller index wavevector Boltzmann's constant Miller index terrace length linear combination of atomic orbitals local density approximation low energy electron diffraction low energy electron microscopy low energy ion scattering low energy positron diffraction linear muffin-tin orbitals long-range order transfer matrix, spontaneous magnetization, atomic or molecular mass, magnification electron mass, lattice mismatch, mirror reflection plane elements of transfer matrix molecular beam epitaxy medium energy ion scattering mean square height deviation number of sites on a lattice, number of atoms xiv
NA
N(E) 1l tli
P e(g) P Q
O q qst R R,r R(Q) RHEED RBS REM si S SCF-LCAO SIMS SOS SPALEED STM STS t
t(g) T T(Q) T TEM THEED U(E) U U
uhv U, V
[uv]
UPS V V
W W
Avogadro's number distribution of secondary electrons index of refraction site occupation variable in lattice gas model pressure, polarization factor pair distribution function, Patterson function primitive lattice partition function scattering vector phonon wavevector isosteric heat gas constant, reliability factor, radius position vectors in real space reflectivity reflection high energy electron diffraction Rutherford backscattering spectroscopy reflection electron microscopy magnetic moment in the Ising model entropy, screw dislocation self-consistent field, linear combination of atomic orbitals secondary ion mass spectroscopy solid-on-solid model spatially analyzed LEED scanning tunneling microscopy scanning tunneling spectroscopy time, reduced temperature, thickness transfer function translation operator instrument response function temperature, tunneling probability transmission electron microscopy transmission high energy electron diffraction inner potential internal energy, vibrational amplitude displacement from equilibrium position ultra-high vacuum coordinates of two-dimensional lattice points expressed in terms of a and b as units indices of a direction in the real space net indices of a form of directions related by symmetry ultraviolet photoelectron spectroscopy volume, potential speed work of adhesion, 2W is the exponent of the Debye-Waller factor interface width XV
XPS x-,y-,z-
x,y Z t~
A 8 Z E
Y
y(hkt) F 11 A P kt V
0 0
s O)
~(R) ~3(Q)
X-ray photoelectron spectroscopy directions of crystallographic axes, x- and y- are parallel to the surface, z- is the outward normal coordinates on any point in the unit mesh, expressed as fractions of a and b units canonical partition function total scattering cross section, mirror transformation matrix critical exponent of specific heat, ratio of scattered amplitudes from successive layers, angular divergence 1/kaT, critical exponent of order parameter vs. T critical exponent of order parameter vs. conjugate field bond length conserving rotation resolution, uniform strain susceptibility binding energy surface energy, angle between unit mesh vectors, critical exponent of susceptibility surface energy pair correlation function, center point of Brillouin zone critical exponent of correlation function potential function, work function attentuation length for elastic scattering wavelength density, density of states chemical potential critical exponent of the correlation length one-half the scattering angle coverage, angle grand potential, solid angle angular frequency, tilt angle correlation length dislocaton line scattered wave amplitude, wave function corrugation function interference function reflectivity
xvi
Part I
Basic Aspects of the Structure of Crystalline Surfaces
This Page Intentionally Left Blank
CHAPTER 1
Surface Crystallography W.N. U N E R T L Laboratory for Surface Science and Technology Sawyer Research Center University of Maine Orono, ME 04469, USA
9 1996 Elsevier Science B.V. All rights reserved
Handbook o.f Su~. ace Science Volume 1, edited by W.N. Unertl
Contents
1.1.
S o m e basic concepts of bulk c r y s t a l l o ~ a p h y 1.1.2.
Structure o f the unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface structure and surface order
1.3.
Surface c r y s t a l l o g r a p h y
1.5.
!.6.
5 5
Lattices, directions, and planes
1.2.
1.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 8 9
1.3.1.
C r y s t a l l o g r a p h y of a plane
11
1.3.2.
Point and space group s y m m e t r y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3.3.
Isolated adsorbed atoms and m o l e c u l e s . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.3.4.
The reciprocal lattice
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.4.1.
W o o d notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.4.2.
Matrix notation
!.4.3.
Classification o f o v e r l a y e r m e s h e s
Unit m e s h t r a n s f o r m a t i o n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vicinal surfaces, steps, facets, and the stereographic projection
.................
31
1.5.1.
Stereographic projection
1.5.2.
Vicinal surfaces and u n i f o r m arrays of steps . . . . . . . . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Disorder
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27 30 31
1.6.1.
Pair distribution function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
1.6.2.
Defects of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
1.6.3.
Defects of the second kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
A p p e n d i x : The 17 t w o - d i m e n s i o n a l space groups
43
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
This chapter introduces the basic terminology and notation that is used in this book. Symbols for physical quantities follow the recommendations of the International Union of Pure and Applied Physics (SUN Commission, 1978). SI units are used throughout, unless explicitly stated otherwise. The chapter begins with a brief review of some basic concepts of bulk crystallography, followed by an extension of these concepts to surfaces. Next, the crystallography of perfect systems in one and two dimensions is presented with emphasis on those concepts most relevant to the structure of surfaces. This leads naturally into a treatment of the relationships and transformations between surface and bulk structures. Then the stereographic projection, vicinal surfaces, and notations for steps and facets are described. Finally, various types of disorder are introduced.
1.1. Some basic concepts of bulk crystallography Many aspects of surface terminology and surface crystallography are simple extensions of those used to describe the structure of bulk materials. Therefore, this chapter begins with a review of the relevant concepts from bulk crystallography. More extensive introductory treatments can be found in solid state physics textbooks such the one by Burns (1985). Vainshtein (1981) gives an advanced treatment. 1.1.1. Lattices, directions, and planes
Crystal structures in three dimensions are defined in terms of their Bravais lattices. A Bravais lattice is an infinite regular array of points that fills all space. This lattice is made up from a basic building block called the unit cell, which is usually specified in terms of three unit cell vectors (a, b, c), whose lengths and directions define the size, shape and orientation of the unit cell. This notation is illustrated for the case of the simple cubic unit cell in Fig. 1.1. The volume of the unit cell is V = la.(b•
(1.1)
Each unit cell can be translated to the position of any other cell in the lattice by the operation of a translation vector T(hkl) = ha + kb + lc
(1.2)
where (h,k,l) are integers. Crystal lattices often have several other symmetry properties in addition to translational symmetry. The symmetry operations that occur in both two and
6
W.N. Unertl
J
lal
= Ibl
=lcl
Fig. 1.1. Unit cell vectors a, b, and c for a cubic unit cell. [ 111 ] is the unit cell diagonal.
three-dimensional crystallography are rotation, reflection and glide plane. Operating on the lattice with any of these, or with combinations of them, always leaves the lattice unchanged. These operations as applied to surface crystallography are discussed further in w 1.3 below. A specific direction in a lattice is specified by the vector R ( u v w ) = ua + vb + wc
(1.3)
which is often written in shorthand using the square bracket notation [uvw]. As examples, [001 ] is parallel to c and [ 111 ] is parallel to the unit cell diagonal R( 111 ) = a + b + c as shown in Fig. 1.1. A line drawn over a character indicates negative values; i.e., [uvw] - [-u,v,w]. In most lattices, several directions are equivalent since at least one of the symmetry properties of the lattice transforms them into each other. A group of such symmetry related directions is called directions o f a f o r m and is denoted by angle brackets; e.g., . For example, the simple cubic lattice, whose unit cell is shown in Fig. 1.1, is left unchanged when rotated about the a, b or c axes by integral multiples of rt/2. Thus, the specific directions [001 ], [001], [010], [0T0], [TOO] and [100] are all members of the form <001>. Once the coordinate system is specified, the direction [uvw] is unambiguous. However, because the form represents more than one direction, its correct interpretation requires that both the unit cell vectors and the symmetry operations of the lattice have also been specified. These two types of notation for directions are n o t interchangeable. The reader should beware however because this important distinction is misused frequently in the scientific literature of surface crystallography. A lattice p l a n e is any plane that contains three or more non-collinear, lattice points. Families of equally spaced parallel lattice planes can be constructed so that together they contain all the points of the Bravais lattice. Each such family of lattice planes is denoted by the notation (hkl), where the indices h, k and l are the Miller indices. A general operational procedure for determining the Miller indices for a particular family of planes is" m
Su~. ace crystallography
7
~
~
1.5c
y =
1.75b
(a)
x
~F
(100)
(110)
{111)
(b)
(c)
(d)
Fig. 1.2. Examples of Miller index notation for lattice planes in a cubic system. 1. Choose the origin so that one of the members of the family passes through it. This is always possible since the location of the origin in the unit cell is arbitrary. 2. Set-up a coordinate system with axes parallel to the mesh vectors a, b and c. Note that this coordinate system will not always be a Cartesian coordinate system. Only three of the fourteen lattice types possible in three dimensions have Cartesian coordinate systems; e.g., orthorhombic, tetragonal, and cubic. 3. Determine the intercepts (x, y, z) of the plane in the family that is closest to the origin; x, y, and z are measured in units of a, b, and c, respectively, and may be either positive or negative. 4. Take the reciprocals: (l/x, l/y, 1/z). 5. Express in terms of the lowest common denominator. 6. Factor out and discard the denominator. Consider the following examples: 1. Figure 1.2 shows a segment of a plane that intersects the x-axis at 2a, the y-axis at 1.75b, and the z-axis at 1.5c. Step 3 above yields (2,1.75,1.5). The reciprocals are (1/2,4/7,2/3) and the lowest common denominator is 42. Factoring this out leaves (21,24,28) which are the Miller indices of the plane shown. 2. Figure 1.2b-d shows some of the planes most commonly discussed in cubic crystals and gives their Miller indices. The (100) plane, for example, intercepts the axes at (1,oo,oo); i.e., the plane is perpendicular to the vector a. As is the case for directions of a form, several families of lattice planes may be related to each other by the symmetry properties of the lattice. Such planes of a form are denoted with curly brackets; e.g., {hkl}. The (100), (010) and (001) planes of a cubic lattice are members of the { 100} form of planes. The direction [hkl] is perpendicular to the (hkl) family of planes.
8
W.N. Unertl
1.1.2. Structure of the unit cell The Bravais lattice provides a useful framework for the discussion of real crystal structures. However, the Bravais lattice should never be confused with the actual atomic structure of a real crystal. For example, Bravais lattices are strictly infinite in extent whereas real crystals are finite. For many problems in bulk crystallography, this distinction is unimportant as long as the region of interest is far enough from the surface. However, the mere presence of a surface means that translational symmetry does not exist in the direction of the surface normal since neighboring atoms that would have been present in the bulk are missing on one side at the surface. Furthermore, the mathematical points that make up the Bravais lattice do not have to coincide with the positions of atoms in the real crystal. In a real crystal, the basic structural unit is the unit cell. There may be more than one atom per unit cell; e. g., in a crystal of protein molecules, the unit cell may contains thousands of atoms. These unit cell atoms form the basis of the cell and their positions are written as (xyz) where x, y and z are measured in units of a, b and c respectively. Again, note that x, y and z need not refer to a Cartesian coordinate system. The location of the origin within the unit cell is completely arbitrary. However, a corner of the unit cell is often selected. The usefulness of the Bravais lattice concept is that it provides a mathematically well-defined geometrical framework which can be used to model the properties of a real crystal. In this book, the terms ideal crystal and ideal surface are used to describe defect-free idealizations of real crystals with translational and symmetry properties which can be completely described by a Bravais lattice with a basis. 1.2. Surface structure and surface order
When an ideal surface is prepared by making a planar cut through an ideal bulk crystal, the resulting surface is identified by the Miller indices of the bulk plane parallel to which the cut was made. Most studies of real crystal surfaces have been carried out on the low-index surfaces (i.e., small values of h, k and l) of crystals with four types of bulk unit cells: face-centered cubic (fcc), body-centered-cubic (bcc), diamond and hexagonal-close-packed (hcp). Figures 1.3 to 1.6 show their bulk unit cells and the ideal structures of some of their low-index planes. In the case of hcp crystals the lattice planes and surfaces are usually, but not always, described by a four digit notation (hktl) where t = - ( h + k ) . The index t contains no new information about the crystal and serves only to identify the structure as hcp. Since the bulk unit cell vectors do not necessarily lie in the surface plane (consider for example, the (111) plane), it is usually more convenient to use a new coordinate system to describe the surface. The coordinate system used throughout this book is shown in Fig. 1.7. The z-axis points outward from the surface and the x- and y-axes lie in the surface plane. Unit vectors along the x, y and z directions are denoted i, j, and k, respectively. The y-axis is usually chosen to lie along a low-index direction in the surface.
Su~. ace crystallography
9
~z
t~
101oi
11101
0o0o|
.0 0 0 0 0 8~[
1~" o 0 o | ._ | 1 7 4 1 7 4 o| |174174 o0o|
aI
0 0
0o0o|
0
o o o o 9 0 0 0
o o
0 0
(I00)
9
[11Ol
o _ o ~ o ~ o ~ o
o?o?o?o?o
|174 o~o~o~o~o
o o
0
o o
0
o
: _
0
o
0 0 0 (II0)
9
0
o~o~o~o~o
I,Oy
O'O'O'O'O ~Ol~l
(111) Fig. 1.3. The face-centered-cubic unit cell and the (100), (110), and (111 ) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, and the filled circles are atoms in the third layer. Nearest neighbor atoms in the unit cell are shown connected by solid lines.
Directions are defined with respect to their polar and azimuthal angles. The polar angle 0 is measured from the surface normal and the azimuthal angle ~ is measured from the positive x-axis with the positive sense corresponding to a counter clockwise rotation when observed by looking into the positive z-axis.
1.3. Surface
crystallography
Surface crystallography is the study of the structure and symmetry properties associated with the atoms and molecules near the surfaces of a crystal. Surfaces may have completely different atomic structure than found in the bulk of the crystal. Even the composition may be different due to structural rearrangements or because atoms and molecules from the environment can interact with the surface. Chapter 2 discusses the role of thermodynamics and statistical mechanics in determining
10
W.N. Unertl
~
z
I
y
>
[OlO]
t
0 @ @
9 0
o o
@ @
@
--.
@ 0
o o
@ 0
@
9
a~
@
o
@
a v
o
0
o
@
0
0
0
~
0
@ o o 0
@
o
0
@ o
O o @ o O o
o
O
o
O o 0 O o O o
@
(100)
0---~ o 111Ol
O
( 11 O}
0 9
0
..oy~
|
0 9
0
~
@
0 9
0
@
[110] 9
0
o (111)
Fig. 1.4. The body-centered-cubic unit cell and the (100), (! 10), and (111 ) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, and the filled circles are atoms in the third layer.
surface structure. Chapters 3, 4 and 11 describe fundamental aspects of the theory of metal, semiconductor, and insulator surface structures. Experimental methods used to determine surface structures are reviewed in Chapters 7 and 8. Examples of the surface structures for various materials and adsorbed layers are given throughout this volume. The final two chapters address the topics of defects on surfaces and surface phase transitions. This chapter focuses primarily on some of the foz'mal aspects of surface crystalIography. A useful starting point for this discussion, which is the topic of this section, is a study of the crystallographic properties of a purely two-dimensional system, the plane. The plane has no thickness and any point (x,y) on it can be described with reference to a two axis coordinate system with x- and y-axes. The reader should keep in mind, however, that the plane provides only an idealized description of a real surface since, in even the simplest cases, a real surface extends into a third dimension and may even be curved.
Su~. ace crystallography
Il
~
z
I
I
['~101
[010----~
i
~i 9" ~
9"0
"~ 9
8 8 8
9
9e. ~. ~. oo..eo..~o..~
~.~8~
(100)
,go
(110) C4
[110] 00000000 ~
;?;?;?;?;
GoOoOoOo G o~o_o_o--o ^~^~o~o @'@'@'@'@ O ~a' ~
~j
[,oTi/r
~ 0 x'~, (111)
Fig. 1.5. The unit cell of the diamond lattice and the (100), (l l 0), and (! l I) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, thc fillcd small circles arc atoms in the third.
1.3.1. Crystallography of a plane The basic crystallographic properties of planes can be described in terms of the s y m m e t r y properties of an ideal t w o - d i m e n s i o n a l Bravais lattice with most quantities defined in direct analogy with their three-dimensional analogs. The definitions and notation used in this book are selected to be consistent with the International Tables 3"or Crystallography (1959). The two-dimensional Bravais lattice is an infinite planar array of points in real space, each of which can be transformed into any other by the translation operation vector
T(hk) = ha I +
kG 2
(1.4)
where the indices h and k are integers. The basic structural unit of the lattice is the
unit mesh defined by the unit mesh vectors a~ and a 2. The convention for c h o o s i n g
12
W.N. Unertl
I
[lOOl
,o,o,
t ~ ~ a ~
@o@o
a
@o@o.
9
I
O.O.O.O.O
C
.O.O'O.O"
*'~176176176
9
@o@o@o@o@
I ,_._, 0 o
0 , 0 o 0 , 0 , 0
.0.0.0.0.
@o@o@o@o@
@o@o@o@o@
(2i3ol
(1230)
@
@
@
@
@----~
o@o@o@o@o
!11Ol
@0 @0 @0 @0 @ 0
d
@0 @0 @0 @0
@o@o@o@o@ ~o~ (ooo~)
m
~
Fig. 1.6. The hexagonal-close-packed unit cell, the (0001) surface, and (1230) and (2130) prism faces of ah ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, the filled small circles are atoms in the third layer.
R.I. ' Rzi z'" \ "-. \ \
.If Fig. 1.7. Cartesian coordinate system used in this book; the z-axis is the outward pointing normal.
Sueace crystallography
13
a~ and a 2 is that a 2 is always longer than al (i.e., lall < Ja2J) and will be used in this book. l Reader beware! Other non-standard conventions are c o m m o n l y encountered in the literature, specifically la~l > la21. The unit mesh is the basic building block of the surface crystal structure. Thus, if one starts with a single unit mesh and applies the translation operator T(hk) to it, every point in the entire two-dimensional lattice will be produced. The unit mesh area is lal• There are only five classes of unit meshes that have the property that they can cover the entire plane of the Bravais lattice when operated on by the translation operator. In crystallography, these meshes are most c o m m o n l y selected to be simple geometrical figures as shown in Fig. 1.8 along with their c o m m o n names: oblique, square, hexagonal, rectangular, and centered rectangular meshes. Each of these meshes can be defined by vectors a~ and a2. For convenience, the angle 7 between a~ and a2 is also specified. The lattice symbol p (always a lower case italic letter) is an optional prefix used for the primitive meshes - - oblique, square, hexagonal and r e c t a n g u l a r - which have lattice points only at their corners; for three-dimensional systems a capital symbol is used. A primitive mesh is always the smallest unit mesh for a given lattice. The only two-dimensional unit mesh which is not primitive is the centered rectangular mesh which, in addition to lattice points at each corner, has one lattice point at its center; i.e., at the location (1/2,1/2). The symbol c (lower case, italic) is used to designate a two-dimensional centered structure. A centered mesh is never the smallest possible unit mesh for its lattice. The primitive mesh for the centered rectangular structure is given by the unit mesh vectors bi and b 2 where b~ = 0.5 (a~ - a2) and b2 = a2. The following convention is used to draw unit meshes for the two-dimensional Bravais lattices: 1. The origin of the mesh is chosen to be in the upper left-hand corner. 2. The vector a 2 is drawn horizontal and pointing to the right. The direction of a 2 and the y-axis, Fig. 1.8, are usually chosen to be the same. 3. The vector as is drawn downward and pointing to the lower left; i.e., the angle , / b e t w e e n a~ and a2 must always be greater than or equal to 90 ~. Reader beware! The opposite convention (7 < 90~ is sometimes used. Failure to realize this can have drastic consequences, particularly in interpretation of diffraction data. 4. la~l _< la21. This convention will be used throughout the book and, where ever possible, results quoted from other sources will be transformed to be consistent with it. Crystallographers tend to use simple geometrical shapes (e. g., squares, rectangles, etc.) for the unit mesh, but this is not necessary. Instead, the unit mesh shape can be constructed to be complex but must still meet the constraint that it fill the plane like a jigsaw puzzle with identical pieces. Additional constraints on the shape are provided by the symmetry properties of the lattice as discussed below. M.C. Escher
1 Anotherwidely used definition of the unit mesh vectors is a = a I and b = a 2.
14
W.N. Unertl
First Brillouin Zone
Reciprocal Space
Real Space
§
§
§
§ o
o
, a1
Oblique
o
~
--,-- lOll
/
+a,']
"x
[tol
9
Square
o
§
a2
~---~ a1
[Ol1
§
§
§
§
(0 1)
.
§
§
9
(l T)
,a2
(lo)
+
+ (Ti)
§
§
§
§
§
§
al +
" ~ l t tl
ltOl
(II) (lo)
§ o
o
o
§ §
§
§
§ §
§
§ §
o
o a ~
o.-,. IYll
IlOl
li ll
[oil
4
--- lo,!
.
+ a l d ~ ' 2 + Ii'll (io)4~ ~. (ol) (2o)§ "d 4{o2)
"v-
o
liOl
§
2
[o 11
o
Hexagonal
§
(z l)
a*
o
9 ai~---~
9
§
)
( 0 0 ~
+
Centered Rectangular
4.
o
1~ol
Rectangular
41-
11o) NF (ll)
§ 9
~(Ol)
41-
11 ll
o
§
(oT)+ + (~176
,
1111
§ §
§ §
§
(ll}
+
+ (Tl)
§
(lo) \ (tl)
Fig. 1.8. Unit meshes of the five two-dimensional Bravais lattices in real and reciprocal spaces.
Su~. ace crystallography
15
Fig. 1.9. (a) A drawing of a two-dimensional p l lattice by M.C. Escher. From MacGillavry (1976). Two possible choices of unit mesh are shown just below the drawing.( 9 1996 M.C. Escher/Cordon Art, Baarn, Holland. All rights reserved.) (b) Top view of the unit mesh structure of the (7x7) surface reconstruction of Si(111 ). After Robinson et al. (1986). This unit mesh is divided in half (shaded and unshaded regions) by an anti-phase boundary. (MacGillavry, 1976) has created many examples of such complex meshes. One of these is reproduced in Fig. 1.9a. It is sometimes useful to subdivide the unit mesh into distinct smaller units. One example, shown in Fig. 1.9b, is the hexagonal unit mesh of reconstructed Si(111) (Robinson et al., 1986), known as the (7• surface m the notation (7• is described in w 1.4.1. The basis associated with this reconstruction, which involves
16
W.N. Unertl
atoms from the first three atomic layers, is shown superimposed on the mesh z. Since the basis is divided into halves by a stacking fault (see Chapters 6 and 12), the mesh can be thought of as composed of two inequivalent triangles. In the case of the Escher print, Fig. 1.9a, the mesh is subdivided into a boat and a fish. The intersection of a three-dimensional plane (h',k',l') with the surface is called a lattice line; it passes through two or more lattice points in the surface plane. Families of equally spaced parallel lattice lines can be shown to contain all the points of the lattice. Lattice lines are denoted by the two-dimensional Miller indices (hk) of the surface vector perpendicular to the line. Since, in general, the unit mesh vectors a~ and a 2 do not coincide with the bulk unit cell vectors a, b, and c, the (hk) will generally be different from any of the (h',k',l') of the plane from which the lattice lines were constructed.
1.3.2. Point and space group symmetry In addition to the translational symmetry described by Eq. (1.2), two-dimensional lattices have other symmetry properties. The possible operations in two dimensions are the identity transformation, rotations, reflections, and glide lines. Groups are made up of combinations of the individual symmetry operations. The groups of interest in this chapter are point groups and space groups. The discussion of groups given here is rudimentary. More information can be found in Ertl and Kuppers (1985), Vainshtein ( 1981 ), and Cotton (1990). A point operation is any symmetry operation which leaves at least one point in the lattice unmoved. Except for the identity transformation E, reflections turn out to be the only possible point operations in one-dimensional lattices. Both reflections and rotations are possible in two dimensions and, in three dimensions, rotations, improper rotations, reflections, inversions and their combinations are possible. The simplest point operation is the identity transformation E which leaves a point (x,y,z) unchanged; i.e.
(x',y',z') = E(x,y,z) = (x,y,z) where
E
(1.4)
/!0!/01
The other point operations in two dimensions are rotations and reflections.
For the purposes of this section, the structure can be thought of as being projected onto a plane. Because of its three-dimensional extent, this structure is not strictly described by two-dimensional crystallography.
Su~. ace crystallography
17
Table 1.1 Two-dimensional symmetry operations Operation
Shorthand notation International
Symbol
Sch6nflies
One-fold rotation
1
C~
none
Two-fold rotation Three-fold rotation
2 3
C2
9
C3
A
four-fold rotation
4
C4
9
Six-fold rotation
6
C6
Mirror reflection
m
G
Glide line
g
In two d i m e n s i o n s , the only rotational axes that are p e r m i t t e d are p e r p e n d i c u l a r to the lattice planet. The angle of rotation about a rotation axis is a - 2rt/N w h e r e N is the order of the axis. The five possible rotations for c r y s t a l l o g r a p h i c s y s t e m s have N = 1,2, 3, 4, 6. T h e s e are listed in Table 1.1 along with their s y m b o l s . In the widely used Sch6nflies notation, rotation axes are specified by CN. In matrix form, a rotation axis passing through the origin is given by
o/COSs,n . . !/ 0 Thus, a general point (x, y, z) t r a n s f o r m s to a new position (x', y', z') as follows
ysin'/"
y' - CN Z'
-
sin0~N+ycoso~ N
(1.6)
Z
N o t e that the z-coordinate is not c h a n g e d by rotations in t w o - d i m e n s i o n a l systems. The lattice is left u n c h a n g e d when it is rotated about the rotation axis by an integral m u l t i p l e of a . As is the case in three d i m e n s i o n s , this r e q u i r e m e n t rules out the e x i s t e n c e of a Bravais lattice that can be t r a n s f o r m e d into itself by a five-fold or
1
In two-dimensional slab systems, i.e., those with finite extent perpendicular to the lattice plane, there may also be 2-fold rotation axes and mirror planes oriented parallel to the lattice plane.
18
W.N. Unertl
seven-fold rotation. The 4-fold rotational operation necessarily contains the 2-fold rotation and the 6-fold rotation contains both the 2- and 3-fold rotations. Mirror-reflection planes, denoted by m or o, are the third two-dimensional point transformation. For example, if the xz-plane is a mirror plane,
o =fo o ~0 0
(1.7) 1)
and
I X///IY/ Y
y' =CYxz Z'
(~.8)
=
For this example, operation on a general point (x,y,z) changes only the y-coordinate. Only mirror-reflection planes that are perpendicular to the surface are required in two-dimensional crystallography. The intersection of a mirror plane and the twodimensional lattice is symbolized by a heavy solid line as shown in Table 1.1. In two dimensions, the mirror-reflection plane is also often referred to as a mirror line. There are only ten unique combinations of the rotation and reflection point operations on a two-dimensional lattice. These are called the ten two-dimensional point groups and are listed in Table 1.2. Each of the two-dimensional point groups can be identified by a three component notation, called the international (or H e r m a n n - M a u g u i n ) notation, which is interpreted in the following way: 1. First position: a numeral indicating the rotation operation (N = 1,2, 3, 4, or 6). 2. Second position: a mirror plane perpendicular to the x-axis plus all other mirror planes related to it by the rotation operation. 3. Third position: any second form of mirror planes. It is not always necessary to specify the second and third positions. The symbols in parentheses in Table 1.2 give a shorthand notation which is sufficieni to uniquely Table 1.2 The ten two-dimensional point groups Bravais lattice Oblique Oblique Rectangular Rectangular Square Square Hexagonal Hexagonal Hexagonal Hexagonal
Point notation I 2 lm 2mm 4 4mm 3 3m 6 6ram
Group
(m) (ram) (4m)
(6m)
Sch6nfleis notation E or C i C2 o C2v C4 Cnv C3 C3v C6 C6v
Su~. ace crystallography
19
identify each point group but does not contain enough information to reconstruct the group. The Schi~nflies notation is another widely used notation for point operations and is given as the third column of Table 1.2. In the Schtinflies notation, the 1-fold rotation is designated by E or by C,. An n-fold rotation axis is described by Cn. The mirror plane m is given by ~v where the subscript v indicates that the mirror plane is normal to the two-dimensional lattice. For two-dimensional point groups containing more than one operation, the notation Cnv is used. Only one other symmetry operation, a space group operation, which is constructed by successive application of a mirror and a translation, is possible in two dimensions. It is called a glide line and is symbolized by a heavy dashed line or the letter g. Specifically, the glide operation consists of a reflection followed by translation parallel to the mirror by one-half the unit mesh length. Note that the glide operation does not contain the mirror operation even though reflection is used in the construction of the glide line. The set of points generated by applying all the symmetry operations of a group to an arbitrary point (x,y) is called the regular point system (RPS). Figure 1.10 gives an example of the RPS for the group 2mm. If (x,y) happens to lie on one of the symmetry elements, it is a point of special position, otherwise it is called a point of general position. There are only seventeen unique combinations of the ten two-dimensional point groups and the glide line. These combinations are called the two-dimensional space groups and represent all the possible two-dimensional lattices which can fill the
t-~.y) $
(x,y)
m
e) (x.-y) m
Fig. 1.10. The regular point system for the two-dimensional space group 2mm.
W.N. Unertl
20
entire plane I. These space groups are fully tabulated in the International Tables of Crystallography (1959). A simplified version of these tables is presented in the Appendix to this chapter. The space groups are arbitrarily numbered from 1 to 17. The space groups numbered 4 and 7 have proven to be important for adsorbate systems are reproduced in Fig. 1.11. Across the top of each space group are listed the Bravais lattice type (rectangular in these cases), the short form symbol for the point group (m and ram), the full space group notation (p 1g 1 and p2mg), the number of the space group (4 and 7), and the short form symbol for the space group (pg and pmg). T w o diagrams are shown are shown for each case. For each, an x,y- coordinate system is assumed with the x-axis pointing along the direction of a~, the y-axis pointing to the right along a2, and the origin in the upper left-hand corner. The left-hand diagram illustrates one unit mesh and shows how a point of general position (x,y), shown by the circled comma, is transformed under application of the space group operations. The right-hand diagram also represents the unit mesh and shows the location of the symmetry operators within it. Centered directly under these diagrams is a statement giving the convention that has been used to locate the mesh origin with respect to the location of the symmetry operations since this location is in general not unique. In the middle of Fig. 1.11 is a drawing of the structure of CO adsorbed on the Pt(110) surface. Individual CO molecules adsorb at atop sites on this surface (see w 1.3.3). However, the molecules are slightly too large to adsorb at adjacent sites and still be vertical to the surface. Instead, alternate molecules tilt to opposite sides of the 1110] rows of Pt atoms. Lambert (1975) recognized that this zigzag structure is described by glide lines along the rows as shown and assigned the structure to the p l g l space group. However, the structure also has two-fold rotations on the glide lines and mirror lines perpendicular to the glide lines. Because of these additional symmetry properties, the structure actually belongs to space group p2mg. This example illustrates the care that must be used to determine the correct space group of a structure. The space group of the (7x7) reconstruction of Si(l l 1), Fig. i.9b, is No. 14, D
p3ml. 1.3.3. Isolated adsorbed atoms and molecules Point symmetry operations are useful in the analysis of vibration and electronic properties of adsorbed atoms and molecules. Individual atoms and molecules are often adsorbed at specific sites in the surface mesh. Figure 1.12 gives examples of adsorption sites on various surfaces of fcc and bcc crystals. On-top sites (A) are directly above a substrate atom. A-sites are also called atop and linear. Bridge sites
In the case of planar layers or slabs of finite thickness, there are 80 symmetrygroups called the diperiodic or layergroups. Additional symmetryelements are needed to transform objects on one side of a singular plane located at the center of the layer into symmetrically equivalent sites on the other side. These additional elements must lie in the singular plane; they include 2-fold rotation and screw axes and glide planes. The layer groups are discussed in detail in the book by Vainshtein ( 1981). The layer groups have not found much use for the description of surface structures.
Su~. ace crystallography
Rectangular
21
m
plgl
Pg
No. 4
|
Origin on g
Rectangular
p2mg
mm
|
ping
No. 7
|
I
I
I
I ,t
I A
I A
| |
|
Origin at 2 Fig. 1.11. Top: The rectangular lattice space group p lgl. Center: An adsorbate structure with p2mg space group. Bottom: The rectangular lattice space group p2mg. (B) involve bonding to two substrate atoms. B-sites typically have two-fold or mirror point symmetry. On some substrates both long bridge LB and short bridge SB sites are present. Finally, there are c e n t e r e d sites (C) in which bonds are formed to 3 or more surface atoms. C-sites are also called h o l l o w sites. C-sites on surfaces with square unit meshes are often referred to as four-fold hollows. Similarly two-fold and three-fold hollows occur on surfaces with rectangular and triangular arrangements of surface atoms. Other more complex sites are possible including bonding to subsurface atoms; see Chapters 9, 11 and 12.
W.N. Unertl
22
fcc
(100)
bcc
B_
B.
i
SB C (110)
LB
SB
7~A C LB
@@_O_~~c-3 (111) A C-I
Fig. 1.12. Various types of adsorption sites on low-index fcc and bcc surfaces. Black dots represent adsorbed atoms. Sites labeled A are atop (on top) adsorption. Centered (Hollow) sites with high symmetry are indicated by C. Bridge sites are labeled B if all possible sites are equivalent, otherwise there are long-bridge LB and short-bridge SB sites depending on the relative distances between the substrate atoms.
In the crudest approximation, the symmetry of a site is determined only by the adsorbed species and those substrate atoms to which it is directly bonded. For example, an atom bonded in an on-top site has Coo symmetry. More generally, the combined s y m m e t r y of the adsorbate-substrate system must be used. Thus, an on-top site for a lattice with square symmetry is reduced from Coo to C4v.
1.3.4. The reciprocal lattice The translation vector, Eq. (1.2), defines an infinite set of points called the direct, or real space, lattice. Another lattice, called the reciprocal lattice, is also extremely useful for describing diffraction, electronic band structure, and other properties of crystals. The reciprocal lattice can be specified in terms of a set of reciprocal lattice vectors G that satisfy the equation
eiCr= 1
(1.6)
for all allowed combinations of the Miller indices h, k, I. Since each Bravais lattice is described by a different set of T, each will also have a different reciprocal lattice. G is measured in units of inverse length.
Su~. ace crystallography
23
In a three-dimensional crystal, the basis vectors a*, b*, and c* that define the reciprocal lattice are related to the real space basis vectors by 2/1; a" - - ~ (b x c)
(1.7a)
b* = ~2~ (c x a)
(1.7b)
271;
c* - ~
(a x b)
(1.7c)
where V is the volume of the real space unit cell. (Note that some authors do not include the factor of 2re in the definitions of Eqs. (1.7).) The basis vectors of the reciprocal lattice have the properties a* . a = b * . b = c * . c = 2r~
(1.8)
and a* . b = a * . c = 0 b* . a = b * . c = 0
(1.9)
c* . a = c * . b = 0 The origin of the reciprocal lattice coincides with the origin of real space and the points of the reciprocal lattice are generated by the reciprocal lattice vectors Ghkt= ha* + kb* +/c*
(1.10)
where (hkl) are the Miller indices. In two dimensions, only two reciprocal mesh vectors are required. In analogy with Eqs. (1.7a-c), these are defined by a2•
a~ = 2rc ~ la I x a21 a 2 ==-2rt
(1.11)
c xa 2
la~ x a21
where c is the outwardly directed unit vector and la~ x azl is the area of the unit mesh. The mesh vectors have the properties a l 9a i = a~ . a 2 = 2rt (1.12) a~
9 a
2
-
a 2 9a~ = 0
W.N. Unertl
24
The two-dimensional reciprocal lattice vectors are Ghk = ha] + ka~
(1.13)
These Ghk define a set of lines that are perpendicular to the two-dimensional lattice plane. These lines are called reciprocal lattice rods and are very important for the description of diffraction phenomena, as described in Chapter 7. Figure 1.8 gives examples of the two-dimensional reciprocal lattices for each of the real space Bravais lattices shown in Fig. 1.8. Each reciprocal lattice point and several important directions are also labeled in Fig. 1.8. Note that the primitive unit mesh has been used for the centered rectangular lattice. The choice of the nonprimitive mesh results in different unit mesh vectors in both real and reciprocal space; e. g., a - a ~ + a 2 and b - -a~ + a 2. This has the consequence that the directions and reciprocal lattice points have completely different names; e.g., [ 10] ~ [ 1,1 ], [11] ~ [10], (11) ~ (10), (01) ~ (1/2,1/2), etc. The first Brillouin zone is a common choice for the primitive cell of the reciprocal lattice and is an essential element in the description of electronic band structure, lattice vibrations and diffraction. Electronic band structure and lattice vibrations are treated in subsequent volumes of this handbook; diffraction is discussed in Chapter 7. The first Brillouin zone, which is the analog of the W i g n e r Seitz cell in real space, is constructed as follows" first, reciprocal lattice vectors are drawn from the origin (00), to the nearby reciprocal lattice points. Next, lines are drawn perpendicular to these reciprocal lattice vectors at their midpoints. The smallest area containing (00) and enclosed by these lines is the called the first Brillouin zone. The line segments defining the edges of the Brillouin zone are called the Brillouin zone boundary. The right hand column in Fig. 1.8 illustrates the Brillouin zones for each of the five two-dimensional Bravais lattices. The shapes of the Brillouin zones for the oblique and centered rectangular lattices will vary somewhat depending upon the precise values of a~, a2, and y. Figure 1.13 shows the relationships between the Brillouin zones l;or the (111), (110), and (001) planes of the fcc and bcc lattices and their respective bulk Brillouin zones. Various points and lines in Brillouin zone are related to each other by the symmetry operations of the lattice listed in Tables 1.1 and 1.2. Thus, when describing the properties of a crystal, it is sufficient to consider only the smallest symmetry independent portion of the zone. The center of the each zone (k - 0) is denoted by the Greek letter F. Other points with high symmetry are denoted by upper case roman letters; e. g., H, K, L, M, N, P, X. Symmetry points of surface Brillouin zones are sometimes written with a bar to distinguish them from points in the bulk zones. The simple square lattice has space group p4 and the triangle 1-"X M is the smallest symmetry independent portion as can be demonstrated by sequential applications of the C4 rotation axis at the zone center and the diagonal mirror lines. See Burns (1985) for a more detailed discussion of Brillouin zones and the notational conventions used to describe them.
Sue'ace crystallography
25 10011
(111j
.
]
IX
...,
(a)
Io~
"~.
"~" x
11001
(001)
',
(b)
,
,
\ (111)
o)
Fig. 1.13. (a) First Brillouin zones for the bulk, (111), (110), and (001) planes of fcc crystals. (b) First Bfillouin zones for the bulk, ( 111 ), (l 10), and (001 ) planes of bcc crystals. (Reproduced with permission of W.N. Unertl.)
W.N. Unertl
26
1.4. Unit mesh transformation In practice, it is sometimes more convenient to use a unit mesh which is different from one of the five Bravais lattices. This is particularly true for ordered overlayers, many of which can have a unit mesh of different size, symmetry and orientation from the substrate surface; see Chapters 9 and 10 for examples. Several notational systems have been developed to name and facilitate transformations between various possible lattices. The Wood notation and the matrix notation are the only widely used of these.
1.4.1. Wood notation In the early 1960s, the noted X-ray crystallographer, Elizabeth A. Wood (1963), developed the first systematic notation for surface crystallography. In addition to formulating one of the most widely used notations for ordered surface structures, she discussed other important concerns for development of a uniform, unambiguous system for describing surface structures which is also consistent with the well established conventions of bulk crystallography, as presented in the International Tables of Crystallography (1959). In W o o d ' s system, the unit mesh of the ideal substrate lattice is taken as the reference for all other lattices and is called the ( l x l ) mesh. The conventions given in the preceding section are used to specify this mesh. The Bravais lattice type of the substrate must be clearly specified in W o o d ' s system. Any other mesh, for example that of an ordered adsorbate on the substrate, can then be described in terms of this ( l x l ) mesh. In the simplest case, the overlayer unit mesh vectors (b~,b2) are parallel to (a~,a2) but have different lengths; i.e. bl = real and b2 = ha2 where m and n are integers. Such an overlayer mesh is identified as an (mxn) structure. Figure 1.14a shows an example of a (3x2) structure on a rectangular lattice. It is extremely important that m, the coefficient ofa~, precede n. For example the (3x2) structure of Fig. 1.14a is obviously very different from the (2x3) structure shown in Fig. 1.14b. The overlayer mesh can also be rotated by an angle 13 with respect to the substrate. In this case n and m are no longer restricted to be integers. There seems to be no general convention for defining the direction of positive 13. In this book, we assume 13 to be the clockwise rotation required to align b~ and a~. The full notation for the rotated overlayer mesh is
(mxn)R~ where 13 is expressed in degrees. An example for a ( ~ - x ~ - ) R 2 2 . 5 7 overlayer is shown in Fig. 1.14c. The full Wood notation for a chemical species A adsorbed onto the (hkl) surface of a substrate of chemical composition S is:
Su~.ace crystallography
27
b2= 2 _
(~)
9
9
9
9
9
9
9
|
~
a2
1t 2
b. I
9 9
9
~ ,
(~)
9
|
.
9
o
o|
9
b 2--3 a 2
Q
v
|
A a2
It
~
~
9 |
|
9
9
,, =28. I w
9
. +
,,
9
|
9
|
. ~ o
_
| 1
|
Fig. 1.14. Examples of the Wood notation.
S( hkl)( m• )R~-rlA where 1"1 is the number of species A in the unit mesh. If the space group is known, it can be incorporated into the Wood notation as follows: S(hkl)(mxn)R~spacegroup-rlA; e.g., Pt(110)(2• Systems that have more than one type of adsorbate do not have a standard notation. We suggest S(hkl)(mxn)R~5-rlA~ where is the number of species B per unit mesh. Unfortunately the Wood notation is unable to describe all physically possible overlayer-substrate combinations. For example, in the case of a hexagonal overlayer on a square substrate lattice, the angles between (al,a2) and (bl,b2) are different. Furthermore, there are several very common structures which have more than one description in the Wood notation. The most widely known of these are the identical c(2x2) and (x/2-x'~t2-)R45 structures illustrated in Fig. 1.14d. The latter description derives from the primitive unit mesh of the overlayer. In spite of these shortcomings, the Wood notation is usually clear and very easily visualized. It is also a neatly compact notation. These attractive features have led to its wide adoption in surface science.
28
W.N. Unertl
1.4.2 Matrix notation
A matrix notation, first introduced into surface crystallography by Park and Madden (1968), overcomes some of the shortcomings of the Wood notation. It is not as easy to visualize as the Wood notation but it facilitates mathematical manipulations involving quantities related to the surface structure. The matrix notation for two-dimensional systems is a special case of that developed for three-dimensional systems as described in the I n t e r n a t i o n a l Tables o f C r y s t a l l o g r a p h y (1959). As was the case for the Wood notation, the substrate unit mesh (a~,a2) provides the reference for all other lattices. Any overlayer mesh (b~,b2) can be described in terms of this substrate mesh by a linear transformation bl = sllal + s12a2
(1.14a)
b 2 = s21a I + $22a 2
(1.14b)
and the inverse linear transformation al = tzlbl
+
tlzb2
(l.15a)
a 2 = t21b I + t22b 2
(1.15b)
where the unitless coefficients sij and t,j are elements of the matrices S and T, respectively. In matrix notation
s ilai I
1 $22
2
(1.16)
2
and
Since these transformations, when applied sequentially, must reproduce the starting mesh ST=I
(1.18)
where I is the unit matrix; i.e., the transformation must be reversible. T is the inverse of S; T = S -~. The matrix S provides a complete description of the new mesh (b~,b2) in terms of the reference mesh (a~,a2). Thus, once (a~,a2) is chosen, specification of S completely determines the overlayer mesh (b~,b2). Once S is given, the t,~ can be determined from Eq. (1.17):
Su~ace crystallography
29
S22 t j~
-
det S --$21
t~2 = det S --S 12
t21 = det S Sll
t22 - det S
( 1.1 9)
where det S is the determinant of S; det S is never negative unless a left-handed coordinate system has been used. The area of the overlayer mesh, Iblxbzl, expressed in units of the substrate mesh area, lal• is equal to det S. Similarly, det T is the ratio of the substrate unit mesh area to the overlayer unit mesh area; i.e. b2-------~l Ib z x - (det T )-~ = det S la~ x a21
(1.20)
The structures shown in Fig. 1.14 provide examples for the matrix notation. The (3x2) structure has b~ = 3a~ and b 2 "" 2 a 2. Thus
(30)
is the matrix equivalent to the Wood notation (3x2). Since det S - 6, the o v e r l a y e r unit mesh area is six times that of the substrate. The inverse matrix is
,t
~,0 2 )
The (,fffx,~-)R22.57 structure in Fig. 1.13c has bl = 2al + a 2 and b 2 --- ---aI + 2a2, so that S=
(')
_21 2
The overlayer unit mesh area is 5 times the substrate unit mesh area and 2 T~
-1
W.N. Unertl
30
In the matrix notation, only the primitive overlayer unit mesh can be described and it is not possible to define the centered rectangular unit mesh. Thus, the ambiguity between the c(2x2) and the (x/2-x'~-)R45 notations does not arise. In this case b~ = a~ + a 2 and b 2 = -gl + a2, so that
':(: '/ 1 1
det S = 2, and 1
-1
Ambiguities do occur in application of the matrix notation since S can have several equivalent forms related by symmetry. For example, each of the four matrices below describes the (x/2xq2-)R45 lattice:
(', (:,
:/
Each can be obtained fi'om the others by sequential application of the rotation operator C4; i.e., Eq. (1.5) with czN = 90 ~ The matrix notation is very useful for the analysis of diffraction patterns where the need to make transformations between the real and reciprocal space lattices becomes important. The Wood notation for overlayer or reconstructed meshes,
S( hkl)( mxn ) R ~3-rlA is modified to incorporate the matrix description by replacing
(mxn)R~ with
,2 I 21 S22
1.4.3. Classification of overlayer meshes Overlayer, or superlattice, meshes can be divided into three general classes. 1. Simple lattices. Simple lattices have all s o as integers and det S is also an integer. The four examples in Fig. 1.14 are all simple lattices. 2. Coincidence lattices. For coincidence lattices, the sij are all rational numbers and det S is a simple fraction. Consider the one-dimensional example shown in
Su~. ace crystallography
31
Fig. 1.15a in which the points on the two lattices are spaced by a and b = (5/4)a. If the lattices coincide at any lattice point, they will coincide again after a distance of 5a or 4b. Simple lattices and coincidence lattices are also often grouped together as coherent or commensurate lattices because of their property that lattice points of the substrate and overlayer repeatedly coincide. 3. Incoherent or incommensurate lattices. If the sij contain irrational numbers, det S is an irrational number, and the two lattices can never coincide at more than a single lattice point. Such lattices are said to be incoherent or incommensurate. This type of lattice plays an important role in the theory of two-dimensional phase transitions as described in Chapter 10. A simple example is an undistorted hexagonal lattice superimposed on a square substrate lattice as shown in Fig. 1.15b. Here b~ = a~ and b2 = - s i n 3 0 a~ + cos30 a2 and
5a or 4b
-I
b 9
9
9
9
9
9
9
9
9
R
(a)
ax
b1
a2.
0
o 9
o .
0 9
0
/.o. o .
o 9
o
0
0
9 0
9 0
9 0
9 0
9 0
0
0
9 0
0
*
0
0
0
0
0
0
9 0
9 0
9 0
0
9
9
0
o .
0
.
0
o 9
0
,
o .
0
.
.
.
.
o
0
0
0
*
0
0
0
9 0
9 0
0
0
0 9
9
0
0
0
0 9
0
0
*
9
9
0
0
0
0 9
9
0
9
0
9
0 9
,
0
b2
9 0
0
0
*
9 0
0
0
*
9 0
0
0
(b) Fig. I. 15. (a) A one-dimensional example of a coincidence lattice with b = (5/4)a. (b) An i n c o m m e n s u r a t e h e x a g o n a l lattice (bl,b2) on a square lattice substrate (al,a2) for the case la I = Ibl.
32
W.N. Unertl
/'
~
S = -sin 30 cos 30 so that det S = cos 30. Reader beware! In the literature, it is common to confuse coincidence lattices with incommensurate structures. Thus, the reader must be cautious before accepting such a reported incommensurate lattice.
1.5. Vicinal surfaces, steps, facets, and the stereographic projection
1.5.1. Stereographic projection A useful method for presenting crystallographic information, including a description of vicinal surfaces, is provided by the stereographic projection. Imagine a small crystallite located at the center of a sphere of unit radius. The points at which low-index directions [uvw] intersect this sphere can be mapped onto a plane. Figure 1.16 shows the principle of the projection. The projection plane is chosen to be tangential to the unit sphere, usually at a point where one of the low index directions intercepts it. The projection point is chosen to be opposite this point. Any point p on the unit sphere is mapped into a point P on the projection plane by constructing the straight line passing through the projection point and p as shown. Figure 1.17a shows the resulting stereographic projection for a face-centered crystal with the tangent point chosen as the [001] direction. The projection of the normal to each major family of low-index planes is labeled by the Miller indices hkl of their normal. Because of the high symmetry of most crystal lattices, the complete projection contains redundant information. The stereographic triangle, Fig. I. 19b, is the smallest segment of the total projection which contains all of the information of a stereographic projection.
projection
Fig. 1.16. The stereographic projection.
33
Su~. ace crystallography
__
210,
310
~]o
too
glo 310
~_lo
110,
tl0
-
,120 130,
,T3o ~Tso
t50q
o?o,
)010
1,~0,
150
1~o
30
~.0
ST0
100
510
310
(a) ! lOO] zone
(001)
(ol])
(012 I
o
(]22)
(lll)
(b) Fig. 1.17. (a) The stereographic projection for a face-centered-cubic lattice with the [001 ] direction as the center. (b) The stereographic triangle for (a).
34
W.N. Unertl
The surface of any crystal is represented on the stereographic triangle in two ways. In the first, the surface normal [hkl] is projected onto a point that is labeled by the Miller indices hkl of the family of lattice planes parallel to the physical surface. In the second, the great circle formed by the intersection of the surface plane with the unit sphere is projected onto the stereographic triangle. This projected line is called the [hkl] zone. Examples of both types of projections are given in Fig. 1.17b.
1.5.2. Vicinal surfaces and uniform arrays of steps A slightly misoriented crystal surface will have atomic scale steps on it whose structure is described by the terrace-ledge-kink model illustrated in Fig. 1.18 where individual atoms are represented as cubes. The large flat regions between individual steps are called terraces and are connected by step edges, or ledges. Kinks occur wherever the ledge changes direction or when there is a vacancy or adatom defect at the step edge. Numerous direct observations of the t e r r a c e - l e d g e kink structure have been made and examples are shown in Chapters 2 and 12. Ordered arrays of nearly identical, equally spaced steps can be formed by cutting a surface at a small angle from a low index plane. Such surfaces are called vicinal surfaces and have been extensively studied because diffraction methods can be used to characterize their average structural properties as discussed in Chapter 7. Figure I. 19 shows perspective sketches of some vicinal surfaces. In many cases, the vicinal surface consists of large terraces with a low index orientation separated by steps that are one or two atoms high. In Fig. 1.19, each vicinal surface is labeled by the Miller indices of the family of bulk lattice planes parallel to the surface. However, the Miller index notation does not provide a clear visualization of the orientation of the vicinal plane. For example, it is not obvious from the Miller indices that the (14,11,10) vicinal surface shown in Fig. 1.19 consists primarily of (111) terraces and is tilted only 8.29 ~ away from the (111) plane toward [725]. A more convenient
Fig. 1.18. The terrace-ledge-kink model of a solid surface with single and double atomic height steps on it.
Su~. ace crystallography
fcc
35
(977)
rcc
17551
rcc
18331
!
fcc
(4431
fcc 114.11.101
fcc
fcc
(3321
( 0,8,7)
fcc
(3311
fcc ( 13. I 1.91
Fig. 1.19. Ball models of the atomic structure of some vicinal surfaces. Steps shown in the upper two rows do not have kinks whereas each step in the lower row has kinks. (After Somorjai, 1994.)
36
W.N. Unertl
notation, which is more descriptive than the Miller indices, gives (1) the Miller indices of the nearest low index plane, (2) the angle between that low index plane and the vicinal plane, and (3) the zone which contains the surface normals of both the vicinal surface and the nearby low index plane; i.e., the direction that the vicinal surface is tilted away from the low index plane. For the example above, the (14,11,10) vicinal surface becomes a (111)8.29~ plane in this notation. Similarly, the top row of vicinal surfaces in Fig. 1.19a are all tilted from (111) in the [211] d i r e c t i o n ; i.e., (977) is e q u i v a l e n t to ( 1 1 1 ) 7 . 0 1 ~ ( 7 5 5 ) to (111)9.45~ and (533) to (111)14.42~ Other notations are sometimes used to describe regularly stepped surfaces. These notations all p r e s u p p o s e a knowledge of the atomic structure of the steps information which is not always available. They also assume the steps to be perfect with no step edge wandering and no statistical spread in terrace widths. Furthermore, some uniformly stepped surfaces, such as those near (0001) surfaces of hcp lattices, may have two different step widths that alternate across the surface. For these reasons, it is recommended that such notations not be used for the description of real stepped surfaces. 1.6. Disorder Real surfaces are never perfect. Fortunately, however, it is often possible to prepare samples that are sufficiently ideal that the effects of their intrinsic defects on an experiment are of secondary importance. On the other hand, imperfections play essential roles in many important processes and are coming increasingly under study as the power of available instrumentation improves. Tringides and Lagally give a detailed description of surface defects in Chapter 12. Schematic examples of most types of surface imperfections are shown in Fig. 1.21. Many bulk imperfections also terminate at surfaces including screw and edge dislocations and grain boundaries. Steps are defects characteristic of surfaces and interfaces and are without bulk analogs. This section summarizes some of the ways used to describe the various types of surface imperfections. In thermodynamic equilibrium, at T > 0 K, even the most perfect low-index surfaces must have defects. The probability of forming a defect at a given temperature is determined by the Boltzmann factor for the energy required to form a single defect from the perfect crystalline lattice. Figure 1.20 shows the results of a computer simulation of the equilibrium distributions of vacancies and adatoms at the (100) surface of a simple cubic lattice. As shown in the figure, the number of imperfections increases dramatically as the temperature is raised. 1.6.1. Pair distribution function
One way to describe the positions of atoms at a surface is to specify the instantaneous position vector of every atom. This is convenient only when the number of atoms involved is small or if they are arranged on a perfect crystalline lattice so that the basis of the single unit mesh specifies the whole structure. However, for many
Su~ace crystallography
37
~0
Fig. 1.20. Some types of surface defects.
0
~
Fig. 1.21. Thermally induced disorder on a surface. The temperature given for each configuration is in units of the vacancy formation energy divided by Boltzmann's constant.
38
W.N. Unertl
important systems, it is neither possible nor useful to attempt to determine the instantaneous atomic positions. Liquids and gases and their surface analogs are obvious examples as are amorphous solids. In these cases, it is usually the statistical properties that are of interest anyway. Furthermore, many real solids and their surfaces, although for the most part highly crystalline, have important imperfections which break their ideal translational symmetry - - many of these imperfections are described in Chapter 12. Finally, several of the most important experimental methods are not able to determine the individual atomic positions; these include all diffraction techniques such as LEED, X-ray diffraction and atom diffraction. An alternative, statistical description of an atomic arrangement is useful in many of the above situations and is provided by the pair distribution function, P(R) ~. If p(r) is a function that describes the probability of finding an atom within dr of r, P(R) is defined as
P(R) - ~p(r)p(r + R)dr
(1.19)
P(R) gives the probability of finding an atom at position R if there is another atom located at r. That is, P(R) is the self-convolution of p(r). P(R) is also called the auto-correlation function or the Patterson function. Figure 1.22 shows some simple, one-dimensional examples. The first, Fig. 1.22a, is an infinite array of equally spaced atoms with spacing a. Once any of these is chosen as the origin, there is zero probability of finding another atom anywhere else except at distances which are integral multiples of a. The Patterson function is simply the average of all such probability distributions obtained when each scatterer in turn is taken at the origin. For this example, P(R) is unity at each integral multiple of a and zero otherwise as shown. The second example, Fig. 1.22b, is also for an infinite array. However, in this case, the array consists of pairs of identical atoms with characteristic spacings a and b. If the left hand member of each pair is chosen as the origin, other scatterers will be found at a, a+ b, 2a + b, 2a + 2b . . . . . Choosing the right hand scatterer at the origin yields the distances b, b + a, 2b + a, 2b + 2a . . . . . These are the only unique choices of origin, so that each contributes equally to P(R) yielding the result shown. Notice that the peaks in P(R), which are at integral multiples of (a + b), have twice the weight. The third example, Fig. 1.22c, is for a finite array of four identical, equally spaced atoms. If the left most one is chosen as the origin, the others are found at a, 2a and 3a. If the second is chosen, the relative locations of the others are - a , a and 2a; for the t h i r d , - 2 a , - a and a; for the f o u r t h , - 3 a , - 2 a and - a . Since each of these choices is equally likely, each contributes equally to produce the P(R) shown. Thus, if one of the atoms is chosen at random, the probability of finding a second at +a is 0.75, at +2a is 0.5 and at +3a is 0.25. In all cases, P(R) is symmetrical about
1 Otherdistribution functions such as the height-height distribution function are also useful, for example, to describe surface roughness. See Chapter 12 for additional discussion of distribution functions.
Su~. ace crystallography
39
m(x) 1
o
o
o
T
0
(c)
~
a
9
2a
T
i
3a
F
4a
X
P(x) 1
--•
-,~{-- a Oe
Oe
Oe
Oe
--
0
b
2b
T
3b
x"'
4b
v~
X
(b) P(x) 1
"'"
0
0
0
0
0
0
0
0
"'"
0
a
2a
3a 4a
5a
(a) Fig. 1.22. One-dimensional linear arrays and their pair distribution functions. (a) An infinite array with equal spacings. (b) An infinite array with two spacings. (c) A finite array with four equally spaced points. the origin. This has important consequences for diffraction as is discussed in Chapter 7. The preceding examples used simple, highly ordered atomic arrangements to illustrate the physical meaning of P(R). P(R) is also extremely useful in the description of disordered systems, including fluids. Most types of defects on crystalline lattices can be divided into two classes. Long range correlations are preserved on average for defects of the first kind and structure is retained in P(R) for large R. For defects of the second kind the long range correlations are not preserved.
1.6.2. Defects of the first kind Thermal vibrations of atoms about their mean positions in the lattice is the most common example of a defect of the first kind. Other examples include localized defects such as vacancies, interstitials, and impurity atoms. The distribution of
40
W.N. Unertl
P(x) 1
0
a
2a
3a
4a
5a
Fig. 1.23. Pair distribution functions for crystalline and fluid phases.
defects of the first kind 9(r) can be described with respect to the atomic distribution of the average lattice <9(r)>; i.e. p(r) - + Ap(r)
(1.20)
where Ap(r) is the deviation from the average lattice and <9(r)> = 0. The average <...> can be an average over time as in the case of thermal vibrations or over space as in the case of static vacancies. Figure 1.23 shows the effect of thermal vibrations on the pair distribution function for an ideal one-dimensional lattice like that in Fig. 1.22a. P(R) still has equally spaced maxima at the average atomic locations. But the vibrational motion causes the atoms to have a finite root-mean-square displacement from equilibrium so that the peaks in P(R) are broadened. 1.6.3. Defects of the second kind
Defects of the second kind do not maintain long range correlation between the atomic positions. There are several important examples. A surface step is created if a bulk screw dislocation S intersects the surface as illustrated for a simple cubic crystal in Fig. 1.24. The screw dislocation in Fig. 1.24 is formed on the (100) surface of the simple cubic crystal by slipping the top and bottom parts of the crystal in opposite directions parallel to the [100] direction to reveal the small shaded ledge of (001) orientation. A screw dislocation can also be visualized as a spiral stacking of crystal planes about a dislocation line. In this example, the screw dislocation line is parallel to [100]. At any point along a dislocation line, the direction is specified by a unit vector ~. For the screw dislocation in Fig. 1.24, ~ points in the [100] direction. Screw dislocations are important in crystal growth because their surface steps provide continually available nucleation sites at which new atoms can be incorporated into the growing crystal. The other type of bulk dislocation is the edge dislocation E formed when an extra half plane of atoms is inserted into the crystal. The edge of this half plane forms the dislocation line. Edge dislocations are also characterized by the vector ~. For the example in Fig. 1.24, ~ is parallel to [010]. The surface defect created by the intersection of an edge dislocation with the surface has locally distorted surface
Su~ace crystallography
41
(ool)
J
J
J J
J J
J J
J J
J
jL ---'qJ r
f
f
I
(olo)
-i
~
TM
(!00)
b
f f
r
f
I
f
r
m
Fig. 1.24. A mixed dislocation ~ in a simple cubic crystal. It emerges at the (100) surface as a perfect screw dislocation S and at the (0T0) surface as a perfect edge dislocation E.
W.N. Unertl
42
structure and will have different mechanical, chemical and electronic properties than the perfect surface. A displacement called the Burgers vector, b, is associated with every dislocation line. A simple procedure for determining the Burgers vector is to compare an atom-to-atom closed circuit which encircles the dislocation line to a similar loop in the perfect crystal. The direction of this loop is chosen to be that which would advance a right-hand screw in the direction of ~,. When compared to the loop in the perfect crystal, the loop around the dislocation requires an extra segment to close it. For example, consider the loop labeled P in Fig. 1.24. A closed loop is obtained by following the path 1 to 2 to 3 to 4 to 5. A similar loop drawn to enclose the edge dislocation is not closed. The Burgers vector is defined as the vector, drawn from the start of the loop at 1 to its end at 5, that closes the loop. In this case b is 1 unit in the [100] direction, or 1[100]. A second example is given for the screw dislocation. When b.~ = 0, the dislocation is said to have edge orientation; i.e., b is perpendicular to ~. When b.~= b, where b is the magnitude of b, it has screw orientation. In this case b is parallel to ~. A single dislocation, such as the one connecting the screw and edge dislocations shown in Fig. 1.24, is characterized by the same value of b at each point along its length even though it has very different characteristics on the two surfaces at which it emerges. Between the two faces, 0 < b.~ < b, and the dislocation is said to be mixed. If more than one dislocation line is enclosed in the loop path, the resulting Burgers vector is the sum of the Burgers vectors of the individual dislocations. Burgers vectors are also useful in describing the diffraction from a crystal with dislocations (Jackson, 1991). A more in-depth discussion of dislocations can be found in Wert and Thomson (1970). Most "single crystals" actually consist of a number of very slightly misoriented crystallites. Such a structure is called mosaic. The lattice mismatch, which occurs at the boundaries between the crystallites, is taken up by arrays of dislocation lines except for the special case of a twin boundary in which the lattices of the two crystals, which make up the twin, are related by a rotation about a crystallographic axis [hkl] called the twinning axis and/or by mirror reflection across the twin boundary plane (hkl). A simple example is a (111) or [111] twin in an fcc lattice. In the perfect lattice, the (111) planes are stacked in a sequence which repeats every three atomic planes; e. g., ...ABCABCABCABC .... The stacking sequence at a twin boundary is altered: ...ABCABCBACBAC... where the boldface C indicates the location of the twin boundary which, in this case, is both a mirror plane and a rotation axis. The second kind of surface defect discussed so far has been that associated with the surfaces of bulk crystals. Other important imperfections occur in layers of adsorbed atoms or molecules. In real systems these ordered structures will not be perfect. Both vacancies and impurities can be present. Furthermore, defects such as
43
Su~. ace crystallography
c~
c"
c~
","
C~
",;
,,
,;
.
"'L . . . . .
".z
c)
~
;
", ~
r
C.s~.
.,~
~,,i
II (I
C;
,","
(',
C;
,";
C~
Fig. 1.25. Domain boundaries for a (2•
C}
C)
C)
overlayer structure on a square lattice.
emerging dislocation lines, which are present on the substrate surface, will influence the structure of the ordered overlayer. Mixed phases also occur in adsorbed layers. For example, if there is an attractive interaction between the molecules, small islands of an ordered phase, which might form a simple, coincident, or an incoherent lattice with respect to the substrate, can nucleate and grow in a surrounding two-dimensional fluid phase. On a perfect substrate, nuclei will form randomly and ordered islands will grow around them in registry with the substrate as illustrated in Fig. 1.25. If any two of these nuclei form at substrate sites which cannot be connected by a translation vector of the overlayer lattice, it will not be possible to cover the entire surface with a single region of order. Instead, when the ordered islands grow together, their edges will not match and a defect, called a domain boundary or domain wall, is created. These boundaries are sometimes also called anti-phase boundaries because their presence can cause certain diffraction beams to be absent as discussed in Chapter 7. Figure 1.25 shows an example of domain boundaries for a (2• adsorbate structure formed on a square lattice substrate by adsorption at atop sites. In this case, there are four equivalent adsorption sites labeled 1, 2, 3, and 4 in the figure; i.e. these sites lie on four independent, interpenetrating (2• sub-lattices. Domain walls form between the various sub-lattices and limit the degree of long range order that can be obtained in an overlayer. The domain walls are said to be light domain walls if the local atom density is lower than that of a perfect surface and heavy domain walls if the density is higher. Domain walls play an important role in surface phase transitions as discussed in Chapters 11 and 13.
References Burns, G., 1985, Solid State Physics. Academic Press, Orlando, FL, Chapters 1-5. Cotton, F.A., 1990, Chemical Applications of Group Theory. Wiley, New York.
44
W.N. Unertl
Ertl, G. and J. Kuppers, 1985, Low Energy Electrons and Surface Chemistry. VCH, Weinheim, Chapter 11. International Tables for Crystallography, eds J.S. Kasper and K. Lonsdale, 1959, Kynoch Press, Birmingham. Jackson, A.G., 1991, Handbook of Crystallography for Electron Microscopists and Others. Springer-Verlag, New York. Lambert, R.M., 1975, Surface Sci. 49, 325. Leamy, H.J., G.H. Gilmer and K.A. Jackson, 1975, in: Surface Physics of Materials, ed. J.M. Blakely. Academic Press, New York, p. 121. MacGillavry, C.H., 1976, Fantasy & Symmetry, Abrams, New York. Park, R.L. and H.H. Madden, 1968, Surface Sci. 11, 188. Robinson, I.K., W.K. Washkiewicz, P.H. Fuoss, J.B. Stark, and P.A. Bennett, 1986, Phys. Rev. B33, 7013. Somorjai, G.A., 1994, Surface Chemistry and Catalysis. Wiley, New York, p. 49. SUN Commission of IUPAT, 1978, Physica 93A, 1. Vainshtein, B.K., 1981, Modern Crystallography I. Springer-Verlag, Berlin. Wert, C.A. and R.M. Thomson, 1970, Physics of Solids. McGraw-Hill, New York. Wood, E.A., 1963, J. Appl. Phys. 35, 1305.
Appendix: The seventeen two-dimensional space groups Oblique
1
pl
0
0
No. 1
pl
0
0
Origin on 1
Oblique 2
No. 2
p211
| |
| |
Origin at 2
|
p2
Su~. ace crystallography
Rectangular
45
plml
m
No. 3
pm
| |
Origin on m
Rectangular
plgl
m
No. 4
Pg
Origin on g
Rectangular
clml
m
No. 5
|
Origin on m
Cm
46
W.N. Unertl
Rectangular
|
p2mm
mm
|
|
pmm
No. 6
| A
|
|
j |
|
|
|
Origin at 2ram
Rectangular
mm
p2 mg
v J
9
I
I
I
6
6~--i
v J
|
~)
pmg
No. 7
I
Origin at 2
Rectangular
mm
Pgg
No. 8
p2gg
L
9
i I , ----f V
Origin at 2
I
9
9
I
9
47
S u # a c e crystallography
Rectangular
C2 m m
mm
@ |
@ @ | |
@ |
cram
No. 9
&
A
A
@ | | | @ |
| |
@ @ | |
,
.
Origin at 2 m m
Square
4
p4
No. I 0
@ @
@ @ @ |
L r
@ @
@ @ @ |
k
@ |
9 9
Origin at 4
Square
4 mm
p4 mm
@ @ | | @@
@ @ | @| @
No. 1 1
'*
// @~) |
|
@ @ | | @@ Origin at 4 r a m
W.N. Unertl
48
Square
4ram
p4 gm
No. 12
A
T',,
9
I /wx---l-~/ m
m
|
i(--
Origin at 4
Hexagonal
pa
3
No. 13
p3
|
Origin at 3
Hexagonal
3 m
p3mi
g
Origin at 3 m 1
No. 1 4
p3ml
Su~. ace crystallography
49
Hexagonal
3m
p31m
p31m
No. 15
@
%
% Origin at 31 m
Hexagonal
6
p6
@ |
@ | |
p6
No. 1 6
|
| @
@
@@
@@
Origin at 6
Hexagonal
6mm
p6mm
No. 17
J m
|
| v
Origin at 6ram
,
v
This Page Intentionally Left Blank
CHAPTER 2
Thermodynamics and Statistical. Mechanics of Surfaces E.D. WILLIAMS and N.C. B A R T E L T Department of Physics University of Maryland College Park, MD 20742-4111, USA
Handbook of Sueace Science Volume I, edited by W.N. Unertl
9 1996 Elsevier Science B.V. All rights reserved
51
Contents
2.1.
Introduction
2.2.
Thermodynamic formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. The surface excess quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
2.3.
Thermodynamics offaceting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Thermodynamics oforientational phase separation . . . . . . . . . . . . . . . . . . .
66 68
2.2.2.
2.3.2. 2.4.
2.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gibbs adsorption equation
Types offaceting transitions
53 55 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Statistical mechanics ofvicinal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Simple stepped surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Expressions for step formation and interaction energies . . . . . . . . . . . . . . . . .
79 80 82
2.4.3.
90
Experimental determination of statistical mechanical parameters
............
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
52
2.1. Introduction Real solid surfaces are seldom in equilibrium, yet the study of surface thermodynamics provides us with a wealth of tools for understanding and working with surfaces. In part this is because we can often use the concept of local equilibration. What this means is that even though an entire sample has not equilibrated with its vapor, on some limited length scale mass transport has occurred (for instance by diffusion across the surface or through the bulk) to a sufficient extent to allow the surface to equilibrate. One example of this phenomenon is adsorption onto a non-equilibrium substrate. Even though the original surface may be far from equilibrium, the adsorbing particles may equilibrate on the arbitrary structure of the surface to minimize their thermodynamic free energy subject to the constraints of their environment. Another example is thermodynamic faceting. While in real thermodynamic equilibrium a solid will attain a crystal shape which minimizes the surface free energy, in most real situations mass transport is not sufficient to allow this to happen. However, on a local scale surfaces with an arbitrary net orientation will rearrange to expose the orientations that would be present on the equilibrium crystal shape. The resulting new facets will grow, reducing the surface free energy until their size becomes so large that mass transport across them becomes negligibly slow. An example of observation of such local facet formation is shown in Fig. 2.1, in which facet growth on a Pt covered W surface is shown to occur at larger and larger length scales following annealing at higher and higher temperatures. In addition to its use in describing equilibrated systems, thermodynamics provides us with the information needed to understand the evolution of surfaces under far from equilibrium conditions (Blakely and Mykura, 1962; Bonzel et al., 1984; Herring, 1951b; Keeffe et al., 1993; Mullins, 1961). Very simply this is because mass flow requires gradients in the chemical potential, which in turn is a thermodynamic quantity. Thus it is worth some effort to understand the thermodynamics of equilibrium systems, and in particular how the thermodynamic quantities vary with parameters of interest in mass transport, such as temperature, concentration, composition and surface orientation. As a result, a large amount of effort is expended in trying to prepare well characterized, thermally equilibrated systems and to measure their properties. The question of whether a system is in equilibrium, or can be described even partially by equilibrium thermodynamics remains difficult. Many non-equilibrium processes result in structures that can look very much like equilibrium structures. A traditional strong signature of thermal equilibrium is the observation of reversible phase transitions. More generally thermal equilibrium can be demonstrated by an experiment which shows that the structure observed is not dependent on the path (for instance, thermal cycling or growth rate) used to 53
54
E.D. Williams and N.C. Bartelt
Fig. 2.1. A W(I 11 ) surface is thermodynamically stable when clean. However addition of Pt changes the free energies to favor formation of {211 }-type facets. Limitations of mass transport prevent the surface from completely rearranging to form three large {211 } facets. Instead, a large number of small faceted mounds form, each exposing the preferred orientations. Annealing at increasing temperatures allows increasing growth of the facets. The figure show pairs of LEED (incident energy 101 eV) and STM patterns of the faceted surface following annealing at 900 K (top panel), 1200 K (middle panel) and 1400 K (bottom panel). The coverage of Pt on each surface is -- 1-2 ML. Figure provided by Prof. T.E. Madey of Rutgers (Madey et al., 1993; Madey et al., 1991; Song et al., 1990; Song et al., 1991 ).
Thermodynamics and statistical mechanics of surfaces
55
reach it. With the advent of powerful surface imaging techniques, it is now also becoming possible to observe thermal fluctuations of systems in equilibrium directly. Such measurements not only confirm the existence of equilibration, but provide a wealth of atomic-scale information about rates and mechanisms. A thermodynamic approach to such information provides a way of condensing out the aspects of the information that have macroscopic physical consequences. Surface thermodynamics and statistical mechanics has been the subject of many comprehensive reviews, from which we will here only condense a focused subset of information. Most of the formalism needed to discuss surface thermodynamics is contained in Gibbs' work (Gibbs, 1961), and the quantitative application of these ideas to problems of surface morphology and faceting was nearly complete by the time of Herring's work on surfaces (Herring 1951a; Herring 1951b; Herring 1953). Excellent reviews exist on the applications of thermodynamics to problems of clean solid surfaces (Blakely, 1973) and interfaces (Cahn, 1977). The fundamental statistical mechanical problem of surfaces, understanding the entropy of step wandering, was first addressed by Mullins (Gruber and Mullins, 1967), and consequences of this problem and the related problem of surface roughening were worked out thoroughly in the 1970s and early 80s (Weeks, 1980; Wortis, 1988). The problems of merging a thermodynamic and statistical mechanical approach to understanding surface processes remains a topic of current research (Nozieres, 1991). We will review this evolution of surface thermodynamics and statistical mechanics briefly, and with an emphasis on the problem of surface faceting. The formal introduction of classical surface thermodynamics and surface excess quantities in w 2.2 should serve as a starting point to allow further investigation of the applications of these ideas to other problems as well (Dash, 1975; Griffiths, 1980). The discussion of surface faceting as an orientational phase separation, which is presented in w 2.3, is a formal and detailed presentation of ideas not generally explicitly addressed in reviews of surface thermodynamics. The overview of surface statistical mechanics is a synopsis of useful forms for quantifying experimental observables in such a way that they can be used in a thermodynamic formalism to predict the stability of surfaces with respect to faceting. A brief overview of some recent experimental observations which demonstrate the ideas concludes the chapter.
2.2. Thermodynamic formalism Before one can discuss surface thermodynamics, one first must have some idea about what one means by a surface. While many geometric, microscopic pictures instantly spring to mind, from the viewpoint of thermodynamics a surface is only what separates two phases in thermal equilibrium. Thus, strictly speaking, surfaces only occur for the special values of pressure and temperature where two phases coexist. The fundamental quantity of surface thermodynamics is the surface tension ~,, which governs the amount of work required to create a surface. It is defined as the change in internal energy when the surface area between two phases in thermodynamic equilibrium is increased at constant entropy S, volume V and particle number of each of the chemical components N/:
56
E.D. Williams and N.C. Bartelt
= Y
~9U -~
. s.v.u,
(2.1)
In thinking about this equation, it is important to emphasize that the numbers being conserved are the total particle numbers in the system, which includes both the condensed and vapor phase as well as the intervening surface. In the light of the discussion which follows, it should also be emphasized that Y is independent of any microscopic ideas about where the surface is actually located. As a simple example of the meaning of the surface tension, consider a simple cubic solid with lattice constant a which is bound together by nearest neighbor bonds of strength E. If the solid is pulled apart at T = 0 K to create two surfaces of (100) orientation one bond is broken per unit cell of area a 2. Thus the energy cost per unit area of creating each of the two surfaces is just E/2a 2. In principle 7 for solid surfaces can be measured directly in a cleavage experiment by determining the work required to separate two parts of a crystal from each other. The difficulties of performing and interpreting such an experiment are described by Blakely (Blakely, 1973). Another way of determining the surface tension directly is the method of "zero creep" (Josell and Spaepen, 1993). In the "zero creep" method the sample, often a thin wire, is subjected to an external force. At high enough temperature that surface diffusion is readily possible, the sample will tend to change its shape to minimize the area. The applied external force which balances this tendency (i.e. for which the rate of change ~ or creep ~ of the surface area is zero) gives an absolute measure of the surface tension. These measurements have limitations, being only suitable to solids at high temperatures, and also where the surface tension does not depend greatly on surface orientation (Blakely, 1973). Alternatively, the surface tension can be obtained by theoretical calculation. Although great progress has been made in the ability to calculate surface energies, this remains a difficult problem due to the complexities of surface reconstructions. Moreover, most calculations provide values only for zero temperature. A survey of representative values of the surface tension from calculation and experiment is presented in Table 2.1. This provides a feeling for the range of uncertainty in the determinations of the surface tension, and of its general magnitude. In spite of limited knowledge of the absolute values of the surface tension of real materials, the concept of surface tension is useful experimentally because changes in the surface tension with temperature, adsorption and orientation govern a range of important processes. To see how this arises, we will consider the inclusion of the surface tension first in the bulk thermodynamic equations, and then we will introduce the concepts of surface thermodynamics. Including the definition of surface tension in the internal energy, we have: U = T S - pV + yA + ~_~ kt~N~
(2.2)
d U = TdS - pd V + ydA + ~_~ ~i dN~.
(2.3)
and
i
Thermodynamics and statistical mechanics of surfaces
57
Table 2.1 Selected values of the surface tension determined experimentally and theoretically are shown here to illustrate the range of values observed. The surface tension is a decreasing function of temperature so the experimental values for the melting temperature should be lower than the theoretical values for absolute zero. More complete tabulations of experimental values can be found in (Bonzel, 1995; Kumikov and Khokonov, 1983; Tyson and Miller, 1977). Tabulations of theoretical values can be found in (Liu et al., 199 lb; Methfessel et al., 1992; Smith et al., 1991). Element
Experimental value: T = Tm (from Tyson and Miller 1977) (Jim 2)
AI Ag
1.02 1.09
Ni
2.08
Pd Mo
1.74 2.51
Theoretical values: T = 0 K Miller index LAPW calculation (from EAM Calculation (from Methfessel et al. 1992) Liu et al. 1991a) (J/m 2) (Jim 2) (100) (100) (111) (100) (111) (llO) (311) (100) (110) (100)
1.21 1.21
1.86 3.14 3.52
0.55 0.70 0.62 1.63 1.49 1.78 1.77 1.45
w h e r e kti is the c h e m i c a l potential o f the ith c h e m i c a l c o m p o n e n t . (The additional c o m p l e x i t y of "wall t e n s i o n " terms is discussed by Griffiths (Griffiths, 1980), but will not be i n c l u d e d here). By standard m a n i p u l a t i o n s a m o u n t i n g to L e g e n d r e t r a n s f o r m a t i o n s (Callen, 1985) we can define other t h e r m o d y n a m i c potentials with different i n d e p e n d e n t variables, such as the H e l m h o l t z free energy, F = U - T S , yielding: d E = - S d T - p d V + ~[dA + ~_~ Ixi dNi .
(2.4)
i
T h e c h a n g e of variables gives alternative definitions o f the d e p e n d e n t variables, such as the surface tension, yielding: OF ~[= - ~
9
(2.5)
r.v.N,
W h i l e the functional f o r m of Eq. (2.1) for the surface tension w o u l d be useful for c o n s i d e r i n g an adiabatic process, Eq. (2.5) w o u l d be useful in an i s o t h e r m a l process. Both Eqs. (2.1) and (2.5) apply to closed systems, that is s y s t e m s in w h i c h the total n u m b e r (mass) o f each c h e m i c a l c o m p o n e n t is constant. A n o t h e r potential w h i c h is useful u n d e r c o n d i t i o n s o f c o n s t a n t c h e m i c a l potential is the g r a n d therm o d y n a m i c potential f l = F - ,Y_,kt~Ni = - p V + ),A, yielding" l
58
E . D . W i l l i a m s a n d N. C. B a r t e l t
dEl = - S d T - pdV + ~ldA - ~_~ N~ d~,
(2.6)
i
and another definition of the surface tension: (2.7) T, V, lai
We once again emphasize that in using the bulk thermodynamic equations such as Eqs. (2.1)-(2.7), we must keep in mind that they refer to the complete system of the solid, the fluid (either gas or liquid) in equilibrium with the solid, and the interface between the two. Thus the quantities held constant (such as entropy, volume and number in Eq. (2.1)) refer to the entire system, not the solid, fluid or surface individually. Because the surface generally represents a small contribution to the total content of the system, it can be difficult to isolate the thermodynamic effects of the surface. To make these points concrete, we can consider as an example a cleavage experiment, which might be used to measure the surface tension. To make such a measurement under the conditions of Eq. (2.1), we would consider a solid in equilibrium with its fluid in a rigid (zero net volume change), closed (zero net mass change), thermally insulated (zero heat flow and thus zero net entropy change) container. The surface tension would be measured by measuring the force needed to separate the solid into two pieces, exposing new surfaces of total new area 2A, as described by Blakely (Blakely, 1973). During this process, gas might adsorb onto the new surfaces, or atoms or molecules might be displaced from the new surfaces into the fluid phase. Thus, the numbers in the solid and fluid phases individually would change. However, the total number in the system would remain fixed, so Eq. (2.1) remains applicable. Thus using the bulk thermodynamic equations allows us to ignore the atomiclevel behavior of the system due to the surface in describing the macroscopic behavior. Of course this also means that using the bulk thermodynamic equations alone does not allow us to develop predictive capabilities for the influence of the interface. A simple example shows this immediately. Suppose that we carry out the adiabatic cleavage experiment. We expect that there will be a temperature change when we do work on a system adiabatically, so we approach this problem using the standard approach (Zemansky, 1968) of writing the differential of the entropy given constant volume and number: o = y, t s =
3S dT + T-ff-~
igS A, V,N i
dA
(2.8a)
T, V,N,
and then using a Maxwell relationship derived from Eq. (2.4):
~S ~A
T. V,N~
~T
(2.8b) VJ,,N,
59
Thermodynamics and statistical mechanics of surfaces
and rearranging to obtain"
as
by
CA,V. N dT = T - ~
A, V,N~
dA,
(2.9)
A,V,N~
where C is the heat capacity. This shows that we need to know the heat capacity of the system as a function of the surface area, and the variation of the surface tension with temperature. Rigorously, we would measure both parameters directly for the real system. For a first estimate, if the solid has compact shape and thus few atoms at the surface, we might assume that the heat capacity is the sum of the heat capacities of the fluid and solid phases, and neglect the influence of the area on the heat capacity. The variation of the surface tension with temperature can be measured, and is typically found to be about (Blakely, 1973): 1
c3y
=-2•
-5 K -l
(2.10)
y(Tm) aT where T(Tm) is the surface tension of the solid just at the melting point. For a specific system, let us consider a solid of 1 mole of Cu at 1200 K. The vapor pressure at this temperature is about 10 -I torr and the heat capacity of the solid is Cp = 30.6 J/mol-K. The surface tension of Cu at the melting point is 1.7 J/m 2. If we have initially comparable volumes of solid and vapor, the contribution of the vapor to the total heat capacity is negligible, and we estimate the change in temperature between the original (T o) and final (Tf) states as: ln(Tf/To) = - 10- 6 m -2 AA ,
(2.11)
where the area is measured in square meters. This illustrates that the cleavage of a compact piece of Cu, which will yield a change in surface area of about 1 cm z, results in a negligible change in temperature. This conclusion is inescapable so long as the conditions which allowed us to neglect the effect of the surface on the heat capacity hold. For an extreme example of a large surface area situation, consider the same mole of Cu arranged as a thin film only 10 atoms thick, which is to be cleaved to give two films each 5 atoms thick. The increase in the surface area in this case will be
Nav a2 _ 6x 1013
AA =---~
- ---if---- (2.6x10 -8 cm)2 = 8•
7 cm 2
(2.12)
and the temperature change would be a decrease of about 0.8% from the original temperature using Eq. (2.11). However in this case, the assumption that the heat capacity of the system is the same as that of bulk Cu is highly questionable. To deal with such problems where the surface is playing an important role, we need some way to address the contribution of the surface to the thermodynamic parameters such as the heat capacity explicitly.
60
E.D. Williams and N.C. Bartelt
2.2.1. The surface excess quantities The question of what is the surface, as opposed to the solid or vapor, contribution to any thermodynamic quantity depends on how one defines a surface atom. This is a microscopic question, and thus there is no unique thermodynamic approach to answering it. Thus some arbitrary decision must be made to define what is the surface. Then, once this decision is made, it is possible to go on and derive an internally consistent surface thermodynamics. One natural starting point for specifying surface functions (taken recently for example, by Nozi~res (Nozi/~res, 1991) is to break up the volume of the coexisting phases into three regions by introducing a pair of dividing surfaces. These dividing surfaces define three regions, the condensed phase, the fluid phase, and the intervening surface region. As originally suggested by Gibbs (Gibbs, 1961; Griffiths, 1980), there is a simpler (although slightly more abstract) way to define surface quantities. The approach is to define a single, flat dividing surface (the Gibbs dividing surface) which divides the system in to two phases c (the condensed phase) and f (the fluid phase). Once one has done this the volumes of the two phases are fixed, and satisfy I,x + Vf = V
(2.13)
This dividing surface is not in any way intended to describe the real surface. The real surface, defined roughly as those atoms whose properties are very different from the bulk phases, could be very broad. Regardless of the width of the physical surface, sufficiently far away from the dividing surface, the densities of all the extensive quantities such as energy, entropy, and atom numbers have the welldefined values of the bulk phases. This allows the extensive properties of each phase to be specified: for example the number of atoms in the condensed phase c is just ~ = 9~V~. In general, sums such as ~ + ~ are not the same as the total number N,. The differences are called the surface excesses. For instance for the number of atoms of chemical component i, the equation: + NI+ N~ = Ni
(2.14)
defines the surface excess N~ of the ith component Ni, and similar equations exist for the other chemical components and the entropy. At this stage, the position of the dividing surface is arbitrary. For a single component system, the choice of Gibbs is to place the dividing surface so that the value of N~ is zero for the one component of the system. In a multi-component system, one can choose any one of the components to set to zero in defining the dividing surface. In using the thermodynamic equations which result from defining surface excesses, two points are important: (1) The placement of the dividing surface defines the area A which is used in the thermodynamic equations, and (2) the placement of the dividing surface is not fixed geometrically. The dividing surface will move physically to maintain/W as the system changes thermodynamically. Thus, for the component chosen as the reference, the intuitive property of a one-to-one correspondence between the surface area and the number of atoms on the surface is lost.
Thermodynamics and statistical mechanics ~r
61
p-gas p-solid
0000 2.75A 1.0000 0000 0000
( ~ p(Z)
2.5A
Fig. 2.2. Schematic cross section of a solid-vapor interface illustrating the choice of the dividing surface. The hypothetical simple cubic solid with bulk lattice spacir~g of 2.5/~ has a top layer which is expanded by 10% in its vertical (normal to the surface) lattice spacing. The zero of the z coordinate system is arbitrarily set through the center of the top layer of atoms. The variation of the density of this "real" system with respect to z is illustrated on the graph to the right of the diagram. The density in the gas phase is effectively zero.
As an illustrative example of the definition of the dividing surface, consider a s o l i d - v a p o r interface for which the spacing between the top two layers of atoms is 10% larger than the spacing in the bulk as shown in Fig. 2.2. As a simple approximation, we treat the atomic density in the solid as a series of delta functions located at the centers of the atoms, We then place the zero of the z axis arbitrarily through the center of the top layer of atoms as shown in Fig. 2.2. Then the overall density profile can be.written as a discontinuous function: P = P,,,,iJ
for
-1.1a z<~
(2.15a)
for
-1.1a 1.1a - - - 7 < z -< ~ 2
(2.15b)
for
1.1a z > --~
(2.15c)
1
P=~
P = Pgas
Pso,~J
-
2
where a is the bulk interplanar spacing, Pgas is the density of the gas phase, and Psolid is the density of the condensed (bulk solid) phase. The surface excess n u m b e r density can then be calculated for any arbitrary choice of the dividing surface ds, using:
62
E.D. Williams a n d N. C Bartelt
N~ = A
p(z)dz
P~olia dz -
-
Zmin
Z min
(2.16a)
pgas dz s
w h e r e Psolid and Pgas, the densities o f the solid and gas phases, are explicitly a s s u m e d to be constant, p(z) is the real density o f the s y s t e m as a function o f position, and A is the area o f the d i v i d i n g surface, which is p e r p e n d i c u l a r to the z axis. P u t t i n g the density profile o f Eq. (2.15) into Eq. (2.16a), we obtain Ns 2.75/~ - ~ = ( - 1.375 A, - ds) Psolid "1- 1.1 Ps,,,~a+ (ds - 1.375)9gas
(2.16b)
To obtain a zero surface excess in this model, the d i v i d i n g surface m u s t be p l a c e d at ds = 1.125 A,, slightly a b o v e the center o f the top layer o f atoms. T h e intuitive c h o i c e o f a d i v i d i n g surface, which w o u l d be t h r o u g h the centers o f the top layers, does not yield a zero surface excess number, and in general will not do so. N e g a t i v e surface e x c e s s e s are o b t a i n e d for dividing surfaces placed h i g h e r than 1.125 A, and p o s i t i v e surface e x c e s s e s are obtained for dividing surfaces placed l o w e r than 1.125 A,. T h e case o f an e x p a n d e d surface layer, used to d e m o n s t r a t e this point is not artificial. Real surfaces have contracted or e x p a n d e d interlayer s p a c i n g s as illustrated in Table 2.2. In m u l t i c o m p o n e n t systems there is no reason to e x p e c t that the density profile o f a s e c o n d c o m p o n e n t will be similar to that o f the first (Blakely and S h e l t o n , 1975). For instance, if the solid is a s t o i c h i o m e t r i c c o m p o u n d , then there is likely to be preferential s e g r e g a t i o n o f one of the other c o m p o n e n t s near the surface. Or, in a n o t h e r c o m m o n situation, one can introduce a large v a p o r pressure o f a s e c o n d c o m p o n e n t which has a negligible solubility in the solid, but which adsorbs in Table 2.2 The interplanar spacings of near-surface layers are often expanded or contracted from the values of the bulk interplanar spacings. Some experimentally determined values of these modified spacings are shown here. The value ~!12 is the relaxation of the first interplanar spacing expressed as a percentage of the bulk interplanar spacing. A positive value indicates an expansion of the first interplanar spacing, a negative value a contraction. The values ~123 and &t34, are the corresponding relaxations of deeper layers. The selected values are taken from (Rous, 1995), where a full tabulation, references, and detailed discussion can be found. Element Mo Mo Ni Ni Ni Ni Ag Ag
Miller index
&l 12
~123
(100) (110) (111) (100) (110) (311) (111) (100)
-9.5+2.0 -1.5+2.0 -1.2+1.2 -1.1+_2.0 -8.6x'-0.5 -15.9+1.0 0.0-&5.0 -7.6+3.0
1.0+_2.0
+3.5_+0.5
+4.2+3.0
~134
-0.4_+0.7
Thermodynamics and statistical mechanics of surfaces
63
appreciable quantities at the surface. For such a multi-component system, there will be an equation like Eq. (2.14) to calculate the excessnumber at the surface. Except for special cases, it will not be possible to find a dividing surface where more than one of the surface excess numbers is zero. Once the extensive values have been defined in terms of their bulk (solid and fluid) components and the surface components as in Eqs. (2.13) and (2.14), we can subtract the contributions due to the bulk components from the thermodynamic potentials of Eqs. (2.2)-(2.6) to obtain thermodynamic functions which describe the surface alone. To do this, we begin with the integral form of the internal energy: U "- TS - p V =
T(S~ +
.-I-'yA + ~,~ ].l,iN i i S f + a s) -- p ( V c -~- V f + V s) +
(2.17)
yA +
~.Li(N~/+
N ~ / + N~/)
i
and subtract the bulk (condensed + fluid phase) contribution to the entropy, volume and number, giving
9 = rs"
(2.18)
vA + is l
In making this definition, we have explicitly (by setting the sum over i ~ l) indicated that the values of N~ are no longer independently variable because we have defined the dividing surface by setting N~ = 0. We can write the corresponding differential form: dU~= TdS"+ 7dA + ~ B,dN~
(2.19)
is 1
where the equation reflects the fact that dN~ is zero because N~ is a constant. The form of the surface excess for the Helmholtz and grand potentials then can be found from the standard expressions, F = U - TS and ~ = U - T S - ~ BiN i to obtain t
F ' = 7,4 + X l'tiN~
and
d F ' = - S" d r + "idA + X Bi dN~
is 1
~2~= yA
(2.20a,b)
ir l
and
d~S=-
S ~ dT + ydA - X ~
dl.t,
(2.21 a,b)
isl
Thus, the Helmholtz free energy per area is equal to the surface tension for the case of a single component system, when the dividing surface is defined to set the number at the surface to zero. However, the grand potential per area is equal to the surface tension in all cases, which makes it useful for addressing changes in the surface properties during adsorption or segregation.
64
E.D. Williant~ and N.C. Bartelt
Once we have defined the surface thermodynamic potentials, we then immediately have all the associated thermodynamic identities which can be used to identify relationships among the various quantities. For the Helmholtz and grand potentials we readily derive the Maxwell relationships relating to heats of cleavage and adsorption, and the effect of adsorption on the surface tension: Helmholtz
Grand
-~9Ss ba r,u~
~9y
~T a .,V~
-~)S s T.A.N~,,
~N~ A.T.N~.,
0~ti /)T
bA
-/9S s ba
~T a.~
-/)S s
,gN~
~[i
~9y
A
r.a,~%
~9T A,~,,~j
,T,~j,,
~A
(2.22a,b)
(2.22c,d)
(2.22e,f)
In all of these equations it is implicitly assumed that i ;el, because the number density of the component used to define the Gibbs dividing surface is not an independent variable. The application of such thermodynamic relationships is most commonly made to problems of adsorption on surfaces. In these cases, the reference component used for defining the dividing surface is the dominant component of the bulk material. It is generally assumed implicitly that the substrate is unperturbed by the adsorption of other components. Under this assumption the shape of the surface is unchanged during adsorption, and the derivatives with respect to area are equivalent to definitions of quantities per area. As we will show later, such assumptions are unnecessary, and the thermodynamic formalism described above is applicable to real processes in which the substrate changes as well.
2.2.2. Gibbs Adsorption Equation The final Maxwell equation shown above for the grand potential (Eq. 2.22f) is also known as the Gibbs Adsorption Equation. It is most commonly expressed as -~),,/
~kl'i A.T.il,,,
_ -3N~
OA
_ N~ - 19 M II,.T.Ilj A A
(2.23)
where the coverage 0 is defined as the ratio of the number of adsorbed particles to the number of available adsorption sites M. In deriving this equation, we again note that it is based on a definition of the dividing surface in which the number of surface atoms of the principle component NI is fixed at zero. Under many conditions it is a good approximation to make the assumption that the fluid phase is an ideal gas, for which the chemical potential is:
65
Thermodynamics and statistical mechanics of su~'aces
kt(T,p) = kt~
+ kTln(p/po)
(2.24)
Then the adsorption equation can be written as
~ln(p,/p,..)
= k T 19
Nmax
(2.25)
A
A ,T,~I,j~i
Thus the surface tension decreases as an increased pressure of the adsorbing gas causes more adsorption. To evaluate this expression quantitatively, we need to know the relationship between the surface coverage and the pressure (or chemical potential) of the adsorbing species. Such relationships cannot be predicted thermodynamically. They must be measured, or calculated using some microscopic model of adsorption. There are a large number of different models for the adsorbing layer (Clark, 1970; Dash, 1975; Hill, 1960; Payne et al., 1991). As an example of the application of Eq. (2.25) we will present just one, the Langmuir adsorption model. In the Langmuir adsorption model, we consider the case of strong adsorption (chemical adsorption) in which adsorption occurs at specific sites on the surface. We also make the simplifying assumption that there are no interactions between atoms at different sites. Then the adsorption of atoms changes the energy via the binding energy E per atom with thesurface, and it changes the entropy via the configurational entropy generated by distributing the adsorbed atoms among the available binding sites of the surface. We can deal with this problem simply in the Canonical Ensemble by writing first the partition function of an individual adsorbed atom (Hill, 1960): q = qo exp (-~kT)
(2.26)
where qo is the vibrational partition function of the atom trapped in the potential well of the binding site. We then use simple combinatorics to determine the number of configurations of N molecules distributed among the M sites of the surface, and write the partition function of the adsorbed layer: M!q(T) N Q(N,M,T) =
(2.27)
N ! ( M - N) !
Then using the standard definition: (2.28)
F(T,A,N) = - kTln Q(N,M,T)
we can derive an expression for the chemical potential, l.t = - k T
~)lnQ 'ON
T,M
=
t" )
(2.29)
66
E.D. Williams and N.C. Bartelt
where 0 = N / M as in Eq. (2.23). Using Eq. (2.26) for the individual atom partition function, we see that the binding energy of adsorption contributes to the chemical potential of the adsorbed species:
)o
(2.30)
Explicitly noting that the area of the surface is Ma, where a is the area per binding site, we can also calculate the contribution of the overlayer to the surface tension: /)lnQ
- _ kTln
vo=-kT Ma
a
T,N
M M-N
_ kTln a
1
(2.31)
1 -0
This contribution to the surface tension is the equivalent of the "spreading pressure" of the adsorbed layer. That is 7~dA is the work done in expanding an overlayer of coverage 0 from area A to area A + dA. We can alternatively derive Eq. (2.31) by integrating the Gibbs adsorption equation, Eq. (2.23), with the relationship between chemical potential and coverage of Eq. (2.30) inserted explicitly: 6)
d T = - - - dlaa
-kT
a
dO 1-0
(2.32)
and integrating to obtain kT
y - y , + ~ In (1 - 0) a
(2.33)
where the constant of integration T0 is the surface tension at zero coverage, i.e. the surface tension of the clean surface.
2.3. Thermodynamics of faceting The reasons for the faceting of surfaces and the mechanisms by which faceting occurs are problems of long standing in materials science and surface physics (Flytzani-Stephanopoulos and Schmidt, 1979; Herring, 1951 a; Moore, 1962). Faceting is defined as the break-up of a surface of some arbitrary macroscopic orientation into a "hill-and valley" structure which exposes surfaces of different orientations. Such a change in surface morphology is illustrated in Figs. 2.1 and 2.3. In recent years, unambiguous identifications of equilibrium faceted surfaces, in sufficient detail for thermodynamic analyses to be applied, have begun to appear (Drechsler, 1985; Dreschler, 1992; Heyraud et al., 1989; Ozcomert et al., 1993; Phaneuf and Williams, 1987; Phaneuf et al., 1988; Song and Mochrie, 1994; van Pinxteren and Frenken, 1992b; Wei et al., 1991; Williams et al., 1993; Yoon et al., 1994). Such
Thermodynamics and statistical mechanics ~su~. aces
67
Fig. 2.3. When clean, stepped Ag surfaces of orientation near the (110) are stable. However, exposure to oxygen changes the surface free energies, causing faceting to expose the Ag(l 10) orientation (with no steps) and another orientation containing a high density of steps. These STM images, each showing a 500 x 500 /~ area, show the evolution of the surface structure from one containing thermally lquctuating steps, through nucleation of facets, to the formation of large scale facet structure during exposure to 1• -9 torr of oxygen at room temperature. The panels show (a) the clean surface with an average step separation (terrace width) of 35/~, (b) after 62 min exposure to oxygen, (c) after 74 min large fluctuations in the terrace width become apparent, (d) after 96 min the nucleation of a facet has occurred, (e) after 102 min facets are growing linearly, (f) after 138 min facet sizes saturate and no further growth is observed. Figure provided by Prof. J.E. Reutt-Robey of the University of Maryland (Ozcomert et al., 1993; Ozcomert et al., 1994).
68
E.D. Williamsand N.C. Bartelt
observations provide an opportunity to obtain information about the anisotropy of the surface tension, and about how the anisotropy is influenced by adsorption, reconstruction and temperature. Furthermore, when such thermodynamic observations are combined with the results of direct imaging techniques, such as STM (Binnig and Rohrer, 1987), LEEM (Bauer, 1985), and REM (Osakabe et al., 1981), it becomes possible to understand faceting from an atomic point of view. In this section, we will describe equilibrium faceting as a thermodynamic phase separation, and develop the appropriate intensive variables to describe the resulting "orientational phase diagram". 2.3.1. Thermodynamics of orientational phase separation
Surface morphology arises on solids (as opposed to liquids) because the surface tension depends on the crystallographic orientation of the surface. As a result, the surface morphology of minimum area (i.e. a flat surface) is not necessarily the morphology which minimizes the free energy. Gibbs (Gibbs, 1961) recognized that surfaces will spontaneously rearrange to minimize their total surface tension, even if this involves an increase in surface area. Herring explicitly addressed the problem of the equilibrium morphology of a macroscopic surface of arbitrary orientation
Fig. 2.4. A surface will be unstable with respect to faceting if the total surface tension decreases in going from the upper panel to the lower panel. Notice that the projected areas Ai' are additive (analogous to volume in fluids, and in contrast to the total areas). The requirement of conservation of macroscopic orientation, given by Eq. (2.34), is illustrated in the insert.
69
Thermodynamics and statistical mechanics of surfaces
(Herring, 195 l a). The important physics of his approach is illustrated in Fig. 2.4. A The requirements for a surface of a given macroscopic orientation no to break up A A into new orientations n a and n b are simply that the net orientation is conserved, and the total surface tension is reduced: AO A
no =
o
A
Aa A
A n,, + A t' nb a
A
(2.34) b
A
A Y(no) > A y(n b) + A Y(nb)
(2.35) A
where y is the surface tension and A i is the area of the surface of orientation n~. Several formalisms can be used to determine the morphology that minimizes the surface tension, the most common being the construction of a Wulff plot to determine the equilibrium crystal shape (Herring, 1951 a; Jayaprakash et al., 1984; Kern, 1987; Rottman and Wortis, 1984; Wortis, 1988). Unfortunately, the relationship between the orientational variation of the surface tension, and the surface stability is cumbersome to apply quantitatively. A more easily applied approach to evaluating the conditions for faceting is to describe faceting as a phase separation: in other words we identify the facets of different orientations (as shown in Fig. 2.4) as phases in equilibrium with one another. To make this analogy explicit, we need to define a free energy for which the standard convexity arguments familiar to phase separation in fluids apply (Bartelt et al., 1991 ; Chernov, 1961 ; Metois and Heyraud, 1989; van Pinxteren and Frenken, 1992a). We recall from Eqs. (2.20) and (2.21) that the surface tension y can be identified with either the Helmholtz free energy in a one-component system, F~(T,A)
= yA
and
F.~ y(T) - ~ (one component system)
(2.36)
or with the Grand potential in a system with any number of components. ~(T,A,I~,,~) = yA
and
y(T,p~,,) - A
(2.37)
We then see that Eq. (2.35) above is equivalent to requiring that either the Helmholtz free energy or the Grand Potential be minimized when a surface breaks into a faceted structure. In what follows, we will use the Grand potential as it allows generalization to multi-component systems. An identical development can be performed using the Helmholtz free energy for a single component system. In a phase separation, the minimization of the free energy is constrained by conservation equations on the extensive variables. (For instance for the case of a two component fluid, the volume and numbers of each of the two components are conserved.) For the case of faceting, this constraint is the constraint that the macroscopic orientation of the surface is conserved, as indicated by Eq. (2.34). The vector form of Eq. (2.34) represents three conservation requirements, which can be expressed in rectangular coordinates as: A z = A "z + A b
where
A
A z = z. A~
(2.38)
70
E.D. W i l l i a m s a n d N.C. B a r t e l t
where
A y "- Ay + A or
where
A x = A~x + A b
Ay = ~. A~t A
A
A x = x. An.
(2.39) (2.40)
Physically, in making these definitions we generally define the z-direction as perpendicular to a low index surface, which serves as the reference surface. We then choose the x- and y-directions as orthogonal high-symmetry directions within the reference surface. The z component of the area Of a facet of arbitrary orientation is then the projection of the area of that facet onto the reference surface. The x and y components give the projections of the area onto planes perpendicular to the reference surface. The magnitude of their vector sum ~/A~ + Ay2 is a measure of the total "rise" of the facet with respect to the reference surface; in other words tan~ = "~A 2 + AI~ /Az.
We now can write the grand potential explicitly in terms of these extensive variables. This in turn allows us to define corresponding intensive variables, which must be equal for different phases (facet orientations) which are in equilibrium: ~ "~ = ~ "~( T , A z , A x , A y , l.t i~ l )
(2.41)
yields
3Az
~pz ~
etc. T . A .A t , l t , ,
(2.42)
,
Here we use the symbol "Pz" to indicate an intensive variable analogous to pressure, and note that there will be two similar variables corresponding to the derivatives with respect to the x- and y-components of the area. The use of these intensive variables is, as in standard problems of phase separation, to define equations which can be used to discuss the criteria for phase separation" explicitly we would write 9 . . A A for co-existence of facets of orientations n,, and n b ,,p,, . . . . =
h,,
Pz
etc.,
(2.43)
However, the use of the rectangular coordinate system leads to unnecessary mathematical complexity in setting up the conditions for equilibrium. It is possible to obtain a much more physically meaningful and mathematically tractable set of equations by working in spherical coordinates. We define the polar angle ~ as the angle between the normal to the surface of interest and the vector normal to the A reference plane, z. (In Fig. 2.4, ~ is shown as an angle between the surface and a horizontal reference surface. This is equivalent to the angle between the normals.) We define the azimuthal angle 0 as the angle between the projection of the surface normal into the reference plane and the x axis. With these definitions, the components of the surface area can be written as:
Thermodynamics and statistical mechanics of su.rfaces
71
A z = A cos t~
(2.44)
Ax = A sin ~ cos 0
(2.45)
A x = A sin ~ sin 0
(2.46)
The physical meaning of the two angles as illustrated in Fig. 2.5 for a surface with an orientation close to (typically within 10 ~ of) the reference surface has been discussed precisely by Nelson et al. (Nelson et al., 1993). The tangent of the polar angle is equal to the density of steps on the surface, and the tangent of the azimuthal angle is equal to the mean one-dimensional density of kinks on a step edge (at T = 0 K). Precise applications of these ideas to real surfaces require careful attention to the crystallographic definition of the step (Eisner and Einstein, 1993; Nicholas, 1965; Van Hove and Somorjai, 1980). Because the tangent of the polar angle is a physically meaningful quantity, we now define an appropriate reduced free energy, which has a simple dependence on tan
f=^
~
I
AzA~Ay ] ~ _f2 s z . A~ - -A~ T , -A- z' -A- z' A~' ['l'i~l
(2.47)
which now contains an explicit dependence on two areal densities:
Px = Ax/Az = tanr
(2.48)
py A y / A z = tant~sin0
(2.49)
=
h
Fig. 2.5. A surface vicinal to a high index surface will be composed of a density of steps and kinks determined by the misorientation angles 0 and t~. The number density of steps tan~ = p =h/, where h = the step height and = the distance between steps. When we define a reference direction in the reference surface for which the azimuthal angle 0 equals zero, the number density of kinks tan0 = an/, where an is the kink depth and is the average distance between kinks. In thermal equilibrium there will be a distribution of step-step and kink-kink separations, leading to step wandering. Perfectly periodic spacings are shown in the figure simply for clarity.
72
E.D. Williams and N.C. Bartelt
The areal densities have the physical meaning of the projected step density perpendicular to the x and y axes as in Fig. 2.5. We then re-write Eqs. (2.34) and (2.35) as s
A~' = A z + Abz
(2.50)
o o b b Azf > Azf ~ + Azf
(2.51)
The convexity requirement thus applies to this reduced free energy just as to the original grand potential. This allows us to use the tie-bar formalism in a plot of reduced free energy vs. either the x- or y-component of the areal density (Eqs. (2.48) and (2.49)). We can illustrate this graphically as shown in Fig. 2.6. The resulting equations describing the phase separation between a flat surface of orientation n o A A and a hill-and-valley structure containing new orientations na and nb are" A~ _ p~ - pO _ py _ py Az
P]~- p~
(2.52)
o O r - p~
In making use of the reduced surface free energy, we retain the original definitions of the intensive variables conjugate to Ax and Ay a s usual"
"Px
,,
Of 2 s OAx
T~4 t.A
v,
B ,, t
O(Ax /Az)
T,A
v,la,,;
0p/
, etc.
(2.53)
T,py,B,,~
f(p)
o
,r
~b
~0 Xb
~a
~
Xa
Fig. 2.6. Orientational phase separation occurs when a "hill and valley" structure has a lower total surface tension than a flat surface, as in Eqs. (2.35) and (2.51). This translates into a convexity requirement on the "reduced surface tension" (which is analogous to a Helmholtz free energy) versus either component (Eq. (2.8)) of the step density, p. The figure illustrates a non-convex surface tension curve which would lead to faceting. The illustrated curve is schematic only: the form shown would occur physically only if there are attractive interactions between steps strong enough to overcome entropic and elastic interactions. The,phase separation is indicated by the tie bar connecting points a and b. For a macroscopic orientation ?io the relative areas of the orientations na and nb are found from 9
,
A
A
Eq. (2.52), as illustrated in the figure with x,, _ p,, - Po _ A'b Here we use the nomenclature illustrated Xb Po -- Pb A'a" in Fig. 2.4 where A'i = A~ in the nomenclature of Eqs. (2.44) and (2.52).
Thermodynamics and statistical mechanics of su~. aces
73
The physical equality of the intensive variables in the two phases (" Px~ " = " pb ,,, , , p y a , , = , , pby ") appears as the equivalent slope of the free energy curves at the points of tangency in Fig. 2.6. In order to obtain a mathematically convenient form of the third independent equation corresponding to the z-component of the "step pressure", " -z P " we can define a different reduced free energy to obtain a thermodynamic function which varies as tanS, where we define p = tans = ~/p2 + p2. We proceed by dividing the grand potential by a combination of the x- and y-components of the area:
Az Ax Az fF _ f~s T, ----------w ~l A x 2 + 2' ~l A 2 2' ~[ A 2 + Ay2' ~[A 2x+ Ay Ay x at- Ay
~L i~ `
)=f P
(2.54)
We then recover the intensive z-component of the step density via: /
~gf~ s
"Pz" -
/)Az T,A z "4 y ,la i,~I
+
aOtal/ # +
A2
~(f/p) T,O,B~,,I
~(1/p)
(2.55) T,O,I.t.,~
As illustrated in Fig. 2.5, the magnitude of thermodynamic density of Eqs. (2.48) and (2.49), p = ~/p2 + P~ = tan~, can be interpreted physically as the step density on a vicinal surface. The two components are the step density projected onto the high symmetry direction (x) and an orthogonal direction (y). The azimuthal angle 0 in this description is the angle between the average direction of the step edge and the direction perpendicular to a high symmetry reference direction on the facet. It is thus physically related to the density of kinks on the step edge (Bartelt et al., 1992). This physical interpretation of the density components becomes particularly useful in interpreting observed orientational phase diagrams in terms of atomic models of the surface.
2.3.2. Types of faceting transitions To make the application of Eqs. (2.43), (2.53) and (2.55) concrete, we will introduce here a statistical mechanical description of stepped surfaces which uses the step and kink densities as the fundamental units of the structure. This equation and its parameters will be discussed in detail in the following section. The statistical mechanical description of the variation of the reduced surface tension with orientation which we will use is (Jayaprakash et al., 1984): f(~,0,T) =if(T) + ~(0,T) Itan~l + g(0,T) [ tan~[ 3 h
(2.56)
wheref"(T) =f(0,0,T) can be seen from Eq. (2.47) to be equal to the surface tension of the reference plane, 13(0,T) is the free energy cost per unit length of creating an isolated step, h is the step height, and g(0,T)ltan~l 3 is the free energy cost per unit area due to step-step interactions. The variations of the step formation and step interaction terms with both temperature and azimuthal angle 0 are governed by the
74
E.D. Williams and N. C. Bartelt
kink energy e. For temperatures well b e l o w the r o u g h e n i n g transition o f the low-index surface, specific functional forms for 13 and g can be derived for specific atomic models of the stepped surface, and used for quantitative analysis of experimental observations. Using this form for the reduced surface tension, we will discuss in general the different conditions which can arise in evaluating phase equilibria i n v o l v i n g faceting. An orientational instability may arise from the intrinsic variation o f the surface tension with orientation, as is illustrated in Fig. 2.6. H o w e v e r , real vicinal surfaces are most often o b s e r v e d to be stable with respect to faceting (Somorjai and Van Hove, 1989; Williams and Bartelt, 1989), so we can expect that a m o n o t o n i c variation of free energy with angle, as described by Eq. (2.56), will usually be appropriate for clean surfaces. In contrast, orientational instability (faceting) is frequently induced by chemical adsorption or by structural phase transitions. As p r o p o s e d by Cahn (Cahn, 1982), this can be understood by considering that the additional process completely alters the orientational variation of the reduced surface tension, resulting in intersecting curves as shown in Fig. 2.7. In this case the phase separation occurs not only b e t w e e n different orientations, but also between different compositions (in the case of adsorption) or structures (in the case of phase transitions). In the following, we will present an analysis of the conditions g o v e r n i n g phase separation in the case of such intersecting curves. B e c a u s e Eq. (2.56) contains a cusp at the origin, there are some possibilities in orientational phase diagrams which do not arise in normal phase equilibria. In Fig. 2.7, we illustrate three types of phase separation that might occur for a surface of arbitrary polar and azimuthal angle (00, 00) near to a low index orientation (~ = 0). The three
Opposite" Fig. 2.7. A change in surface composition (e.g. due to adsorption or segregation) or structure
(e.g. due to a phase transition) can change the variation of the surface tension with orientation. If the curve for a "perturbed" surface, labeled b, intersects that of the "unperturbed" surface, labeled a, then the convexity requirement illustrated in Fig. 2.6 will lead to^orienta~ional phase separation. The new ^ surface orientations exposed will have the vector densities p,, and Ph. The surfaces of orientation na ^ will have the composition or structure of the "unperturbed" phase, and those of orientation nb will have the composition or structure of the "perturbed" phase. The figure illustrates schematically how intersecting free energy curves lead to faceting. In the left hand column are shown the variation of the projected surface tension with one component of the vector density, Eqs. (2.48) or (2.49). Because the intersecting curves lead to a projected surface tension curve which is not convex, there is a region of unstable orientations, leading to step rearrangement to form facets. The corresponding projections of the phase diagrams into the 0--9 plane are shown in the right hand column. The three general types of tie lines one might expect from the faceting of vicinal surfaces: (a) No cusps are involved. The convexity requirement is applicable to the variation in reduced surface tension with both components Px and py of the step density. (The variation with lay is shown.) Thus Eqs. (2.66) and (2.67) determine the misorientation angles of the two phases. (b) If there is a knife-edge cusp along a high symmetry direction, one of the ends of the tie line can intersect the cusp, removing one of the tangency requirement, and replacing it with the inequality of Eq. (2.68). (c) If there is a deep cusp in the surface tension at the low-index surface (~ = 0), one of the ends of the tie line can intersect the cusp, leading to phase separation in the polar angle, leaving the azimuthal angle fixed. In this case the convexity requirement on the free energy is expressed in terms of the magnitude of the step density, rather than its vector components, as in Eq. (2.69).
Thermodynamics and statistical mechanics of surfaces
fl'py)
75
a
P~ po
p~ Oa
I/I ////
::,"
I
Py
'
I b I p, p~
', p~
Y
P~
Y
(a)
X
f(Py)
pa pO ,"" P~x Oo ,'" ,"'"
/
fa(O,l~a,T)fb(o,C'v T)-
O(bl// j lIt
t I p~ po p~
Py
P~
(b)
f~p)
X
a
pa
p~ Pb
I
Po
I
Oo
l
Pa
(c) Fig. 2.7. Caption opposite.
p~
p~
Ii
Y
76
E.D. Williams and N.C. Bartelt
cases are: Case 1, separation to two arbitrary orientations, ~,, 0a and Cb, 0b for which the reduced surface tension is smoothly varying (differentiable); Case 2, separation between an arbitrary orientation Ca, 0a and an orientation %, 0b = 0 where there is a singularity in the reduced surface tension for 0b -- 0; and Case 3, separation between an orientation ~,,, 0,, = 00 and the orientation of the low index surface Cb = 0 when there is a cusp in the reduced surface tension for the low index surface. The projection of the tie-bar into the tanr plane for each of the three cases, and the variation of the reduced surface tension of the two phases with the components of the step density are shown in Fig. 2.7.
Case 1 In order to have a tie-bar between points a and b when the reduced surface tension has no cusps, as shown in Fig. 2.7a, there must be a plane which is tangent to the two reduced surface tension curves at both points a and b. This requirement results in three equations which specify that the two components of the slope must individually be equal, and that a plane of the overall slope actually intersects both points. If we define Px and Or axes, corresponding to 0 = 0 ~ and 90 ~ respectively, as in Fig. 2.7a, the first two equations determined by these conditions are that the slopes (or the x- and y-components of the step pressures) at the two points are the same:
afo(oo,r
aL(oo,r p,: ap~
(2.57)
and
afo(oo,r Op,~,
o:
ap~
(2.58)
b
P,
The evaluation of these derivatives requires some care (Williams et al., 1993). If the free energy function is expressed in terms of variables tan~ and tan0, then the expressions to be used in evaluating the derivatives are:
af
af
Opx
Otanr
Pv
cos0 + tan0
-tan0 /
af 0tan 0
tanr
(2.59)
tanO cos 0
and
af
af
~f ~)Py
o,
Otanr
tan0
sin0 + 0tan0
'-"nO
/tant~ lc~
"
(2.60)
The requirement that the tie-bar actually intersects both points results in the third equation"
77
Thermodynamics and statistical mechanics of surfaces
L(o~,,~) =L(O~162176 + (px~ - pa)
+ (py~- p~)
afa(Oa'r
2p~
afa(O~,r p~
(2.61)
The geometrically derived equation can be replaced with the thermodynamically derived requirement of equal z components of the "step pressure". (Arduous algebra confirms that the simpler form of the step pressure equation can be expressed as a linear combination of Eqs. (2.57), (2.58) and (2.61).) From Eq. (2.55), the equality of the z-"step pressures" is:
fo-po&
T,O,,,la i, i
=fb -- Pb ~--~p b
(2.62) T, Ob,lai, I
When we have a specific expression for the variation of the reduced surface tension with orientation, we can evaluate these expressions explicitly. Using Eq. (2.56) in Eqs. (2.57) and (2.58) allows us to obtain the relationships:
f3i = "~/cosOi + 3g i tanZt~i cosOi-
/
/r176 LcOSOi1 tan20)(' )
1 ()~._._.._+.L_atanO ()gi tan2t~i hi/)tanO
5P~ o. = ~'/sin0i + 3g, tan2~i sin0i + -hi ~c)tan0 + atan 0
cos O;
(2.63)
(2.64)
In the simplifying case where there is no change in the polar angle 0 during the phase separation, we can define the angle 0 = 0 to be orthogonal to the step edges, and then Eqs. (2.57), (2.58) and (2.61) give a very simple result:
& D
o~tanr
T,O,,,M ,,t
atanCb T.0h.la,,~
(2.65)
Upon inserting Eq. (2.56) for the angular dependence of the surface free energy, and rearranging, this results in: ~ h ( T ) - [5,(T) = 3g,(T)h, tan2~,- 3gb(T)h o tan2~b
(2.66)
Using Eq. (2.56) in Eq. (2.61) directly gives the general result: f,~'-f~' = 2g,(0,,,T)ltan~,l 3 - 2gh(0h,T)ltan~bl3
(2.67)
where the difference in the surface tension of the two phases at ~ = 0 is Aft(T) = fo"-f~'. These results show that, within the formalism of Eq. (2.56), Case 1 step phase separation can occur if a perturbing process causes changes in step interactions, which could arise for instance from adsorption-induced changes in surface stress. However these changes must match in rather restrictive ways the corresponding changes in the zero-angle reduced surface tension f o and the step free energies [3. Phase separation between two low-symmetry orientations has been observed for
78
E.D. Williams and N.C. Bartelt
vicinal Pb(111) (van Pinxteren and Frenken, 1992a; van Pinxteren and Frenken, 1992b). In this case, the perturbing process is surface melting, so that phase separation occurs between an unmelted surface of low step density, and a melted surface of higher misorientation angle. In this case, of course, Eq. (2.56) is not appropriate to describe the free energy of the melted surface. However, the driving mechanisms for the phase separation can still be understood in general physical terms as due to changes in the initial slope and curvature of the surface tension curve. Case 2 In the second type of phase separation, illustrated in Fig. 2.7(b), we imagine that the reduced surface tension for one of the phases, fb, has a knife-edge singularity along the high-symmetry direction, 0b = 0. In this case we cannot require that the y-component of the tie-bar be tangent to the reduced surface tension curve at Oh = 0, and we lose the requirement of Eq. (2.58). However, for phase separation to Oh = 0 to occur, the slope of the reduced surface tension curve as it approaches Oh = 0 must be greater than the slope of the tie-line. Thus Eqs. (2.58) can be replaced by an inequality corresponding to
(2.68)
c)P>,
b
b
p,pv ~ ()
Equation (2.57) remains valid. Because the slope of the tie-bar must still match the tangent of free energy curve for phase a the form of Eq. (2.67), which describes the difference in the surface tension between the two phases, is also unchanged. The derivatives of the reduced surface tension of phase b, Eqs. (2.63) and (64), are simplified in form as Oh = 0. Case 3 In the final type of phase separation, illustrated in Fig. 2.7(c), we imagine that the reduced surface tension for one phase has a cusp-like singularity at Ch = 0. Physically, this corresponds to phase separation between a uniformly stepped surface, and a surface with a hill-and-valley structure consisting of a low-index facet and a bunches of steps. In this case we have no information about the behavior of the system with azimuthal angle, so we are reduced to two independent equations describing the system, rather than three. Furthermore, because of the cusp, the requirement on the slope becomes an inequality which can be written most usefully as:
&
I a
/)tan r
(2.69)
I 0, = o, ,~ -, ,,
The physical meaning of this equation is that the initial slope of curve 2 must be steeper than the slope of the tie bar (which is in turn tangent to curve 1). The initial slope is, of course, set by the energy cost of isolated steps, so this mathematical requirement tells us how much the perturbing process must change the step energy in order to cause faceting. The resulting inequality, using the form of Eq. (2.56) for the reduced surface free energy, is:
Thermodynamics and statistical mechanics of surfaces
132(T) - I]l(T) > 3gl(T)hl tan 2 ~1
79 (2.70)
where the step heights are the same in both phases. As in Case 1 phase separation, this shows that the magnitude of the step-interactions sets the energy scale for step bunching to occur. Equation (2.67) still remains valid because of the requirement that the slope of the tie bar matches the tangent to the free energy of curve a, but it is simpler in form since ~h = 0: Af" =f~' - i f , = 2ga(0a,T)ltan~al 3
(2.71)
The magnitude of the change in the surface tension and the step energy required to allow orientational phase separation can be estimated using the formalism described in the following section and calculated or measured energetic values of step or kink energies and the surface stress (see below). The result is that approximately a 0.1% change in the facet energy coupled with approximately a 10% change in the step energy is needed to cause phase separation between a flat surface and a step bunch. Such a mechanism has been proposed for the orientational phase separation of vicinal S i ( l l l ) (Williams and Bartelt, 1991), Pb(111) (Metois and Heyraud, 1989; Nozieres, 1989) and for Au(111) near the melting temperature (Bilalbegovic et al., 1992; Bilalbegovic et al., 1993; Breuer and Bonzel, 1992).
2.4. Statistical mechanics of vicinal surfaces The thermodynamic description above allows us to think about changes in macroscopic parameters such as orientation, and to discuss those changes in terms of related thermodynamic parameters such as the surface tension. Intellectual curiosity, as well as the hope of someday being able to predict behavior such as faceting from an understanding of atomic properties, cause us to want to understand the governing thermodynamic principles in terms of underlying atomic structure and energetics. The link between thermodynamics and atomic properties is statistical mechanics. To capture the essential physics of the role of entropy in determining the surface behavior above T = 0 K, we use a simple lattice model of the solid as the basis for the statistical mechanics of the surface. In this model, called the solid-on-solid (SOS) model, atomic positions are restricted to discrete lattice sites and interact with one another via pair-wise interactions. To obtain a qualitative understanding, we use the simplest possible model, in which we neglect atomic vibrations at the lattice sites and restrict the model to only near-neighbor interactions. A surface structure calculated using a Monte Carlo simulation within the solid-on-solid model is shown in Fig. 2.8a. The effect of finite temperature is to create disorder in the structure in the form of vacancies and extra atoms on the terraces, and in the form of "kinks" at the edges of the steps. Within this model we find that the natural parameters which appear in the thermodynamic equations, the step density tan~, and the kink density tan0, are also the natural atomic scale structures whose energies define the behavior of the stepped surface.
E.D. Williamsand N.C. Bartelt
80
I
a)
1.2
kT=O.8E
. . . .
'
. . . .
'
. . . . .
'
()j_
1.0 0.8
"'--
O
~- 0.6
0
0.4 4
0.2 0.0 0.0
. . . . . 0.5
1.0 kT/e
1.5
2.0
Fig. 2.8. (a) Monte Carlo simulation of a stepped surface in the solid-on-solid (SOS) model at kT = 0.8[;,
where • is both the nearest-neighbor interaction energy and the kink energy. (Figure from the work of Bartelt et al. (1991 ).) (b) Calculation of the variation of the surface tensionf ~ and step energy ~(T) (solid line) for the body centered solid-on-solid model, based on the work of van Beijeren (van Beijeren, 1977; van Beijeren and Nolden, 1987).
2.4.1. Simple stepped surfaces The first stage in d e v e l o p i n g a description of the surface tension based on the properties of steps is to a s s u m e that there are cusps in the surface tension at a relatively small n u m b e r of low index orientations, and that between these orientations,
Thermodynamics and statistical mechanics of suqaces
81
the surface tension varies smoothly. One can then describe changes in orientation in terms of changes in step density, and introduce the temperature dependence of the surface tension by treating the entropy of the steps correctly (Gruber and Mullins, 1967; Voronkov, 1968). The variation of the surface tension with step density, tanr should then have a leading term which is due to the contribution of the terraces between the steps, a second term due to the free energy cost of the steps, and a third term due to any step-step interactions (Wortis, 1988). The inadequacy of zero-temperature calculations of the surface tension is immediately apparent if one considers the temperature dependence of these terms separately. The energy cost of creating an excitation at a step edge is much lower than that of creating an excitation on a terrace. For the simple-cubic solid-on-solid model with only near neighbor interactions of energy ~, for instance, the energy cost of moving an atom from a position in a straight step edge to a site along the edge is 2E. However, to create a vacancy and an extra atom on the terrace requires energy 4~. As a result, at a given temperature (below the roughening temperature) thermal excitations at the edge of a step are more prevalent than those on a terrace, as illustrated in Fig. 2.8a. Correspondingly, the contribution to the temperature variation of the surface tension of vicinal surfaces due to the steps is much larger than that of the terrace, as shown in Fig. 2.8b. Thus, one cannot simply extrapolate relative zero-temperature energies for high- and low-index surfaces to non-zero temperature. The correct treatment of the problem of the entropy of steps as a function of step density is not obvious, and was first addressed explicitly by Gruber and Mullins (Gruber and Mullins, 1967). They noted that at non-zero temperature, a step can wander due to the thermal excitation of kinks, as illustrated in Fig. 2.8a, thus generating configurational entropy which lowers the free energy of the isolated step, as illustrated in Fig. 2.8b. However, when there is a train of steps, the wandering is constrained because one expects that a step crossing, which would create an overhang, is energetically unfavorable. Thus, the entropy of step wandering makes the largest favorable contribution to the surface tension when the step density is lowest, an effect which is referred to as the "entropic step repulsion". This entropic repulsion has the same effect as a true energetic repulsion which falls off in strength as the square of the distance between the steps. Subsequent work provided a quantitative description of how this entropic repulsion (as well as any energetic interaction which is not of shorter range) influences the surface tension (Jayaprakash et al., 1984; Nozieres, 1991). The result is that the step interaction term is proportional to the third power of the step density and the expansion for the reduced surface tension contains no second order term in step density. The form of the reduced surface tension thus is, as previously stated in Eq. (2.56) (Jayaprakash et al., 1984): f(O,T) - ~ ' ( T ) + 13(T)Itan~] + g(T)ltan~l 3
(2.72)
where ~~ is the surface tension of the terraces between the steps, 13(T) is the free energy cost per unit length of an isolated step, h is the step height, and g(T) Itan3~l is the free energy cost per unit area due to step-step interactions. It is worth
82
E.D. Williams and N.C. Bartelt
emphasizing the physical reason for the absence of a second order term in this expansion. The "entropic" interaction between steps is the entropy cost associated with the fact that adjacent steps cannot pass through each other (Fisher, 1984; Fisher and Fisher, 1982): Each time a step wanders next to another step, the number of choices of directions the step can wander is reduced by a factor of 1/2. Thus, for each step collision, the entropy of the surface is reduced by kin2. The density of step collisions, and thus the total entropy cost, can be estimated from simple random walk arguments (Fisher, 1984; Fisher and Fisher, 1982). The average distance between step collisions of one step is roughly given by the distance required for a step to wander a distance equal to the average distance l between steps. This distance is simply determined by the step diffusivity as apl2/b 2 (see Eq. (2.84) below). The number of step collisions per unit area is thus kTb2/13ap, and thus the total free energy cost due to the entropy loss of step collisions is kT(ln2)b2/13af, = kT(ln2)b21tan3t~l/(h3ap), reproducing the form of Eqs. (2.56) and (2.72). The simplicity of this argument suggests that the form of Eqs. (2.56) and (2.72) is very general: thermal wandering only contributes a term proportional to Itan~l 3 in the surface free energy. Within a lattice model for the surface, as illustrated in Fig. 2.8, exact expressions are available for the temperature dependence of both the step formation energy (van Beijeren and Nolden, 1987; Williams and Bartelt, 1991) and the step interaction energy (Jayaprakash et al., 1983) in terms of the kink energy (Jayaprakash et al., 1984). Thus given a tabulation of calculated zero-temperature energies for the facet, step and kink, as illustrated in Table 2.3, one can estimate both the thermal and orientational variation of the surface tension. Using the equations presented in the next section at temperatures typical of experiments, we find that the magnitudes of the three parameters are approximately 0.1 eV/]k 2 for the surface tension ~(T), approximately 0.05 eV//~ 2 for the step formation term fS(T)/h, and 0.02 eV/A 2 for the step interaction term g(T).
2.4.2. Expressions for step formation and interaction energies Extracting specific values for atomic energies from thermodynamic observations is always fraught with problems of uniqueness. For the particular case of stepped surfaces, the thermodynamic behavior is completely defined by the tendency of steps to wander, with the result that one can obtain unique information about the step "stiffness" (defined below) which characterizes step wandering (Bartelt et al., 1992). To proceed from a value of the stiffness to a determination of the characteristic energies of the system, one must choose a reasonable microscopic model using either assumptions or additional information available about the system. It is sensible to begin this process by invoking the simplest possible microscopic model that can produce the phenomena of interest. One then uses the observations to fit the microscopic energies of this model. In evaluating how well the energies thus determined describe the true physical energies of the system, consistency of all observations with the model is necessary of course, but not sufficient.
83
Thermodynamics and statistical mechanics of surfaces
Table 2.3 The energetics of steps on Ag(100) and Ag(111) as obtained from the embedded atom method (EAM) (Nelson et al., 1993) and the equivalent crystal theory (ECT) (Khare and Einstein, 1994). The value listed first in each entry is the EAM value, the number listed second is the ECT value. The staircase direction is the vector perpendicular to the average step edge, pointing in the downhill direction. Surface orientation
(100)
Facet e n e r g y ](~ = 0) (meV//~2) 44 98.6
Step-staircase direction
{0111 {001}
Step energy 13(T= 0) (meV/A)
Kink energy
36 56 49
102 213
(meV)
m
(111)
39 76.0
m
{211} {211} {110}
65 170 66 161 76
102 213 99 255
m
In this spirit, we present a specific formulation of the parameters of Eqs. (2.56) or (2.72) in terms of a nearest-neighbor square lattice model, with the addition of a long-range repulsion between steps. There are three energetic parameters in this model: the energy cost 13(0,0) per unit length of creating a step at T = 0 in the high symmetry (0 = 0) direction, the energy e of creating a single kink of depth an (depth is defined normal to the step edge), and the magnitude of the step-step repulsions. More complex models, including kink corner energies (Bartelt et al., 1992; Swartzentruber et al., 1990), non-linear variation of kink energy with kink size (Bartelt et al., 1992), kink-kink interactions (Zhang et al., 1991), honeycomb symmetry, and different forms for the step-step interactions (Frohn et al., 1991; Joos et al., 1991 ; Redfield and Zangwill, 1992) can be considered with correspondingly more complex forms of the equations below. The leading term in the expression for the surface tension as a function of orientation, Eq. (2.72), is the linear variation of the surface tension with step density. The coefficient of this term, I3(0,T), is the free energy per unit length of an isolated step. For arbitrary temperatures, the step free energy can be calculated using forms for the interface energy for the Ising model with both isotropic (Rottman and Wortis, 1981) and anisotropic (Avron et al., 1982) near neighbor interactions. The use of isotropic interactions is equivalent to assuming a square lattice in which the zero-temperature energy cost of steps equals the kink energy per unit length, 13(0,0) = e/ak, where ak is the length of the kink edge (ak > an, with the equality occurring when the kink is perpendicular to the step edge). The square-lattice model is a reasonable zeroth-order approach for high-symmetry surfaces. However, for surfaces of lower symmetry, such as Si(100) (Dijkkamp et
84
E.D. Williams and N.C. Bartelt
al., 1990; Swartzentruber et al., 1990), the kinks and steps have different bonding configurations, and thus a rectangular lattice model, for which one would use the anisotropic calculation, would be more appropriate. The results of the Ising model analogy give the step free energy for steps in the high symmetry direction, 0 = 0, as (Avron et al., 1982) 1 3 ( 0 ' T ) = 1 3 ( 0 ' 0 ) - kapT l n / c ~
2 ~~ /
(2.73)
where ap is the minimum separation between kinks along the step edge. We can derive the low temperature limit of this expression rather simply. In the case where the temperature is low compared to the kink energy, we can safely assume that only kinks of depth one will be excited, and that these will be excited rarely enough that they can be treated as independent. In this case we write an independent particle partition function q for each element of the step edge. Because each element can be straight, kinked out, or kinked in, this partition function has three terms:
q(T) = exp(-ap~(O,O) /kT) [1 + 2exp (-e/kT)]
(2.74a)
where the energy cost of a straight step element is just the length of the element at, times the energy per length of the straight step at zero temperature [3(0,0), and the energy of a kinked element (either in or out) is the energy of the straight element plus the kink energy ~. We then assume a step of N elements, or length L = aN, write down the step Helmholtz free energy and derive the step free energy as the line tension of the system:
F = - k T In qN
13(o,T)-
(2.74b)
~)F
(2.74c)
Then using a series expansion of the remaining logarithmic terming assuming e > kT, one obtains:
2kT ~(0,T) = [3(0,0) - ~ exp ap
"_s (2.74d)
Analytical expressions for the variation of 13(0,T) with tan0 for all temperatures less than the Ising critical temperature (kT c -- 1.13c) can be obtained for the symmetrical case with the substitution of ~ = 2J in the formulas given in (Rottman and Wortis, 1981 ), and for the anisotropic case with the substitutions, [3(0,0) = 2J/ap and ~ = 2J x (Avron et al., 1982). The full isotropic formula is: _a_v_ 13(0,T) = Icos01sinh-~(a(0)lcos01)
kT
+ IsinOIsinh-~(cz(O)lsinOI)
(2.75a)
Thermodynamics and statistical mechanics ~'surfaces
85
where 0;(0) = _2 1 1 -c 2 c [. 1 + (sin20 + c 2 cos 2 20)~]
(2.75b)
and
c=
2sinh (s/kT)
(2.75c)
cosh 2(s/k 73
At T = 0, the step energy reduces to
13(0,0) = 13(0,0) cos0 + -
13
ap
Isin01
(2.76)
where 13(0,0) = 13/ak in the isotropic case. The form of Eq. (2.76) indicates a cusp in the step energy at 0 = 0. At any non-zero temperature the entropy contribution due to the random distribution of kink sites removes the cusp at 0 = 0. However, differentiation of Eq. (2.75) with respect to 0 shows that the transition from zero slope to a large positive slope occurs at extremely small values of 0 for values of 13/kT> 5. This behavior which appears experimentally very much like the knife edge singularity discussed in w 2.2 (Case 2) may be observed for large kink energies or low temperatures. The variation of the step free energy with temperature and with kink density, tan0, calculated using Eq. (2.75), is shown in Fig. 2.9. The slopes of the [3-tan0 curves, which are important in evaluating Eqs. (2.63) and (2.64), can be calculated numerically. The final term in the expression for the reduced surface tension is due to the interactions between steps. The form and magnitude of this term are determined by step wandering due to thermal excitation of kinks, as well as any energetic interactions between the steps such as elastic or dipolar repulsions. This wandering can be described by the "step diffusivity", b2(0,T), which can be calculated as the mean squared displacement of the step edge when there are no kink-kink interactions (Bartelt et al., 1992). The general expression for the diffusivity for a step of zero misorientation angle 0 is:
2a 2 y_~ n 2 exp [-e(n)/kBT] b2(T) =
"--~
(2.77)
I + 2 ~ exp [-s(n)/kBT] n=l
Closed forms for the step diffusivity can be calculated by summing Eq. (2.77) for special cases of the dependence of the kink energy on kink depth, 13(n). The results are shown in Table 2.4 for four cases where no overhanging kinks exist, the
E.D. Williams and N.C. Bartelt
86
I
1.2-
I
I
I
...... " " .
~0--
.........- , . .
0~
_ _ O= 1 0 ~
.............
..... 0 - 2 0
~
1.0-
~'* (D v a3.
0.8-
o
0.6-
0.4-
(a) I
0.2
0.0
(a)
1.4-
1.3-
I
I
0.4
I
I
kT/E
I
0.6
I
0.8
I
E/kT = 2 ~ 5 .......... 10 ....... 20 . . . .
I
. ........ :."
. ...... ;.-:22...... : : - ' " .........
30 .......
...-:;':"
....
1.2--
" "
. ...............
..................
.__
P <E 1.10
1.0
0.9-
(b) 0.8-
(b)
o.o
oI,
oI~
oI~ tane
o14
oI~
Fig. 2.9. The free energy per unit length of a step 13(0,T) calculated for the square nearest-neighbor lattice model, using Eq. (2.75). (a) Step energy vs. temperature for several values of the net azimuthal angle. (b) Step energy vs. tangent of the azimuthal angle for several values of the temperature. When the temperature ishigh relative to the kink energy e, step wandering lowers the step free energy, and makes it relatively isotropic. When the temperature is low, there is a significant energy cost to changing the orientation away from the high-symmetry direction (Williams et al., 1993).
u n r e s t r i c t e d T S K ( t e r r a c e - s t e p - k i n k ) m o d e l in which the kink e n e r g y varies linearly with kink depth, for the T S K m o d e l with a extra e n e r g y cost for the c o r n e r of a kink, for a discrete G a u s s i a n model, and for a m o d e l in which kinks are restricted to a depth o f o n e unit only. For steps along the h i g h - s y m m e t r y direction 0 = 0, these
87
Thermodynamics and statistical mechanics of su~. aces
e q u a t i o n s a r e r e a s o n a b l y a p p l i e d for k T < e, t h e k i n k e n e r g y . W i t h i n the u n r e s t r i c t e d T S K m o d e l , the e x p r e s s i o n for the step d i f f u s i v i t y is s h o w n to b e ( B a r t e l t et al., 1992) b2(T) 2z0 ----r- = ~ a, (1 - z0) 2
(2.78)
w h e r e z0 = e x p ( - e / k T ) . ( T h i s e x p r e s s i o n is e q u i v a l e n t to t h e f o r m s h o w n in T a b l e 2.4 via the s i n h i d e n t i t y . ) T h e t e m p e r a t u r e d e p e n d e n c e o f t h e s t e p d i f f u s i v i t y is s h o w n in F i g . 2.10, for the T S K m o d e l , a n d the o t h e r m o d e l s o f T a b l e 2.4.
Table 2.4. Temperature dependence of the step diffusivity b 2 for different models of the energy cost of forming a kink as a function of the kink depth, E =fiE,n). The table is reproduced from (Bartelt et al., 1992). Model
E(n)
b2/a2
TSK
InlE
I sinh-2(e/2kBT) 2
InlE + (1 - 8(n,o))E c
b2sK/(1 + [exp(e./kBT)- 1] tanh(eJ'kst))
n2E
not a simple function
E(0) = 0, E(+I) = 1
2/[2 + exp(e/kBT)]
TSK + corner Discrete Gaussian Restricted
'
c4~ --.-t~..O0
2
'
'
I
.
.
.
.
1
'
"
'
9
!
.
~
0 0.0
.
.
.
,
, ";
............... ;"; 0.5
1.0
.
.
.
.
.
. 1.5
.
/
.
.
'
' "
. 2.0
kT/c
Fig. 2.10. Temperature dependence of the step diffusivity, b2(T) for the four models of kinks at step edges shown in Table 2.4. (a) solid curve: TSK model in which a kink of length na costs energy E(n) = E; (b) dotted curve: TSK model with an additional corner energy 3E; (c) short-dashed curve: discrete Gaussian model, E(n) = n2E; (d) dash-dotted curve: restricted model, with only n = 0, 1 allowed; (e) long-dashed curve: restricted model in which the step is misoriented away from the high-symmetry direction (Bartelt et al., 1992).
E.D. Williams and N.C. Bartelt
88
For steps with a misorientation angle 0, there is an intrinsic kink density proportional to tan0. Within the TSK approximation, the derivation of the step diffusivity including the dependence on azimuthal angle yields: b2(0,T) 4z~)+ (1 + z~) ~/4z~]+ (1 -z~) 2 tan20 a--------~= tan20 +
(2.79)
(1 - z~):
(Note that due to typesetting errors, incorrect versions of this equation have appeared previously (Bartelt et al., 1992; Williams et al., 1993). Also note that in the case where a~, ~ea, (see Eq. 2.73) tan8 should be replaced by (a~ / tan8 in Eq. 2.79.) This equation is valid only for small values of 8, as overhangs quickly become important for 8 > 5 ~ The diffusivity is related to another important quantity, the step-edge "stiffness" [3 (Bartelt et al., 1992; Fisher, 1984; Fisher and Fisher, 1982): -kTa, 13(0,T) = b2(0,T ) cos30
(2.80)
The step edge stiffness, which measures the free energy cost to deform a step, is determined by the orientational dependence of the step free energy 13(0,T) through (Fisher, 1984; Fisher and Fisher, 1982): --
I3(0,T) - 13(0,73 +
~9~13(0,T)
002
(2.81)
For steps with overhangs, Eqs. (2.90) and (2.81) generalize the definition of the diffusivity in Eq. (2.77). The step edge stiffness diverges at T ---) 0 for high-symmetry (0 = 0) steps, as the diffusivity becomes small. The variation of the step diffusivity with angle, calculated using Eqs. (2.75), (2.80) and (2.81), is shown in Fig. 2.11. Once an expression for the step stiffness or diffusivity is obtained, ?t can be used to evaluate the step-step interaction term, g(0,T) in Eq. (2.56) or (2.72). The general form for this term, valid when the step diffusivity is much smaller than the square of the average step separation is (Williams et al., 1994): I/2
[ 4Aa 1 'Jt
g(O,T) = rc2kTb2(O'T) I + 24ap h3 L kTb2(O, T)
2
(2.82a)
for steps of height h, and for step-step interactions in which the strength of the interaction falls of as the square of the distance between steps, U(x) = A/x z. The inverse square form of energetic step-step interactions is expected physically for interactions which are due to elastic interactions or to dipole-dipole interactions (Einstein, 1995; Marchenko and Parshin, 1980; Voronkov, 1968). The temperature dependence of this term depends on two characteristic energies, the kink energy, and the interaction energy. The variation of the step interaction term with kink
Thermodynamics and statistical mechanics o/'su~. aces I
89
I
I
I
I
1.0-
....-S.-
30
~
/
0.8-
i
0 -,0.6
-
% 0.4-
0.2-
0.0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
tanO
Fig. 2.11. The step diffusivity b2(0,T) as a function of azimuthal angle, calculated from Eqs. (2.80) and (2.81 ), using the values of the step free energies shown in Fig. 2.9. The diffusivity increases with kink density and with temperature (Williams et al., 1993).
I 2.0
,
!
I
I
I
-
1.8-
1.6to
o 1.4-
~1.2 c-
"
~~~
1.0-
..
0.80.60.0
I--" -
Jl ...................................
..jJ~5.... ............. '."_~~~~---:-..::..:::.::::::: ............... 1
0.1
I
0.2
I
0.3
tonO
1
0.4
I
0.5
Fig. 2.12. The contribution of step interactions to the reduced surface tension coefficient, g(0,T) as a function of the azimuthal angle, calculated using Eq. (2.82) with the values of the diffusivity shown in Fig. 2.11. The strength of the step-step interaction used in the calculation is A/apE - 0.22. The magnitude of the interaction coefficient increases with increasing step wandering and with temperature (Williams et al., 1993).
density, tan0, is shown in Fig. 2.12 for a specific ratio of the step interaction strength A to the kink energy e. A useful limiting form for the s t e p - s t e p interactions for the case where there are only entropic interactions, i.e. when A = 0, is,
90
E.D. Williams and N. C Bartelt
g(0,T) =
rc~kTb~(0,T) 6aph 3
(2.82b)
For low temperature and misorientation angle, the step diffusivity goes to zero and another simple form results: rI:2A g(0 ---) 0,T--~ 0 ) = 6h 3
(2.82c)
Finally, we will also need to consider the variation of the step interaction term g(0,T) with tan0. If all the 0-variation enters through bZ(0,T) (in other words, if the step-step interaction strength A is independent of angle, as it should be for elastic interactions on surfaces with three-fold or higher symmetry), we find easily that ~)g ~~0at0=0. 0tan0
2.4.3. Experimental determination of statistical mechanical parameters The advent of experimental techniques which allow direct imaging of steps allows the statistical mechanical description outline above to be tested directly. In particular, it is now possible to determine a partition function experimentally by compiling statistical distributions of the kink structure and the terrace widths. From such measurements one can test whether the step-wandering is consistent with thermal equilibrium and deduce the energetics governing surface morphology. A particularly beautiful example, in which a static kink structure was measured with atomic resolution, was performed by Swartzentruber et al. (1990) for kinks on stepped Si(100). They showed that the kink structure obeys Boltzmann statistics, and deduced the kink energy. Such studies can be performed if the steps can be thermally equilibrated at temperatures where a large number kinks are present and then quenched to a temperature where there is no real time kink motion to confuse the STM imaging process. Under these conditions, by simply counting the number of times each different type of kink structure is observed, one can obtain the relative energies via the Boltzmann distribution:
N(n) 2N(0)
- exp (-e(n)/kT)
(2.83)
where N(n) is the number of observations of a kink of depth n, N(0) is the number of step sites where there is no kink, and E(n) is the energy cost of a kink of depth n. The factor of two arises when the edge of the step has a mirror symmetry plane perpendicular to the step so that in- and out- kinks are equivalent. For the simple TSK model, one expects E(n) = InlE. For the case of Si(100), the best fit to the experimental data included a corner energy, E(n) = InlE + ~c (Swartzentruber, 1993;
Thermodynamics and statistical mechanics of surfaces
91
Swartzentruber et al., 1990). While direct measurement of kink energies is desirable for comparison with atomic scale calculations of surface energetics, all that is really needed to predict the thermodynamic behavior of steps is the step stiffness, or the diffusivity. Generally, even if steps cannot be imaged with atomic resolution, the diffusivity can be determined from direct measurements of the mean-square displacements of steps, g(y) = ([x(y) - x(0)] 2)
(2.84)
where, as in Fig. 2.4, x is the displacement perpendicular to the average step direction and y is the position along the step edge. In the limit of low temperature or small y, the mean square displacement varies linearly with the distance along the step edge according to (Bartelt et al., 1990):
g(y) = b2(T)y/a
(2.85)
where a is the unit cell length parallel to the step edge. Measurements of the diffusivity in principle involve taking instantaneous "snapshots" of the step edge configuration, and then calculating the mean-square displacement by suitable averaging along the step edge. Such measurements can be made using electron microscope techniques such as REM, as have been performed for steps on S i ( l l l ) (Alfonso et al., 1992). When using a scanning technique such as STM, however a static image can only be obtained if the surface is at a low temperature, where the thermal motion of the steps is so slow as to be unobservable. Such conditions are often found at room temperature for semiconductor surfaces. However, if the activation energy for step motion is low enough, thermal motion of the steps can occur at room temperature. In this case, the images of steps will appear "frizzled" because the step edge position will move due to fluctuations during repeated scans of the STM tip. Examples of such analyses are the STM measurements of "frizzled" step edges on Ag and Cu surfaces (Ozcomert et al., 1995; Poensgen et al., 1992), and on Au surfaces (Kuipers and Frenken, 1993). Direct measurements of the terrace width distribution contain information about both the thermal excitations of the step edge and the nature of the step-step interactions. As mentioned above, bringing steps near to one another decreases the amount of wandering, and thus the configurational entropy, leading to an effective entropic repulsion between steps, even when there is no direct energetic interaction. This entropic repulsion also manifests itself in the spatial distribution of the steps. A step which is midway between its neighbors has more configurational entropy than one that is near a neighboring step, and thus the distribution of step-separations will be peaked near the average step-separation (Joos et al., 1991; Kariotis, 1991; Wang et al., 1990). The distribution for the case where all the steps are wandering simultaneously can be solved exactly for moderate temperature, and is shown as the solid curve in Fig. 2.5 (Joos et al., 1991). This scaled curve is universal: in other words is will be the same for any system regardless of the average step separation and the kink energy. Deviations from this universal form thus can be used to deduce
E.D. Williams and N. C. Barter
92
. 4
,L
,
.
.
i
.
.
.
.
f
,
9
,
9
i
.
.
.
.
i
.
.
.
.
1.2 J
1.0 .
! is
~
~" 0.8
v~ " 0.6-
~,,,. ,,"
"/7
'
X'
0.4 0.2 0.0
'~,,,'"' '.J-.~-',
0.0
,
,
0.5
. . . . .
1.0
e/(e)
1.5
2.0
-
2.5
Fig. 2.13. Distribution of step-step separations calculated using the free-fermion approximation (figure from B. Joos et al. (1991)). P(/) is the probability of observing two steps separated by a distance l. The behavior of freely wandering steps is shown by the solid curve. The distribution for steps with a repulsive energetic interaction in addition to the entropic behavior is shown by the dashed curve. The distribution for steps with an attractive energetic interaction in addition to the entropic behavior is shown by the dash-dot curve. The distributions change only slightly with temperature up to temperatures near the roughening transition of the terraces.
the presence of true energetic interactions between the steps, as illustrated by the dashed curves in Fig. 2.13 (Joos et al., 1991). If there are attractive interactions (of insufficient strength to overcome the entropic repulsion and cause step coalescence), the distribution becomes broader and peaked at a value smaller than the average step separation. If there are repulsive interactions, the distribution becomes narrower, and the peak moves to slightly larger value than the average step separation. Quantitative measurement of the shape of the distribution function can thus provide information about the nature of the interaction. Unfortunately, except for a few special cases, analytical expressions relating the shape of the distribution to the strength of the interaction do not exist. However, for repulsive interactions, a gaussian distribution is an excellent approximation (Bartelt et al., 1990) to the analytical form. The result is generally true for any repulsive interaction (Alerhand et al., 1988; de Miguel et al., 1992; Swartzentruber, 1993), but assumes an especially simple form if the interaction between steps is of the type (2.86)
U(x) = Ax-"
where x is the distance between steps measured perpendicular to the average step edge direction. In this case, the distribution of step separations is approximately a Gaussian of width w -
8 n ( n + 1)Aa
I-7-
(2.87)
Thermodynamics and statistical mechanics of surfaces
93
where l is the average step-step separation, b2(T) is the diffusivity which is measured from observations of the step wandering, and a is the minimum kink-kink separation (Bartelt et al., 1990). A measurement of the width of the distribution as a function of the average step-step separation thus in principle allows a determination of both the form (value of n) and magnitude (value of A) of the step interactions. Terrace width distributions have been measured for several systems using STM and REM. Distributions characteristic of repulsive step-step interactions (Alfonso et al., 1992; Rousset et al., 1992; Swartzentruber, 1993; Wang et al., 1990), attractive step-step interactions (Frohn et al., 1991), and non-interacting steps (Yang et al., 1991) have been observed. The most extensive data set has been obtained by Alfonso et al. for steps on Si(111) using REM (Alfonso et al., 1992), which allows a large range of step separations to be studied by virtue of its large (compared to STM) field of view. These results suggested that the repulsion between steps falls off monotonically with the square of the distance (n = 2 in Eqs. (2.4) and (2.5)) (Alfonso et al., 1992; Balibar et al., 1993), with a magnitude of A -- 0.2 eV-A. In contrast, there is also evidence for oscillatory interactions between steps, that is step separations whose form varies from repulsive to attractive as a function of the step separation (Pai et al., 1994). The physical origin of step-step interactions, and in particular the origin of oscillatory interactions is discussed in Chapter 11 of this volume (Einstein, 1996). While it is difficult to prove the form of the step-step interaction using Eq. (2.87), because the dependence on the exponent n enters as n/4, physical predictions for step-step interactions through stress (Blakely and Schwoebel, 1971 ; Marchenko and Parshin, 1980) or dipole form (Redfield and Zangwill, 1992" Voronkov, 1968) all suggest a repulsive inverse square relationship. Stress mediated interactions are expected to be comparable or larger in magnitude than dipole interactions, and this provides a useful method for estimating the magnitudes of step-step interactions. The physical basis for stress-mediated interactions is illustrated in Fig. 2.14. Due to the different number of neighboring atoms for surface and bulk atoms, the lattice constant which minimizes the energy will generally be different for the surface than the bulk. (This can be shown intuitively by considering a Lennard-Jones interaction (Shuttleworth, 1950; Wolf, 1990).) The surface atoms can accommodate this difference by relaxation in the direction perpendicular to the surface, but are forced geometrically to maintain the lateral periodicity of the bulk. As a result, the surface is under a stress, which is the effective force holding the atoms in registry with the bulk. When there is a step on the surface, the atoms near the step can relax towards the preferred lattice constant, creating a strain (or displacement) field as shown in Fig. 2.14. This relaxation lowers the energy of the surface. However, when a second step (in the same direction) is created on the surface, the atomic displacements which lower the energy, oppose the displacements of the first step, and the net gain of relaxation energy is reduced. Thus there is an effective step-step repulsion. While this is an atomic picture, the form and magnitude of the stress-mediated step-interactions can be calculated using bulk elasticity theory, and are accurate down to atomic length scales (Poon et al., 1990; Wolf and Jaszczak, 1992). The relationship between the step interaction strength and the elastic parameters is:
94
E.D. Williams and N.C. Bartelt
+ p /...~
.
9
+
e ...,
I i +.,.+,,.
i
9
+ ,
+
J
.,o.~.,., ,,,,i.,.,
9 .
4
:+
. , 0 , ~ 0 0 o o e e o e ~ 6 . o o , * * . , ~ o o . .
'''t
-~
'? +
+
_+
_
' I
I
t
'
t .....
! ....
I
Fig. 2.14. The atomic displacements near two steps on a surface, calculated in a simple bali and spring model of atomic interactions are shown (a) as real-space displacements magnified • for clarity, and (b) contours of constant log-displacement, showing the extent of the strain field into the bulk.
2(1 - 0 2) z2 h2 A= rtE
(2.88)
where c~ is Poisson's ratio, E is Young's modulus, x is the surface stress and h is the step height (Marchenko and Parshin, 1980). The values of Poisson's ratio and the Young's modulus are bulk parameters which can obtained from various tabulations (Frederiske, 1972). The surface stress depends on the details of the surface structure, and must be measured or calculated directly. Increasingly there are good calculations (Meade and Vanderbilt, 1989; Needs et al., 1991) of the surface stress, and measurements of changes in the surface stress during chemical adsorption or deposition (Martinez et al., 1990; Sander and Ibach, 1992; Sander et al., 1992; Schnell-Sorokin and Tromp, 1990). There is also the possibility of determining stress from direct observations of the strain field (Pohland et al., 1993; Sato et al., 1992). Some of the available values are listed in Table 2.5, along with the corresponding values of the step-interaction coefficient A. There is good agreement between values of the step interaction coefficient determined from measured step distributions (Alfonso et al., 1992; Wang and Lagally, 1979; Wang et al., 1990; Williams and Bartelt, 1992; Williaml et al., 1992) and values determined from the surface stress using Eq. (2.88). Thus preliminary estimates of the strength of step-step interactions can be obtained from the tabulations of surface stresses
95
Thermodynamics and statistical mechanics of surfaces
Table 2.5 Calculated values of the absolute surface stress are listed in the upper section, along with the corresponding step-interaction coefficient (as in Eq. (2.88)). Measured changes in surface stress due to adsorption are listed in the lower section of the table. Surface
Stress (eV/]k)
Reference
Step interaction coefficient A, eV-A (calculated Eq. 2.88)
Si(l 11)(7x7)
0.186
0.18
Si( 111)( 1x 1)-As Si(ll 1)(2x2) AI(I 11) lr( 111 ) Pt( 111 ) Pb( 111 ) Au(l 11)
0.178 0.130 0.078 0.331 0.349 0.051 0.173
Martinez et al. (1990); Meade and Vanderbilt (1989) Meade and Vanderbilt (1989) Meade and Vanderbilt (1989) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991)
Change in stress
Reference
--O.41 -0.47 -0.53 +0.02 -0.45
Sander et al. (1992) Sander et al. (1992) Sander et al. (1992) Sander and Ibach (1992) Sander and Ibach (1992)
Surface + Adsorbate Ni( 100)-c(2x2)-S Ni( 100)c(2x2)-O Ni( 100)(2x2)-C Si(100)-O Si( 111 )-O
0.16 0.09 0.04 0.08 0.26 0.05 0.13
which are b e c o m i n g increasingly available ( C a m m a r a t a , 1992; G u m b s c h and Daw, 199 l; M e a d e and Vanderbilt, 1989; N e e d s et al., 1991 ; Vanderbilt, 1987; Wolf, 1990; W o l f and Jaszczak, 1992). T h e f o r m for the step interaction s h o w n in Eq. (2.88) is the interaction e n e r g y for two isolated steps. W h e n there is a periodic array of steps o f finite spacing, the total interaction on any one step due to all o f its n e i g h b o r s is ~2/6 larger than this ( K o d i y a l a m et al., 1995). W h e n the steps are w a n d e r i n g thermally, the effective step interaction_ to_be used in Eqs. (2.82) and (2.87) will be a value A i n t e r m e d i a t e b e t w e e n Aand rtZA/6.
2.5. Summary Surfaces are far m o r e interesting than their simplest description as infinitely periodic t w o - d i m e n s i o n a l structures w o u l d suggest. The realization of this fact, and its i m p o r t a n c e in all real surface processes, dates back to the earliest days of surface science. T h e r m o d y n a m i c s provides the f o r m a l i s m to describe the c o m p l e x i t y o f surfaces within a f r a m e w o r k of m e a s u r a b l e m a c r o s c o p i c p a r a m e t e r s . Statistical m e c h a n i c s p r o v i d e s the f r a m e w o r k to interpret those p a r a m e t e r s in terms o f under-
96
E.D. Williams and N.C. Bartelt
lying atomic properties. As a result of many years of experimental and theoretical effort, it is now possible to measure and understand the complexity of surface morphology within a simple and quantitative formalism based on the physical properties of steps on the surface. An increasing physical understanding of the properties of surfaces now make it possible to estimate the relatively small numbers of important parameters governing step behavior. Using such estimates, it is possible to approach any given problem with a reasonable qualitative understanding of what role morphology and changes in morphology can be expected to play. Conversely, direct observations of the thermodynamic behavior of surfaces can be used to deduce the physical quantities governing step behavior, and use them to predict behavior under varying conditions.
Acknowledgements The authors thank the DOD, ONR, NSF-MRSEC and NSF-FAW for support during the preparation of this manuscript, and also very gratefully acknowledge useful discussions with Profs. J.D. Weeks, T.L. Einstein, Dr. Hyeong Jeong and Mr. S. Khare on the topics of the manuscript. We also gratefully acknowledge Prof. W.N. Unertl's help and patience as editor.
References Alerhand, O.L., D. Vanderbilt, R.D. Meade and J.D. Joannopoulos, 1988, Phys. Rev. Lett. 61, 1973. Alfonso, C., J.M. Bermond, J.C. Heyraud and J.J. Metois, 1992, Surf. Sci. 262, 371. Avron, J.E., H. van Beijeren, L.S. Schulman and R.K.P. Zia, 1982, J. Phys. A 15, L81. Balibar, S., C. Guthmann and E. Rolley, 1993, Surf. Sci. 283, 290. Bartelt, N.C., T.L. Einstein and C. Rottman, 1991, Phys. Rev. Lett. 66, 961. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1990, Surf. Sci. 240, L591. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1991, Surf. Sci. 244, 149. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1992, Surf. Sci. 276, 308. Bauer, E., 1985, Ultramicroscopy 17, 51. Bilalbegovic, G., F. Ercolessi and E. Tosatti, 1992, Europhys. Lett. 18, 163. Bilalbegovic, G., F. Ercolessi and E. Tossati, 1993, Surf. Sci. 280, 335. Binnig, B. and H. Rohrer, 1987, Rev. Mod. Phys. 59, 615. Blakely, J.M., 1973, Introduction to the Properties of Crystal Surfaces. Pergamon Press, New York. Blakely, J.M. and H. Mykura, 1962, Acta Metall. 10, 565. Blakely, J.M. and R.L. Schwoebel, 1971, Surf. Sci. 26, 321. Blakely, J.M. and J.C. Shelton, 1975, Surface Physics of Materials. Academic Press, New York. 189 pp. Bonzel, H.P., 1995, Surf. Sci. 328, L571. Bonzel, H.P., E. Preuss and B. Steffen, 1984, Appl. Phys. A35, 1. Breuer, U. and H.P. Bonzel, 1992, Surf. Sci. 273, 219. Cahn, J.W., 1977, lnterfacial Segregation. American Society of Metals, Metals Park, OH. Cahn, J.W., 1982, J. de Phys. C6(suppl.), 43, 199. Callen, H.B., 1985, Thermodyamics and an Introduction to Thermostatistics. Wiley, New York. Cammarata, R.C., 1992, Surf. Sci. 279, 341. Chernov, A.A., 1961, Sov. Phys. Usp. 4, 1116.
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97
Clark, A., 1970, The Theory of Adsorption and Catalysis. Academic Press, New York. Dash, J.G., 1975, Films on Solid Surfaces. Academic Press New York. De Miguel, J.J., C.E. Aumann, S.G. Jaloviar, R. Kariotis and M.G. Lagally, 1992, Phys. Rev. 46, 10257. Dijkkamp, D., A.J. Hoeven, E.J. van Loenen, J.M. Lenssinck and J. Dieleman, 1990, Appl. Phys. Lett. 56, 39. Drechsler, M., 1985, Surf. Sci. 162, 755. Dreschler, M., 1992, Surf. Sci. 266, 1. Einstein, T.L., 1996, in: Handbook of Surface Science, Vol. 1: Physical Structure, W.N. Unertl (ed.). Elsevier, Amsterdam, Chap. 11. Eisner, D.R. and T.L. Einstein, 1993, Surf. Sci. 286, L559. Fisher, M.E., 1984, J. Stat. Phys. 34, 667. Fisher, M.E. and D.S. Fisher, 1982, Phys. Rev. B25, 3192. Flytzani-Stephanopoulos, M. and L.D. Schmidt, 1979, Prog. Surf. Sci. 9, 83. Frederiske, H.P.R., 1972, American Institute of Physics Handbook. McGraw-Hill, New York. Frohn, J., M. Giesen, M. Poensgen, J.F. Wolf and H. Ibach, 1991, Phys. Rev. Lett. 67, 3543. Gibbs, J.W., 1961, The Scientific Papers of J. Willard Gibbs. Dover, New York. Griffiths, R.B., 1980, Phase Transitions in Surface Films. Plenum Press, New York. Gruber, E.E. and W.W. Mullins, 1967, J. Phys. Chem. Solids 28, 875. Gumbsch, P. and M.S. Daw, 1991, Phys. Rev. B44, 3934. Herring, C., 1951a, Phys. Rev. 82, 87. Herring, C., 1951 b, The Physics of Powder Metallurgy. McGraw-Hill, New York. Herring, C., 1953, Structure and Properties of Crystal Surfaces. University of Chicago Press, Chicago, IL. Heyraud, J.C., J.J. Metois and J.M. Bermond, 1989, J. Cryst. Growth 98, 355. Hill, T.L., 1960, An Introduction to Statistical Thermodynamics. Addison-Wesley, Reading, MA. Jayaprakash, C., C. Rottman and W.F. Saam, 1984, Phys. Rev. B30, 6549. Jayaprakash, C., W.F. Saam and S. Teitel, 1983, Phys. Rev. Lett. 50, 2017. Joos, B., T.L. Einstein and N.C. Bartelt, 1991, Phys. Rev. B43, 8153. Josell, D. and F. Spaepen, 1993, Acta Metallurgica et Materialia 41, 3007. Kariotis, R., 1991, Surf. Sci. 248, 306. Keeffe, M.E., C.C. Umbach and J.M. Blakely, 1994, J. Phys. Chem. Solids 55, 965. Kern, R., 1987, Morphology of Crystals. Terra Scientific Publishing Co., Tokyo, p. 79. Khare, S.V. and T.L. Einstein, 1994, Surf. Sci. 314, L857. Kodiyalam, S., K.E. Khor, N.C. Bartelt, E.D. Williams and S. Das Sarma, 1995, Phys. Rev. B 51, 5200. Kuipers, L. and F.W.M. Frenken, 1993, private communication Kumikov, V.K. and K.B. Khokonov, 1983, J. Appl. Phys. 54, 1346. Liu, C.L., J.M. Cohen, J.B. Adams and A.F. Voter, 1991a, Surf. Sci. 253, 334. Liu, C.L., J.M. Cohen, J.B. Adams and A.F. Voter, 1991b, Surf. Sci. 253, 334. Madey, T.E., J. Guan, C.-Z. Dong and S.M. Shivaprasad, 1993, Surf. Sci. 2771278, 826. Madey, T.E., K.-J. Song and C.-Z. Dong, 1991, Surf. Sci. 247, 175. Marchenko, V.I. and A.Y. Parshin, 1980, Soy. Phys. JETP 52, 129. Martinez, R.E., W.M. Augustyniak and J.A. Golovchenko, 1990, Phys. Rev. Lett. 64, 1035. Meade, R.D. and D. Vanderbilt, 1989, Phys. Rev. Lett. 63, 1404. Methfessel, M., D. Hennig and M. Scheffier, 1992, Phys. Rev. B 48, 16. Metois, J.E. and J.C. Heyraud, 1989, Ultramicroscopy 31, 73. Moore, A.J.W., 1962, Metal Surfaces. American Society for Metals, Metals Park, OH, p. 155. Mullins, W.W., 1961, Phil. Mag. 6, 1313. Needs, R.J., M.J. Godfrey and M. Mansfield, 1991, Surf. Sci. 242, 215. Nelson, R.C., T.L. Einstein, S.V. Khare and P.J. Rous, 1993, Surf. Sci. 295, 462. Nicholas, J.F., 1965, An Atlas of Models of Crystal Surfaces. Gordon and Breach, New York.
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CHAPTER 3
Surface Reconstruction" Metal Surfaces and Metal on Semiconductor Surfaces
C.T. CHAN Physics Department Hong Kong University of Science and Technology Clear Water Bay, Hong Kong
K.M. HO Ames Laboratory-USDOE and Department of Physics and Astronomy Iowa State University Ames, Iowa 50011, USA
K.P. BOHNEN Forschungszentrum Karlsruhe Institut fiir Nukleare Festk6rperphysik Karlsruhe, Germany
Handbook (?fSu~ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
lOl
Contents
3.1.
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2.
M e t h o d s of c a l c u l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.
3.4.
3.2.1.
Classical models
3.2.2.
Tight-binding models
3.2.3.
First p r i n c i p l e s m o d e l s
3.2.4.
Forces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2.5.
New advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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H o w to m o d e l a surface
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3.3.1.
Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.2.
G r e e n ' s function
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3.3.3.
Slab
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W h a t c a l c u l a t i o n s can tell us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.1.
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.2.
Surface energy
3.4.3.
S u r f a c e band structure
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114
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3.4.4.
W o r k function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.5.
C h a r g e density
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.
C l e a n metal s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.6.
S e m i c o n d u c t o r surfaces
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3.7.
Metal o v e r l a y e r s on s e m i c o n d u c t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
3.7.1.
Si( 111)(ff3-3•
. . . . . . . . . . . . . . . . . . . . . . . . .
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3.7.2.
O t h e r p h a s e s of noble metals on Si or G e . . . . . . . . . . . . . . . . . . . . . . . .
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3.7.3.
B,AI, Ga, In on Si(100) and S i ( l l l )
. . . . . . . . . . . . . . . . . . . . . . . . . .
128
3.7.4.
As, Sb and S i ( l l l ) - B i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.7.5.
Alkali metals on s e m i c o n d u c t o r s
Ag and Si(l 1 l ) - A u
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.8.
Epilog
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3.1. Introduction The positions of the atoms at the surface is one of the most basic questions one can ask in surface science. The atomic geometry is an important factor affecting the various behaviors of the surface such as chemical reactivity, surface energy, surface work functions, surface vibrations, and surface electronic states. With the advent of ultra-high vacuum technology and the development of experimental methods sensitive to the surface atomic structure, it rapidly became apparent that the arrangement of atoms at the surface of a crystal can be much more complicated than previously realized: the surface atomic arrangement can exhibit complicated deviations from the situation one would expect if one simply removed half of the atoms of the crystal to form a surface. Such rearrangements, called surface reconstructions, are driven by the tendency of the surface to reform bonds broken by the formation of the surface. However, such atomic rearrangements often occur at the expense of disruption of the bulk bonding and the creation of surface stresses and the resultant surface geometry is given by a compromise between the two competing interactions. As such, surface reconstructions offer an interesting view into the microscopic interactions present at the surface, and our understanding of surface structures is intimately connected with our understanding of the microscopic behavior of the surface. Over the past two decades, impressive progress has been made in the experimental field in both the amount of information acquired as well as in the number of techniques developed from which we can obtain information about the detailed geometry of the surface atoms. This began with the refinement of electron diffraction techniques and culminated with the more recent development of surface X-ray diffraction. Especially remarkable is the development of the scanning tunneling microscope (STM) which has achieved atomic-scale imaging of the surface in real space. However, even with all the powerful techniques available to date, it is still not possible to determine the atomic geometry by any one method alone. Reliable information can only be obtained by piecing together information from different sources. These include various experiments, and in many cases, input from theoretical calculations is also essential in differentiating between competing structural models. The richness and variety of surface reconstruction behavior is an indication of the delicate balance between the various energies involved at the surface. Although various simple theories have been developed to model the interatomic interactions at the surface, the accuracy needed for a proper description and the subtlety of the interactions at work make it necessary to use quite sophisticated calculations to get reliable answers and to provide accurate data for use in more 103
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approximate theories. The progress in this direction in the last two decades has also been very substantial, fueled by the rapid advance in the computing power of modern computers and developments in computational techniques and numerical algorithms. A detailed discussion of these advances is given in the next section. In addition to yielding reliable data, these microscopic calculations provide detailed information on the surface electronic structure, electronic wavefunctions and charge density distribution at the surface allowing us to examine and understand the basic physical forces in operation on the various surfaces. Such information is very useful in predicting and understanding the behavior of more general systems including chemical trends in surface geometries and the influence of various external fields such as stress or electric charging effects on the surface geometry. The basic concepts of the theories that have resulted in this area are the subject of this chapter. The discussion in subsequent sections are organized into different areas, involving different types of surfaces, specifically: metal surfaces, semiconductor surfaces and metal-semiconductor interfaces. The rich and vast field of chemisorption will not be covered in detail in the present article; instead, we will focus on clean crystalline surfaces. Our discussion of adsorbed surfaces will be limited to the case of metal-semiconductor interfaces, which is still in an early stage of development, and for systems which are directly related to the physics of the clean surface, such as alkali-metal-adsorbed surfaces.
3.2. Methods of calculation In this and the following sections, we will discuss (a) the three most popular methods for modeling inter-atomic interactions, (b) the three most popular ways to model a surface, and (c) what kind of information we can extract from a surface calculation. Some of these methods are also discussed in the following chapter in the context of their application to surfaces of insulating materials. To study surface structure and surface phenomena at atomic length scales, we have to deal with interatomic interactions at the atomic level. Atomic interactions can be handled with a hierarchy of approaches: "classical" models, tight-binding models, and first-principles methods, in increasing order of accuracy and computational difficulty. 3.2.1. Classical models
In classical models, the total energy of the surface system is written in some simple (usually analytic) functional forms that depend on the bond lengths and bond angles and have a short interaction range, so that energy and forces can be evaluated very quickly. The parameters in the energy functions are usually determined by fitting to some known experimental quantities, such as the cohesive energy, bond lengths and elastic constants. Recently, it has become increasingly popular to fit the parameters to high quality theoretical results. The simplest form of classical potentials are of the Lennard-Jones type (Kittel, 1976), in which the energies are written as simple
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functions of bond lengths. These simple models can only give generic, but not atom specific information for surface systems of interest. More sophisticated models have to include explicitly the physics of the bonding; e.g. changes in electronic states. For metals, the most popular and reasonably accurate classical method is the embedded-atom-model (EAM) (Daw et al., 1993). In this approach, the total energy of the system is written as:
Etota,= '~ F(pi)+ ~ ,(r~)' i
ij
p, = ~
f(rii )
(3.1)
j~i
where F(p;) is an analytic many-body term dependent upon the local electron density Pi, which is calculated from a tabulated function fir). rij is the distance between two atoms at sites i and j, and ~(r) is usually a simple two-body potential. We note that the form resembles an energy expression of the density functional formalism (see below) although the energy depends only on the "charge density" at the atomic sites, rather than the charge density in the whole space. The energy is a non-linear functional of the bond lengths, and is thus more sophisticated and flexible than simple pair-wise interaction potentials such as Lennard-Jones potentials. We note that the EAM potentials are not explicitly volume dependent, which makes them very suitable for surface calculations. The energetics of the system depends only on the distances between the atoms, but not on the bond angles. This class of theories are thus more applicable to metallic systems where the atoms are almost "spherical" such as transition metals and noble metals where the d-shells are essentially filled. Examples are Au, Ag, Cu, Pt, Pd, Ni. Other methods such as effective medium theories (Norskov, 1989; Stave et al., 1990), glue models (Ercolessi et al., 1986), and Finnis-Sinclair models (Finnis and Sinclair, 1984) are similar in spirit to the EAM method and have the same range of applicability. For systems with directional bonding, such as semiconductor surfaces, realistic classical models must allow for the explicit dependence of the total energy on the bond angles. The simplest approach is to include potential functions that have "three-body" terms of the form V(1,2,3) = V(rl2,rl3,cos(O)), where rl2, rl3 are the distance between atoms 1,2 and 1,3, and 0 is the angle formed by the bonds r~2, rl3 (see, e.g., Keating, 1966; Stillinger and Weber, 1985; Biswas and Hamann, 1985; and Tersoff, 1986). There are also methods such as the equivalent-crystal-theory (Ricter et al., 1984) that are intended to work for metals as well as semiconductors. The advantage of these classical models is that modern supercomputers can handle systems with up to millions of atoms on the dynamical level and can be used readily with simulation techniques such as molecular dynamics to model fairly complex surface phenomena. The disadvantage is that there is always the risk of extending these models beyond their intended range of validity. The electronic degree of freedom is not included explicitly, and in situations where band structure effects, Fermi surface effects and charge transfer effects are important, these models should not be expected to give reliable results.
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3.2.2. Tight-binding models In the tight-binding approach, the potential energy of a system is written in the form: occupied
Etota, = Z I~i-k-Z O(Pij) (3.2) i 0 where r o is the distance between atoms i and j; and ~(r) is usually a classical potential. The first term, ~ei, is the sum of the eigenvalues for the occupied states obtained by diagonalizing, a Hamiltonian matrix H: /4~ = e S ~
(3.3)
The matrix S is called the overlap matrix. If S = I (identity matrix), the tight-binding orbitals are assumed to be orthonormal, and the tight-binding model is called an "orthogonal" tight-binding model; otherwise, the tight-binding model is called non-orthogonal. The Hamiltonian and overlap matrix elements are generated from a set of empirical parameters that are obtained by fitting to the band structure of the elements under consideration. Although more sophisticated approaches are possible, tight-binding models are usually based upon the Slater-Koster two-center approximation (Slater and Koster, 1954). A comprehensive account of the tightbinding method can be found in the book by Harrison (Harrison, 1980). In surface calculations, we may need to supplement Eq. (3.2) with a term of the form ~i l/2Ui(qi _ Q)2, where U is frequently of the order of 1 eV, and qi is the charge of the atom at site i and is determined self-consistently in the calculation, and Oi is the number of valence electrons of that atom. As is obvious from its form, this term can control charge transfer in the system and is frequently important for surface systems if we need to maintain a certain degree of charge neutrality. Since the tight-binding model contains electronic structure information, it is good for semiconductors, where covalent bonding prevails, and in bcc transition metals where directional d-bonding governs the structural and cohesive properties. It is also superior to classical models whenever Fermi surface or band structure effects are important. Tight-binding models have enjoyed considerable successes when applied to semiconductor surfaces (see e.g., Chadi, 1994), and have also been used to describe the surface reconstruction of bcc transition metals such as Mo(001) and W(001). Tight-binding parameters for many elements are published (see, e.g., Papaconstantopoulos, 1986), but the parameters frequently require modification for surface situations. New and more sophisticated tight-binding models are fit to a broad data base and can describe the energetics of covalent systems quantitatively under different physical and chemical environments (see e.g., Goodwin et al., 1989; Xu et al., 1992). There are now also methods that can extract tight-binding parameters directly from first-principles linear muffin-tin orbitals calculations (Nowak et al., 1991; Stokbro et al. 1994). Tight-binding models are usually more difficult to implement and computationally more intensive than classical models. The computational cost is dominated by the need to diagonalize Hamiltonian matrices of the order N = Natom
Su~ace reconstruction
107
)< Norbita b where Natom and Norbita ! a r e the number of atoms and the number of orbitals
per atom respectively. Since diagonalizing a matrix is an N 3 problem, we are limited to a few hundred atoms if the traditional diagonalization schemes are used. However, it has been realized that the energy and forces in a tight-binding model can be formulation as an N 1 problem with methods such as the recursion method and density matrix methods (see the discussion below). A few thousand atoms can now be handled dynamically with tight-binding models. Although first principles techniques, as will be described in the next section, have made progress in leaps and bounds in recent years, there is always an important place for empirical and semi-empirical techniques for surface physics, simply by virtue of their capacity to treat a much larger ensemble of atoms, and to provide answers for more complex systems in a more timely manner. This is especially true in situations where dynamical and finite temperature information are needed. A sophisticated LDA surface calculation can take months to complete. First principles techniques, on the other hand, can provide definitive answers for problems that require higher accuracy. Available experimental information seldom provides enough information for fitting uniquely the parameters of a microscopic model and first principles methods can fulfil the purpose of generating an otherwise unobtainable data base, with which we can fit more reliable and transferable empirical parameters.
3.2.3. First principles models In surface calculations, "first principles methods" usually mean the local-densityapproximation (LDA) to the density functional formalism (Hohenbergand Kohn, 1964; Jones and Gunnarsson, 1989; Kohn and Sham, 1965; March and Lundqvist, 1983; Phariseau and Temmerman, 1984) for almost all occasions. We seldom go beyond LDA because LDA provides results that are good enough for many purposes, and in most surface calculations, we cannot afford the computer time to go beyond that. LDA almost always gets the correct structural model, in the sense that the structural model found in experiments also has the lowest energy in LDA, and the bond lengths are usually good to a few percent. We should remark that chemists have more stringent requirements for terms such as "first principles" or "ab initio", although LDA is gaining more acceptance in that community too. This is because a new generation of exchange-correlation functionals incorporating gradient corrections can now treat small molecules more accurately (Langreth and Mehl, 1981; Perdew and Yue, 1986; Perdew, 1986). One nice feature about LDA is that it includes exchange and correlation energies in the mean field sense, but the complexity of the problem is only that of a Hartree-type calculation, and is more economical than Hartree-Fock calculations, which do not include correlation energies. It is thus computationally very attractive. As originally formulated, LDA is meant for systems that have smoothly varying charge densities, and charge density variations at surfaces cannot be considered smooth. However, over a decade of computation has shown that the results are very
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good and in quite a few cases, theoretical predictions actually preceded subsequent confirming experiments. Within the frame-work of the local density formalism, there are different implementations, such as LMTO (linear muffin-tin orbitals) (see, e.g., Fernando et al., 1986), FLAPW (full potential linear augmented plane wave) (Wimmer et al., 1981), pseudo-potential-plane-waves (Ihm et al., 1979; Pickett, 1989), and various forms of LCAO (linear combination of local orbital) approach (see, e.g., Chelikowsky and Louie, 1984). These approaches basically differ in the way the electronic wave-functions, the charge density, and the potentials are represented, and also in the way the core-electrons are treated. Most of the approaches that are applicable to bulk calculations are also applicable to surface calculations, although plane-wave based methods have had many early successes. Traditionally, pseudopotential methods are applied to semiconductor surfaces and simple metals, while methods with local orbitals like FLAPW are more frequently applied to transition metals. With rapid advances in algorithm developments, these traditional boundaries are fading away. For a surface LDA calculation, the reliability of the results should be judged by its "convergence" (completeness of the basis, number of k-points sampled), whether the atomic positions are fully relaxed, and whether any constraints are imposed on the form of the potential or charge density. In density functional theories, the total energy is expressed as a functional of the electronic charge density 9(r), and can be written as , tlxc + E.xc[9(r)] + E~w~,lj
-
(3.4)
i
where ] ~ is the sum of eigenvalues of an effective Hamiltonian (the index i goes over band and k-point indices), p(r) is the charge density, VinpUtis, .xc the input Hartree and exchange-correlation potential, with the suffix "input" emphasizing that the screening potential is the input potential that determines the Hamiltonian. The first two terms in the right-hand-side of Eq. (3.4) give the kinetic energy of the electrons and the potential energy due to the electron-ion interaction respectively. The third term, Euxo is the electron-electron interaction energy and is given by the sum of the Hartree energy and exchange-correlation energy and is a functional of 9(r). E~w,,~dis the Coulombic ion-ion interaction energy of the ionic cores. It is called the "Ewald" term because we usually compute this term with an algorithm suggested by Ewald (Ziman, 1972). The eigenvalues and charge densities in Eq. (3.4) are obtained by solving an effective Schroedinger equation, usually called the KohnSham equation,
]
+ Vi,,,(r)+ VHxc(r) ~gi(r)- ei~lli(r)"
-~m
p ( r ) - Z f/vi(r)~gT(r)
(3.5) (3.6)
i
wheref~ is the occupancy of an eigenstate, ka~ the ionic potential, and VHxc is the screening potential due to the electrons, and is given by Vnxc = V. + Vxc, where
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VH = ~ d3r p(r') . I r - r'l'
Vxc _ ~9Exc[p(r)] 5p(r)
(3.7)
If we do not make any approximation to the exchange-correlation energy Exc [p(r)], we can obtain the exact ground state energy for a many-electron system. However, the exact functional form of Exc[p(r)] is not known, and it is approximated by a local functional of the form, Exc[p(r)] - ~ d3r~xc[p(r)] p(r), where exc(p(r)) is the exchange-correlation energy density of a homogeneous electron gas with density p(r) (Ceperly and Alder, 1980). This is called the local-density approximation. We note that in the K o h n - S h a m equation, the potential VHxc depends on the charge, which in turns depends on the wavefunctions ~(r), so the solutions to Eqs. (3.5) and (3.6) have to be obtained self-consistently. Nearly all the computational effort in a LDA calculation goes into the solution of Eq. (3.5), and there are a few ways in which it can be achieved. The traditional method transforms the K o h n - S h a m equations into a Hermitian matrix eigenvalue equation and the matrix (of the same form as the tight-binding Hamiltonian problem in Eq. (3.2)) is then diagonalized to yield the eigenvalues and eigenfunctions. Very robust diagonalization routines are readily available for this purpose. However, as we have mentioned before, the complexity of the diagonalization procedure goes like N 3, where N is the order of the matrix and the storage goes like N 2. This puts severe limits on the number of atoms that can be handled. The situation here is much worse than the case with tight-binding calculations, where the number of basis functions per atom are much smaller, and hence have smaller matrices to diagonalize. First principles calculations can handle only about 50-100 atoms if we use matrix diagonalization, depending on the details of the method used. In some methods, particularly plane-wave based methods, the number of occupied eigenstates is only a small fraction of the order of the Hamiltonian matrix. In those cases, iterative diagonalization procedures such as the "Davidson" scheme (Davidson, 1975) can save a lot of computational effort if we only need a small percentage of the eigenfunctions from the Hamiltonian matrix. There are methods that bypass the diagonalization and the self-consistency procedures, and solve for the ~ ( r ) ' s by directly minimizing the LDA energy functional with respect to a set of trial wavefunctions, subject to the orthonormal constraints. A very efficient algorithm for this purpose is the preconditioned conjugate gradient scheme proposed by Teter and co-workers (Teter et al., Allan 1989). Another very important development is the Car-Parrinello method (Car and Parrinello, 1985), which treats both the electrons and ions as classical dynamical systems and recasts the electronic structure problem as a molecular dynamics problem. The equations of motions for the electronic and ionic degrees of freedoms are ~lli = - Hltli + E
AijlllJ "
Mi'gi= - ddR E[R,gtl
(3.8)
J
where the A's are Lagrangian multipliers needed to keep the wavefunctions orthonormal, ~ is a (fictitious) mass for the electronic degrees of freedom. These
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1 10
equations are integrated in the time domain by standard molecular dynamics methods. The Car-Parrinello scheme is the first scheme that allows us to perform first principles (LDA) molecular dynamics simulations. Although the details are not the same, these new algorithms (iterative methods, conjugate gradient, Car-Parrinello) rely on the fact that H ~ (a trial wavefunction ~ operated on by the Hamiltonian operator H) can usually be computed very efficiently. In many cases, the Hamiltonian matrix elements are not computed and need not be stored, H is treated as an operator. This is particularly the case for plane wave basis sets, where the operations can be performed partly in real space (potential energy) and partly in momentum space (kinetic energy). It is thus not surprising that the conjugate gradient and Car-Parrinello methods are mainly associated with pseudo-potential plane wave methods and most of the ab-initio molecular dynamics simulations on surface systems to date are performed with plane waves. The newer algorithms can handle a few hundred atoms if the system under consideration can be represented reasonably well with plane waves.
3.2.4. Forces For static structural problems (surface relaxation and reconstruction) and for dynamical calculations such as molecular dynamics simulations, the importance of calculating forces accurately and efficiently cannot be understated. For classical potentials, the force calculations are trivial to implement. When there is an electronic degree of freedom, the forces due to the electronic part of the total energy can be computed by the Hellmann-Feynman theorem (Hellmann, 1937; Feynman, 1939), which can be formally written as: OH (V, ~--R Vi)
F~-Z i
(3.9)
"-"'ta
Here, Fo is the electronic force acting on atom It, and the index i runs over the occupied bands. Equation (3.9) may look trivial, but it has important implications. It means that there is no need to move the atoms to compute all the forces, otherwise it will take M calculations to get the forces if there are M degrees of freedom. In tight-binding calculations, this can be evaluated very easily. In first principles calculations, the calculation of the forces by the Hellmann-Feynman theorem can be tricky. If the basis set used is independent of the atomic positions (such as plane waves), the formal form of Eq. (3.9) holds and computation of forces is straightforward. If the basis set contains functions that are centered on the atomic coordinates, extra terms must be included before the forces calculated represent the true gradient of the energy surfaces. These terms are related to dCp/dR, where ~ is a local orbital. They are called the Pulay forces (Pulay, 1969), and are well known in quantum chemistry. In physics, their importance and formal formulation have also been known for some time (Bendt and Zunger, 1983), although the implementation has proved to be non-trivial. The mixed-basis pseudopotential approach, which uses a mixed representation of plane waves and local orbitals, has these "Pulay" terms coded (Ho et al., 1992) and thus has been rather successful for the structural studies
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of transition metal surfaces. Recently, almost all local-orbital based methods can perform force calculations (Yu et al., 1991, Methfessel and van Schilfgaarde, 1993; Jackson and Pederson, 1990). In first principles self-consistent calculations, the total energy is a variational quantity so that it is rather forgiving in the self-consistency of the charge and the potential. This is not the case for the forces, which require highly self-consistent potentials and charges before they can be computed accurately. There are methods that can alleviate this problem (Chan et al., 1993). Of all the first-principles methods, the pseudopotential plane-wave methods have made early and important contributions to surface structural studies, especially in semiconductor surfaces. The main reason for this success is the ability to compute forces easily and accurately. In recent years, pseudopotential plane-wave techniques have been even more popular because of the introduction of conjugate gradient and Car-Parrinello methods that elevated LDA from the statics to the dynamical level. These methods are essentially plane-wave based. There are however some difficulties with pseudopotential plane wave methods. The plane wave basis and momentum space formulations are efficient for those elements which have smooth pseudopotentials and valence (pseudo-) wavefunctions that can be represented easily by a plane-wave expansion. Examples are alkali metals (Li, Na, K), simple metals (AI, Mg, Be), and semiconductors (Si, Ge). Systems like first row elements and transition metals (especially 3d metals) have fairly sharp local orbitals, and require many plane waves to converge, and hence demand substantially more computing time for reasonably converged results. Many authors have introduced "smooth" pseudopotentials to alleviate this problem (see, e.g., Rappe et al., 1990; Troullier and Martins, 1991; Vanderbilt, 1990). By "smooth", we mean that the Fourier transform of the pseudopotential decays rapidly in momentum space, and are thus easier to converge with plane waves. We can also supplement the basis with local orbitals, such as the mixed-basis technique (Louie et al., 1979). There have also been recent attempts to improve the efficiency of the plane wave representations by employing wavelets, multigrids and adaptive grids, and some of these techniques may soon find applications in surface calculations. 3.2.5. New advances
Recent advances in "order(N)" methods may have significant impact on theoretical surface studies in the near future. As mentioned in previous paragraphs, modern electronic structure algorithms (such as conjugate gradient and Car-Parrinello) circumvent the diagonalization of the Hamiltonian matrix by taking advantage of the fact that we just need some (the occupied states) but not all of the eigenvectors; but these algorithms are still N 3 methods because the number of occupied states has to be proportional to Natomand making the eigenvectors mutually orthogonal is 3 already an N~,tom process. Recently, it has been recognized that we do not need to know any of the individual eigenvalues or eigenfunctions to obtain energies and forces. We only need to compute objects like the Green's function (Resolvent), the density matrix, or a sub-space spanned by localized orbitals that are unitarily equivalent to the subspace spanned by the eigenvectors. These can be computed
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with operations proportional to N ~, and are thus called "order(N)" methods. These new algorithms are still under rapid development (see, e.g., Li et al., 1993; Daw, 1993; Ordejon et al., 1993; Baroni and Giannozzi, 1992), and have already showed good promise in tight-binding calculations.
3.3. H o w to m o d e l a surface
In numerical computations, we need to model a solid/vacuum interface on the computer. There are a few ways that it can be achieved, and we will discuss them in the following.
3.3.1. Cluster Cluster methods use a "large" cluster (large here means 50 or so atoms for first principles calculations) to construct a fragment of a solid which has a surface with a particular orientation (see, e.g., Ye et al., 1989). This method is most popular among chemists. The advantage of this approach is that quantum chemistry codes are readily usable. Quantum chemistry codes are generally more "standardized", better documented and also readily available from many sources; and they usually have the ability to go beyond LDA, so that correlation effects can be accounted for in a more controlled manner (if one can afford to do so). Cluster methods are good for modeling isolated atoms on surfaces or low coverage situations in chemisorption problems. In this approach, many atoms that are not intended to represent surface atoms are also exposed and are not bonded. These atoms are usually "passivated" by bonding H atoms to them. It is difficult, if not impossible, to obtain information pertaining to the extended nature of the surface, such as the surface states; and modeling an extended system with a cluster of atoms may run the risk of obtaining results that are qualitatively different from the truly extended system. The convergence of the results relative to cluster size is usually slow.
3.3.2. Green's function Surface properties can in principle be determined if the Green's function of the entire surface system can be obtained by surface Green function matching techniques. Such techniques have been used for obtaining surface electronic structures, surface states and other elementary excitations, especially for simplified model Hamiltonians (see, e.g., Garcia-Moliner and Flores, 1979). Although Green's function methods offer beautiful formal solutions and have a lot of nice analytic properties, it is always tedious to construct and manipulate the Green's function for real systems with real potentials. It is especially difficult if we need to solve the problem self-consistently and have to compute forces to take care of relaxation and reconstructions of the surface (or interface) atoms. However, recent efforts by a few authors, especially the work of Feibelman (Feibelman, 1986), have demonstrated that these difficulties can be overcome. Green's function methods will be very important for studying adsorbate effects in low coverage situations. It is better in
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principle than cluster approaches because, once the Green's function of the substrate is known, the effect of the embedding environment can be taken into account essentially exactly, while for the cluster approach mentioned in the last paragraph, the cluster used to represent the surface is not embedded in the correct medium and does not have the right boundary conditions. Green's function methods are difficult to implement. 3.3.3. Slab
The most popular method to date is to use a thin slab, which by definition is bounded by 2 surfaces, to model a surface. As a rule, results are more reliable with thicker slabs. However, meaningful results 6an sometimes be obtained with slabs as thin as 5 or 7 atomic layers. The slabs are either repeated (separated by vacuum) to regain 3D periodicity or not repeated, depending on the method chosen. In Fourier space based techniques, such as the pseudopotential approach, a repeated slab geometry is required, while for local orbital based methods, the repeated slab configuration is optional. Though not necessarily the most elegant formulation, this is by far the most popular method because it is straightforward to implement. In many cases, codes developed for bulk calculations can be adapted with little or no modification for surface calculations. We just need more memory and more computer time. The method is limited to systems with small surface unit cells, including high-coverage ordered surface overlayer structures.
3.4. W h a t c a l c u l a t i o n s c a n tell us
The results of a first-principles total energy calculation for a clean or adsorbate covered surface can tell a lot about the structural and electronic properties of a surface. Once the surface structure has been determined, the same calculation usually provides more information than just the positions of the atoms. We will attempt to give a brief account of these various types of information and how they can be obtained from an electronic structure calculation. This information should be viewed as an integral part of the theoretical surface structure determination, because in many cases, it is these results rather than the atomic coordinates that can be compared directly with experiments. More extensive discussions of these nonstructural aspects of surfaces are given in Volume 2 of this handbook. 3.4.1. Structure
Surface structural parameters such as bond lengths and interplanar distances can be obtained directly from the atomic coordinates after the atomic degrees of freedom are fully relaxed. These quantities can be directly compared with experimental data from techniques such as LEED, ion-scattering and surface X-ray scattering. Theoretical calculations determine surface atomic arrangements by searching for a "zero-force" configuration with the "lowest" energy. In reality, it is very difficult to tell whether a "zero-force" configuration is just a metastable state (local minimum)
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or a ground state (global minimum); and in some surface systems, metastable states can be observed, and are as important as the global minimum configuration. This is particularly true in semiconductor surfaces with strong covalent bonds where energy barriers are high, and the n-bonded chain Si(111)(2• system is a good example of a metastable surface configuration. We need subsidiary evidence, such as the surface band structure, work function change, etc., to compare with experimental information before a strong case can be established.
3.4.2. Surface energy The surface energy can be determined in a total energy calculation. From a slab calculation, it can be determined by either one of the two methods described below, with the former method somewhat more popular. In the first method, we need to compute the total energy of the slab, subtract the total energy of the corresponding number of atoms in an ideal bulk environment, and divide the difference by the total number of surface atoms on both sides of the slab. The only difficulty with this method is that we have to obtain the same convergence in the k-point sampling for both the surface and the bulk calculations. The second way is to calculate the total energy of a slab as a function of the slab thickness, and then fit the results in a least square manner to the equation E(n) = nE(b) + 2E(s); where E(n) is the energy of a slab n layers think, E(b) is the energy in the ideal bulk environment for the atoms in one plane, and E(s) is the surface energy. In this manner, no separate bulk calculations are required. However, since E(b) is much larger than E(s) in magnitude, even small numerical errors can result in larger errors in E(s). It also requires several slab calculations (at least 2) (Ricter et al., 1984). When we make comparison with experimental results, we should note that the numbers quoted in experiments are usually surface tension measured at elevated temperatures, and usually measured for a surface with a mixed orientation; whereas theoretical calculation gives the surface energy of a surface of a specified orientation at T-- 0. In principle, the surface energy can also be determined if we can define and calculate the total energy density E(r) at any point r; which will allow us to compare the energy of the surface region with the interior. This is possible (Chetty and Martin, 1992) but this method has seldom been used.
3.4.3. Surface band structure The surface band structure can be of great use in characterizing the surface if photoemission results are available. After a tight-binding or first-principles surface calculation is performed, the eigenvalues and the eigenstates for each k-point of the surface Brillouin Zone (BZ) can give the dispersion of the surface states and surface resonances. Surface states are those electronic states that are created by the formation of the surface, and their amplitudes are localized near the surface region, decaying exponentially into both the vacuum and the bulk. We need to distinguish the surface states from the bulk states in the calculation. For "thin" slab calculations, that distinction can sometimes be a fairly tedious procedure. There are usually
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three criteria from which we can distinguish the surface states from bulk states in a calculation. First, bona-fide surface states exist in gaps or symmetry gaps (on high-symmetry lines) of the bulk band structure projected onto the 2D surface BZ (called the projected band structure). Second, surface states in a slab calculation normally come in nearly degenerate pairs. This is because there are two surfaces in a slab and the surface state can be on either side. If the slab is infinitely thick, the two surface states should be exactly degenerate, but the degeneracy is lifted by their interaction for a slab of finite thickness. The third and the operationally most useful criteria is to examine the charge profile of the state along the "z-direction" (the direction parallel to the surface normal). A surface state, by definition, should be localized near the surface layers. There is no universally accepted criteria concerning the degree of localization of a state on the surface layer before we label it a surface state, but when combined with the first two criteria, surface states can usually be identified by examining their charge profile in the z-direction. For slab calculations, the thicker the slab, the easier it is to identify surface states and surface resonances. Once we obtain surface states and surface resonances, results can be compared with photoemission or inverse photoemission experiments. In some cases, such comparisons can either rule out or provide strong support for a structural model. This is especially true for chemisorption systems, since the surface states of the substrate are strongly perturbed. In the older days, where only electronic but not total energy information was available from calculations, the comparison between theoretical and experimental surface bands had been used to determine the site of adsorption of foreign atoms on metal surfaces (Louie, 1979). However, when we compare the eigenvalues from a LDA calculation with the results of a photoemission experiment, we should note that the eigenvalues from a density functional theory (a ground state theory) should not be used directly to interpret electron excitation energies. For metals, the comparisons of LDA with spectroscopic data are in general reasonably good. In many cases, the eigenvalues from a LDA calculation can be used directly to map out surface states, with good agreement with photoemission results. For systems with a band gap, the gaps are usually underestimated by the LDA eigenvalue spectrum but there are well-formulated remedies to this problem, such as the GW approximation to the self-energy operator (Hybertsen and Louie, 1985; Louie, 1994) which was successful in computing quasiparticle energies for semiconductors and insulators. 3.4.4. Work function
The work function is a very useful piece of information in chemisorption systems where large changes can be induced, and the work function is strongly dependent on the position and coverage of the adsorbate. The work function can be obtained by comparing the Fermi level with the vacuum level in a LDA calculation. If a repeated slab geometry is used, care should be taken to ensure a large enough vacuum region so that the vacuum level is well represented. In many cases, an accuracy of about 0.1 or 0.2 eV can be obtained from LDA calculations.
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3.4.5. Charge density The charge density from a fully self-consistent LDA calculation can give us insight into the bonding of surface atoms as well as offering information for a quantitative interpretation of certain experimental results. For example, the interaction between He atoms and a surface is approximately proportional to the charge density, and so, the calculated charge density outside the surface can be of use in simulating He-scattering experiments. We can also compute the charge density contributions from electronic states that fall within a specified energy window, using E+5
p(E,r) = f dEIt~e(r)l 2
(3.10)
E-8
where ~e(r) is an electronic state with energy E, and 5 is a small number (in practice, it is taken to be a fraction of an eV). In the lowest order approximation, the charge density contour of p(E,r) at a small distance (a few A) above the top atomic layer can be used to approximate STM images with a tip-to-sample bias potential equal to E-EF (Tersoff and Hamann, 1985). This can be of great use in interpreting STM images (see e.g. Ding et al., 1991), which represent a convolution of structural and electronic information, as discussed in Chapter 9. Direct computation of tunneling signals are also possible, although it is rarely done.
3.5. Clean metal surfaces
We begin our discussion with the surfaces of simple metals. Simple metals here refer to those metals that have s,p electrons as their valence electrons and are reasonably well described by free-electron models. Typical examples are alkali metals and AI. Most of the low index simple metal surfaces do not reconstruct. They reduce the surface energy primarily by relaxation, i.e. by changing the inter-layer distances in the first few layers, and keeping the same surface periodicity as an ideal truncated bulk crystal. Transition metal and noble metal surfaces may exhibit reconstruction in addition to relaxation. In most cases, there is a contraction of the top interlayer spacing, and this seems to be a general trend for metal surfaces, transition metal and simple metal surfaces alike. The relaxation of compact closepacked surfaces such as fcc(111) are usually very small, of the order of 1% of the ideal interlayer spacing. The relaxation of the interlayer distances are of bigger magnitude for more open surfaces. For example, the top layer inward relaxation of AI(110), Cu(110) and A g ( l l 0 ) are about 7-9%, and Au(110)(1• has a top layer contraction of about 18%. Similar behavior is observed in bcc systems. Relaxations are typically small for the more close-packed bcc(110) surface, while the more open (100) surface usually has substantial relaxation. For example, the top layer of Mo(100) contracts by about 10%. Relaxation can extend several layers into the bulk, usually exhibiting an oscillatory pattern of contraction and expansion, with
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diminishing magnitude in the change of interlayer distances as it go deeper into the bulk. Tables of surface relaxation from theoretical calculations and experiments for many metal surfaces can be found in other review articles (see e.g., Bohnen and Ho, 1993). For the more open surfaces, the inward top-layer relaxation and the oscillatory relaxation pattern can be qualitatively explained by the redistribution of electronic charge at the surface. Following the picture as first discussed by Smoluchowski (Smoluchowski, 1941; Finnis and Heine, 1974), imagine that the crystal is partitioned into Wigner-Seitz cells, and the electronic charge remains in the unrelaxed position in the Wigner-Seitz cells when a surface is formed. The electron distribution would then have a very rough profile, leading to high kinetic energy and the system can reduce its energy by bringing the electrons into a smoother distribution parallel to the surface. This redistribution would tend to relocate electrons away from the ridge regions to the hollow regions on the top layer. Thus the ridges become slightly positively charged and the hollows slightly negatively charged. This charge redistribution on the top layer will produce electrostatic forces on the inner layers. The directions of the forces depend on the stacking sequence of the layers and for most cases, will lead to a contraction of the top interlayer spacing and an oscillatory relaxation pattern. A schematic picture of this mechanism is shown in Fig. 3.1. The few exceptions occur only for densely-packed faces where the relaxation effects are usually small. Because the computational effort grows as the size of the surface unit cell, most LDA calculations are restricted to the low-indexed crystal surfaces, which have been studied rather comprehensively (Bohnen and Ho, 1993). The structure and the energetics of higher indexed surfaces and stepped surfaces have also been investigated with classical potentials (see, e.g., Rodriguez et al., 1994). In general, atomic relaxations of higher-indexed faces involve substantial lateral atomic displacements in addition to normal displacements found in low-indexed faces. In surface reconstructions, in addition to the loss of translation periodicity in the normal direction, the surface atomic rearrangements also reduce the surface periodicity in the surface plane. Quite a few noble and transition metal surfaces reconstruct. The surfaces of transition metals and noble metals have been much studied because of their involvement in various catalytic processes. In particular, the noble metals are very popular for experimental studies because of the ease of preparing and maintaining a clean surface, they are also important electrodes for electrochemical studies. In this section, instead of giving a comprehensive review
I2/
\.
I -/
N ~ I -/
\
- I -/
\
- I -/
\
- I -/
\
- ! -/
\
- I
Fig. 3.1. A schematic picture showing the electron redistribution mechanism of Smoluchowski smoothing.
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[11]
t
Top View
f[ll]
Fig. 3.2. The top view and side view of the W(001) reconstruction. For the top view, the empty circles and the dark circles represent the positions of the surface W atoms before and after the reconstruction.
of the various reconstruction processes appearing on transition metal surfaces, we describe a few prototypical cases. These can be classified into reconstructions of the displacive type, the missing-row type, and the contractive type. In d i s p l a c i v e r e c o n s t r u c t i o n s , the surface periodicity is disturbed by static distortions of the atomic positions occurring in the surface and near surface layers. The most well-studied prototypical example of displacive reconstruction is the c(2• reconstruction of W(001 ), where the surface atoms are displaced in the [ 11 ] direction to form zig-zag chains (Debe and King, 1977, 1979). A schematic figure is shown in Fig. 3.2. The displacements are mostly confined to the top layer. A similar, although more complex reconstruction has been observed in Mo(001), where recent experiments give a c(792-x~/2-) unit cell (Smilgies and Robinson, 1993; Daley et al., 1993). First principles calculations for both W(001) (Fu et al., 1985" Fu and Freeman, 1988) and Mo(001) (Wang et al., 1988) found that the top layer atoms are indeed unstable with respect to a zig-zag type displacement on the surface, and the magnitude of the displacement (about 0.2/~) agrees well with the experimental observations (Debe and King, 1977, 1979). Tight-binding calculations (Wang and Weber, 1987" Wang et al., 1988) indicate there is a softening of a surface phonon branch along the <11> direction for both W and Mo (001) surfaces. For W, the maximum instability of the surface phonon branch occurs at the zone boundary of the surface Brillouin Zone, and is thus consistent with a c(2x2) unit cell. For Mo, the maximum instability happens to be close to, but not exactly at the zone boundary, leading to a much larger surface unit cell. The difference in behavior of the W(001 ) and the Mo(001 ) surface can be traced to relativistic effects which lowered the s bands relative to the d bands in W, leading to different surface wavevectors for the Fermi-surface-nesting giving rise to the instability. The symmetry of the undistorted ( l x l ) surface means that it is equally probable for the distortion to occur along the [ 11 ] direction as in the [ 11 ] direction. Thus, one would expect to find different domains to exist on the reconstructed surface where the displacement would point in different directions. From symmetry arguments, the potential energy of the surface as a function of the two dimensional displace-
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ment amplitudes of the zig-zag surface phonon modes should exhibit a Mexicanhat-like behavior: maximum at zero displacements, minima with displacements along the [11] and equivalent directions and saddle points along the [10] and equivalent directions. For such a potential, one would expect the c(2x2) long-range order of the reconstruction to be destroyed by randomizing the directions of the displacements of the atoms at different points of the surface rather than by reducing the atomic displacements to zero. Thus, we expect the high-temperature ( l x l ) surface to correspond to a disordered reconstructed surface (Debe and King, 1979) rather than an unreconstructed surface. This is in agreement with observations that, even in the high-temperature ( l x l ) surface, surface atoms still have large displacements from the truncated bulk positions comparable with the reconstructed phase (Stensgaard et al., 1979; Robinson et al., 1989). Theoretical calculations also indicate that the energy difference between the reconstructed and unreconstructed surface is much larger than kB To, where T c is the transition temperature (Singh and Krakauer, 1986; Yu et al., 1992; Roelofs et al., 1989). Details of the order-disorder transition have been studied both with molecular dynamics (Wang et al., 1988) as well as Monte Carlo simulations (Han and Ying, 1993). We note that displacivetype reconstruction is more common in bcc metals where directional d-bonding dominates the energetics. In displacive reconstructions mentioned above, the density of surface atoms remains the same as in the ideal surface and the reconstructions are achieved by small local displacements of the surface atoms from their ideal positions, without any long-range atomic transport. This is not the case for m i s s i n g - r o w and c o n t r a c tive reconstructions found in fcc metal surfaces. In the missing-row reconstructions, which occur on the (110) face of some fcc transition metals and noble metals, alternate rows of atoms on the surface are removed to form a (lx2) reconstruction of the surface periodicity (Moritz and Wolf, 1985; Copel and Gustafsson, 1986). Prime examples are the (lx2) reconstruction on the Au, Pt, and Ir (110) surfaces. In these cases, the surface (110) layer become less dense than the bulk (110) layers, with half the atoms missing. The resulting structure can be regarded as exposing micro-facets of the (111) surface, which has a lower surface energy than the (110) surface. First principles theoretical calculations (Ho and Bohnen, 1987) found a (lx2) ground state geometry with a missing top layer configuration that has substantial first layer contraction, lateral displacements in the second layer and buckling of the third layer, all in very good agreement with available experimental data (Moritz and Wolf, 1985; Copel and Gustafsson, 1986). A schematic figure of the (lx2) reconstruction is shown in Fig. 3.3. The calculations found that the (lx2) is stabilized relative to the ideal ( l x l ) since the more open (lx2) geometry allows the lowering of the kinetic energy of the s and p electrons without disrupting the d-bonding. Analogy with the case of the jellium surface indicate a threshold electron density above which the missing-row reconstruction is favored. For the 4d and 3d fcc transition metals, the clean (110) surfaces do not reconstruct because the bonding is weaker and hence the interstitial electron concentration is below the threshold. However, the (lx2) reconstruction can be induced on these surfaces by increasing the surface electron density (Fu and Ho,
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[110]t~ [ooi]
(
)
(
)
Fig. 3.3. A side view of the missing row reconstruction of the Au(110) surface. The arrows indicate the direction of the atomic displacements from their ideal positions.
1989). Experimentally, induced (1• reconstruction is observed with the deposition of a small amount of alkali metals (Behm, 1989). From this picture, we would expect the missing-row reconstruction to be further stabilized on the Au(110) surface when alkali atoms are added to the surface. This is in agreement with experimental observation (Behm et al., 1987). With higher coverages, the Coulomb repulsion between alkali ions destabilize the missing-row geometry in favor of a c(2• structure, observed when 0.5 monolayer of K is adsorbed on the Au(110) surface, which can be regarded as the formation of a surface alloy (Ho et al., 1989; Haberle and Gustafsson, 1989). In principle, larger facets of the (111) surfaces can be formed by removing more atomic rows, leading to (lxn) reconstructions of the (110) surface. Theoretical results (Bohnen unpublished) found that the ( I x3) reconstruction is very close in surface energy to the (1• Experimentally, (1• reconstructions are indeed observed upon the absorption of small quantities of alkali-metals (Behm, 1989) or in electro-chemical cells (Ocko et al., 1992). The disruption of the surface periodicity is even more severe with surfaces undergoing reconstructions of the contractive type. In this case, there.is a contraction of the surface layer atomic spacing leading to a denser incommensurate or almost incommensurate close-packed surface atomic layer over the bulk substrate layers. We will discuss below the prototypical cases of Au(100) (Fedak and Gjostein, 1967; Van Hove et al., 1981 ; Binninget al., 1984) and Au(111) (Van Hove et al., 1981; Perdereau et al., 1974; Melle and Menzel, 1978; Heyraud and Metois, 1980; Takayanagi and Yagi, 1983, Harten et ai., 1985; EI-Batanouny et al., 1987; Woll et al., 1989). If we imagine that we have a single layer of fcc metal atoms suspended in vacuum, the layer should prefer to be in a hexagonal structure because that is the close-packing arrangement in 2 dimensions. If we view the surface of an fcc metal as being composed of a top layer residing on top of a substrate of the same species, only the top layer in a (11 l)-oriented surface is in the preferred close-packing arrangement. Would the top layer of say the (100) surface of an fcc metal reconstruct from a square net to a more compact hexagonal arrangement? It turns out that such top layer rearrangement does happen on 5d transition and noble metal surfaces. The unreconstructed (100) surface of Au should be a square lattice, but
Su~. ace reconstruction
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experiments indicate that the top layer becomes a contracted quasi-hexagonal overlayer on top of a square substrate. A schematic figure is shown in Fig. 3.4a. Since the reconstructed top layer is no longer commensurate with the substrate layers, it takes a big unit cell to describe the reconstruction if we insist on giving a surface unit cell. Earlier experiments (Fedak and Gjostein, 1967) indicated a (1• unit cell and later LEED measurements suggested a larger c(26• unit cell. The main driving force behind the Au(100) reconstruction is the strong tendency for the top layer to go to a more compact arrangement, so strong that it can overcome the energy loss by losing registry with the substrate underneath. For Ag in the 4d series, theoretical calculations (Takeuchi et al., 1989) indicate that the top layer also wants to transform to the more compact hexagonal arrangement, but the energy gained in such a transformation is not large enough to overcome the substrate potential which pins the top layer, so the surface stays unreconstructed. The bigger gain in energy
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in the case of Au upon contraction is due mainly to a stronger participation of the d-orbitals in the bonding in Au than Ag, which in turn can be traced to the fact that Au has a higher atomic number, so that it has a bigger core and stronger relativistic effects (Takeuchi et al., 1989, Fiorentini et al., 1993). These differences are not specific to Ag and Au, hence it is not surprising that similar reconstructions occur in the (100) surfaces of the 5d fcc metals Ir and Pt, but not in the corresponding (100) surfaces of the 4d metals Rh and Pd. It is interesting to note that Au(111) reconstructs, and apparently, .it is the only fcc(111) clean metal surface that exhibits a reconstruction. We have mentioned that as a single 2-dimensional layer, the top layer would like to be hexagonal-closepacked. The (111) face of Au is already hexagonal but the top layer, as a single layer, would prefer to have a higher atomic density than that of the fcc (111 ) planes as dictated by the bulk lattice constant. Apparently for most fcc elements, the energy gained in the contraction from a hexagonal to a more compact hexagonal structure is not big enough to compensate for the loss of registry with the substrate lattice, but the energy gained by the Au 2-dimensional layer is large enough to overcome the energy loss due to the mismatch of the substrate. Under the influence of the substrate pote_ntial, the contraction is not uniform. The to__p layer atoms contract along the <110> directions by about 4-5%, forming a (22x~/3) superlattice. A schematic figure is shown in Fig. 3.4b. Since there are 3 equivalent <110> directions on the surface, domains with different orientations that are degenerate in energy can co-exist, and mesoscopic ordered domains called "herringbone" patterns have been observed directly by STM (Barth et 1990; Chambliss et al., 1991). These mesoscopic patterns are consequences of the anisotropic surface tensile stress and can be explained by simple models (Narasimhan and Vanderbilt, 1992). The above discussions are centered around several prototypical reconstructions that have been studied with LDA calculations. Relaxation and reconstruction for clean metal surfaces have also been studied comprehensively with classical force models such as the embedded atom method and the closely related effective medium method. For surface relaxation, the results are almost always qualitatively correct, and in many cases, the results compare rather well with experimental results and first-principles calculations. Some of these results have been tabulated in (Ricter et al. (1984) and Bohnen and Ho (1993). For the more demanding case of surface reconstruction, these models do not always yield results that are consistent with experimental findings. For example, EAM potentials that are fitted mainly to bulk properties have incorrectly predicted an instability of Ag(110) towards the missing row reconstruction (Einstein and Khare, 1994) and have failed to predict the contractive construction of Au(100) (Haftel, 1993). However, EAM or EAM-like potentials (such as the "glue" model) that are specifically fitted to consider surface properties give correctly the structure and the energetics of the Au(100) (Tosatti and Ercolessi, 1991 ; Haftel, 1993). These models are of course indispensable if we want to study more complex surface phenomena such as diffusion mechanisms, energetics of surface defects, steps and kinks, faceting, surface roughening, surface melting, and the kinetics and dynamics of surface reconstructions.
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3.6. Semiconductor surfaces
Because of its relevance to device physics, semiconductor surfaces have been intensively studied ever since the advent of ultra-high vacuum technology enabled experimentalists to obtain reproducible results on surfaces. Early theoretical calculations were focused on the (111) and (100) surfaces of Si, and the GaAs(110) surface. These systems have also attracted a lot of attention from theorists, and have been treated by various computational techniques from tight-binding methods to first principles calculations. A comprehensive review on these surfaces can be found in Duke's article (Chapter 6). Here, we briefly discuss some prototypical systems (Si(111) and Si(100)) which will be referred to in the section concerning metals on semiconductor. S i ( l l l ) is a favorite surface for experimental studies since it is the common cleavage face for silicon. The freshly cleaved surface exhibits a (2x1) reconstructed structure (Lander et al., 1963) which converts to a (7x7) reconstruction (Schlier and Farnsworth, 1959) upon annealing. Some of the earliest first-principles self-consistent surface calculations were performed for the Si(111) (2x l) surface. While the (2• reconstruction was initially supposed to be caused by a buckling of the surface layer (Haneman, 1961), this model was found to be in conflict both with experimental data and total energy calculation results. A number of possibilities were considered before the Tt-bonded chain model due to Pandey (Pandey, 1981) was generally accepted to give a good description of the surface geometry. This structure is shown in Fig. 6.3 in Chapter 6. It is interesting to note that this is one of the first examples in which a theoretical effort correctly predicts surface atomic arrangements. The (7x7) reconstructed surface is so complex, that full first principles calculations of its atomic geometry are possible only very recently using the latest development in both computational techniques and massively parallel computers with novel computer architectures (Brommer et al., 1992; Stich et al., 1992). The currently accepted DAS (dimer-adatom-stacking-fault) model (Takayanagi et al., 1985) has a massive reconstruction, with a (7x7) unit cell containing a stacking fault, 12 adatoms and 9 dimer bonds. The driving force behind the reconstruction is a compromise between strain relaxation and the healing of the broken bonds at the surface. The Si(100) surface is of considerable technological interest since this is the surface adopted for silicon wafers in chip and device fabrications. The clean surface is reconstructed with a (lx2) structure in which the atoms in the surface layer pair up to form dimers (Lander and Morrison, 1962), as illustrated in Fig. 6.6 in Chapter 6. The detailed geometrical structure of the surface has been investigated by several calculations. Both tight-binding (Chadi, 1979) and first principles calculations (Batra, 1990) have contributed significantly to the understanding of the geometry of these systems. 3.7. Metal overlayers on semiconductors
There is a lot of interest in the study of the metal-semiconductor interface (Batra, 1991; Batra, 1987) because of its importance in various technology-related issues
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such as the quality and stability of metallic contacts in semiconductor devices, the formation of Schottky barriers, and heteroepitaxial growth on semiconductor substrates. Although much effort has been devoted to experimental studies of these systems, it is quite evident that our understanding of the metal-semiconductor interface lags considerably behind our understanding of the clean surface systems. The situation here is complicated by the intrinsic reactivity of the deposited metal layers with the semiconductor substrates, leading to various possibilities involving intermixing and compound layer formation. Because of the strength and the variety of chemical bonding in these systems (metal-metal, semiconductor-semiconductor, metal-semiconductor), various metastable phases can exist, the occurrence of which depends on details of the deposition procedures as well as on the initial condition of the substrate surface since these factors can influence the extent to which diffusion of atomic species is allowed to take place. Theoretical study of these systems by first-principles calculations is still in its infancy and the few available calculations are concentrated on cases with simple geometries: for example, when there is a sharp interface between the substrate and a silicide layer (Hamann, 1988), and for monolayer or submonolayer coverages of the metal overlayer in cases when intermixing or penetration of the metal atoms into the substrate can be ignored. We will focus on a few cases in which the surface structure is relatively well characterized experimentally and where reliable calculations have been performed. We also mention a number of systems whose structures have not yet been completely determined. These illustrate the level of complexity that is currently being attacked theoretically and experimentally. Chapter 6 contains additional examples. Since a relatively large proportion of the detailed experimental studies have been devoted to the adsorption of various overlayers on the S i ( l l l ) and Si(100) surfaces, it is not surprising that these are also the popular surfaces for theoretical studies. For the Si(l 11) surface, surface reconstructions with (~(3-x~-)R30 surface periodicity are very common among systems with adsorbed metal overlayers (Kono et al., 1994). Many calculations have been performed for this case, including the geometries for AI, Ga, In, Ag, Au, and Sb adsorption. The case of ordered vacancies has also been considered (Ancilotto et al., 1991). However, it should be cautioned that the coverage of the metal overlayers are not the same in all of these q3-x~- structures" the AI calculations considered a coverage of 1/3 monolayer while the Ag calculations considered both 2/3 and one monolayer. The Au and Sb calculations are for one monolayer coverage. Other ordered patterns such as (lx3) are also frequently observed. For the case of Si(100), the (lx2) dimerized periodicity of the clean surface is often carried over when a monolayer of metal overlayer is adsorbed. The Si dimers frequently remain intact upon metal adsorption. Theoretical investigations of adsorption on this surface include the cases of AI, As and alkali metals. Calculations for the adsorption of overlayers on GaAs have been restricted to the cases of AI and Sn on GaAs(110) (Batra, 1984; Pandey, 1989). These are still preliminary studies and much more work is required to obtain a clear and detailed picture of metal adsorption in the system. In general, it can be said that there is a big imbalance between the theoretical effort and the experimental effort in the
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investigation of metal-semiconductor interfaces. Much more theoretical work is needed to help interpret and organize the large amount of experimental data available on these systems and to obtain a good understanding of the physical mechanisms at work. It should be mentioned that in many cases, although the periodicity of the surface structure is known fairly accurately, the coverages of surface metal atoms are more uncertain and often form the subjects of controversy between different experimental groups.
3.7.1. Si(l l l)(~[3•
and Si(l l l)-Au
The cases of noble-metal adsorption on silicon and germanium surfaces have attracted a lot of interest because noble metals are thought to be less reactive than transition metals with unfilled d-bands and hence the interfaces should have simpler structures more amenable to analysis. A survey of earlier experimental results can be found in a comprehensive review by LeLay (Lelay, 1983). The (~/3-x43-)R30 structure is observed on both surfaces after deposition of the metal at high temperatures. The Ag/Si interface has been regarded as the prototype of an unreactive interface with sharp transition between the two materials. However, even so, the deduction of the atomic arrangements on these structures has proved to be quite non-trivial. For a long time, there were nearly as many models proposed as there were papers published on this system (Ding et al., 1991). There were controversies even on whether the metal coverage in the system is l or 2/3 monolayer. The geometry of these structures have been studied with a formidable arsenal of experimental techniques including STM, ion scattering, LEED, RHEED, and surface X-ray diffraction (van Loenen et al., 1987; Wilson and Chiang, 1987a,b; Nicholls et al., 1986; Takahashi et al., 1988; Vlieget al., 1989; Kono et al., 1986; Bullock et al., 1990). Numerous conflicting models were proposed, sometimes even from similar data information. Recent first principles total energy calculations show that most of these models are unsatisfactory in that the surface formation energies of the system are so high that the Ag layer would not wet the Si surface. It was found that most of the models were misled by the honeycomb pattern observed in STM experiments. The actual geometry deduced from calculations and also from X-ray diffraction data is shown in Fig. 3.5a. This so-called "honeycomb-chained-trimer" or HCT model consists of a top layer of Ag atoms with one monolayer coverage arranged in a so called "honeycomb-chained-triangle" pattern located on top of a "missing-top-layer" (MTL) Si(111) substrate. For a "missing-top-layer" substrate, the top layer Si atoms have three broken bonds per atom, and the top Si layer is distorted in a (~/3-x~-)R30 periodicity, forming trimers. With two of the broken bonds satisfied by the formation of the Si trimers, the remaining bond is satisfied by a metallic-type of bonding with the Ag atoms in the top layer. The calculations indicated that the honeycomb pattern observed by STM c a n be explained by tunneling into an empty surface band whose electronic wavefunction has a maximum between three Ag atoms, so that the bright spot in the STM pictures represent a trimer of Ag atoms, rather than an individual Ag atom. Tunneling into the
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(a) Ag/Si(111)
et al.
(b) Au/Si(111)
Fig. 3.5. Top views comparing (a) the "Honeycomb-Chained-Trimer" (HCT) model for Si(111)-Ag and the (b) "Conjugate-Honeycomb-Chained-Trimer" (CHCT) model for Si(111 )-Au. The large dark
dots are the metal atoms, and the rest are the Si atoms, with sizes decreasing from the surface to deeper layers. Note that for the case of Ag, the Si atoms form trimers and for the case of Au, the Au atoms form trimers. occupied states should yield a pattern reflecting the atomic geometry of the top layer. This has been verified by recent STM studies (Wan et al., 1992a). For the Au adsorbed surface, the surface structure obtained depends on the initial state of the clean Si(ll l) surface. If one starts from a (7x7) substrate and deposit the Au layer at high temperatures, the surface exhibits a (~r3-x~-) pattern. Intriguingly, STM pictures for this surface show a different image from the Si(11 l )-Ag case. Instead of observing a honeycomb pattern of two bright spots per ( ~ - x ~ ) unit cell, STM images for Si(111)-Au show a triangular pattern of one bright spot per (~x'4~-) cell. Medium energy ion scattering (MEIS) experiments (Chester and Gustafsson, 1991) indicated a Au coverage of one monolayer and a twisted-trimer geometry quite similar to that of Si(l 11 )-Ag. Surface X-ray scattering also found Au trimers (Dornisch et al., 1991). First principles total-energy calculations were performed and the results show that the ground state geometry consists of a monolayer of Au on a missing-top-layer S i ( l l l ) substrate as in the case of Si(11 l) (~/3-x~-)R30-Ag. However, the lateral arrangements of the Au and Si atoms are reversed from that of S i ( l l 1)-Ag: the Au atoms form trimers while the Si atoms form honeycomb-chained-trimers. Such an atomic arrangement in the S i ( l l l ) ('~-• is called the "conjugate" honeycomb-chained-trimer (CHCT) model (Ding et al., 1992). A top view is shown in Fig. 3.5b. Unlike the results from the MEIS experiment, first principles calculations found no twist in the orientations of the Au trimers. This finding is in agreement with the results of surface X-ray diffraction and LEED data which indicate the presence of a mirror plane perpendicular to the surface for the ground state geometry. It is possible that the analysis of the MEIS data is complicated by the presence of domains of different registry for the Au layer on the substrate. Recent analysis of LEED data provide
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strong support for the calculated ground state model (Quinn and Marcus, 1992). The different atomic geometries found in Ag and Au on Si(111) can be explained by the difference in energy of the metal-metal and metal-semiconductor bonds. In the S i ( l l l ) (,]3-x,~-)R30-Ag case, both the Ag-Ag and Ag-Si bonds are weaker than the Si-Si bonds. The primary reconstruction process thus involves the rebonding of the surface Si atoms to form trimers, which in turn prompts the Ag atoms into a honeycomb-chained-trimer arrangement. Two of the three Si dangling bonds are satisfied by the trimer formation, and the remaining dangling bond is satisfied by the Ag adatom. For the case of Au, we note that Au has a higher cohesive energy than Ag (by about 0.9 eV/atom). The Au-Si bond is also stronger than the Ag-Si bond. In this case, the primary process involves the trimerization of Au atoms, forming trimers with bond lengths close to the bulk values. In this way, the Au atoms can optimize their own bonding, while keeping commensurate with the substrate Si to form strong Au-Si bonds. 3.7.2. Other phases o f noble metals on Si or Ge
When the Si(111 ) (,~-x,13-)-Ag phase is annealed at a high temperature (T = 550~ so that the Ag partly desorbs, it undergoes a transformation to a (1 x3) phase, which converts to a (lx6) upon cooling to room temperature (Lelay, 1983; Wan et al., 1993). The atomic arrangement has not been completely resolved yet, although it is believed that the (Ix6) structure is very similar to the (lx3) structure. Most of the older models assume that the Ag coverage is 1/3, with rows of Ag atoms sitting on 3-fold sites of an ideal S i ( l l l ) surface, aligned in the <110> directions. Recent experiments suggest that the Si substrate has a missing-top-layer arrangement, and the top layer Si atoms are probably forming r~-bonded chains; and the Ag coverage may actually be 2/3 monolayer (Wan et al., 1993). More experimental and theoretical work is needed to determine the exact stoichiometry of Si surface layers and the exact location of the Ag chains. A 1/3 monolayer model recently proposed for (lx3) alkali metal/Si(111 ) may also be applicable to this case (see the discussion in w 3.7.5 below). The (,13-x4-3-)R30 structure is also observed when Au and Ag are deposited on Ge(l 11), although these systems have received less attention than the corresponding systems with Si as the substrate. Recent experiments found that the saturation coverage for the metal is also about one monolayer, and the atomic arrangements are very similar to the case with Si(111) substrate. For Ge(l 11) (,~-x~4-3-)R30-Au, surface X-ray diffraction (Howes et al., 1993) found that the structure is similar to that of S i ( l l l ) (,~-x,~-)R30-Au and can basically be described by the CHCT model, with the metal atoms forming trimers. The case of Ge(11 l)(,]3-x,~-)R30-Ag is more controversial, with surface X-ray diffraction (Dornisch et al., 1991) favoring the CHCT model, while a very recent LEED analysis (Huang et al., 1994) favors the HCT model. The basic physics that governs the surface structure should be very similar to the case of Si(111)-Ag and the ground state structure depends on the relative strength of the Ge-Ge bond. Since the CHCT model is found for Ge(111) (4-fxx/3-)R30-Au, it indicates that the Au-Au and Au-Ge bonding is more important
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than the Ge-Ge bonding, just as in the case of Si(111)-Ag. The LEED results (HCT model) for Ge(111) (q-3-xf3-)R30-Ag suggests that the Ge-Ge bonding is stronger than the A g - A g bonding, while the surface X-ray data (CHCT model) for the same system suggests just the opposite. A first principles calculation would be useful in providing a second opinion. 3.7. 3. B,A l, Ga, In on Si(100) and Si(l 11)
The growth of the first monolayer of the Group III elements AI, Ga, In on Si(100) shows common behavior. These systems are regarded as well characterized and the agreement between theory and experiment is very good, especially for the case of A1 which has been studied carefully by first principles calculations (Northrup et al., 1991). Although the phase diagram seems complex, the structure can basically be accounted for by the following construction: the substrate Si(100) reconstruction is almost the same as the clean surface, characterized by Si dimers. The metal atoms form chains of metal dimers, with the chains orientated perpendicular to the Si dimer chains. First principles calculations indicate that the metal dimers should be aligned parallel to the Si dimers, as shown schematically in Fig. 3.6. The metal-dimer chains repel each other so that at low coverages, larger unit cells such as (2x3) are observed, reaching an "ideal" arrangement of a (2x2) phase as shown in Fig. 3.6 when the coverage reaches half a monolayer. The atomic arrangements in the (2x2) phase are such that all the metal atoms are bonded to 3 nearest neighbors and all the substrate Si dangling bonds are saturated. On the S i ( l l l ) surface, the geometry and surface electronic structure of the (~-xq-3-) phase observed in the adsorption of A1, Ga, and In are also well characterized. At 1/3 monolayer coverage, a (q3-xq-3-)R30 unit cell is observed. The substrate Si(11 1) is believed to be an ideal truncated Si (11 1) surface, with the AI ad-atoms forming a (q3-x'~-) overlayer. Two adsorption sites were discussed, both
AI
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Fig. 3.6. A top view of Si(100)(2x2)-A1, the black dots are AI atoms and the empty circles are Si atoms.
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0
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Fig. 3.7. A schematic figure showing the/43 and 7"4sites on a Si(11 1) surface. lying in the three-fold sites above the surface Si(111) layer: the 7'4 site that lies directly above a Si atom in the second Si layer, and the H 3 site that is laterally displaced so that the AI atom lies above a hollow in the second Si layer. These two adsorption sites are illustrated in Fig. 3.7. For all three elements, the ?'4 site was shown to be energetically more favorable than the H 3 site by first principles calculations (Northrup, 1984; Nicholls et al., 1985, 1987). The calculated surface band structures obtained agree well with subsequent angle-resolved photoemission and inverse photoemission measurements. Recent STM investigations for the Si(l 11) ('~-x'~-)-In structure also support the results of the L D A calculations. Note that complete saturation of the dangling bond is achieved by such an atomic configuration at 1/3 monolayer A1 coverage and the AI atoms are also bonded to 3 nearest neighbor Si atoms. The bonding configuration for the Group III metals on Si(100) (2x2) and Si(111 ) (~3-x'~-)R30 are thus very similar. The smallest of element of the group, B, shows different behavior. Although B also induces a (,~-x'~-)R30 phase with 1/3 monolayer coverage, LDA calculations found that it is most favorable for the B atoms to substitute for the Si atoms directly below Si adatoms at a 7"4 site (Bedrossian et al., 1989; Lyo et al., 1989), in agreement with many experiment observations. When the coverage is higher than 1/3 monolayer, AI/Si(I 11) shows other ordered structures such as (47--x47-) and (7x7). Ga and In can also induce a variety of complex reconstructions on Si(l 11). There are no detailed theoretical studies on these more complex systems yet.
3.7. 4. As, Sb and Bi/Si(l 1 1) As adsorption produces an ideal termination of the Si(111) surface. The As atoms substitute for the Si atoms in the top layer forming a ( l x l ) structure in which all the Si atoms in the system are four-fold coordinated. Each As atom is bonded to three Si atoms in the second layer, leaving a lone pair which fully occupies the dangling bond orbital at the surface. This passivation of the surface is so effective that the surface is very chemically inert, even to oxygen exposures. The geometry and electronic structure of the system has been well studied in first principles self-consistent pseudopotential calculations (Uhrberg et al., 1987). This system is also quite thoroughly examined experimentally because of its relevance in the epitaxial
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growth of GaAs on Si(111). The heavier Sb and Bi atoms induce ({3-x'~-) reconstructions at one monolayer coverage. These group V atoms are believed to form trimers (Martensson et al., 1990; Shioda et al., 1993) on top of the Si double layer. Si(111)-Bi also has a 1/3 ML coverage ('~-x~/3-) phase, which has one Bi atom per ('#3-x'~-) unit cell, absorbed at the T4 site on the Si double layer (Shioda et al., 1993). 3.7.5. Alkali metals on semiconductors
The adsorption of alkali metals on semiconductors has attracted a lot of experimental attention (see, e.g., Soukiassian and Kendelewicz, 1988) and has been studied as a prototypical system for the metalization of semiconductor surfaces and Schottky barrier formation. These systems also received early attention from first principles calculations (Soukiassian and Kendelewicz, 1988; Batra, 1987, 1991), mainly because these systems can be treated by plane wave pseudopotential approaches. Unfortunately, there are still controversies about many aspects of these systems, such as the nature of the metal-semiconductor bonding, the absorption sites and the saturation coverage and more experimental work is probably needed to clarify these systems. For alkali metals on Si(100), the dimer rows in the Si substrate are believed to remain intact and the dimers are slightly elongated upon alkali metal absorption. At high coverages, the alkali metal atoms form one-dimensional chains parallel to the Si dimer rows. However, consensus has not been reached concerning the exact absorption site and the saturation coverage (i.e. how many alkali metal atoms are there for each surface Si atom). There are several inequivalent sites in which the alkali metals can reside, and they are depicted in Fig. 3.8. It seems that theoretical results using the plane wave pseudopotential approach (Kobayashi et al., 1992; Zhang et al., 1991) agree that the coverage should be one monolayer for Na and K (which means one alkali metal atom per Si atom, or equivalently, two metal atoms per Si dimer), and two inequivalent sites (the so-called "pedestal" site and "valley bridge" site) are occupied. Since these two sites are at a different level (the pedestal sites are above the dimer rows and the valley bridge sites are in the troughs between the dimer rows), this class of model is called a double layer model (Abukawa and Kono, 1988). Cluster calculations favors absorption at the cave site with half monolayer coverage (Spiess et al., 1993). Experimental results are divided between one monolayer and half monolayer saturation coverage for the case of Na and K, and various absorption sites have been proposed (see, e.g., Soukiassian et al., 1992; Wei et al., 1992). A ( l x 3 ) structure is also observed when alkali metals are deposited on Si(111) (Daimon and Ino, 1985). STM experiments (Jeon et al., 1992; Wan et al., 1992b) observed ordered rows of double chains of bright spots, separated by 3 times the lattice constant of the ( 1• 1) surface unit cell. There are also controversies regarding the coverage of the alkali metals (although more recent results seem to favor 1/3 monolayer coverage (Weiteringet al., 1994)), and the structure of the substrate, and quite a few models have been proposed. First principles calculations (Morikawa, 1994) found none of the proposed (1• alkali-metal/Si(l 11) models to be satisfactory. The most promising model seems to be the new model that has 1/3 monolayer
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~176 (
---
5-.O+(3
9 O+O
~ - bridge site X__ k___ pedestal site (H) cave site (B) valley bridge site (C) top site (T)
Fig. 3.8. A schematic top view of Si(100) showing various possible sites for alkali metal adsorption. The names of these sites are those most commonly used in the literature. alkali metal coverage with the Si substrate reconstructed similar to the Pandey n-bonded chain model, but with 5 and 7-fold Si rings separated by a 6-fold ring to form (Ix3) surface periodicity (Erwin, 1995). More theoretical and experimental study will be needed to characterize these systems.
3.8. Epilog Since the physics and the mechanism underlying various surface reconstructions are tied to the properties of the constituent elements, which differ drastically from one element to another, it is very difficult to give one single rule of thumb as to how a surface should reconstruct or why it should do so. There is perhaps only one common tautological reason: surfaces reconstruct to lower the surface energy, and different classes of systems reduce their surface energy in very different ways. In any case, we will try to present some general observations from a theorist's point of view. Semiconductor systems are by definition those that have a gap in the eigenvalue spectrum. In the process of surface formation, states will be introduced in the gap, generally leading to a higher band energy. Most of the surface relaxation and reconstruction phenomena in semiconductor surfaces are consistent with the reduction or removal of these "defect" states in the gap, leading to a lowering of the sum
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of the eigenvalue term in the total energy (see e.g. Eq. (3.4)). This can be achieved by rebonding or relaxation. On a more intuitive level, we may say that for semiconductor systems which have directional covalent bonding, dangling (broken) bonds are produced by cutting a crystal to form a surface and the main purpose of the surface reconstruction is to reduce the number of dangling bonds. The surface atoms can u n d e r g o m a s s i v e r e a r r a n g e m e n t s to rebond t h e m s e l v e s , with Si(111 )(7• being a famous example. Since the covalent bonding is usually strong, such massive rearrangements of surface atoms may require a sizable activation energy, and thus may not be attainable unless the surface is annealed. There may be metastable configurations where the atoms can reduce the number of dangling bonds to a lesser extent but which can be achieved more easily by local rearrangement of atoms. Si(111)(2x l) with the r~-bonded chain configuration is a good example. These concepts are discussed further by Duke in Chapter 6. For bcc metals such as Mo and W, the energetics are dominated by the bonding of the electrons in the d-shells which are only half-filled. The bulk density of states exhibit a dip or minimum at the Fermi level, and this fact is usually used to explain why half-filled d-shell systems prefer the bcc structure over the more compact fcc or hcp arrangement of atoms. When a low index surface is formed, especially for the (100) surfaces which amounts to cutting away half of the nearest neighbors, the local density of states of the surface atoms actually have a peak at the Fermi level and the reconstructions of these surfaces can be regarded as the rebonding of the surface atoms to reduce the density of states at the Fermi level. From this point of view, bcc metal surface reconstructions share the same mechanism as elemental semiconductors. The fcc transition and noble metal atoms are nearly "spherical" since their d shells are full or nearly full, they reconstruct by changing the density of the surface atoms. The main effect is to reduce the kinetic energy of the s,p electrons and increase the binding due to the d-electrons. The complex reconstructions of the 5d low index fcc metals surfaces can be rationalized this way: The missing-row type reconstruction reduces the kinetic energy of the s,p electrons without decreasing the d-electron binding; while the contractive-type reconstructions increases the d-electron binding without increasing the kinetic energy of the s,p electrons. For simple metals, which can be regarded as ion cores sitting in a sea of nearly-free valence electrons, there is little the system can gain by changing the symmetry of the surface unit cell, so these systems typically relax by changing the interlayer distances. The contraction of the top layer can be explained by the "Smoluchowski" smoothing effect. The situation is much more complex with adsorbate-induced reconstructions. For the special case of metal overlayers on semiconductors, there are a few competing factors: the system will try to minimize the number of the dangling bonds in the semiconductor substrate, maximize the metal-semiconductor interaction, and at high coverage, optimize the metal-metal bonding. The final atomic configuration, most favored adsorption sites and the saturation coverage depend very much on the relative strength of the metal-metal, semiconductor-semiconductor and the metal-semiconductor bonds.
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Acknowledgements We thank Drs. N. Takeuchi, T.C. L e u n g , Y.G. Ding, B.L. Z h a n g , and M. Y a m a m o t o for c o l l a b o r a t i o n in the area of surface physics. A m e s L a b o r a t o r y is o p e r a t e d for U.S. D e p a r t m e n t of E n e r g y by I o w a State U n i v e r s i t y u n d e r C o n t r a c t No. W - 7 4 0 5 E N G - 8 2 . O u r work on surface physics has been s u p p o r t e d by the D i r e c t o r of E n e r g y R e s e a r c h , Office of Basic E n e r g y Sciences, i n c l u d i n g a grant of c o m p u t e r time on the C r a y c o m p u t e r s at the National E n e r g y R e s e a r c h S u p e r c o m p u t e r C e n t e r at Livermore.
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CHAPTER 4
Theory of Insulator Surface Structures J.P. L A F E M I N A Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory* Richland, WA 99352, USA *Operated for the US Department o f Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830
Handbook of Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
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Contents
4.1.
Introduction 4.1.1.
4.2.
4.3.
4.4.
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Basic definitions
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F u n d a m e n t a l principles of surface structures . . . . . . . . . . . . . . . . . . . . . .
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Purpose, scope, and o r g a n i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Computational methods
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
4.2.2.
Quantum mechanical methods
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4.2.2.1.
SCF-LCAO methods
4.2.2.2.
Density functional methods . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.2.3.
Tight-binding methods
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. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3.
C l a s s i c a l potential m o d e l s
4.2.4.
C o m p u t a t i o n a l a p p r o a c h e s to surfaces
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156
154
4.2.5.
C o m p a r i s o n of the m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
T h e structure of clean surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
160
4.3.1.
Diamond(Ill)
surface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
160
4.3.2.
R o c k s a l t (001) surface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
4.3.3.
Rutile (110) surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166
4.3.4.
P e r o v s k i t e (100) surfaces
4.3.5.
C o r u n d u m surfaces
4.3.6.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171
Silica surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
T h e structure of surface defects
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
4.4.1.
Introduction
4.4.2
Defects on the rocksalt (001) surface . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.
T h e structure of adsorbates
4.6.
Discussion
175
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
177
4.6.1.
R e h y b r i d i z a t i o n of d a n g l i n g bond charge density
. . . . . . . . . . . . . . . . . . .
178
4.6.2.
S u r f a c e stress and the i m p o r t a n c e of surface t o p o l o g y . . . . . . . . . . . . . . . . .
178
4.6.3.
A r e a s for future research
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
179 180
4.1. Introduction
4.1.1. Basic definitions The Webster's II New Riverside University Dictionary (1988)defines an electrical insulator, or dielectric, as "a nonconductor of ... electricity". A more precise (and useful) definition can be found in Van Nostrand's Scientific Encyclopedia, which states that an insulator is a material which "when placed between conductors at different potentials, will permit a negligible current ... to pass through it." (Considine, 1976). (At sufficiently large potentials, however, the insulator will allow current to flow as the material undergoes a phenomenon known as dielectric breakdown.) Even with this definition, however, the difference between insulators and semiconductors, for example, is far from straightforward. How negligible need the current be before a material is classified as an "insulator"? The closest thing to a perfect insulator is a vacuum, and quantitative measures of the insulating nature of materials are usually referenced to the vacuum. For example, in an ideal parallel plate capacitor, the capacitance between the plates in vacuum, Co, can be written as ~oA/d, where to is the dielectric constant of a vacuum (also called the permittivity of free space), A is the area of the plate, and d is the distance between the plates. If the vacuum is replaced by an insulator, the capacitance, C, is now given by eCo where e is the dielectric constant, or dielectric permittivity, of the insulator (Considine, 1976; Lorrain and Corson, 1970). Many factors can affect the value of the dielectric constant, including temperature and humidity. As can be seen in Table 4.1, materials with a wide range of dielectric constants are available, and they span almost every type of material known, including crystalline, amorphous and glassy ceramics and porcelains, metal oxides, minerals, and organic polymers and resins. The industrial and environmental applications of the materials listed in Table 4.1 result from far more than their inability to conduct electricity. In fact, the chemistry that occurs at the surfaces of these materials is important in diverse areas such as catalysis (Campbell, 1988), corrosion (Fehlner, 1986), ceramic synthesis and chemistry, (Dufour and Nowotny, 1988; Nowotny and Dufour, 1988) the formation of insulating layers in microelectronic devices (Pantelides, 1978; Pantelides and Lucovsky, 1988), groundwater transport of contaminants (Hochella and White, 1990; Reeder, 1983), glass formation (Paul, 1982), and the design of nuclear waste storage material (Northrup, 1987). As a result, there is an enormous literature on the surface physics and chemistry of "insulating" materials. Unfortunately, the bulk of this work has been performed on poorly characterized samples, such as powders, amorphous, and polycrystalline materials. Consequently, the results of 139
140
J.P. LaFemina
Table 4.1 The dielectric permittivityof some representative materials. Valuestaken from Van Nostrand's Scientific Encyclopedia (Considine, 1976). Material Alumina (ceramic) Alumina (crystalline) Pyrex Muscovite Calcite Magnesium oxide Barium titanate Polyethylene Nylon Cellophane
8.1-9.5 10.0 5.1 7.0-7.3 9.2 8.2 4100 2.3 4.0--4.6 6.6
these studies are difficult to interpret in terms of the relationships between the surface atomic structure, chemical bonding, and process chemistry. The basis for understanding these relationships, is a detailed knowledge of atomic structure of the surface or interface of interest. "...there is no doubt that the atomic geometry of a surface or interface is its most fundamental characteristic. Any serious scientific effort to determine the electronic structure and properties of an interface must start with a realistic description of its atomic geometry..." (Duke, 1982) This chapter will present a simple set of chemical and physical principles that can be applied to insulator surfaces, and used to understand the qualitative features of surface structure, including surface stoichiometries, relaxations, and reconstructions (Duke, 1992; LaFemina, 1992). These principles have emerged from many years of detailed theoretical and experimental work on semiconductor surfaces. For insulating surfaces, this process is in its infancy. As stated above, the vast majority of the experimental work has been performed on poorly characterized material (Henrich, 1985). Sample preparation difficulties, coupled with the fact that many insulator surfaces fracture rather than cleave, make the preparation of surfaces with well-defined stoichiometries difficult (Henrich, 1985). In addition, there are many problems, such as surface charging, and the decomposition of surfaces under charged-particle beams, associated with the application of the traditional surface structural probes to insulator surfaces. Computationally, because of the complexity of many of these materials relative to metals and semiconductors, much effort has gone into the formulation of classical potential models (Colbourn, 1992). These models, while making many contributions to the understanding of insulator surface structures, are inherently limited. The classical nature of these potentials precludes any knowledge of the surface electronic structure; knowledge critical to the elucidation of the driving forces behind surface reconstructions and to the formulation of surface structure-activity
Theory of insulator surface structures
141
relationships. At the opposite end of the spectrum are the first-principles, or ab initio, methods that, even with the most recent advances in computer hardware and software, are impracticable for most of the systems which we will examine. In between these two extremes lie the semiempirical and empirical quantum-mechanical methods, such as the tight-binding models, which have been enormously successful in providing quantitative descriptions of semiconductor surfaces and interfaces. (See Chapter 6 and reviews by Duke (1992) and LaFemina (1992).) The development of these models for many insulating systems has been inhibited, however, by lack of the detailed experimental information needed for their formulation. At this point it is useful to review some of the basic nomenclature used in this chapter. A surface is "relaxed" if the surface atomic geometry exhibits the same symmetry as the truncated bulk solid. Relaxed surfaces are referred to as (1• even though the atoms at this surface may lie as much as an Angstrom away from the truncated bulk lattice sites. If the symmetry of the surface is different than that of the bulk, then the surface is "reconstructed", and the Wood notation described in Chapter 1 for surface overlayers is used. 4.1.2. Fundamental principles of surface structures
Insight into the structure of insulator surfaces can be obtained, as stated above, by applying a simple set of chemical and physical principles. In fact, these principles can be cast as a set of five simple rules (which roughly follow the presentation in Chapter 6). (1) Saturate the dangling bonds: The creation of the surface creates dangling bonds: that is, bonds which used to bind surface atoms to their, now missing, bulk neighbors. This is in an energetically unfavorable situation, because these dangling bonds are only partially filled with electrons. Therefore, a driving force exists at the surface to redistribute the dangling bond electrons (charge density) into a more energetically favorable configuration. In general, this occurs in a way which satisfies the local chemical valences of the surface atoms. (2) Form an insulating surface: The most energetically favorable way to eliminate the dangling bonds is to create an "insulating" surface (i.e., open a gap between the occupied and unoccupied surface states). This can be done by forming new bonds at the surface, either between surface atoms or between the surface atoms and adsorbates. The surface also can be insulating as a result of a surface structural rearrangement (i.e., a surface relaxation or reconstruction) that transfers electrons between surface atoms. Finally, the energy of the surface can be lowered electronically, through strong electron correlation effects which open a gap between the occupied and unoccupied surface states. The bottom line, however, is that the opening of a gap between the occupied and unoccupied surface states lowers the energy of the occupied states while raising the energy of the unoccupied states, resulting in a net energy lowering for the surface. (3) Do not forget about kinetics: In general, and especially for the fracture surfaces of the crystalline metal oxides, the structure exhibited by any surface is dependent upon the processing history of the sample. That is, the "structure ob-
142
J.P. LaFemina
served will be the lowest energy structure kinetically accessible under the preparation conditions" (Duke, 1993). This applies to the cleavage surfaces as well, where the exhibited surface structures are "activationless" in the sense that the activation energy for the relaxation or reconstruction is less than the energy provided by the cleavage process. (4) Form a charged neutral surface: The most energetically favorable way of pairing up dangling bond electrons is to fully occupy the anion dangling bonds and completely empty the cation dangling bonds. Alternatively, this can be thought of as completely filling the valence band orbitals while completely emptying the conduction band orbitals, which is, of course, what happens in the bulk. This rule, also referred to as autocompensation, or electron-counting, is simply a consequence of the fact that a surface which carries a net charge produces long range electric fields which cost a significant amount of energy. Anything which eliminates this surface charge will also greatly reduce the surface energy. This includes the formation of non-stoichiometric surfaces through the creation of surface and subsurface defects, and/or the adsorption of atoms or molecules onto the surface. This is an enormously powerful principle since it allows for the identification of a few, most likely, surface stoichiometries. (5) Conserve bond lengths: All of the factors described above provide potential driving forces for the surface atoms to move away from their bulk atomic positions and lower the surface energy. If this movement creates local strain in the surface or subsurface region, (that is, strain which results from the distortion of the local bonding environment) then the surface energy will be raised and the movement of the surface atoms resisted. Not all surface strains are created equal, however. For example, distortions of bond angles typically cost an order of magnitude less energy than distortions in near-neighbor bond lengths. Consequently, surface atomic motions which move the atoms into electronically favorable conformations, while (nearly) conserving near-neighbor bond lengths are most favorable. It is the topology, or atomic connectivity, of the surface that controls which atomic motions will be energetically favorable. Hence, it is the balance between the surface energy lowering due to the elimination of surface dangling bonds and the creation of an insulating surface, and the energy cost due to induced local strain that primarily determines the nature of surface relaxations and reconstructions.
4.1.3. Purpose, scope, and organization In the remaining sections of this chapter these five principles will be applied to a variety of insulator cleavage, fracture, and growth surfaces. Looking back at Table 4.1, a complete discussion of all of the material types listed is clearly beyond the scope this chapter. What this chapter will do, however, is to focus on the subset of these materials for which the most detailed understanding of surface atomic structure exists, namely the surfaces of diamond and of the crystalline metal oxides. In many cases, the knowledge gained from examining these surfaces is directly applicable to other materials of the similar structural types (e.g., the crystalline metal sulfides and selenides, as well as the rocksalt structure alkali halides). In
Theory of insulator su~'ace structures
143
addition, other chapters in this book will cover some of the materials not examined in this chapter. In Chapter 6 the structures of elemental and compound semiconductor surfaces are examined from both an experimental and computational point-ofview using these same principles as a guide. In Chapter 5 the structures of crystalline ceramics and insulators (oxides, carbides, nitrides, and halides) are examined from an experimental viewpoint. Other relevant chapters include Chapter 3, which details the computational methods used in surface structure computations, Chapters 7 and 8, which review the experimental methods of surface structure determination referred to in this chapter, and Chapters 10-13 which will examine the structure of adsorbed layers and surface defects. As indicated by the title, the emphasis of this chapter is primarily on the computation of the atomic structure of insulator surfaces. Most importantly, a major goal of this chapter is to demonstrate the applicability of the simple chemical and physical principles, summarized above, to insulator surfaces. Experimental surface structure determinations by low-energy-electron-diffraction (LEED) (Van Hove et al., 1986) and low-energy-positron-diffraction (LEPD) (Canter et al., 1987) intensity analyses, X-ray photoelectron diffraction (XPD) (Chambers, 1992), and ion scattering (van der Veen, 1985) will be used, where available, to evaluate the computational predictions of surface and interfacial atomic structure. In addition, the results of X-ray photoemission spectroscopy (XPS), ultraviolet photoemission spectroscopy (UPS), and electron-energy loss spectroscopy (EELS) will be used to evaluate the electronic structure predicted for the structural models that emerge from the computations (Woodruff and Delchar, 1986). The litmus test of success is that the same model predicts both the surface atomic and electronic structures in agreement with experimental measurements. The presentation in this chapter is such that readers familiar with fundamental concepts in solid state atomic structure and chemical bonding at the following level should have no difficulty: "Introduction to Solid State Physics" by C. Kittel (Kittel, 1976), "Chemistry in Two Dimensions: Surfaces" by Gabor A. Somorjai, (Somorjai, 1981) and "Physical Chemistry" by P.W. Atkins (Atkins, 1978). For readers looking for more in-depth treatments of these topics, several excellent graduatelevel texts are available: "Physics at Surfaces" by Andrew Zangwill (Zangwiil, 1988), "Solids and Surfaces: A Chemist' s View of Bonding in Extended Structures" by Roald Hoffmann (Hoffmann, 1988), "Atomic and Electronic Structure of Surfaces: Theoretical Foundations" by M. Lanoo and P. Friedel (Lanoo and Friedel 1991), and "Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond" by W.A. Harrison (Harrison 1989). This chapter is organized as follows. In w 4.2 the various computational models commonly used for the computation of insulator surface structures are explored; the basic assumptions outlined, and the relative strengths and weaknesses evaluated. In w 4.3, the results of surface structure computations for the most common surfaces of insulators are examined. Sections 4.4 and 4.5 look at the structure of surface defects and surfaces with chemically adsorbed atoms respectively. Finally, w 4.6 contains a discussion of the major issues both clarified and raised by the computations and the chapter concludes with some speculations about promising areas for future research.
144
J.P. LaFemina
4.2. Computational methods 4.2.1. Introduction This section describes various computational methods used to determine the atomic structure of insulator surfaces. The approximations associated with each of the methodologies will be presented and their advantages and disadvantages discussed. The methodologies can be roughly classified into two groups: those that use quantum-mechanical potentials and those that use empirically determined classical potentials. The quantum-mechanical methods can be further sub-divided into three major types w self-consistent field, linear-combination of atomic orbitals (SCFLCAO); density functional; and tight-binding B depending upon how the interactions between the electrons are treated. The empirical classical-potential models have primarily been developed, and applied to, "ionic" insulators because of the presumed predominance of the Coulombic interactions between the ions in the lattice (Madelung, 1909, 1910a,b). The various applications these methods to the study of surfaces also will be discussed. These techniques can be subdivided into three categories as well: Green's function techniques, slab calculations, and cluster calculations. The main difference between these methods is the boundary condition used to model the semi-infinite surface. The Green' s function techniques treat the semi-infinite nature of the surface exactly (and as a result are the most complex). Slab models treat the infinite (in the two dimensions parallel to the surface) two-dimensional nature of the surface properly, but have a finite thickness in the third dimension perpendicular to the surface. Finally, the cluster methods model the surface with a finite set of atoms. This section will emphasize the basic terminology; the goal is to enable the reader to evaluate computational results. For the details of a particular method, the reader is referred, primarily, to the review literature and to graduate-level texts. One feature common to all of these methods is the way in which the equilibrium surface atomic geometry is obtained. That is, once the method is chosen, and the total-energy functional specified, atomic forces (Feynman, 1939) are computed, and the total energy (or atomic forces) minimized as a function of the nuclear coordinates. The methods differ, of course, in the way in which the total-energy functional is specified. It is this specification, along with the associated assumptions, that will be addressed in the remainder of this section. The section is organized to proceed from the quantum to the classical; moving approximately from ab initio, or first-principles, methods to the empirical methods through the introduction of successive approximations. Section 4.2.2 begins with a description of the fundamentals common to all quantum mechanical methods. The two different ab initio approaches that have been developed B SCF-LCAO and density-functional B are then examined in w 4.2.2.1 and 4.2.2.2, while the most common (for surface structure computations) empirical quantum-mechanical method w tight-binding m is described in w 4.2.2.3. Section 4.2.3 covers the empirical classical potential models, which are, by far, the most widely used. The
Theory of insulator su~. iwe structures
145
Table 4.2 Summary of the advantages and disadvantages of the various theoretical methods discussed in w4.2 Method (Section)
Advantages
Quantum-Mechanical Methods SCF-LCAO (w 4.2.2.1) Excited state properties can, in principle, be computed.
Disadvantages
Unscreened exchange at HartreeFock level; correlation can only be included a posteriori; very computationally intensive (tens of atoms).
Density Functional (w 4.2.2.2)
Both exchange and correlation included in zeroth-order; first principles yet less computationally intensive than LCAO-SCF; can deal with tens of atoms.
Ground state theory, no excited state properties can be computed. Exchange and correlation interactions are approximated with no clear way to improve.
Tight-Binding (w 4.2.2.3)
Computationally simple; able to deal with tens to hundreds of atoms; description of bonding in terms of atomic-like orbitals; quantitative descriptions of surface structure possible.
All interactions are parameterized; non-self consistent; needs to be benchmarked against experiment or first principles computations.
Classical Potentials ( w4.2.3)
Computationally very simple; able to deal with hundreds of atoms.
Non-quantum mechanical; all interactions are parameterized.
different c o m p u t a t i o n a l a p p r o a c h e s to surface m o d e l i n g are d i s c u s s e d in w 4.2.4, and a c o m p a r i s o n of all of the m e t h o d s is given in w 4.2.5. Finally, a s u m m a r y of this i n f o r m a t i o n is given in Table 4.2. 4.2.2. Q u a n t u m m e c h a n i c a l m e t h o d s All of the q u a n t u m m e c h a n i c a l m e t h o d s c o n s i d e r e d in this section begin by m a k i n g the B o r n - O p p e n h e i m e r (Born and O p p e n h e i m e r , 1927) a p p r o x i m a t i o n , which ass u m e s that the m o t i o n of the electrons is fast c o m p a r e d to the m o t i o n of the nuclei, or ion cores. In this way the motion of the nuclei may be n e g l e c t e d when c o m p u t i n g the e l e c t r o n i c structure, in effect, separating the electronic and nuclear d e g r e e s of f r e e d o m . In this a p p r o x i m a t i o n the t o t a l - e n e r g y of a s y s t e m of N atoms, w h o s e n u c l e a r positions are g i v e n by R ~N~, can be written as the sum of an e l e c t r o n i c (EEl.) and a n u c l e a r (ENuc) c o n t r i b u t i o n . The nuclear c o n t r i b u t i o n , n o w i n d e p e n d e n t of the e l e c t r o n c o o r d i n a t e s , is the repulsion b e t w e e n ion cores ~t and v with c h a r g e s Z, and Z,,, s e p a r a t e d by a distance R~v
N ENUCIR(N)Ile2 It'"I-- 4--~C(I ~' ~,vY--~Z~ZVR~v
(4.1)
J.P. LaFemina
146
The electronic energy is determined by solving the many-electron, time-independent Schr6dinger (1926a-d) equation A
H~(r)
=
E~P(r)
(4.2)
where ~P(r) is the electronic wavefunction of the system of n electrons whose positions are given by r. At the set of nuclear coordinates R (N), tP(r) is a function not only of the electron positions but also of the electron spin. E is energy of the system. H is the Hamiltonian operator for the system, and represents the kinetic, T, and potential, V, energies of the system of n-electrons moving in the field of fixed nuclei. The kinetic energy term, then, represents the kinetic energy only of the n-electrons, and can be written as n
T-
vy
(4.3)
i=1
where h is Planck's constant, m is the mass of the electron, and V~ is the gradient operator applied to electron wavefunction i. The potential energy term
n
v-
N
-ZZ-+Z
n
•r o
/ (4.4)
describes the electrostatic interaction between the electrons and the fixed nuclei, or ion cores (an attractive interaction given by the first term in Eq. 4.4) and the electrostatic interaction between electrons (a repulsive interaction given by the second term in Eq. 4.4). The Hamiltonian described by Eq. (4.3) and (4.4) is incomplete,.omitting the effects of electron spin. These effects are typically small compared to the energies involved in determining surface structures, although, as we shall see in w 4.3, they may be important for some diamond surfaces. Also, because most insulators (and semiconductors) possess an even number of electrons (i.e., a closed shell), the ground state electronic structure is well described by pairing electrons of different spin. Consequently, the presentation of the various formalisms will be for closedshell, spin-paired systems, with no explicit consideration of electron spin. This Hamiltonian is also non-relativistic, and, hence, is not appropriate for systems which contain high atomic number atoms where the velocities of the electrons can approach the speed of light. In practice, it is the second term in Eq. (4.4), the electron-electron interaction, that makes Eq. (4.2) so difficult to solve. This is a many-body problem since the wavefunction depends upon the coordinates of all of the electrons in the system. This problem is often simplified by recasting it as an effective single-body problem, in which the total wavefunction for an n-electron system is expressed as a product
Theory of insulator su~ace structures
147
of n-wavefunctions, gt,,(r), (one for each electron in the system). This leads to a system of n one-electron SchrOdinger equations A
Hqt.(r) = E. gt.(r)
(4.5)
where E,, are the one-electron eigenvalues of tff,,(r). Each of these n-wavefunctions depend only upon the coordinates of a single electron (hence the name one-electron wavefunctions). It is in the specification of this effective single-particle potential (resulting from the one-electron wavefunctions) that the various quantum-mechanical methods differ. Before we discuss the differences between the methods, however, there are many features they have in common. Typically, the one-particle wavefunctions are expressed as a linear combination of some set of basis functions, ~i,
qt,(r) = ~_~ c,,,(r)
(4.6)
i
where c~ are the expansion coefficients (Lowe, 1978). Using Eq. (4.6), the system of n one-electron SchrOdinger equations given by Eq. (4.5) can be rewritten in matrix form as He = SeE
(4.7)
where E is the (diagonal) matrix of one-electron eigenvalues and c is the matrix of expansion coefficients (which define the eigenvectors). S is the overlap matrix, accounting for the spatial overlapping of the basis functions, whose elements are given by
So - I d?~(r) t~j(r ") dr dr"
(4.8)
Finally, H is the Hamiltonian matrix, whose elements are given by Ho- I r
HCj.(r ') dr d r '
(4.9)
The idea is to approximate the spatial character of the one-electron wavefunctions with as few functions as possible to minimize the computational expense. The proper choice of basis set is one of the critical aspects of practical computations. For example, the choice of an orthonormal set of basis functions can greatly simplify the problem since, for orthonormal functions, the overlap matrix is simply the identity matrix. Many different sets of basis functions have been used. They can be Slater-type orbitals, as in the empirical and semiempirical SCF-LCAO methods (Pople and Beveridge, 1970); contracted Gaussian functions as in the ab initio SCF-LCAO (Hehre et al., 1986; Pisani et al., 1988) and density-functional (Dunlap et al., 1990) methods; plane waves as is typical of the density-functional methods at all levels (Cohen and Chelikowsky, 1988); or finally a numerical basis derived from sophis-
J.P. LaFemina
148
ticated atomic computations (Delly, 1986, 1990). Considerable effort has gone into investigations on the construction of basis sets, and the literature available on this subject is enormous (Feller and Davidson, 1990). For the empirical tight-binding methods, the question of basis set is, in practical terms, a question of parameter set since these methods have no basis set p e r se. That is, basis functions are never actually specified, and Hamiltonian matrix elements are never computed. Instead, the "basis" is typically assumed to be a set of valence atomic-like orbitals which are orthonormal, eliminating the overlap matrix. The Hamiltonian matrix elements for this "basis" are then treated as adjustable fitting parameters. Once specified, Eq. (4.7) is solved for the eigenvector coefficients. Although the determination of these parameter/basis sets is dependent upon the details of the particular system under investigation, two particular parameterizations have emerged as the most widely used. The reader is referred to these for more information (Harrison, 1976, 1981, 1989; Vogl et al., 1983). Quantum mechanical computations can be all-electron (i.e., computations which take explicit account of every electron in the system) or valence-electron only. The valence-electron only calculations, assume that the valence-shell electrons dominate the interesting chemical and physical processes in the system. A further approximation is then made to the single-particle potential as these valence electrons are then considered to move in an effective, or pseudo-, potential that includes the interactions of the nuclei and core electrons. The pseudopotential can be empirical, semiempirical, or ab initio, depending upon the way in which it is derived. Empirical pseudopotentials are not self-consistent and are derived from fitting to experimental data. Self-consistent, semiempirical psuedopotentials assume a parametric form for the ionic core potential a priori and are then fit to either experimental data or the results of an all-electron computation. A b initio psuedopotentials, on the other hand, assume no a p r i o r i form for the potential, but construct the potential to most closely match the all-electron wavefunctions. More detailed discussions on the construction and use of pseudopotentials are available in Szasz (1985) and Zunger (1979). As stated previously, the differences between the various quantum mechanical methods lie in the way in which the many-body electron-electron interactions are reduced to effective one-particle, or one-electron, interactions. Perhaps the most straightforward way of illustrating these differences is to partition the one-electron electronic energy into component parts as follows: E E L I 'IR~NII -- F. lh,~N~l + rz IP~N~I + E X [I~ '" " I --KE-el t'" I ~ " ' t t L''~ I
1
+ rz IO~N~I " - " C t' ' ~ J
(4.10)
ERE_el is the one-electron kinetic energy and electron-ion attraction which results from Eq. (4.3) and the first term in Eq. (4.4). The remaining terms in Eq. (4.10) arise from the second term in Eq. (4.4), the electron-electron interaction. EH represents the two-electron (because it depends upon the coordinates of two electrons) Hartree (or Coulomb) interaction; Ex represents the two-electron exchange (or Fock) interaction, which arises from the indistinguishability of the electrons, which must be reflected by the wavefunctions; and Ec is the electron correlation energy. The
Theory of insulator su~ace structures
149
correlation energy corrects for the neglect of the correlated nature of the electron motion in recasting the many-electron problem into an effective one-electron problem. In the following subsections, the way in which the SCF-LCAO, density-functional, and tight-binding methods determine each of the terms in Eq. (4.10) will be presented, discussed, and contrasted. It is important to note that the presentation in the following sections will be for the ab initio variant of the SCF-LCAO and density functional methods. That is, the interactions that are included by the method will be assumed to be computed explicitly. Empirical and semiempirical variants, for which some subset of the interactions (i.e., Hamiltonian matrix elements) in the system are either neglected or parametrized, also exist for each method type. The details of these variants can be found in the following references (Pople and Beveridge, 1970; Parr and Yang, 1989; Cohen and Chelikowski, 1988). 4.2.2.1. SCF-LCA O methods Typically, SCF-LCAO methods simplify the computation of the electronic energy by restricting the one-electron wavefunction to a single Slater determinant (Slater, 1929). This requirement is one way to assure that the wavefunction is antisymmetric and describes electrons as indistinguishable particles. It also results in the neglect of electron correlation effects (although a limited amount of correlation is included since the method also obeys the Pauli exclusion principle). This level of calculation is called the "Hartree-Fock" level. In molecular systems, electron correlation effects can be included a posteriori in a variety of ways (Hehre et al., 1986). Unfortunately, these same methodologies are not applicable to periodic systems. (Some estimate of the effect of electron correlation on the energy of the system, however, can be obtained from electron density functionals.) At the Hartree-Fock level, then, Eq. (4.10) reduces to
IR(N~I
IR(N~IJ + 1= I.(N~IJ + E X/'" I.(N,I l'Ht'"
Evl. . t-- ; - EKE-el,'"
I
(4.11)
where all of the terms on the right-hand side of Eq. (4.11) are now evaluated explicitly. To do this, the one-electron wavefunctions are computed using the effective one-electron Hamiltonian operator (referred to in this context as the Fock (1978) operator) obtained from the variational principle A
F-
A
A
HKE_ei 4-
A
H. + Hx
(4.12)
where HKE_el represents the one-electron kinetic energy and electron-ion (nuclear) attraction, N
HKE-el =
87z2m
4rtc0
I r - R.I Ia=l
H , represents the two-electron Hartree screening (or Coulomb) interaction
(4.13)
J.P. LaFemina
150
HHVi(r) =
92
4nr
dr' y__, J=,
Ivj(r )1 vi(r) Ir-r'l
(4.14)
and Hx represents the two-electron exchange interaction dr' Z Vj (r)Vi(r )Vj(r) j=l Ir - r'l
Hx~i(r) = 4rr176
(4.15)
Note that the one-electron wavefunctions, ~i(r), appear in the definition of the Fock operator, requiring an iterative, or "self-consistent" solution. Once the one-electron wavefunctions are found, the energy terms in Eq. (4.11) can be evaluated as follows: n/2 E
Ii?(N)I_/ e2 ) KE-eiI'" I 4riCo
n/'2
Z i=1
n/2
2Ei--y___, Z i=1 j=l
(2dij - Kij )
(4.16)
where E; is the one-electron eigenvalue of ~i(r), Jij represents the two-electron Hartree (or Coulomb) interaction J,j - f dr dr"
I~(r)12 I~j(r')12 Ir - r'l
(4.17)
and Ki/represents the two-electron exchange interaction, 9(
9
t
i
K~j = ~ dr dr' ~d, r)vj (r )~dj(r)~di(r ) Ir - r'l
(4.18)
The remaining terms in Eq. (4.11) can be expressed in terms of the Coloumb and exchange interactions as b?(N)~ =
E"F" '
rt/2
n/2
"=
j=i
n/2
n/2
e2
4~Eo
(4.19)
and EXl/~N)It--
,
( e2 / 4rt~ ~
~_., Z
-Kij
i=1 1=!
yielding the total electronic energy as
(4.20)
Theory of insulator surface structures
9"ELt'" ,
/
4II:E:,, E
I 2Ei -- E
i=' L
151
q /
(2Jo- KO) I
(4.21)
One interesting aspect of Eq. (4.21) is that the electronic energy is not simply the sum of the occupied one-electron eigenvalues, Ei. (The factor of 2 accounts for two electrons, one spin up, the other spin down, in each occupied orbital.) This sum overestimates (or "double counts") the electron-electron interactions. This extra energy, given by the second summation term in Eq. (4.21), must be removed. To summarize, the SCF-LCAO methods neglect electron correlation effects by limiting the wavefunction to a single Slater determinant. The remaining energy terms of Eq. (4.11) m the kinetic, electron-ion, Coulomb, and exchange energies are then evaluated explicitly.
4.2.2.2. Density functional methods The density functional methods differ from the SCF-LCAO methods in that the electron density, p(r), is used as the variable of interest (Jones and Gunnarsson, 1989). It has been shown (Hohenberg and Kohn, 1964) that the ground state electronic energy can be expressed as a functional of the external (or nuclear) potential, v(r), and the electron density as,
/e2/(
E[p(r)] = 47t13o Tip(r)] + I p(r)v(r)dr + Vr162
(4.22)
The first term in Eq. (4.22), T[p(r)], is the kinetic energy. The second (integral) term is the electron-ion energy. The third term, Vee, represents all of the electron-electron interactions, including the Hartree, exchange, and correlation energies. In this way, Eq. (4.22) is simply a restatement of Eq. (4.5). The only constraint on the electron density is that it be "N-representable" (i.e., that it can be obtained from an antisymmetric wavefunction) and thereby have the electrons behave as indistinguishable particles. Most importantly, this electron density can be a "one-electron" density, that is, an electron density constructed from some set of one-electron functions. This is a remarkable result since it exactly transforms the many-body problem into a one-electron problem provided that the terms in Eq. (4.22) can be evaluated. This, of course, is the difficult part: the exact form of the kinetic energy and electron-electron interaction terms are unknown. The most commonly used approach is the Kohn-Sham method (Kohn and Sham, 1965). In this approach, the kinetic energy term of Eq. (4.22), T[p(r)], is replaced by the kinetic energy of a system with no electron-electron interactions, T.,[p(r)], but at the same ground state electron density of the original system (with electronelectron interactions). In this way the (newly defined) kinetic and electron-ion interaction energies can be computed from the one-electron eigenvalues, ei, of a system noninteracting electrons moving in the new external potential, v.,(r), of the noninteracting system,
15 2
J.P. LaFemina
n/2
L ' K E - e l I*"
J
4rter
T,[p(r)] +f "
9(r)v~(r)dr = "
4xE~
.=
2e i
(4.23)
with p(r') v~(r) = v(r) + S irr'i,,dr' +
Vxc(r)
(4.24)
The electron density is computed from the associated one-electron eigenfunctions, ~i(r), n/'2
9(r)-
2 ~
(4.25)
I~i(r)l z
i=1
and the Hartree energy (also referred to as LCAO methods,
E,I,RCN) I -
J[p(r)])
e2 o1 . -2-1 f Ir-r'l Jl o(r) ] - I 9(r)p(r')drdr'4rte
is computed as in the SCF-
(4.26)
This leaves only the exchange and correlation energies to be evaluated, along with the correction needed to account for the neglect of the electron-electron interactions in the kinetic energy computation. All of these terms are collected together as an effective one-electron term, referred to as the "exchange-correlation" energy, and given by Ex~,l-- i - T i p ( r ) ] - T, lp(r)! + V ~ l p ( r ) l - Jlp(r)l
f Vxcip(r.)lp(r)dr (4.27)
where Vxclp(r)] is the exchange-correlation potential. The specification of the exchange-correlation potential is, arguably, the most difficult aspect of applying the Kohn-Sham density functional method. One widely used form of this functional, based on the uniform-density electron-gas model, is 1/3
I/3
(4.28)
If or is set to 2/3 then Eq. (4.28) is the K o h n - S h a m exchange potential (Kohn and Sham, 1965), and contains no electron correlation effects. If or is set to 1, then Eq. (4.28) is the Slater exchange-correlation potential (Slater, 1974) which contains both average exchange and correlation interactions. The parameter o~ can also be
Theory of insulator su~. ace structures
153
treated as an adjustable parameter during the computation as is done in the so-called Xc~ methods (Parr and Yang, 1989). This form for the exchange-correlation potential has worked well for many systems. The development of new, sophisticated exchange-correlation potentials has also continued. The details of these developments can be found in the review of Salahub and Zerner (1989). To summarize, the density functional methods use the electron density, rather than the wavefunction, as the system variable of interest. As a result, it is possible to exactly transform the many-body problem into an effective one-electron problem. In this effective potential, the kinetic energy is computed for a system of non-interacting electrons. The kinetic energy correction (due to the fact that real electrons interact) is then lumped together with the exchange and correlation interactions into an effective, one-electron exchange-correlation potential. The Hartree energy is computed explicitly as in the SCF-LCAO methods.
4.2.2.3. Tight-binding methods The spirit of the empirical tight-binding methods (Slater and Koster, 1954) is simple: none of the terms in Eq. (4.10) are evaluated explicitly. Instead, the SchrOdinger equation (Eq. 4.2) is recast in matrix form (Eq. 4.7), and the elements of the Hamiltonian matrix (Eq. 4.9) are treated as adjustable parameters, fit at the high symmetry points of the first Brillouin zone to either experimental information, or the results of ab initio calculations for the bulk system. These parameters are then assumed to be transferable for use in computing the properties of surfaces. The range of these interactions is usually assumed to be nearest-neighbor or next-nearest-neighbor only, and the interaction matrix elements are assumed to have some parametric dependence upon the internuclear separation d (commonly a d -2 dependence for sp-bonded semiconductor systems) (Harrison, 1976, 1981). The assumption of transferability (from bulk to surface) for the Hamiltonian matrix elements will be valid provided that the charge density at the surface is not significantly different than the bulk. The success of this assumption (see reviews by LaFemina (1992) and Duke (1992)) for the covalently bonded semiconductor systems is an a posteriori justification for its use. For the more "ionically" bonded insulating systems, this assumption should be equally valid. The most important aspect of the empirical tight-binding methods, however, is that the electron-electron interactions are never computed, but are included empirically through the parameterization of the Hamiltonian matrix elements. This has consequences in the way the total energy is evaluated, since the electronic energy can only be expressed as the sum of the one-electron eigenvalues. As we have seen in the previous sections (Eq. 4.21), this sum overestimates the electronic energy by double-counting the Hartree and exchange interactions. Because this extra energy cannot be explicitly accounted for, it is usually lumped together with the nuclearnuclear repulsion, and the total energy is rewritten as E TOT
where
I~
J -" z..~b,~lll
j"~"
l
J
(4.29)
Ebs is the sum of the occupied one-electron eigenvalues (commonly termed
154
J.P. LaFemina
the "band structure" energy) and U is a pair potential (i.e., a potential that depends only upon the pairwise interactions between the atoms in the system) representing the nuclear repulsion and electron-double counting terms. Many forms have been proposed for this pair potential, but the most widely used is the form proposed by Chadi (1978, 1979, 1983, 1984), based on a harmonic, short-range, force-constant model, UIR t'."-(N~I J = ~ (U IE/j + U2c2)
(4.30)
i~j
where Eij is now defined to be the fractional change in the internuclear distance between atoms i and j. The constants U~ and U2 are determined by imposing the following conditions on the total energy functional (Eq. 4.29), 0EToT Ov
- 0
(4.31)
and c)2E,ro y
(4.32)
V - - - B OV 2
where V is the volume and B is the bulk modulus. What these conditions require, in essence, is that the minimum in the total energy functional (Eq. 4.29) occur at the correct bulk lattice constant, and that the curvature of the total energy functional near the minimum be correct. To summarize, in the empirical tight-binding methods none of the interactions in the system are computed explicitly, in contrast to the SCF-LCAO and density functional methods. Instead, they are included empirically, through the parameterization of the Hamiltonian matrix and the pair potential, U. 4.2.3. Classical potential models
The empirically determined classical potential models are, by far, the most widely used in the study of "ionic" insulator surface and interface structure (Colbourn, 1992). Typically, these potentials are central force, two-body potentials composed of three parts; a Coulombic interaction, a short-range interaction, and an interaction that accounts for the polarizability of the atoms. In some cases (for example, zeolites) a three-body term describing bond angle distortions has been found to be important and added to the potential (Colbourn, 1992). The Coulomb potential is simply the interaction between ions g and v with charges Z, and Zv
le /
Vc ...."'mb{R(N)}-- 4g~,,
N
Z laity
Ru v
(4.33)
Theory ofinsulator sud'ace structures
155
The short-range potential is typically given by the Buckingham potential which comprises a Born-Mayer repulsive, and a van der Waals attractive term (Baetzold et al. 1988; Born and Mayer 1932, Colbourn 1992) lh?(N)~= y__~ A~ exp
Vsh,,,. t
-
(4.34)
where A0v, P0v, and C~,vare empirical constants to be fit to either experimental data or the results of more sophisticated computations. Finally, the polarizability of the atoms is usually represented by a shell model (Dick and Overhauser, 1958), in which the atom is modeled as a spherical shell of negatively-charged electron density that is harmonically coupled, with a force constant k (also empirically determined), to a positively charged ionic core. The polarization occurs when a differential displacement, W, occurs between the core and the charged shell. The polarization potential is then given by N ....
=
Z
1
2
(4.35)
IJ~v
In theory the sums in Eqs. (4.33) and (4.34) are over all atom pairs in the system. In practice, however, the short-range potential (Eq. 4.34) is typically limited to nearest- or next-nearest-neighbors only (Colbourn, 1992). The long-range character of the Coulomb potential (Eq. 4.33), however, is more problematic since the Madelung (1909, 1910a,b) sum shown in Eq. (4.33) is conditionally convergent. That is, the answer you get depends upon how you truncate the summation. This can be avoided, however, by using the summation techniques of Ewald (1921) and Parry (1975, 1976). Nevertheless, the computation of the Coulombic interactions is easily the most computationally intensive process for these models. The question of how best to determine the empirical parameters that appear in Eqs. (4.33)-(4..35) is an interesting one. Equation (4.33) is included in this discussion since the issue of whether it is best to use formal charges (i.e., +2 for MgO) or some other "effective" charge remains open. There are essentially two approaches that can be taken in determining the parameters for these classical potentials: fitting to bulk experimental data (the more widely used) or fitting to the results of some "more sophisticated" computations for the bulk material. As in the empirical quantum-mechanical models, these parameters are then assumed to be transferable to the computation of surface properties. In the first approach, a variety of experimental information can be used, but typically the potentials are required to reproduce the correct bulk lattice parameters, along with selected bulk elastic constants, and phonon (crystal vibration) energies and dispersion curves (Colbourn, 1992). Even with the impressive agreement with bulk experimental data for these potentials, the results for surface properties remain very sensitive to both the exact form of the potential and the procedure used for its parameterization. This point will be discussed further in w 4.3.
156
J.P. LaFemina
The second approach, fitting the potential parameters to the result of "more sophisticated" computations, has also had mixed success. This is, in part, because the "more sophisticated" computations were not sufficiently accurate to provide an adequate potential. Typically, these computational results emerged from some variant of the electron-gas model, and had, at best, only moderate agreement with experimental data (Colbourn, 1992). Recently, however, there have been efforts to fit the empirical potentials to the results of ab initio computations (Colbourn, 1992; Kunz ! 988). While the preliminary results of these efforts are promising, much work remains to be done before this becomes a proven method for parameterizing classical potentials.
4.2.4. Computational approaches to surfaces There are three commonly used approaches for computing the properties of surfaces: Green's function (or embedding) methods, slab computations, and cluster models. As stated in the introduction to this section, the main difference between these methods is the boundary condition used to model the semi-infinite surface. An ideal surface is infinite (extending out forever) in the two spatial dimensions parallel to the surface. The surface is also infinite in the third dimension (perpendicular to the surface) but only in one direction (from the surface into the bulk). Thus the surface is said to be semi-infinite. The surface is also periodic in these directions (again only in one of the directions perpendicular to the surface). That is, some collection of atoms (called the surface unit cell) can, through translations, map out the entire semi-infinite system. The Green's function, or embedding, techniques treat the semi-infinite nature of the surface exactly (and as a result are the most complex). Slab models treat the infinite (in the two dimensions parallel to the surface) two-dimensional nature of the surface properly, but have a finite thickness in the third dimension perpendicular to the surface. Finally, the cluster methods model the surface with a finite set of atoms. In principle, any of these approaches may be used in conjunction with any of the methods, quantum or classical. In practice, however, the Green's function techniques are usually used with quantum mechanical potentials, while the slab and cluster models have been used with both quantum and classical potentials. One embedding (though non-Green's function) technique that has been extensively used with the classical potentials in the study of defects is the Mott-Littleton approach (Mott and Littleton, 1938). In this method, the system of interest is divided into regions that are treated at various levels of approximation. Figure 4.1 illustrates one example of this type of approach. In Region I, which contains the defect (or surface, or step, or whatever feature is being modeled) the atomic interactions are computed explicitly for a given potential. Region II contains ions whose positions are fixed, but are allowed to polarize in response to the feature in Region I (Baetzold et al., 1988). Finally, the material in Region III is treated as a continuum dielectric. The main point, however, is that only the atoms in Region I are treated with the full potential and allowed to relax in response to this potential. Region I typically numbers several hundred atoms, which is why this approach is almost exclusively used with classical potentials.
T h e o ~ ~?i' insulator su~. ace structures
157
Region III (Continuum Dielectric) F!g. 4. I. Schematic indication of the Mott-Littleton approach for the computation of detect structures using empirically-determined classical potentials. Region I contains the defect of interest, and the atomic interactions in this region are computed explicitly for a given potential. Region II contains ions whose positions are fixed, and whose distortion from the ideal lattice positions is determined by the defect induced polarization. In region III the material is treated as a continuum dielectric. Green's function methods (Applebaum and Hamann, 1976; Fisher, 1991; Ellis et al., 1991), are typically a more complicated form of embedding techniques, in which the properties of some region containing the feature to be modelled, are computed with the explicit inclusion of the boundary conditions, in this case the semi-infinite nature of the surface. The Green's function contains all of the information about the eigenfunctions and eigenvalues of the Hamiltonian. Recalling the one-electron Schr(~dinger equations (Eq. 4.5), /X
H ~ , , ( r ) - E,,~,,(r)
(4.36)
where ~,,(r) are the one-electron eigenfunctions of the Hamiltonian, H, and E,, are the one-electron eigenvalues of ~n(r), the Green's function is defined as /k
HG(r,r',E) - E G(r,r',E) - 8(r,r')
(4.37)
It can be shown that the solution of Eq. (4.37) is
G(r,r',E) - ~
vn(r)v~(r') E - E.
(4.38)
n
Using complex analysis, the eigenvalues (or the poles of the Green's function) and eigenvectors (or the residues of the Green's function) can be obtained. Hence,
158
J.P. LaFemina
anything that can be learned from the eigenvectors and eigenvalues can be obtained from the Green's function. The Green's function contains all of the information about all of the possible solutions to the differential equation (Eq. 4.37). The great advantage of this approach is that the relevant subspaces of the Green's function can be manipulated, isolated, and solved for. As a result, complex boundary conditions (such as a semi-infinite surface) can be dealt with. These methods are, by their very nature, computationally complex (a feature which has limited their application). For surface studies, however, they allow for the unambiguous identification of surface bound states and resonances; i.e., those states which have their electron density localized at the surface. In slab computations, a finite number of atomic layer, periodic within the layer, are used to simulate the surface (Fig. 4.2) (Hirabayashi, 1969; Lieske, 1984). The slab, by definition, has two surfaces (at the top and the bottom of the slab) which are typically made equivalent by symmetry. The slab thickness is a critical parameter for determining surface properties. Because the slab has two surfaces (top and bottom), the computed surface properties are converged when the addition of layers in the center of the slab does not affect the results. This critical thickness will also be different for different properties since not all properties converge at the same rate. For example, surface energies can typically be computed with slabs 3-4 atomic layers thick. For quantum mechanical potentials, the computation of surface states (states whose electron density is localized at the surface), typically requires on the order of 8 atomic layers. This minimum thickness is defined as the number of atomic layers needed to make the surface states on the top and bottom surfaces non-interacting. The equilibrium surface structures are computing by relaxing the atoms (i.e., computing the forces on each atom from the potential and then moving the atoms so as to minimize the forces) in the vicinity of the surface. Depending upon the complexity of the potential, this may include only the atoms in the top few atomic layers or every atom in the slab. Finally, in cluster computations, the surface is modeled by a finite cluster of atoms, with no periodic boundary conditions whatsoever (Tsukada et al., 1983). The advantage to this approach was that the absence of translational symmetry, i.e., having a finite system, greatly simplified the computation and allowed for the use Infinite
and Periodic./"" ,, . /
/
Surface
~
~
-
-
z ~ . L/"y
iiii~i~ii~iii~i.~.~.~ii~liiiiiii-xiiiiiiiiiiiii~ii!i~i!iiii~i Surface
Fig. 4.2. Schematic illustration of the slab model used to simulate a surface. The slab is infinite and periodic in the two directions parallel to the surface (x and y) and finite in the direction perpendicular to the surface (z). Note that the slab has, by construction, two surfaces which are typically made symmetry equivalent.
Theory of insulator su.rface structures
159
of ab initio quantum-chemical potentials, for which computer codes already existed. Over the past decade, however, ab initio density-functional and SCF-LCAO codes which explicitly incorporate the effects of translational periodicity, have been developed. Therefore, although clusters are interesting in their own right, their use as models for surfaces is no longer state-of-the-art. Other disadvantages to this approach include the presence of edge effects (i.e., artifacts in the computation resulting from how the edges of the cluster are terminated) which makes the examination of the surface electronic structure nearly impossible. 4.2.5. Comparison o f the methods
Of all the methods described in this section the use of empirical classical potentials (Eqs. 4.33-4.35) is, by far, the most widespread. The potential is sufficiently simple so that large numbers of atoms (on the order of hundreds) may be treated explicitly, with several hundred more treated implicitly through the use of Mott-Littleton type approaches. As a result, these potentials have been used to study a wide variety of complicated defects, grain boundaries, interfaces, stepped surfaces, and adsorbed surfaces. Moreover, these potentials have been used in molecular dynamics simulations to investigate the time-dependent nature of these systems. The primary disadvantage of these potentials is their obvious neglect of the quantum-mechanical nature of chemical bonding. No electronic states are computed, and no localized (either at surfaces or at defects) electronic states can be identified. In addition, the sensitivity of the computed results to the form and parameterization of the potential makes the general applicability of these potentials questionable. The empirical tight-binding method, however, is based upon quantum mechanics. It is also computationally efficient, and able to treat a large of number of atoms (on the order of one hundred atoms). It is also sufficiently transparent, being formulated in terms of atomic-like orbital interactions, so that the description of surface atomic movements in terms of how these interactions change at the surface is possible. This, in turn, permits the examination of surface reconstruction mechanisms across homologous systems. And because the method is parametrized to bulk optical and photoemission data, the extrapolation from bulk to surface properties can be both reliable and quantitative (Duke, 1992; LaFemina, 1992). The major disadvantages of this method are that it is not self-consistent, and therefore its predictions must be calibrated against more rigorous methods which explicitly include electron-electron interactions; and that the investigation of new systems requires the determination of new basis/parameter sets. This can be particularly troublesome when the system of interest contains interactions for which there are no bulk experimental data (such as in the study of adsorbates), or for which no usable higher-level computation exists. In these cases, the empirical d -z scaling law is usually invoked to obtain a first-order approximation to the parameters. The remaining methods, semi-empirical and ab initio local-density functional and SCF-LCAO, are all self-consistent. As such, however, they all suffer from the problem of being computationally intensive because of the need to explicitly compute the electron-electron interactions. This typically results in the treatment of
160
J.P. LaFemina
systems with a limited number of atoms (on the order of tens of atoms). This implies either the use of a thinner slab, in which case the identification of the surface bound states and resonances becomes more complicated, or the use of a Green' s function technique or cluster calculation. The advantage to these methods is that, while computationally more difficult, they provide a first-principles determination of the surface structure independent of possible parameterization biases.
4.3. The structure of clean surfaces
4.3. I. Diamond (l I 1) surface Cubic carbon (i.e., diamond) has a band gap of approximately 5.5 eV, making it a good insulator. Diamond is also unusual because it exhibits covalent bonding, yet has a large bandgap. The study of diamond surfaces also provides an interesting contrast to the other Group IV elemental surfaces (i.e., Si and Ge) because of its propensity for forming multiple, Tt-bonds, which, as described in Chapter 6, are a common structural motif in semiconductor surface structures (Duke, 1992; LaFemina, 1992). Hence, understanding the surface and interface properties of diamond may yield fundamental and general insights into the factors which control semiconductor surface structure and chemistry (Pate et al., 1990). From a practical point of view, the structural and chemical properties of diamond surfaces are important to the growth of synthetic diamonds, and attempts to incorporate diamond interfaces into working electronic devices (Pate et al., 1990). The detailed study of diamond surfaces has been inhibited by a variety of experimental difficulties, including problems associated with the preparation of clean surfaces (Haneman, 1982). As most of the experimental and computational work has focused on the diamond (111) cleavage surface, we also will focus on this surface. Following cleavage and low temperature (<900~ annealing, a (Ix l) structure is formed which has been interpreted as being a hydrogenated surface, with each surface carbon atom saturating its dangling bond (Principle 1) by bonding to a hydrogen atom (Pate, 1986; Pate et al., 1990). Upon high temperature (>900~ annealing, a (2x2) LEED pattern is observed which has been interpreted as resulting from a superposition of patterns from the three distinct (2x l) orientations (Pate, 1986; Sowa et al., 1988). As indicated above, the formation of 7t-bonds, in conjunction with sp2-hybrid ized chains, is common to the (2• structures on the (100) and (111) surfaces of Si and Ge (see Chapter 6). Moreover, similarities in the surface electronic structure as probed by angle-resolved photoemission spectroscopy and angle-resolved twophoton spectroscopy suggest a common (2x l) structure for the Group IV (111) surfaces (Himpsel et al., 1981; Kubiak and Kolasinski, 1989; Pate, 1986). The stability of these structures is easily understood using Prihciple 1 of w 4.1: Saturate the dangling bonds. The formation of these structures allows the surface atoms to saturate their valences by forming three bonds to neighbors in the surface and near
Theory o.f"insulator surface structures
161
surface region (sp2-hybridization), and then forming a double, or r~-bond, between the surface atoms. In addition, Principle 3, F o r m a c h a r g e n e u t r a l s u r f a c e , is satisfied trivially by this elemental surface. Two different r~-bonded chains, shown schematically in Figs. 4.3 and 4.4, have been proposed [originally for Si( 111 )] and investigated for the C(111 )(2x 1) surface (see Derry et al., 1986; Iarlori et al., 1992; Zheng and Smith, 1991). The primary difference between these structures is that they originate from different ways in which the cleavage process is proposed to occur. Figure 4.3 illustrates the Pandy rt-bonded chain which arises from cleaving the material in the (l 1 l) plane that contains a single bond per unit cell; the so-called "single-bond scission" (Pandy, 1981). If, however, the surface is formed by cleaving along the plane which contains three bonds per unit cell (the "triple-bond scission"), the resulting rtbonded chain is illustrated in Fig. 4.4 (Haneman, 1987).
C (111)- (2 x 1) Single-Bond Scission
Fig. 4.3. Ball-and-stick model of the C(111)2xl Pandy r~-bondedchain structure resulting from single bond scission. From Duke (Chapter 6).
C (111)-~2x 1) Triple-Bond ~cission
Fig. 4.4. Ball-and-stick model of the C(I 1l)2xl Haneman7t-bonded chain structure resulting from triple bond scission. From Duke (Chapter 6).
162
J.P. LaFemina
The qualitative, n-bonded chain, nature of the C(111)(2• 1) surface is generally agreed upon. And based upon comparisons with Si(111)(2• and Ge(111)(2• surfaces it is generally believed that the single-bond scission model is correct (see Chapter 6 for a more complete discussion). The details of this structure, however, are contentious. In particular, whether (and to what extent) the n-bonded chain is either dimerized, buckled, or both dimerized and buckled (Iarlori et al., 1992). These issues arise because for the unrelaxed n-bonded chain models (i.e., where all of the C atoms in the chain lie in the same surface plane and all C-C bond lengths within the surface chain are equal to the bulk C-C bond lengths) the surface is metallic, contrary to the photoemission data (Himpsel et al., 1981; Pate, 1986) and to Principle 2: Form an insulating surface. Buckling of the chain results in a charge transfer between chain atoms and opens a gap in the surface state eigenvalue spectrum. Dimerization, in which the C-C bond lengths along the chain alternate in length (... short-long-short-long ...) will also make the surface insulating and is analogous to the Peierls distortion in one-dimensional systems (Peierls, 1955). Lastly, strong electron correlation effects (i.e., antiferromagnetic ordering of the chains n-electrons) can open a gap ion the surface state eigenvalue spectrum with neither buckling nor dimerization. Carbon behaves differently in small molecules (and polymers) from the other Group IV elements, favoring the formation of multiple bonds and disfavoring large charge transfers. Hence, an unbuckled and dimerized n-bonded chain is expected for the C ( l l l ) ( 2 x l ) surface. This is in contrast to the S i ( l l l ) and G e ( l l l ) ( 2 x l ) n-bonded chains which are thought to be buckled and undimerized (Duke, 1992; LaFemina, 1992). Experimentally, the structure of the C(111)(2• surface has been limitedly probed by both LEED intensity analysis (Sowa et al., 1988) and by medium-energy ion scattering (Derry et al., 1986). In both of these analysis only the Pandy n-bonded chain was considered, and the LEED analysis considered only a limited range of chain relaxations. The results of these studies are inconclusive, with the LEED analysis indicating a slightly dimerized n-bonded chain, and the ion-scattering results indicating a strongly dimerized chain. Computationally, this issue is no clearer. Ab initio density functional pseudopotential calculations (Vanderbilt and Louie, 1984a,b) find unbuckled and undimerized (and metallic) chains, while recent ab initio density functional pseudopotential molecular dynamic computations (Iarlori et al., 1992) find an unbuckled and dimerized chain. Semiempirical Hartree-Fock computations find either the unbuckled dimerized chain (Dovesi et al., 1987) or the buckled dimerized chain (Zheng and Smith, 1991) to be most stable. Finally, self-consistent tight-binding computations (Chadi, 1989) find buckled and undimerized chains, while non-selfconsistent tight-binding computations (Chadi, 1989) find unbuckled and strongly dimerized chains. In most of these studies, with the exception of the C-C dimer bond length, near-neighbor bond lengths are found to be within 10% of their bulk values, in agreement with Principle 5: Conserve bond lengths. Lastly, we briefly discuss the C(111)(2x l) surface in the context of Principle 3: Don't forget about kinetics. The Si(ll l)(2xl) and G e ( l l l ) ( 2 x l ) structures are metastable (Duke, 1992; LaFemina, 1992). They form because the atomic motions
Theory of insulator su~. ace structures
163
necessary for their formation are kinetically accessible during cleavage conditions. Upon annealing at moderate temperatures (ca. 400~ however, these surfaces reconstruct to form what are believed to be their thermodynamically stable structures, the Si(l 11)(7x7) and the Ge(111)c(2x8). For the C(111) surface, however, the (2x 1) structure is believed to be the thermodynamically stable surface, provided the surface is free of hydrogen. This difference is also believed to be related to the propensity for C to form multiple bonds and avoid large charge transfers, since both the Si(l 11)(7x7) and Ge(111)c(2x8) structures involve large charge transfers between surface and near-surface atoms. In summary, while the qualitative, rt-bonded chain, nature of the C ( l l 1)(2x1) structure is established, and in accord with our five principles of surface structure, much work, both experimental and computational, is needed before the structure is unambiguously and quantitatively known.
4.3.2. Rocksalt (001) surface The rocksalt lattice is perhaps the most simple crystal structure. It is also one of the most prevalent, with the alkali and silver halides and pseudohalides; alkaline earth oxides, sulfides, selenides, and tellurides; and a variety of transition metal monoxides crystallizing in this lattice (Wyckoff, 1963). In addition, many other materials (e.g., II-VI and III-V compounds) exhibit the rocksalt structure as a high-pressure phase (Liu and Bassett, 1986). Unfortunately, the amount of detailed experimental or theoretical information available about these surfaces is scarce. Several excellent reviews by Henrich, (1983, 1985, 1989) are available for oxides while the state of surface structure computations has been recently reviewed by Colbourn (1992) and LaFemina (1994). In the bulk rocksalt structure the atoms are octahedrally coordinated. The most stable surface of this lattice is the nonpolar (001) cleavage face shown schematically in Fig. 4.5. The rocksalt structure materials present an interesting problem in the theoretical interpretation of the chemical bonding. These materials are presumed to be ionic, that is comprising a lattice of positively charged cations and negatively charged anions, held together by Coulombic forces. The reason for this conceptual framework is that it is not possible to construct two-electron bonds for these materials using only the valence electrons. If we consider MgO, for example, the Mg atom contributes two electrons per MgO unit while the O atom contributes its six valence electrons for a total of eight electrons per MgO unit. Each MgO unit in the crystal contains six bonds, leaving each bond with only 4/3 electrons. In the ionic bonding framework, however, the Mg atom's two electrons are transferred to the O atom giving it a filled shell, "Ne-like", electronic structure and leaving the Mg atom also with a "Ne-like", filled-shell electronic structure. In this framework, concepts such as "dangling bond" charge density would seem to be inappropriate. Yet, recently, these concepts have been shown to be useful in understanding a wide range of oxide surface structures (LaFemina, 1994). The reason is simply that in the limit of a "completely ionic" material (defined here as one for which two-electron bonds cannot be constructed) the "dangling bond" charge density
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J.P. LaFemina
Rocksalt (001) Surface
T
[OOl]
Fig. 4.5. Ball-and-stick model of the rocksalt (001) surface relaxation. The large open circles are anions and the smaller filled circles are metal cations. The surface rumpling shown has been exaggerated for clarity.
is localized on the anion, which has a filled shell. The results of numerous electronic structure computations bear this out (LaFemina, 1994). Consequently, for the nc}n-polar faces, the dangling bond saturation and insulating surface principles are satisfied. The non-polar faces are charge neutral and autocompensated (satisfying Principle 4). The driving force for surface relaxation, or reconstruction, then comes from the energy stabilization associated with the rehybridization of the anion charge density in response to the lowered coordination at the surface. This can only {}ccur, however, if the surface atoms can move into more electronically favorable hybridizations without significantly distorting local bond lengths (Principle 5). At the rocksalt (001) surface (Fig. 4.5) the atomic coordination is reduced from six to five, with each surface atom bonding to four neighbors in the surface plane and to one atom in the plane directly below the surface. The symmetry of the (001) surface dictates that any relaxation (preserving the (1• symmetry of the surface) of the surface atoms occur perpendicular to the plane of the surface. It has long been thought (Verway, 1946) that because the anions and cations in the lattice have different polarizabilities (i.e., the ease with which the charge densities surrounding the cations and anions are able to distort is different) the surface should rumple. This rumple would consist of the anions and cations exhibiting a differential relaxation perpendicular to the surface plane. Because the surface atoms are fivefold coordinated, they cannot move very far before some near-neighbor bonds become strained. As a result of this inability of the surface atoms to undergo approximately bond-length conserving motions, the relaxation is inhibited. Very few rocksalt structure materials have had the structure of the their (001) surfaces determined quantitatively. The MgO (001) surface, however, has been extensively studied, both experimentally and theoretically and the results for this surface are representative of this class of material. Table 4.3 contains a compendium of the experimental and theoretical results for MgO (001). The relaxation parameters in Table 4.3 are given only for the surface layer.
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Table 4.3 Comparison of the structural parameters for the relaxed MgO(001) surface. R is the rumple, or differential displacement of the surface anion and cation in the direction perpendicular to the (001) surface. A positive R indicates the surface anion is displaced outward from the surface while the surface cation is displaced inward towards the surface (as shown in Fig. 4.5). C is the contraction (if negative) or expansion (if positive) of the first interlayer spacing. Units are percent of the ideal interlayer spacing (2.1/~,). R +3 +1 +5 to +9 +2 to +3 +ll +3 0 to +5 0 to +5 +2 _+ 2 0 0 to +5 +6 0 +8 _+ 1 +0.5 _+ 1 (a) (b) (c) (d) (e)
C
Method (year)
Reference
-2 0 0 -2 to 0 +1 -0.5 -3 to 0 -3 to 0 0+ 1 0+2 0 _+ 1 0 0 to +3 -15 + 3
TBTE (a) (1991) ab initio HF (b) (1986) Shell Model (1978) Shell Model (1979) Shell Model (1985) Shell Model (1985) LEED (c) (1976) LEED (1979) LEED (1982) LEED (1983) LEED (1991 ) RHEED ~d) ( 1981 ) RHEED (1985) He Diffraction (1982) ICISS Ce)(1988)
LaFemina and Duke Caus?a et al. Welton-Cook and Prutton Martin and Bilz Lewis and Catlow de Wette et al. Kinniburgh Prutton et al. Welton-Cook and Berndt Urano et al. Blanchard et al. Gotoh et al. Maksym Reider Nakamatsu et al.
Tight-binding total energy computation. Ab initio Hartree-Fock computation. Low-energy-electron-diffraction intensity analysis. Reflection high-energy electron-diffraction. Impact-collision ion-scattering spectroscopy.
The first thing that is clear from results in Table 4.3 is that the MgO(001 ) surface relaxation - - a surface rumpling and a change in the first interlayer spacing is small; typically on the order of a few percent of the lattice spacing. The second observation is that it is difficult experimentally to determine changes in surface atomic positions on this order; the uncertainties in the measurements being the same order of magnitude as the displacement. Finally, the data in Table 4.3 reveals the sensitive dependence of the computed surface relaxation parameters computed via classical potential (i.e., "shell") models on the form and parameterization of the potential (Colbourn, 1992), in particular the short-range part of the potential (Tasker, 1979). This is precisely the part of the classical potential that suffers most from the neglect of the quantum mechanical nature of the bonding. Consequently, there is a need for both reliable, quantum-mechanically derived, short range potentials for use in classical dynamics simulations of complex structures (such as grain boundaries), as well as for reliable, quantum mechanical computations of equilibrium surface structures. Of course, the need for quantitative, experimental information is the most pressing, and its gathering is likely to be the rate-determining step in the entire process of developing quantum mechanical descriptions of these materials.
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4.3.3. Rutile (110) surface The rutile crystal structure is tetragonal, comprising octahedrally coordinated cations and three-fold coordinated anions. The most widely studied futile materials are TiO2 and SnO 2. Both materials have wide-ranging technological uses for which surface interactions play a dominant role, including catalysis (Berry, 1982; Wold, 1993) and chemical sensor (G6pel et al., 1988; Semancik and Cox, 1987) applications. Consequently, the discussion in this section will focus on these two materials. The most stable, and well-characterized, rutile surface is the ( l l 0) surface, and we will direct the discussion in this section towards this surface. Unfortunately, rutile crystals, unlike diamond and rocksalt materials, do not cleave, but fracture (Henrich, 1985). As a result, the surface stoichiometry and structure are highly dependent upon the processing conditions used to prepare the surface (Principle 3). Low-energy-electron diffraction and Auger spectroscopy studies of the SnO 2 (110) surface indicate that, depending upon the annealing temperature, the surface O/Sn ratio can vary from 0.49 to 0.76, and the surface can display p ( l x I ), c( I x I ), p(4x2), p ( 4 x l ) and amorphous structures (de Fr6sart et al., 1983). Recipes, however, for preparing stoichiometric, nearly perfect SnO2 surfaces have been published (Cox and Fryberger, 1990). The (110) surfaces of rutile TiO2, on the other hand, forms stc~ichi~metric, p( Ix l) structures under a variety of experimental conditions (Henrich, 1985). As a result, much more work has been performed on single-crystal TiO 2 (I 10) surfaces. A schematic indication of the (110) surface is given in Fig. 4.6. Two different p ( l • ) terminations have been studied. The stoichiometric surface, shown in panel (a) of Fig. 4.6 has a stoichiometric (two oxygens for every tin) surface unit cell. The surface anions, commonly referred to as the "bridging" anions (see Fig. 4.6), are two-fold coordinated with one dangling bond. There are two kinds of surface cations in the surface unit cell, one of which is five-fold coordinated with one dangling bond. The other surface cation maintains its bulk-like octahedral coordination. The surface is insulating (Principle 2) and autocompensated (Principles I and 4). The topology of the surface, however, does not allow the atoms to move significantly away from their bulk positions without straining near-neighbor bonds (Principle 5). Consequently, the p( 1• ) relaxation is expected to be small, as was found by the tight-binding total energy computations of Godin and LaFemina (1993) for the stoichiometric SnO 2 (110) surface, illustrated in panel (a) of Fig. 4.7. No other computational or experimental surface structural determinations for these surfaces have been published. The second termination considered in the literature is commonly referred to as the "reduced", or oxygen deficient, surface. This surface results from the bombardment of the stoichiometric surface with argon ions, which preferentially removes the oxygen atoms (Henrich, 1985; Rohrer et al., 1992). p ( l x l ) LEED patterns have also been obtained from this surface, which is believed to comprise the stoichiometric surface minus the bridging oxygen atoms. This reduced surface termination is illustrated in panel (b) of Fig. 4.6. There are several problems with this simple interpretation of the reduced surface. In this interpretation half of the surface
Theory of insulator su~. ace structures
167
[i 10][001] S= ['110]
(a)
[li~ [ool] v _~[ilo]
(b) Fig. 4.6 . Ball-and-stick models of the truncated bulk (a) stoichiometric and (b) reduced rutile (110) surface. Open circles are anions, filled circles are metal cations. From Godin and LaFemina (1993).
cations are four-fold coordinated with two dangling bonds, and half are five-fold coordinated with one dangling bond. The surface anions, however, retain their bulk, three-fold coordination, with no dangling bonds. As a result the cation dangling bonds are partially occupied, violating the autocompensation principle (Principle 4). As a result, the surface is not charge neutral, but carries a net negative charge since it is oxygen deficient. The question is then: Why is surface stable? The answer
168
J.P. LaFemina
[iI~[ooI] v _[iio] -
(a)
[iI~[ooI] v _-[iio]
(b) Fig. 4.7. Ball-and-stick models of the relaxed (a) stoichiometric and (b) reduced rutile (110) surface. Open circles are anions, filled circles are metal cations. From Godin and LaFemina (1993).
is that the structure shown in panel (b) of Fig. 4.6 is simplistic, and that the true structure of the surface which results from argon ion bombardment most likely contains subsurface defects which autocompensates the surface making it charge neutral. This idea is supported by experimental data which find an appreciable density of occupied states in the bandgap for the reduced TiO2 (110) (Henrich, 1985; Zhang et al., 1991) surface. Data for the reduced SnO2 (110) surface, however, shows no density of occupied states in the gap (Cox et al., 1988; Egdell et al., 1986). No computations of defected "reduced" rutile (110) surfaces have been published.
Theory of insulator sueace structures
169
4.3.4. Perovskite (100) surfaces The cubic perovskite crystal structure has the general formula ABC3 (e.g., SrTiO 3, or KMnF3). It comprises small A atoms surrounded by twelve C atoms, and larger B atoms octahedrally coordinated by the C atoms. The most studied surface is the (100), which, for the perovskite oxides, is widely used as a substrate for thin films (Hikita et al., 1993; Liang and Bonnell, 1993). Materials in this structure, like the rocksalt structure materials, are ionic in the sense that it is not possible to form two-electron bonds from the valence electrons of the constituent atoms. The (100) surface has two possible terminations, illustrated in Fig. 4.8. Both terminations are non-polar (Principle 4) and insulating (Principle 2). As a result the valences of the surface atoms are satisfied, saturating the dangling bonds (Principle 1). These stoichiometric, (1• terminations are typically the result of vacuum fracture, followed by annealing in oxygen and so depend upon the processing history of the sample (Principle 3). The exact nature of the surface relaxation or reconstruction therefore will depend upon whether the atoms can move into some electronically more favorable local bonding environment (due to their reduced surface coordination) while approximately conserving near neighbor bond lengths (Principle 5). The two (100) terminations are classified by the type of atoms which reside in the surface plane. The first termination, shown in panel (a) of Fig. 4.8 is the BC2 or Type I termination. It comprises five-fold coordinated B cations (four neighbors in the surface plane and one in the plane directly below the surface) with one dangling
Fig. 4.8. Schematic illustration of the ideal ABC3 perovskite (100) surfaces. Two {100} steps to (100) terraces are shown. The uppermost terrace represents the Type II AC (100) surface, while the second uppermost terrace represents the Type I BC2 (100) termination. Large open circles are type C anions, large shaded circles are type B cations, and small circles are type C cations. The degree of shading indicates the depth of the atoms below the surface plane. From Henrich (1985).
170
J.P. LaFemina
bond, and four-fold coordinated C anions (two B ligands in the surface plane and two A ligands in the plane directly below the surface) with two dangling bonds. The second termination is the AC or Type II termination, and is illustrated in panel (b) of Fig. 4.8. It comprises eight-fold coordinated A cations (four neighbors in the surface plane and four in the plane directly below the surface) with four dangling bonds, and five-fold coordinated C anions (four A ligands in the surface plane and one B ligand in the plane directly below the surface) with one dangling bond. It is clear from the topology of these surfaces that there are no approximately bond-length-conserving motions of the surface atoms. Consequently, we expect the surface to undergo small atomic motion relaxations. Moreover, the symmetry of the surface dictates that ( I x I ) relaxations be restricted to motions perpendicular to the surface plane, hence we expect a rumpling of the surface analogous to the rocksalt (001) surface relaxation. This is consistent with recent reflection high-energy electron diffraction (RHEED) studies of the TiO2 (Type I) and SrO (Type II) terminations of SrTiO3 (100). The RHEED determined structures for both surface terminations are shown in Fig. 4.9. As indicated in Fig. 4.9, for both the TiO2 and SrO terminated (100) surfaces the relaxations comprises a movement of the surface anions out from the surface and a movement of the surface cations in towards the bulk. A similar relaxation has been computed by Reiger et al. (1987), using a classical potential model, for the (100) surfaces of KMnF 3 and KZnF 3. Because the stoichiometry and structure of these surfaces are dependent upon the sample processing techniques used to prepare them, a variety of structures a
o.lo A t--
1.95 + 0.07/~[ 1.95 + 0.05/~,m (a) Sr 9 9 Ti 0.16 A
1.95 + 0.10 ' ~ I
)
1.95 + 0.05/~ 1 (b)
Fig. 4.9. Ball-and-stick models of the relaxed (a) TiO2 and (b) SrO terminated (100) surfaces of perovskite structure SrTiO3. From Hikita et al. (1993).
Theory qfinsulator su~. ace structures
171
Fig. 4.10. Ball-and-stick model (top view) of the (2• ordered oxygen vacancy-Ti 3§ structure on the TiO2 terminated (100) surface of perovskite structure SrTiO3. From Matsumoto et al. (1992).
possible. For SrTiO3, a (2x2) structure has been identified on the TiO 2 terminated (100) surface and associated with the formation and ordering of surface defects of the type Q-Ti3+-O (Q: oxygen vacancy) as illustrated in Fig. 4.10 (Cord and Courths, 1985; Henrich et al., 1978; Matsumoto et al., 1992). The change in the charge state of the two surface Ti atoms is a direct result of surface autocompensation (Principle 4). The topology of this reduced surface still prohibits any bond-length-conserving motions of the surface atoms, so the relaxation is expected to be small (Principle 5).
4.3.5. Corundum surfaces The corundum oxides, led by corundum itself, ~-AI203, are perhaps the most technologically important class of oxide materials. Some are excellent catalysts, while cz-alumina is one of the most widely used substrates for the growth of thin metal, semiconductor, and insulator films for both basic research and microelectronic applications. The trigonal corundum lattice, M203, comprises cations, M, in a distorted octahedral environment, and oxygen anions in a distorted tetrahedral environment. Ilmenite, ABO3, is a related structure in which cations A and B replace the metal ions, M, equally. Of the corundum (or Ti203, V203, ~-Fe203) and ilmenite (LiNbO3, LiTaO3) oxides that have had their surface properties examined, only the surfaces of or-alumina have been studied in any detail both experimentally and computationally. The discussion in this section, therefore, will focus on the ct-alumina surfaces, although the conclusions concerning surface relaxations should be generally applicable. A variety of ct-alumina surfaces, exhibiting a variety of surface structures, have been studied experimentally (Guo et al., 1992; Henrich, 1985). These structures are
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J.P. LaFemina
of course a function of the sample processing conditions (Principle 3). For example, the (0001) basal plane exhibits a (1• structure in both air and vacuum when annealed at low temperatures (<1250~ Annealing to higher temperatures results in the more complex (ffx~f3-)R30 ~ and (3~x~/-31)R9 ~ structures. The (IT02) (1012) and (1123) faces, on the other hand, exhibit the simpler (2• and (4• structures following high-temperature anneals (Guo et al., 1992; Schildbach and Hamza, 1993). The (1120) surface has been studied by low-energy-electron-diffraction and transmission electron diffraction (Schildbach and Hamza, 1992" Susnitzky and Carter, 1986), as well as by a variety of reflection techniques (Yao et al., 1989). (I• (1• and (12• reconstructions have been reported. As discussed for in the previous sections, the question of surface termination is an appropriate one for all of the cz-alumina surfaces, and we can use the autocompensation principle (Principle 4) to guide us. Like many of the oxides discussed in this review, cz-alumina fractures, hence the question of surface stoichiometry and surface termination is relevant. Unfortunately, no definitive experimental work on determining surface terminations for any cz-alumina surfaces has been reported. Theoretically, there have been few studies of these surfaces, and only in the density-functional work of Guo et al. (1989) was the question of surface termination explicitly considered by computing the cleavage energies for the three possible, symmetry-inequivalent, (0001) and (1102) terminations shown in Fig. 4.11. These cc~mputations indicated that for the (0001) surface (panel (a) of Fig. 4.1 I ), cleavage between the AI planes (labelled C in Fig. 4.1 l(a)) was most stable. For the (I 102) surface (panel (b) of Fig. 4.11), the most stable cleavage was computed to be between O planes (labelled A in Fig. 4.1 l(b)). It is interesting to note that of all the possible surface terminations, the ~nes c~mputed t~ be m~)st stable are those for which the cleavage produces two equivalent, charge-neutral surfaces (Principle 4). On this basis it is easy to understand why these are the most energetically favorable cleavages. Guo et al. (1989) embodied this idea in their cc~ncept of surface building blocks, which comprise charge-neutral, repeating units. They note that the most stable cleavages occurred at the boundaries of these building blocks. This is a conceptual framework for surface stoichiometries similar tc) that of autocompensation. One shortcoming of these computations, however, is that the atomic positions c)l the surface atoms were not optimized, but constrained to remain at the bulk values. The computed energetics, as recognized by the authors, will undoubtedly be affected by any surface relaxation or reconstruction. The magnitude of this effect, however, is unlikely to change, qualitatively, the energetic ordering of the different surface terminations. Turning to the question of what possible surface relaxations or reconstructions may occur, we need to consider the topology and local chemical bonding since we have already determined that these surfaces are autocompensated (Principle 4) and insulating (Principle 2). For the (1102) surface, the surface A1 atoms are five-fold coordinated and the surface O atoms two-fold coordinated. In this respect, the surface topology is similar to that of the rutile ( l l 0) surface (see w 4.3.3) with no bond-length conserving motions of the surface atoms possible. Hence, a similar u
m
173
Theory of insulator su~. ace structures
l[
ooo~] [1 TOO]
A B C
(a)
[1102]
T _
v
A B C
[11'>0]
(b) Fig. 4.11. Illustration of the side views of the corundum (0:-A1203) (a) (O001) and (b) ( 1- 102) surfaces. The top view is of one surface building block comprising (a) three atomic layers of AI-O-AI for the (()()() 1)surface and (b) five atomic layers of O-AI-O-AI-O for the (1-102)surface. Open circles label O atoms, and the large and small filled circles label A1 atoms on the top and bottom layers. The solid lines mark the surface unit cells. The three unique ((~01) and (1-102)cleavage planes are indicated by arrows and labelled A, B, and C. From Guo, Ellis and Lam (1992).
r u m p l i n g relaxation is expected. The (000 l) surface is m o r e interesting since both the surface AI and O atoms are three-fold c o o r d i n a t e d . This t o p o l o g y allows a b o n d - l e n g t h c o n s e r v i n g m o t i o n in which the surface AI atoms can m o v e d o w n t o w a r d s the bulk into an a p p r o x i m a t e l y s p 2 hybridization, while the surface O atoms " p u c k e r " out from the surface into a distorted p y r a m i d a l hybridization. A b i n i t i o H a r t r e e - F o c k c o m p u t a t i o n s (Caush et ai. 1989), of thin slabs ( 1-4 layers) of the (0001) surface, in which the positions of the surface atoms p e r p e n d i c u l a r to the surface were o p t i m i z e d , have d e m o n s t r a t e d that the surface AI atoms relax to positions on the o r d e r of 0 . 4 - 0 . 5 ~ away from their t r u n c a t e d bulk positions. U n f o r t u n a t e l y , the positions of the surface O a t o m s were not o p t i m i z e d . R e c e n t local density functional p s e u d o p o t e n t i a l c o m p u t a t i o n s ( M a n a s s i d i s et al., 1993), also on thin slabs (3 layers), find a similar relaxation, a l t h o u g h the m o v e m e n t o f the o x y g e n atoms is significant.
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J.P. LaFemina
4.3.6. Silica surfaces
The interest in silica (SiO2) and silicate materials is widespread. Geologically, silica and silicate minerals (including zeolites) make up the greater part of the earth's mantle (Liebau, 1985; Liu and Bassett, 1986). Interest in silica glasses is motivated by their potential application as a nuclear waste storage medium (Susman, 1990) and as catalytic substrates (Garofalini, 1990). Amorphous silica is of obvious importance to the microelectronics industry (Pantelides, 1978; Pantelides and Luckovsky, 1988). The structural complexity of these materials and the intense interest in their macroscopic chemical and physical properties has also led to the development of classical force fields for molecular dynamics simulations (Silvi, 1991 ) Crystalline SiO2 exists in nine different allotropes (Wycoff, 1963): c~- and ]3-quartz (Wright and Lehmann, 1981), oE- and [3-cristobalite (O'Keefe and Hyde, 1976), c~- and ]3-tridymite (Leadbetter and Wright, 1976), coesite (Geisinger et al., 1987; Smyth et al., 1987), keatite (Shropshire et al., 1959), and stishovite (Hill et al., 1983; Stishov and Popova, 1961). All of the allotropes, with the exception of stishovite, consist of four-fold coordinated silicon atoms, and two-fold coordinated oxygen atoms. Stishovite, on the other hand, has the futile crystal structure and consists of distorted silicon octahedra and three-fold coordinated oxygen atoms. Many computations of surface electronic structure, and a few computations of surface atomic structure for a variety of silica allotropes have appeared in the literature (LaFemina, 1994). None of the surfaces studied, however, were autocompensated (Principle 4). They are, therefore, unlikely to be the experimentally c~bserved surface. The details of these investigations can be found in the recent review by LaFemina (1994). Finally, no experimental determinations of silica surface structures have been published.
4.4. The structure of surface defects 4.4. I. Introduction
Arguably, the most interesting and technologically important chemistry that occurs ~n surfaces occurs at defect sites: steps, kinks, grain boundaries, and point defects (Campbell, 1988; Colbourn, 1992; Stumm, 1990). Because of the reduced coordination and unsaturated valences at defect sites, the active sites for most chemisorption and catalytic reactions are thought to involve surface defects (Bermudez, 1981 ; Henderson, 1980; Henrich, 1985, 1989). Further discussions on surface defects are contained in Chapters 1 and 12. Surface defects can be broadly categorized as either extended or point defects. Extended defects include steps, kinks, and grain boundaries, while point defects include atomic vacancies and substitutions (Henrich, 1985, 1989). By far, the most studied are the point defects, and in particular, oxygen vacancies on oxide surfaces. The reasons for this are simple. Experimentally, extended defects exist on every surface, cleaved or fractured, and are difficult to prepare in a controlled manner
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(Henrich, 1985). Point defects, and in particular, oxygen vacancies on oxide surfaces, can be prepared via electron bombardment, ion bombardment, or high-temperature annealing and quenching (Henrich, 1987). Theoretically, the treatment of defects either requires the use of large unit cells in periodic computations, so that defects in neighboring cells are effectively non-interacting, or the use of clusters to treat truly isolated defects (Gibson et al., 1993a,b). The need to treat a large number of atoms has limited the application of quantum mechanical models to defect computations. The lack of detailed experimental information, just as with the ideal surfaces, also has limited the development of empirical and semiempirical models. In a practical sense, the simplest defect to treat computationally is the point defect; extended defects require many more atoms. Because the chemistry that takes places on defected surfaces is technologically important, there is an enormous literature characterizing the chemical and catalytic properties of these surfaces which has been extensively reviewed (Bermudez, 1981 ; Henderson, 1980; Henrich, 1985, 1989; Stumm, 1990). Less is known about the detailed atomic structure of these defect sites, however, than is known about the structure of clean, perfect, surfaces. Three defected surfaces that have been studied both experimentally and theoretically are the reduced rutile (! 10) surface (discussed in w 4.3.2), the (2x2) oxygen vacancy perovskite (100) surface (discussed in w 4.3.4), and the rocksalt (001 ) surface for which a variety of defect structures have been considered. We shall focus on these rocksalt (001) defects in this section.
4.4.2. Defects on the rocksalt (001) surface There have been numerous studies on the electronic structure of point defects on the (001) surface of a variety of alkali halides, and alkaline earth and transition metal monoxides (Bermudez, 1981; Henrich, 1985). In particular, these studies have focused on anionic vacancies (commonly referred to as F centers in the bulk, and F, centers at the surface). In general, there is a good agreement between the computed and observed electronic properties. Since these calculations are carried out for an unrelaxed bulk-like structure, they suggest that relaxations effects are not important for these defects. The important surface structural principle to consider in these cases is whether the surface atoms surrounding the defect can undergo approximately bond-length conserving motions (Principle 5). For a surface vacancy defect (cationic or anionic) the surrounding atoms become four-fold coordinated. The defect topology still inhibits large atomic motions, and so it is not surprising that relaxation effects seem to unimportant (Gibson et al., 1993a,b). No total-energy minimization computations of surface vacancy defect relaxations have been reported, however. The energetics of point defect formation and migration has been studied extensively using classical potentials and the Mott-Littleton approach (see w 4.2.3 and 4.2.4) (Colbourn, 1992). These computations have given insight into the problem of surface segregation of impurities, indicating that the driving force for segregation is the reduction of the local elastic strain (Principle 5) associated with the impurity atoms (Colbourn, 1992).
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Turning to extended defects, computations on the energetics and atomic structure of steps, kinks and corners have been carried out also using classical potentials and the Mott-Littleton approach (Colbourn, 1992). For a perfect, monoatomic step the atomic relaxations are on the order of a few percent of a lattice spacing, similar to the perfect rocksalt (001) surface relaxations. This is not unexpected since the coordination of the atoms at the step edge is reduced from five to four, and the topology of the step inhibits large atomic motion relaxations (Principle 5). At a step corner, however, the situation is qualitatively different: the corner atom having its coordination reduced from five to three. This atom can undergo a much wider range of approximately bond-length-conserving motions, and the atomic structure computations indicate that the relaxation of the corner atoms is on the order of 2 0 - 2 5 % of the lattice spacing: an order of magnitude larger that the relaxation at the perfect surface, or at the step edge (Colbourn, 1992).
4.5. The structure of adsorbates
The interesting chemistry which occurs at surface defects usually involves the chemisorption, or chemical bonding, of a species to the surface. Chemical interactic~ns between adsorbates and insulatc~r surfaces are important in many diverse areas, including, catalysis, corrosion, chemical sensors, and electrochemistry (Henrich, 1985, 1988" Janata, 1989" Kn6zinger, 1976). Much of this work has been performed on powders or polycrystalline materials, making atomic-level interpretatic)ns difficult. For example, because of the importance of water-electrode reactitans in electrochemical applications, there is a large body of work on the interacti~)n between water and a wide variety of metal, semiconductor and insulator surfaces which has been recently reviewed by Thiel and Madey (1987). Unfortunately, very little of this work has dealt with determining the atomic structure of the surface reactive site. In fact, there is little overlap between the systems that have been extensively studied experimentally and those which have been treated theoretically. This is mostly due to the fact that most non-defected, single crystalline, insulatc~r surfaces are chemically inert" it is the interactions between the adsorbates and the surface defects which control the surface chemistry. Given the limited understanding of surface defects (see w 4.4), it is not surprising that our understanding of the structure the active sites for surface chemistry is also limited. For a more complete discussion on surface chemisorption see Chapters 9 and 11 and reviews by Colbourn (1992), Henrich ( 1983, 1985, 1987, 1988, 1989), and Henrich and Cox (1994). One system which has had an appreciable amount of experimental and computational work performed is the dissociative chemisorption of molecular hydrogen on defective MgO(001) surfaces. Based upon isotopic exchange equilibria for H2 and D2 on MgO powders Boudart (1972) proposed that the surface V~ center (Fig. 4.12), in which a cationic vacancy at a step corner results in (111)-O microfacets, was the surface active site. Electronic structure computations for this defect indicate energetically favorable interactions between hydrogen and the three O atoms
Theory of insulator su~. "ace structures
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9
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........
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....
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4.6. Discussion As discussed throughout this chapter, there exists a set of simple chemical and physical principles which, when applied to crystalline insulators, can be used to begin to understand their surface atomic relaxations and reconstructions. The application of these principles has both clarified and raised important issues such as the primary importance of autocompensation and the rehybridization, or
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redistribution, of surface dangling bond charge density, the role of microscopic stress, and the role of surface topology in determining the nature of the surface atomic structure.
4.6.1. Rehybridization of dangling bond charge density Perhaps the most interesting result to emerge is the applicability of concepts such as autocompensation and the rehybridization of surface dangling bond charge density in understanding the surface relaxations and reconstructions of these materials. It is astonishing that these concepts are useful even for systems which cannot be thought of in terms of traditional two-electron chemical bonds (i.e., "ionic" materials, such as the rocksalt and perovskite structure materials). In covalent insulators, the surface dangling bonds can be rehybridized in several ways, New bonds can be formed at the surface, such as for the C(I 11)(2xl) reconstruction (w 4.3.1) in which the topological reconstruction of the surface layer moves unsaturated surface atoms from second-near-neighbors to first-near-neighbors. This allows the a new surface bond to form between the unsaturated surface atoms, thereby satisfying their chemical valences (Principle 1). For this surface it is also predicted that the rehybridized surface charge density, following the topological reconstruction, can further reduce its energy by becoming insulating (Principle 2) via a structural relaxation (i.e., buckling or dimerization). This structure is formed upon cleavage (Principle 3) and because diamond is an elemental insulator the surface is charge neutral (Principle 4). Finally, the relaxation following the topological reconstruction approximately conserves bond lengths (Principle 5). For the binary (and higher-order) materials in which the bonds are polarized, the situation is complicated by the fact that the surfaces fracture and hence can exhibit several surface stoichiometries depending upon surface processing conditions (Principle 3). These stoichiometries can be rationalized and predicted using autocompensation and surface charge neutrality arguments (Principles 2 and 4). This is true even for materials which cannot be though of as forming traditional two-electron chemical bonds, such as the perovskite SrTiO3 (100)(2• structure (w 4.3.4). Because the dangling bond charge density for these materials is localized on the anion, which has a filled shell, the chemical valences of the surface atoms are satisfied (Principle 1). As a result the topology of the surface (i.e., whether the surface atoms can undergo approximately bond-length-conserving motions m Principle 5) emerges as the primary factor in determining the nature of the surface relaxation or reconstruction.
4.6.2. Surface stress and the importance of surface topology Surface topology (Principle 5) becomes important because the driving forces for surface relaxations and reconstructions are to move the surface atoms into electronically favorable local environments, lowering the electronic energy of the surface. If, however, because of the connectivity of the surface atoms, these movements significantly distort near-neighbor bond-lengths, large local strain
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fields will be created. The creation of the strain fields costs energy, a great deal of energy when near-neighbor bond lengths are distorted. Consequently, these two effects are in opposition. When the surface topology precludes bond-length conserving motions the surface typically undergoes small atomic motion relaxations (e.g., rocksalt (001 ), or rutile (110)). On the other hand, when the topology of the surface allows such motions, large atomic motion relaxations and reconstructions can occur (e.g., diamond ( 111 ), corundum (0001 ), the step corners of rocksalt (001)). 4.6.3. Areas f o r future research
As stated in the introduction to this chapter, a microscopic understanding of the relationship between the atomic structure and chemical bonding at insulator surfaces and process chemistry will allow for profound improvements in many important technologies. Advances in the computations, however, are intimately tied to needed experimental advances. Two of the more critical areas involve the preparation of well-characterized, single-crystal, stoichiometric surfaces. The other involves the development of new techniques for examining the surface atomic and electronic structure of insulating materials that will overcome surface charging and surface decomposition problems. The development of simple quantum-mechanical models, such as the empirical tight-binding models that have been used so successfully to describe semiconductor surface and interface structure, is an essential step along the path towards achieving a firm understanding of the factors which govern insulator surface reconstructions. The application of the simple chemical and physical principles derived from the semiconductor studies looks promising. However, only for a few specific surfaces (e.g., MgO (001)) is there sufficient experimental and theoretical information available so that a reasonably complete understanding of the surface atomic and electronic structure can be claimed. The extension of these concepts to encompass the full range of insulating systems, including polar as well as nonpolar surfaces, surface defects, and chemisorbed surfaces represents an enormous opportunity. The development of structure-processing relationships will also require a thorough understanding of the surface energetics for the growth faces of these materials. This includes detailed knowledge of the surface potential energy function, including the characterization of local minima; the transition states between the various minima, and the mechanism of structural transformation; and the role of surface dynamics. In fact, the identification of the processing conditions necessary to reach a particular structural minimum is a major frontier in total-energy computations. Acknowledgements
The author is indebted to Dr. Charles B. Duke for collaborations over the past years that have led to many of the themes discussed in this chapter. I am also grateful for his constant interest and encouragement. I would also like to thank Drs. Don Baer, Andrew Gibson, Tom Godin, Victor Henrich, Roger Haydock, and Andrew Skinner for many interesting and informative discussions.
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T h e P a c i f i c N o r t h w e s t L a b o r a t o r y is o p e r a t e d for the D e p a r t m e n t o f E n e r g y by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. T h e a u t h o r g r a t e f u l l y a c k n o w l e d g e s the s u p p o r t o f the P a c i f i c N o r t h w e s t L a b o r a t o r y d u r i n g the p r e p a r a t i o n o f this c h a p t e r .
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J.P. LaFemina
CHAPTER 5
Surface Structure of Crystalline Ceramics R.J. L A D Laboratory for Surface Science and Technology University of Maine Orono, ME, USA
Handbook of Surface Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B.V. All rights reserved
185
Contents
5.1.
I n t r o d u c t i o n to c e r a m i c surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
5.2.
I n t e r - a t o m i c b o n d i n g in c e r a m i c s and bulk crystal structures . . . . . . . . . . . . . . . . . .
189
5.2.1.
Local coordination geometries
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
5.2.2.
C r y s t a l structure classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
191
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
E x p e r i m e n t a l structure d e t e r m i n a t i o n of c e r a m i c surfaces
. . . . . . . . . . . . . . . . . . .
194
5.3.1.
Diffraction m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
194
5.3.2.
Scanned probe microscopies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
5.3.3.
Electron m i c r o s c o p i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
198
O x i d e surface structures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
5.4.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200
Rocksalt (MO)
5.4.2.
Wurtzite (MO)
5.4.3.
Rutile (MO2)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
202
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
202
5.4.4.
Spinel ( M 3 0 4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205
5.4.5.
C o r u n d u m (M203)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
207
5.4.6.
Perovskite (MRO3)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
210
5.4.7.
O t h e r oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
212
C a r b i d e s u r f a c e structures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
212
5.5.1.
T r a n s i t i o n metal carbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
213
5.5.2.
Silicon c a r b i d e
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
Nitride surface structures 5.6.1.
T r a n s i t i o n metal nitrides
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2.
G r o u p I I I - V nitrides
218
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3.
219
Silicon nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
220
D e f e c t m i c r o s t r u c t u r e on c e r a m i c surfaces
. . . . . . . . . . . . . . . . . . . . . . . . . . .
221
5.7.1.
Point defects and defect clusters
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
222
5.7.2.
S u r f a c e steps and e x t e n d e d defect structures . . . . . . . . . . . . . . . . . . . . . .
223
5.7.3.
Facets and p h a s e b o u n d a r i e s
224
General conclusions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
224
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
225
186
5.1. Introduction to ceramic surfaces
This chapter deals with the surface structure of crystalline ceramic materials, defined here as inorganic non-metallic solids containing oxygen, carbon, or nitrogen. Ceramics have been important technological materials throughout civilization, and have traditionally been comprised of oxide-based clays used in high temperature or insulating applications such as pottery, bricks, and tiles. In the last few decades, so-called "advanced ceramic materials" have been developed which rely on careful synthesis and processing of oxides, carbides, and nitrides. The field of ceramics now encompasses high performance materials used in a wide range of applications including micro- and opto-electronic components, corrosion- and wear-resistant coatings, engine components, catalysts and sensors (Brook, 1991). Because of the recent trend and interest in replacing many traditional metals and alloys with carefully processed ceramics and ceramic composites, the understanding of surface and interface properties has become extremely important. Unfortunately, progress in surface science of ceramic materials has lagged considerably behind metals and semiconductors because of the many experimental difficulties encountered during their study. Also, challenges in understanding the structure of ceramic surfaces arise due to their complex bonding and stoichiometry, topics that are addressed in this chapter. The current status of the field is comparable to the understanding of metal and semiconductor surface crystallography 20-30 years ago; there is substantial speculation, contradictory results, and few accurate determinations of the atomic coordinates. As an example of the immaturity of the field, Table 4.3 in Chapter 4 by LaFemina provides a listing of quantitative determinations of the surface structure on MgO (100); despite the fact that MgO (100) is one of the simplest oxide surfaces, there is considerable disagreement about the exact atomic coordinates. The lack of an extensive and well-established database for ceramic surfaces makes this chapter quite different from other chapters in this volume. Table 5.1 summarizes the problems surface scientists face when studying ceramic samples and some possible ways to solve or minimize them. The first and foremost challenge is to reproducibly create a stoichiometric surface for study. As simple as this sounds, the inability to prepare and characterize the same ceramic surface from one laboratory to another has led to several discrepancies and disagreements in the literature. Traditional ion bombardment and annealing treatments used for metals and elemental semiconductors simply do not work in most cases. Also, because ceramics have high melting points and correspondingly small atom diffusivities at temperatures below 1000~ kinetic limitations often dictate the realized structure, which may be far from equilibrium. Therefore, as a general rule 187
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R.J. Lad
Table 5.1 Experimental complications encountered when studying ceramic surfaces
Problems
Possible solutions
Difficult to reproducibly create stoichiometric surfaces.
Cleavage in ultra-high vacuum. Thermal treatments in gaseous environments.
Unavailability and poor quality of single crystal samples.
Small area analysis techniques. Thin film samples.
Charging with electron and ion based analysis techniques.
Low energy electron flood guns. Reduce the bulk composition or add dopants to make semiconducting samples. Thin film samples on conducting substrates.
Electron and photon beam induced damage.
Extremely low particle fluences. Large gain in detectors.
and word of caution to the non-specialist, the surface structure of ceramics is strongly dependent on the exact synthesis and processing history. The other problems listed in Table 5.1 are also very serious and inherently limit the applicability of surface analysis techniques as discussed in more detail in w 5.3. The common structural feature in all ceramics is the localized bonding geometry which usually involves tetrahedral or octahedral coordination polyhedra between the atomic constituents, although other coordination geometries also occur in some cases. Bonding can range from being highly ionic to covalent to metallic which is responsible for the diverse electrical and thermal properties of ceramics. Indeed, it is these electronic interactions between anions and cations which determine the detailed arrangement of the bulk as well as the surface structure. This chapter provides a basic introduction to the surface structure of ceramics, and due to the experimental infancy of this field and lack of extensive data, many highly idealized structures will be presented. However, since real ceramic surfaces are often very complex, idealized models are the necessary starting point for understanding them. Oxide surfaces have by far received more attention from a basic science and technology point of view than carbides and nitrides. This fact reflects the emphasis on oxides in this chapter. To supplement the information presented here, the reader is referred to other reviews of ceramic surfaces including Chapter 4 in this volume by LaFemina which emphasizes the theory of insulator surfaces, an overview of basic and practical aspects of many ceramic surfaces and interfaces (Dufour et al., 1989), and a recent book which concentrates entirely on oxide surfaces (Henrich and Cox, 1994). Chapter 6 in this volume by Duke, which discusses surface structures of compound semiconductors, also contains many concepts that are applicable to tetrahedrally coordinated ceramics. This chapter is organized in the following manner. Details of bonding configurations and structure classifications are presented in w 5.2. Successes and difficulties
Su~. ace structure t~'ceramics
189
of surface analysis techniques applied to ceramics are discussed in w 5.3. Then, w 5.4-5.6 examine in detail the surface structures of oxides, carbides, and nitrides, respectively, and classify their surface structures into several categories. Finally, w 5.7 illustrates the types and properties of defects present on real ceramic surfaces and discusses their ramifications on surface behavior.
5.2. Inter-atomic bonding in ceramics and bulk crystal structures In this section, the basic crystal geometries present in ceramics are examined both in terms of localized molecular bonding and global crystal symmetry. These concepts for bulk ceramics are then applied to surface structure in w 5.4-5.6. Oxides generally have ionic or mixed ionic-covalent bonding and can be described by a sub-lattice of 02- anions interleaved with a sub-lattice of metal cations. Carbides and nitrides, on the other hand, tend to be covalent or even metallic. In most cases, the carbon or nitrogen atoms are appropriately viewed as being interstitial atoms located in the crystal lattice made up by the other constituents. Despite this vastly different bonding between oxides, carbides, and nitrides, their coordination geometries and crystal structures have many common features. 5.2.1. Local coordination geometries
A major criteria that determines the atomic packing arrangement in ceramic materials is the relative size or ionic radius of the atoms. In ionic ceramics, charge transfer occurs from the cations to the anions, and hence the ionic radius of the cations is much smaller than the anion radius. The Pauling law of electronegativity is a simple way to determine the degree of ionicity of a given bond and leads to a consistent set of ionic radii for all of the elements (Pauling, 1960). The oxygen anion radius is quite large compared to that for the metal cations; for example, the ionic radius is 0.140 nm for O 2- compared to 0.080 nm for Zr 4§ and 0.050 nm for AI 3§ For carbides and nitrides, the atomic radius, defined as half the distance of closest approach of atomic centers in their elemental state, is more relevant; the atomic radius is 0.071 nm for both carbon (graphite) and nitrogen (N2) which is smaller than the atomic radius of most metallic elements. All ceramic structures maximize the number of nearest neighbors between the cations and anions in order to maximize the magnitude of the cohesive energy of the crystal. The shapes of the four basic coordination polyhedra that occur in ceramics are shown in Fig. 5.1; often the polyhedra are slightly distorted. The classes of crystal structures are based on the presence of one or more of these coordination geometries. The lowest energy configuration is that which maximizes the coordination number under the constraint of the sizes of the atomic radii. Table 5.2 shows the coordination numbers and bonding polyhedron geometries for various ranges of the cation to anion ionic radius ratio (Rc/R a) determined simply by the packing of rigid spheres. For example, if Rc/Ra is between 0.414 and 0.732, the cation maximizes its coordination by residing in the center of an octahedron with anions at all the corners. If the cation size is smaller than the critical radius of 0.414,
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R.J. Lad
triangular
tetrahedral
octahedral
cubic
Fig. 5.1. The basic types of coordination polyhedra that make up the structures of ceramic materials. Often they are slightly distorted.
Rc+R~l=d=
~__ R a
R c / R a = ~f2 - I = I).414 d
Fig. 5.2. A cut through the (100) plane of the rocksalt structure showing that when the cation to anion ionic radius ratio is 0.414 the cations are small enough that the anions just begin to touch. Table 5.2 Relationship between cation coordination number, shape of the coordination polyhedron, and the cation to anion size ratio Cation coordination #
3 4 6 8
Coordination polyhedron geometry triangle tetrahedron octahedron cube
Ratio of ionic radii Rc/Ra 0.155 0.225 0.414 0.732
~ --~ ~ ~
0.225 0.414 0.732 1.000
it c o u l d still r e s i d e in this g e o m e t r y , but at this p o i n t e a c h o f the a n i o n s t o u c h e a c h o t h e r a n d no f u r t h e r gain in e l e c t r o s t a t i c s t a b i l i z a t i o n e n e r g y o f the c r y s t a l can be o b t a i n e d . T h u s , the c a t i o n r e d u c e s the c o o r d i n a t i o n n u m b e r to a t e t r a h e d r a l g e o m e try to a c h i e v e i n c r e a s e d e l e c t r o s t a t i c e n e r g y via s m a l l e r c a t i o n - a n i o n b o n d dist a n c e s . If the c a t i o n size is l a r g e r than 0 . 7 3 2 , the c a t i o n can h a v e 8 - f o l d c o o r d i n a t i o n in a c u b i c a r r a n g e m e n t .
Sudace structure o.fceramics
191
Figure 5.2 illustrates this geometry for the case of MgO which crystallizes in the rocksalt structure. Both the cations and anions have octahedral coordination, and a (100) cut through the ion cores shows that the critical Rc/Ra ratio equals 0.414 when the cation size is small enough that anions just begin to touch. Similar geometrical considerations lead to the other Rc/Ra ratios listed in Table 5.2. 5.2.2. Crystal structure classes
In oxides, the 0 2- anions are larger than the cations and the various oxide crystal structures are generally derived from different occupancy of the cation sites within M2
A
,~. r~l 2
()2-
/ :! w
Rocksalt (MO)
Wurtzite (MO) Q ...........170
0 ....
o. _ . . . . ~ " o o
O... ~ d
~
"T"O I ....7 " " 0 lo i ........" ........"t~
....o
o...Z...~"-o o... j....~-.-o .... 0
~
"Y9
I ~-"
]
"""
""" "".
e--
oC)..:
. "'-0
M2
9
""0
Rutile (MO_~)
Spinel (M.~04)
{ )2-
I~I 3
~
,M 4
t.'
il
('ortlilcltlm (MeOw)
Perovskite (MNO;)
Fig. 5.3. Bulk lattice models ol~common oxide structures (adapted from Kelly and Groves, 1970).
192
R.J. Lad
a close-packed anion sub-lattice. For example, in Fe~_xO the Fe 2§ cations occupy essentially all of the octahedral interstices of an fcc anion sub-lattice forming the rocksalt structure, whereas in Fe304, the Fe 2§ cations occupy octahedral sites and the Fe 3+ cations reside in both the octahedral and tetrahedral sites making up the inverse spinel structure (Lindsley, 1976). Figure 5.3 shows lattice models of this spinel structure and other common oxide structures. The crystal structures of carbides and nitrides are derived from close-packed arrangements of metal atoms with the smaller carbon or nitrogen atoms inserted into the interstitial sites, exactly opposite from oxides where the metal atoms fill the interstices of the close-packed oxygen lattice. However, because both contain the same types of local coordination polyhedra, many of the crystal structures are the same. The nature of the metal-nonmetal interactions is manifested by having carbon or nitrogen located in either an octahedral interstitial site or in the center of a trigonal prism, since the tetrahedral sites are too small to accommodate the C or N atoms (Toth, 1971). In fact, the majority of carbides and nitrides form either the simple rocksalt structure with C or N located in octahedral interstices of the fcc metal sub-lattice (e.g. TiC, NbC, TaC, VN, ZrN) or the tungsten carbide structure with C or N occupying the trigonal prism sites in a simple hexagonal metal sub-lattice (e.g. WC, 7-MoC, 6-TAN) as shown in Fig. 5.4. If the metal atom radius is too small to accommodate the C or N interstitials, many equilibrium phases can L
w
fcc metal carbide (NaCl-type) structure
tungsten carbide (WC) structure Fig. 5.4. Latticemodels of the two most common carbide and nitride structures (adapted from Toth, 19"/I).
193
Su~. ace structure of ceramics
e x i s t d e p e n d i n g on the e x a c t s t o i c h i o m e t r y , e a c h with a c o m p l e x s t r u c t u r e s u c h as in the C r - C a n d T i - N s y s t e m s ( T o t h , 1971). T a b l e 5.3 s u m m a r i z e s the b u l k c r y s t a l s t r u c t u r e s o f s e v e r a l o x i d e s , c a r b i d e s , a n d n i t r i d e s , p a r t i c u l a r l y t h o s e w h o s e s u r f a c e s h a v e b e e n s t u d i e d . F r o m this t a b l e , s e v e r a l o f the f a c t o r s that lead to c o m p l e x s u r f a c e s t r u c t u r e s o f c e r a m i c s are e v i d e n t . First, in c o n t r a s t to e l e m e n t a l m e t a l s a n d s e m i c o n d u c t o r s , the c o o r d i n a t i o n g e o m e t r y n e c e s s a r i l y i n v o l v e s b i n a r y and t e r n a r y c o m p o n e n t s . C o n s e q u e n t l y , t h e r e are s e v e r a l p o s s i b l e ideal s u r f a c e t e r m i n a t i o n s for e a c h s p e c i f i c ( h k l ) p l a n e . S e c o n d ,
Table 5.3 Crystal structure classes of ceramics Ideal stoichiometry a
Structure prototype
Bravais lattice
Examples
Bulk cation coordination number b
rocksalt
fcc
6
wurtzite
hexagonal
tungsten carbide sphalerite
hexagonal fcc
MgO, NiO, CoO TiC, VC, TaC TiN, VN, ZrN ZnO AIN, GaN WC l-SiC, c-BN
6c 4
MX 2
rutile or-quartz fluorite
tetragonal trigonal fcc
TiO 2, SnO 2 ot-SiO2 c-ZrO 2
6 4 8
MX 3
distorted rhenium oxide
monoclinic
MoO 3, WO 3 CrO 3
6d
M2X3
corundum
trigonal
ot-AI203, ot-Fe203 Cr203, Ti203, V203
6d
MRX 3
perovskite
cubic
12 and 6
ilmenite
trigonal
SrTiO 3, BaTiO 3 NaxWO3 FeTiO3, LiNbO 3
spinel
fcc
MgAI204, Fe304
4 and 6
M3X4
phenacite
trigonal
[3-Si3N4
4
M2X5
niobium pentoxide vanadate
monoclinic ortho-rhombic
Nb205, Ta205 V205
6 6d
MX
MR2X4
4
6
a X = oxygen, carbon, or nitrogen and M, R = other elements; significant deviations from stoichiometry can exist. b d = coordination polyhedron is distorted. c Trigonal prism instead of octahedral coordination.
194
R.J. Lad
most ceramics can exhibit large deviations from ideal stoichiometry. These deviations result in defects in the ideal lattice and also have a profound effect on surface behavior as discussed in w 5.7.
5.3. Experimental structure determination of ceramic surfaces Determining the surface structure of ceramics is challenging not only because they often have a complicated atom basis within the surface mesh, but because many of the established surface analytical techniques used to study surface structure rely on the interaction of ion or electron beams with the surface. These interactions with insulating surfaces can be particularly troublesome. Depending on the total electron yield, i.e. the sum of the secondary electron yield and the backscattering coefficient, the surface can charge negatively or positively. Ichimura et al. (1989) review several practical ways to compensate for the charging including varying beam energies and fluences, sample temperature, and auxiliary flood gun parameters. Another problem, which is especially severe with oxides, is electron stimulated desorption (ESD) and photon stimulated desorption (PSD) upon exposure to the incident probes. The well known mechanism proposed by Knotek and Feibelman (1978) for ESD of oxygen ions from TiO2 has been realized experimentally using electrons and photons on TiO2 surfaces (Kurtz, 1986) as well as other oxides. Even low-density beams have been shown to create noticeable defect concentrations (Wang et al., 1994). Generally, these effects are recognized by most investigators but are assumed, sometimes incorrectly, to be small. The poor quality of single crystals used to prepare samples also hinders experimental studies of ceramic surface structure. In cases where technologies have driven the need for high quality crystals, such as cx-AI203 (sapphire), TiO2, and tx-SiO2 (quartz), samples are readily available. However, for many oxides and most carbides and nitrides, single crystals are scarce and extremely small. Furthermore, both atomic-scale and large-scale defects are abundant and markedly influence experimental results, particularly since most analysis techniques average over large sample areas. The remainder of this section illustrates several examples of successful surface structure determination of ceramics. It also gives a flavor of the type of information that can be attained with different experimental techniques applied to ceramic surfaces, at what resolution, and with what limitations.
5.3.1. Diffraction methods Diffraction methods including LEED, RHEED, LEIS, and MEIS have all been used to measure ceramic surface structure; details of these techniques are reviewed by Conrad in Chapter 7 and by Unertl and Kordesch in Chapter 8. LEED is routinely applied to ceramic surfaces primarily to determine the symmetry of the surface unit mesh. Even when used as a qualitative tool, much can be learned about surface reconstructions as a function of thermal treatment. A good example is the reconstructions observed on o~-Fe203 (0001) and (1012) surfaces (Lad and Henrich, m
Su~. ace structure ~'ceramics
195
1988a). To determine exact atomic positions, quantitative LEED intensity measurements combined with multiple scattering calculations are required. This has been carried out for the (100) surfaces of several rocksalt monoxides including MgO, NiO, CoO and CaO (Prutton et al., 1979; Blanchard et al., 1991) but has not yet been done for oxides with more complicated surface meshes. Accurate measurements of RHEED rocking curves have proven useful for extracting the surface relaxation parameters of MgO (100) (Maksym, 1985) but other examples of quantitative RHEED are few. Photoelectron diffraction measurements also have been used to test surface termination models of oxides (Tran and Chambers, 1994; Galeotti et al., 1994). Low energy and medium energy ion scattering take advantage of channeling and blocking effects of near-surface ion trajectories and are very well established techniques for determining surface structure. In LEIS, typically 4He§ ions with energies on the order of 1 keV are scattered from the surface and measurements of the scattered ion yield versus azimuthal or polar angles of incidence or scattering are correlated with the shielding of surface atoms. Because of the kinematics of scattering, there is a different energy loss in the scattered beam for cations versus anions giving rise to separate scattering signals and shadow cones for each species. This method has proven extremely sensitive for measuring atomic vacancy concentrations on TiC (100) (Aono et al., 1983) and the mechanism of oxygen chemisorption on (100) surfaces of TiC, VC, ZrC and TaC (Souda et al., 1991). In MEIS, ion beams in the range 50-300 keV are used. Zhou et al. (1994) studied the rumpling of the MgO (100) surface using 97 keV H § ions as the probe. Because of the insulating nature of MgO, surface charging was significant and it distorted the ion trajectories near the surface. These effects were minimized by using a low energy electron flood gun to neutralize the charge, but the use of the flood gun complicates the direct measurement of the ion current. Both ion scattering techniques are also strongly influenced by surface roughness and large scale defects which may limit their applicability to a wide range of ceramic surfaces. 5.3.2. S c a n n e d probe microscopies
The ST.M and AFM have made a significant contribution towards understanding the surface structure of ceramics; these and other methods for direct imaging of surfaces are discussed by Unertl and Kordesch in Chapter 8. The STM is applicable to ceramic surfaces which have a high enough surface electron density to yield a measurable tunneling current. As a general rule, the surface must have a small enough bandgap (<1.5 eV) or contain surface states due to reconstruction or bulk impurity induced bandgap states which act to pin the Fermi level (Bonnell, 1989). Contrast between the anions and the cation species arises from differences in surface electronic structure as discussed in Chapter 8.1. For example, when tunneling into the cation-derived empty conduction band states of a transition-metal oxide, the tip is sensitive primarily to the metal cations. However, in some oxides such as V205, there is a significant contribution of O 2p states near the top of the conduction band and tunneling into these states yields images dominated by oxygen
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R.J. Lad
Fig. 5.5. Constant current STM image and surface structure model of V205 (100) (Smith et al., 1995). In the model, the small black spheres are vanadium cations and the large spheres representing oxygen anions are white at the top surface and shaded in subsurface layers. anions as shown by the example in Fig. 5.5 reported by Smith, Lu and Rohrer (1995). Using the STM in the tunneling spectroscopy mode to probe the surface electronic structure often helps in interpreting image contrast between the different elemental surface species. For highly insulating ceramic surfaces, atomic force microscopy is the probe of choice to study details of surface structure. Several reports in the literature show atomic-scale images of crystalline ceramic surfaces obtained by AFM; for example, images of the molecular scale structure of o;-Fe203 have been reported by Johnsson et al. (1991). However, these data exhibit perfectly ordered periodic structures produced by "constant force mapping" of the surface, and do not show any atomic scale defects. Ohnesorge and Binnig (1993) have claimed true atomic resolution by showing an atomically resolved kink along a monoatomic step on calcite (CaCO3).
Su~. ace structure of ceramics
197
The resolution could be achieved only by imaging under water using attractive contact forces on the order of 10 -~1 N or less. This atomic resolution imaging is far from being routine. In fact, in many ceramic systems long-range electrostatic forces are present which prevent imaging with these very small contact forces. The A F M is extremely valuable for observing defect microstructure on ceramic surfaces at length scales ranging from 10 nm to 1 lam. Features which are often prevalent on ceramic surfaces such as steps, facet planes, reconstructions, and ion beam induced d a m a g e are easily discernible and cannot be easily observed by any other technique. For example, Fig. 5.6 shows an imaged step array on a polished
Fig. 5.6. AFM images of single crystal oxide surfaces (Antonik and Lad, 1992). (a) A step--terrace array on a mechanically polished and vacuum annealed o~-A1203 (10T2) surface. The terraces are about 80 nm wide and the steps are 0.35 nm high, which corresponds to a surface miscut of ~1/4~ (b) Facets on vacuum annealed TiO2 (100). The facets are about 300 nm across and 10 nm high and contain ridges spaced 20 nm apart running perpendicular to the facet apices.
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R.J. Lad
Fig. 5.7. A HREM image of a vicinal (011) surface of La203 showing surface faceting into (010) steps and (001) terraces (Zhou et al., 1989). and annealed r-cut ~ sapphire (o;-A1203) surface, and an image of a faceted TiO 2 (100) surface prepared by annealing the sample in 02 at 550~ (Antonik and Lad, 1992). Although methods have been developed and are routinely used to prepare stoichiometric ceramic surfaces for fundamental studies, large-scale structures such as those in Fig. 5.6 are still present on these surfaces but are often ignored in analysis of experimental results. The AFM plays a major role in characterizing these defects and understanding their influences on processes such as film growth.
5.3.3. Electron microscopies Two modes of electron microscopy, High Resolution Electron Microscopy (HREM) and Reflection Electron Microscopy (REM), can provide near-atomic resolution direct observation of ceramic surfaces. In the HREM technique, surface profile images are obtained by creating a specimen whose surface structure can be projected along the beam direction (Smith et al., 1986). Although no depth discrimination is possible in this one-dimensional projection, surface features such as steps, terraces, and surface reconstructions can be viewed and compared to computer simulations. An example of a HREM surface profile image showing the step and terrace structure on a La20 3 surface is displayed in Fig. 5.7 (Zhou et al., 1989). To obtain accurate information about atomic positions, a very careful analysis including assumptions about the potential near the surface is required. In spite of the fact that it is limited to special sample orientations and relies on detailed comparisons between experimental and simulated images, this surface profile imaging has great potential for imaging dynamical processes on surfaces. However, electron radiation induced surface damage by both ballistic surface erosion and electron-stimulated reactions under the presence of HREM beam is a major problem, particularly with oxides (Buckett and Marks, 1990). In REM, electrons reflected from the surface of a specimen are used to form an image of the surface, and hence the technique is very complementary with RHEED.
1 See Table 5.4 in w5.4.5 for definition of r-cut.
Su~. "ace structure of ceramics
199
Fig. 5.8. Various step configurations imaged by REM on different areas of an annealed c~-A1203 (0001) surface (Kim and Hsu, 1991). (a) Straight steps oriented parallel to [ 1TOO](vertical) and [ 11~01 (horizontal) directions; (b) curved steps; and (c) coexisting curved and straight step geometries. Each of the images are foreshortened in the vertical direction by a factor of 30.
Surface features such as steps, reconstructions, dislocations, stacking faults, and superlattices can be imaged with strong contrast and high spatial resolution (Cowley, 1986; Hsu, 1992). The major limitation is that the entire image is obtained by reflecting the electron beam at typically angles less than 5 ~ with respect to the surface and therefore the REM image is severely foreshortened (typically by a factor of 30) along the beam direction. As an example, Fig. 5.8 shows REM images of various step structures observed on an o~-AI203 (0001) surface (Kim and Hsu, 1991). An understanding of the contrast mechanisms often requires a complete dynamical diffraction theory (Yao and Cowley, 1990). Atomic height steps can be resolved in the vertical direction, and lateral resolution is typically in the 10 nm range. Direct images are also complemented by structural information from the accompanying RHEED patterns.
5.4. Oxide surface structures
The surface structures of single crystal oxides are of interest since they provide a fundamental understanding of the surface properties of these materials which then provides a basis for understanding more complicated polycrystalline thin films and powders. Furthermore, from a technological point of view, oxide substrates are very important in thin film technology as the starting materials for heteroepitaxial
200
R.J. Lad
structures and devices. As recently pointed out by Phillips (1995), the progress in optimizing oxide substrate materials has been much less impressive than the progress in the films themselves. Oxide substrates typically undergo mechanical polishing and chemical cleaning but are rarely annealed at high enough temperatures (above ~800~ to obtain true equilibrium surface structures. There is a dearth of experimental effort in this area, and more experimental attention is needed. In this section, discussion of oxide surface structures primarily concentrates on atomic scale arrangements including relaxations and reconstructions. However, it is important to keep in mind that surface microstructure, such as that shown in Fig. 5.6, at length scales ranging from 10 nm to 1 ~tm (e.g. step distributions, facets, and defect structures) is very pronounced on real oxide single crystal surfaces (Antonik and Lad, 1992) and this structure often dictates surface properties as discussed in w 5.7. The starting point for visualizing models of a specific oxide surface is to think of cutting a plane through the bulk oxide lattice. The resulting surface is not strictly two-dimensional but rather has cations and anions situated above and below an average two-dimensional imaginary plane. Collectively, a set of these cations and anions satisfy charge neutrality of the exposed surface, and as a unit they more appropriately describe a given (hkl) surface. Often different surface terminations can be envisioned which satisfy the charge neutrality requirements, and surface energy considerations determine the equilibrium surface termination. If the energies for different terminations are similar, two types of terraces can coexist on the oxide surface. As shown in the remainder of this section, some oxide surface structures are known in detail, while others unfortunately have not yet been determined and our understanding is limited to the idealized surface models. This section overlaps considerably with Chapter 4.
5.4.1. Rocksalt (MO) Oxides with the rocksalt structure such as MgO, NiO, MnO, CoO, and Fel_xO are conceptually the simplest of the oxides since the crystal lattice consists of interpenetrating fcc cation and anion sub-lattices as shown in Fig. 5.3. The (100) surface is extremely stable and is the cleavage surface resulting from fracture. Other surface orientations such as (I10) and (111) prepared by mechanical polishing restructure into {100} facets upon heating; an excellent example of this faceting on MgO is reported by Henrich (1976). A common phenomenon on the (100) rocksalt monoxide surfaces is relaxation and rumpling of the surface atoms with respect to the bulk atomic positions as discussed in detail by LaFemina in Chapter 4. Figure 5.9 shows schematic top and side views of the (100) surface of MgO indicating a slight contraction of the first layer spacing with respect to the bulk lattice parameter along with inward anion and outward cation rumpling (Zhou et al., 1994). The surface is non-polar with a net surface charge of zero and both the anions and cations have five-fold coordination, one fewer than in the bulk. MgO has been studied extensively and the general consensus is that surfaces cleaved in UHV and in air have the same surface structure (Henrich and Cox, 1994).
201
Su~'ace structure of ceramics
MgO (001) ( 01))
(I'i'O)
( )oOoM
0o
(
o
)o
Top View (
o0
(
,/
~
~
~
0 o
~
-(olo)
0 M~ Rumpling
Side View (01 O) plane
~ 0 o
Oo 0 d
Fig. 5.9. Top view and side view of the MgO (100) lattice showing rumpling of the top layer and contraction of the first and second layer spacing, d~2,compared to the bulk inter-layer spacing, d (Zhou et al., 1994).
He scattering experiments suggest, however, that air cleaved surfaces are irreversibly damaged by water vapor since a large density of point defects is observed after annealing in vacuum (Duriez et al., 1990). Another problem with annealing MgO surfaces is that Ca impurities, present in the purest crystals at a concentration of a few hundred parts per million, segregate to the surface above 1000~ and change the distribution of surface steps (Gajdardziska-Josifovska et al., 1993). Above 1200~ electron microscopy has shown that small MgO2 crystallites having the fluorite structure form on the surface, although they also may be related to Ca as well as Cr impurity segregation (Wang et al., 1992). The (100) surfaces of CaO, NiO and CoO have been studied to a limited extent. LEED indicates nearly bulk truncations with less than 3% rumpling for the (100) surfaces of all three of these oxides (Prutton et al., 1979), but the reported cleavage quality in these systems is variable. In the case of NiO, the surface can be easily reduced by CO or H 2 gas exposure at room temperature to form metallic Ni nuclei (Boudriss and Dufour, 1989). MnO and Fe~_xO crystals are scarce and are generally not good enough to provide good cleavage surfaces (Lad and Henrich, 1988b, 1989). Also, since they are sub-oxides, they are difficult to prepare by a polishing, Ar § bombardment, and O2 annealing treatment. A recent approach to preparing high quality rocksalt monoxide surfaces has been to use various thin film molecular beam epitaxy techniques. For example, direct electron beam evaporation of MgO even at temperatures as low as 140 K
202
R.J. Lad
(Yadavalli et al., 1990) yields very high quality single crystal films. NiO films (Lind et al., 1992) and MgO films (Chambers et al., 1995) deposited using electron cyclotron resonance (ECR) oxygen-plasma-assisted growth also show indications that their surface structures approach those of single crystal material. A word of caution, however, is that the surfaces of most oxide films contain a wealth of defects. Characterization by in situ growth diagnostics and/or atomic scale imaging is required to ensure that well-defined surfaces are produced. For example, high quality NiO (100) films have been synthesized by monitoring RHEED oscillations during epitaxy (Peacor and Hibma, 1994). Also, NiO (100) films with few well-defined defects were grown by B~iumer et al. (1991) and characterized using SPALEED and STM.
5.4.2. Wurtzite (MO) The wurtzite structure consists of a hexagonal close-packed array of oxygen anions with half of the tetrahedral interstices occupied by metal cations (Fig. 5.3); oxides having the wurtzite structure include ZnO and BeO. There are several important wurtzite surfaces: the cation-terminated (0001) basal surface, the oxygen-terminated (0001 ) basal surface, the (1010) prism surface, and the (1120) prism surface. Models of these surface structures and their surface reconstruction behavior are shown and discussed in length by Duke in the next chapter. The surface structure of all four of these surfaces of ZnO have been studied experimentally (Lubinsky et al., 1976). Polishing and cyclical Ar § bombardment / thermal annealing treatments are successful in that (1• LEED patterns are produced, but recent LEED results from ZnO (0001) suggest surface pit formation by a thermal etching mechanism (Mr et al., 1994). The cleavage plane is (1010) while (0001) is the predominant growth habit plane. The Zn-terminated (0001) surface can be easily distinguished from O-terminated (0001) via an HCI etch treatment; the (0001) is inert while the (0001) is rapidly etched (Gay et al., 1980). w
m
m
5.4.3. Rutile (M02) In the rutile structure, the cations make up a body-centered tetragonal cell and are surrounded by six oxygen anions at nearly equal distances (Fig. 5.3); each oxygen anion has three cation nearest neighbors. Two important rutile oxides are TiO 2 and SnO 2. The low index (100), (110) and (001) surfaces of these oxides have been perhaps the most studied single crystal oxides because Ar § bombardment and O2 annealing preparation methods are very successful in creating nearly perfect stoichiometric surfaces (Henrich and Cox, 1994). The standard technique is to reduce the bulk composition to be slightly oxygen-deficient by annealing in vacuum thereby creating a semiconducting bulk substrate upon which a stoichiometric layer is regrown by annealing in O2. Through the appropriate choice of processing variables (temperature, time, O2 partial pressure), either nearly-perfect surfaces or surfaces with controlled point defect concentrations can be generated.
Su~. ace structure of ceramics
203
Fig. 5.10. Two different models of the TiO2 (110) surface both of which would yield a (1• surface mesh (Rohrer et al., 1992). (a) Surface containing the rows of"bridging" oxygen anions and (b) surface with the "bridging" oxygens removed to expose all the "in-plane" oxygen anions.
The TiO 2 (110) surface is very stable and a (1• LEED pattern can be easily obtained by preparing a sample using the standard annealing-oxidation treatment (Henrich, 1985). Figure 5.10 shows two models of the surface which would yield a (1• surface mesh, one created by a simple bulk truncation of the (110) plane leaving rows of so-called bridging oxygen anions and the other with the bridging oxygens removed (Rohrer et al., 1992). The Ti cations covered by the bridging oxygens are six-fold coordinated to oxygen anions, as in the bulk lattice, while the exposed Ti cations are five-fold coordinated. Self-consistent total energy calculations suggest that the bridging oxygen terminated surface has the lowest energy and has a slight puckering of the outermost atoms (Ramamoorthy et al., 1994), but the magnitude of the relaxation is at the resolution limit of surface diffraction techniques and thus has not been experimentally observed (Maschhoff et al., 1991). Low concentrations of oxygen vacancies on TiO2 (110) induced by controlled thermal treatments or electron stimulated desorption are believed to be primarily associated with missing atoms from the row of bridging oxygens (GOpel et al., 1984; Wang et al., 1994). The location of oxygen vacancies on SnO2 (110) surfaces is more definitive; ISS has been used to show that annealing below 523~ causes removal of essentially all the bridging oxygen anions, whereas at higher temperatures in-plane oxygen vacancies are produced (Cox et al., 1988).
204
R.J. Lad
Fig. 5.11. Model of the (lx3) reconstruction on TiO2 (100) showing {110} microfacets containing rows of oxygen vacancies (Murray et al., 1994). The (Ix3) surface mesh is shown and the labels x,y indicate rows of exposed Ti cations (small spheres). Recent STM studies of the TiO 2 (110) surface suggest that the structure is more complex than previously thought and furthermore reveal that the structure strongly depends on the exact processing parameters. Atomically resolved images of a ( Ix 1) surface mesh from stoichiometric TiO 2 (110) have been acquired (Novak et al., 1994), but a (Ix2) reconstructed surface mesh (Sander and Engel, 1994; Szabo and Engel, 1995) and even a (2• reconstructed phase (Murray et al., 1995) have also been observed in addition to the ( l x l ) structure, particularly when the surface is annealed in reducing environments. The (lx2) phase corresponds to a superlattice of missing bridging oxygen rows with in-plane Ti cations displaced towards the missing row. The (2x2) phase contains extra rows of oxygen atoms along directions which act to stabilize the missing rows of bridging oxygens. Often several of these surface phases coexist on the TiO2 (110) surface suggesting local variations in stoichiometry. The first attempts to prepare a TiO2 (100) surface by polishing and annealing (Chung et al., 1977) yielded a series of surface reconstructions with increasing temperature as evidenced by (lx3), (lx5) and (lx7) LEED patterns. Grazing incidence X-ray diffraction and LEED analysis of the (Ix3) surface has revealed that the surface reconstruction can be viewed as being due to { 110} microfaceting (Zschack et al., 1992) as shown in Fig. 5. l l, which has been further confirmed by STM imaging (Murray et al., 1994). By removing several alternating rows of atoms along the [001] direction on a bulk terminated (100) plane, {110} type facets emerge, although the X-ray scattering results do indicate significant displacements from the ideal termination of { 110} planes. ( l x l ) LEED patterns can be obtained by annealing in a high enough partial pressure of O2 (Murray et al., 1994; Henderson, 1994). Murray et al. (1994) also compared results between oriented and vicinal surfaces and suggest that the earlier reports of (Ix5) and (Ix7) reconstructions may have been due to a slight sample miscut. The (001) surface of TiO2 is highly unstable and a polished surface easily becomes faceted upon heating either in vacuum or in O2 (Firment, 1982; Poirer et al., 1992; Antonik et al., 1992). According to LEED, STM and electron stimulated desorption ion angular distribution (ESDIAD) results, the surface rearranges into {011} facet planes as shown by the model in Fig. 5.12 (Kurtz, 1986). In this
Surface structure of ceramics
205
Fig. 5.12. Models of the facet structure on TiO2 (001) consisting of {011 } type facet planes (Kurtz, 1986). The Ti cations (small spheres) are 5-fold coordinated on the {011 } faces, and the facet edges contain alternating 4-fold and 5-fold cation coordination.
configuration, the surface contains five-fold coordinated Ti cations except at the facet edges where there is lower coordination. On a real surface, it is expected that atomic relaxations would exist at the facet edges but a quantitative study of the structure has not yet been made. Annealing above 923~ produces { 114 } and { 1 1 1 } oriented facet planes in addition to the {011 } facets.
5.4.4. Spinel
(M304)
In the spinel structure, the oxygen sub-lattice has fcc packing and the cations occupy both octahedral and tetrahedral interstices (Fig. 5.3). In fact, the spinel lattice can be thought of as a variant to the rocksalt lattice in which some of the cations normally situated in octahedral interstices in rocksalt are moved into tetrahedral interstices to make spinel. Many oxide minerals exist with this structure and at least 30 different elements with valencies ranging between +1 to +6 are known to occupy cation sites in various spinels (Lindsley, 1976). The chemical formula is most appropriately written as MR204 where M and R represent cations of different valence which may or may not be the same chemical element. Some oxides such as FeAI204 are 'normal' spinels whose cation occupancy can be notated [Fe2+]tet[Al3+Al3+]oct[O2-]4, while others such as Fe30 4 are 'inverse' spinels with cation occupancy [Fe3+]tet[Fe 3+ Fe2+]oct[O2-]4. Despite the abundance of spinel minerals in the earth's crust, very little is known about their surfaces. Fe30 4 (magnetite) is a technologically important spinel both as a corrosion byproduct as well as a ferrimagnetic material. The (110) surface is the best cleavage face although the resulting fracture surface is curved rather than being perfectly flat
206
R.J. Lad
(a)
(b)
o-Fe=O= (0001) 9= s . 0 3
~
(c)
Fe:304 (111) a = s.92
~
Fe 1.,,0 (111) a -_ 3 . 0 4
Fig. 5.13. Lattice projections of iron oxide structures showing a close-packed layer of oxygen anions with Fe cations on either side of it (Lad and Henrich, 1988). The smaller solid (dashed) circles represent cations lying above (below) the oxygen plane. The cations are: (a) octahedrally coordinated Fe 3§ (b) octahedrally coordinated Fez+and Fe-§ (open circles) and tetrahedrally coordinated Fez+(solid circles)" and (c) octahedrally coordinated Fez+ (drawn for the case x=0).
(Lad and Henrich, 1989). The ideal ( 111 ) surface is conceptually easy to understand and a projection of the ( i l l ) plane also illustrates the similarities in structure between Fe:~O4 (111), Fe~_xO (111), and c/.-Fe203 (0001) surfaces. As shown in Fig. 5.13, all three contain close-packed planes of oxygen anions but with different Fe cation distributions on either side of it. The planes of oxygen follow an A B C A B C packing sequence in Fe304 and Fe~_xO and an ABABAB sequence in c~-Fe203 but the anion-anion distances are essentially the same. A quantitative LEED analysis on Fe:~O4 (111) by Barbieri et al. (1994) shows considerable relaxation of both surface anions and cations which is not unexpected for a polar surface. Atomic scale imaging of polished and annealed Fe304 (001) surfaces by STM revealed several atomic surface reconstructions after different sample treatments (Tarrach et al., 1993). This strong dependence on processing history is often a complication particularly for sub-maximal valent oxides such as Fe304 and considerable work is needed before a clearer picture emerges. An important thin film ceramic with a defect spinel lattice is y-alumina. This phase of aluminum oxide is generally synthesized in thin film form by wet oxidation of aluminum followed by a dehydroxylation high temperature annealing step. A range of y-alumina phases exists depending on the degree of dehydroxylation (Cocke et al., 1984), but the pure "/-alumina structure is completely free of hydroxyl ,,roups. The unit cell consists of a cubic close-packing of oxygen anions with aluminum cations occupying both octahedral and tetrahedral interstices, as in the spinel structure. However, one ninth of the aluminum sites are vacant in order to attain the correct 2:3 stoichiometry. Detailed quantitative structure determinations have not yet been made, but atomic models of the (100), (110) and (112) surfaces are postulated by Ealet et al. (1994) to exhibit different stoichiometries, which may be responsible for their useful adsorption properties in the area of catalysis.
207
Su~. ace structure of ceramics 5.4.5. C o r u n d u m (M203)
Several oxides including o~-A1203, ~-Fe20 3, CrzO 3, Ti203, and V2O 3 crystallize in the corundum structure (Fig. 5.3). The lattice has trigonal (rhombohedral) s y m m e try but is c o m m o n l y described in terms of a structural unit cell based upon a set of hexagonal Miller indices (Lee and Lagerof, 1985). The oxygen anions form closepacked hexagonal planes with the ABAB... stacking sequence and cations occupy two-thirds of the octahedral interstices to yield the proper stoichiometry. The octahedral coordination polyhedra are slightly distorted by electrostatic repulsion between adjacent cations along the c-axis direction. Thus, each interstitial plane between the oxygen planes contains 1/3 of the sites occupied by cations that are relaxed upward, 1/3 relaxed downward, and 1/3 vacant. The structural unit cell is complex and contains a sequence of 12 layers of atoms along the c-axis direction which can be written" ActB[3ATB~AI]B"/, where the ABAB... sequence represents packing of close-packed planes of the oxygen sub-lattice and the c~13"/.., sequence represents planes of cations and empty interstices cycling through each of the three possible types of interstitial sites as shown in Fig. 5.3. A detailed discussion of the bulk crystallography for the corundum system can be found in Kronberg (1957) and Jackson ( 1991 ). The three most important corundum oxide surfaces are the (0001) basal plane, the (10]-2) pyramidal plane, and the (1 120) prism diagonal plane. The various notations used to denote these and other surfaces in the corundum system are listed in Table 5.4. Sapphire (12-A1203) surfaces have been studied extensively because of their use as substrates in microelectronics, and 'superpolishing' techniques to Table 5.4 Different notations used to describe various surfaces in the corundum system Plane t y p e
Mineralogical notation
Basal Pyramidal
c-axis r-axis
Prism diagonal
a-axis
Prism
m-axis
Rhombohedral
s-axis
Miller-Bravais indices for the hexagonal unit cell a
Millerindices for the trigonal unit cell b
(0001) (1012) (0112) (1i02) (1120) (1210) (21-]-0) (lOlO) (0110) (1'100) ( 101'1) (0111) (1'1Ol)
(111) (411) (110) (101) (101) (011) (llO) (2l-l-) (112) (121) (100) (001) (OLO)
" 4-axis, 4-index system following conventions for hexagonal basis vectors defined in the International Tables for Crystallography (1983). b RefmTed to a trigonal basis vector set defined by lattice points 0,0,0; 2/3, 1/3, 1/3; and 1/3, 2/3, 2/3 in the hexagonal reference system. Often other basis vector conventions are used (see Jackson, 1991).
208
R.J. Lad
(0001)
(a)
(b) )
t~ Hl~Ol (10q'2)
(c)
(d)
Fig. 5.14. Lattice models of the 0~-A1203 (0001) surface (a,b) and o~-A1203 (10T2) surface (c,d) showing both top views (Gillet and Ealet, 1992) and side views (Guo et al., 1992a). The small (large) spheres correspond to AI3+cations (02- anions). The top views contain examples of anion vacancies.
produce atomically flat and damage free surfaces have been developed (Hader and Weis, 1989). The discussion here will focus primarily on this system, but many concepts are expected to be applicable to the other less-studied corundum oxide surfaces. The (0001) surface has the easiest surface crystallography to visualize since it has a hexagonal surface mesh derived from the plane of close-packed oxygen anions. However, this surface turns out to be very complex since it undergoes several non-reversible reconstructions depending on temperature. Figures 5.14a,b show models of the ( l x l ) mesh that results if the ideal bulk lattice is cleaved parallel to the (0001) basal plane at a position in the unit cell that places both types of cations (those relaxed upward and downward) on top of the surface plane of oxygen anions. A similar model with only half as many cations on the surface, viz. only those relaxed downward, would also have the same ( l x l ) mesh (Guo et al., 1992a). LEED has indicated that (0001) surfaces exhibit a ( l x l ) mesh when annealed below 1200~ but at higher temperatures complex reconstructions including ( ~ - x f f ) R 3 0 , (3"~-x 3 f f ) R 3 0 , and ( ~ x ~ 1 )R9 have been observed (Chang, 1968; French and Somorjai, 1970). Although detailed structural determinations have not been made, these structures are believed to be superlattices associated with different occupancies of the AI surface sites. Total energy calcula-
209
Su~. ace structure of ceramics I
~
1.299 nm
-J-=
,,.._
[oool]
Fig. 5.15. A perspective view of the hexagonal o~-A1203unit cell oriented to show a model of an oxygen-terminated (11~20) surface (Hsu and Kim, 1991). tions using the embedded cluster method indicate that the (1• termination with one-third of the A1 sites occupied is the lowest energy configuration (Guo et al., 1992a). Empirical as well as ab initio calculations based on pseudopotential methods suggest that there can be additional very large relaxations of the surface atoms which gives a further major reduction in surface energy (Manassidis et al., 1993). The (1012) is the cleavage plane in the corundum system since it intersects several of the vacant cation interstices (Fig. 5.14) requiring fewer bonds to be broken during fracture. ~-A1203 and o~-Fe203 single crystals do not cleave very well, but Cr203, Ti203 and V203 all provide excellent cleaves (Henrich, 1985). Polished surfaces of (1012) sapphire can easily be prepared with large flat terraces separated by single height steps (Fig 5.6a) by annealing in air or vacuum (Antonik and Lad, 1992). The surface created by an ideal bulk termination of the ( 1012) plane is shown in Fig. 5.14c,d. The oxygen cations sit above the cations and form 'zigzag' rows. The ( I x I ) surface mesh for sapphire is rectangular with dimensions 0.512 nm along the rows and 0.476 nm between rows. LEED studies have shown ( I x 1), (2xl), and (~/2-x~-)R45 patterns depending on the thermal history (Gillet and Ealet, 1992). The (1• surface is stable to at least 1000~ At higher temperatures, the reconstructed patterns become evident with an associated increase in the AI/O ratio suggesting that these reconstructed phases are due to ordering of vacancies along the 'zigzag' oxygen rows on the surface. The (1120) prism diagonal plane of sapphire has received less attention than the other surfaces" however, it has been used a substrate in silicon-on-sapphire technology since the surface mesh has a close epitaxial match with Si (111) (Nolder and Cadoff, 1965). Figure 5.15 shows the atomic arrangements on a bulk-terminated (1120) plane. REM studies of air-annealed (1120) sapphire surfaces indicate that the clean surface undergoes a (I• reconstruction (Hsu and Kim, 1991), although it may be a consequence of the sample preparation procedure. At this point, additional work is needed to clarify the surface structure of (1120) as well as several of the other planes listed in Table 5.4. The ilmenite structure displayed by FeTiO 3 and LiNbO 3 is essentially the same as the corundum structure except that the 'upward' and 'downward' cations are two different atom types. The surface of LiNbO3 is important for surface waveguide applications. A recent RHEED study of the (1012) surface indicated that annealing m
m
210
R.J. Lad
above 950~ leads to defect ordering on the surface caused by selected outdiffusion of lithium oxide species (Rakova, 1994).
5.4.6. Perovskite (MR03) The perovskite structure is cubic and is common to many oxides which have two types of cations, one having a much larger ionic radius than the other. The bulk unit cell is constructed by placing the larger 'M' cation at the body-center of a cube, the smaller 'R' cation at each of the corners, and the 02- anions at the midpoint of all of the cube edges (Fig. 5.3). Equivalently, a unit cell can be constructed with the 'M' cations at the corners of the cube, the 'R' cation at the body-center, and the O z- anions at the centers of the cube faces, as shown for SrTiO3 in Fig. 5.16. For (100) surfaces, two types of surface terminations can exist: a SrO plane and a TiO2 plane. Both planes are charge neutral, and one would expect that on a real surface, there would be a distribution of both types of terraces depending on the surface treatment and relative surface energies. Bulk-terminated (110) and (111 ) surfaces are polar and hence, as with rocksalt oxides, these surfaces tend to facet to include { 100} type surfaces. SrTiO3 (100) surfaces are technologically useful as ferroelectrics, supports in photocatalysis, and lattice-matched substrates for epitaxial growth of high temperature perovskite-based superconductors such as YBa2CuOT. Quantitative LEED studies have shown that the top surfaces of both the SrO and TiO2 terraces are rumpled, with the oxygen anions displaced outward from the ideal bulk termination by 0.08 and 0.16 angstroms, respectively, causing a distortion in the top two interplanar distances (Bickel et al., 1989). A recent energy minimization simulation which considers the competition between long-range Coulomb and short-range repulsive interactions suggests that the SrO-terminated layer may reconstruct to become ferroelectric via lateral displacements of the 02- and Sr 2§ rows, which give rise to a net surface dipole moment (Ravikumar et al., 1995). These calculations also predict that the TiO2-terminated surface should not reconstruct but retain its four-fold rotational symmetry. Experiments have yet to confirm these predictions. Experimental study of the surface structure of SrTiO3 (100) is complicated by the fact that the surface stoichiometry, and hence structure, is very dependent on Sr-O plane bulk
9
Ti-02 plane
Go
q,
O Sr
9
Fig. 5.16. Two different unit cells of the SrTiO3 perovskite structure showing two possible (100) surface terminations: a SrO plane and a TiO2 plane (Hirata et al., 1994).
Su~ace structure of ceramics
211
the exact surface preparation treatments and can be affected by electron-beam induced damage. Although there is some controversy in the literature about whether the lowest energy configuration surface contains both SrO and TiO 2 terminated terraces or predominantly TiO2-terminated terraces, EELS studies (Hirata et al., 1994) as well as STM observations (Matsumoto et al., 1992; Liang and Bonnell, 1994) have indicated that, for surfaces annealed in 02, the TiO2 termination is preferred. A SrO termination can be easily generated by depositing a monolayer of Sr from an effusion cell in an oxidizing atmosphere (Hikita et al., 1993; Hirata et al., 1994). This SrO surface is free of electronic surface states in contrast to TiO2-terminated SrTiO 3 (100) surfaces which contain intrinsic and defect surface states (Hirata et al., 1994). Annealing SrTiO3 in ultra-high vacuum leads to a loss of oxygen from the surface. STM imaging of reduced (100) surfaces has revealed ordering of oxygen vacancy defects into (2x2) (Matsumoto et al., 1992) or ('f5-x~ -) (Tanaka et al., 1994) overlayer structures; these structures are imaged via tunneling from localized surface states within the bulk band-gap created by the O-vacancy defects. Extended annealing in vacuum leads to the formation of reduced phases with composition Srn+lTinO3n+l as a result of the different sublimation rates for Sr, Ti, and O at the surface, and these phases give rise to row-like surface structures (Liang and Bonnell, 1993). Similar surface structures have been reported for BaTiO3 (100). Qualitative LEED observations indicate that ( l x l ) patterns can be produced by short term annealing treatments in vacuum, but long term vacuum annealing leads to ordered (2x2) (Cord and Courths, 1985) or (~-x,ff) (Aberdam et al., 1971) O-vacancy structures as a result of oxygen deficiency at the surface. An important class of so-called layered perovskite oxides are the high-Tc copper-oxide-based superconductors such as YBa2Cu307_ x and related systems. These oxides are orthorhombic and can be viewed as being an oxygen-deficient perovskite structure made up of three perovskite-like unit cells stacked in the c-direction: two 'BaCuO3-1ike' and one 'YCuO3-1ike' cells containing oxygen vacancies (Brook, 1991 ). Several of the surface studies of these materials that were carried out shortly after their discovery in 1986 were plagued by carbonate contamination and nonstoichiometry. LEED studies of well-defined YBa2Cu307 (100) surfaces prepared by cleaving single crystals in UHV indicate a ( lxl ) periodicity consistent with bulk termination (Halet and Hoffmann, 1989), but the surface easily loses oxygen when heated in vacuum. The (100) surface of Bi2Sr2CaCu2Os_~ is more stable and also is believed to have the termination predicted by cleaving the bulk structure (Claessen et al., 1989). A large emphasis has been given to epitaxial thin film growth in these complex systems, and the surface structures, when known, tend to be very dependent on synthesis and processing conditions. Another class of oxides having perovskite-related structures are the cubic tungsten bronzes, RxWO3 (x
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R.J. Lad
(100) surfaces prepared from single crystals with compositions ranging from 0.5 < x < 0.9 have shown the presence of ( l x l ) , (2xl), and ( 3 x l ) patterns, with the reconstructed phases appearing as x decreases (Langell and Bernasek, 1980). These results suggest that the Na cations order into rows on the surface, and the spacing between the rows increases as the Na concentration decreases. An alternative interpretation of the reconstructed LEED patterns is surface relaxation involving the tilting of the W-centered oxygen octahedra (Peacor and Hibma, 1993). 5.4. 7. Other oxides
Several other technologically important oxides do not fit into any of the oxide crystal structure classes mentioned in this section and have not yet been discussed. These systems include silica, silicates, zirconia, tantalates, and vanadates. Owing to the wide stoichiometry ranges in some of these systems, the bulk as well as surface structures can be quite complex. Silica (SiO2) is very abundant in the earth's crust and is the basic building block of a variety of silicates and glasses. A discussion of the surface structures of vitreous silica and amorphous silicates is beyond the scope of this chapter; the interested reader is referred to a paper by Garofalini (1990) reporting molecular dynamics calculations of various surface structures. Crystalline silica can exist in several forms including quartz, cristobalite, and tridymite. Similarly, many silicates are crystalline (e.g., olivine, muscovite, and zeolites). The only crystalline silica surface to receive a large amount of attention has been c~-quartz due to its importance as a substrate material. The Y-cut (0110) plane of the trigonal or-quartz lattice can be prepared with a ( lxl ) structure, but annealing to 900~ in air leads to a ( 3 x l ) or (Ix3) reconstruction (Bart et al., 1992). Furthermore, annealing in vacuum yields a disordered surface and a significant oxygen-deficiency in the top several layers. The surface is also extremely sensitive to low energy Ar § ion bombardment; a 1 keV Ar + beam can easily reduce the surface to elemental Si to a steady-state depth of 8.5 nm (Collart and Visser, 1989). ZrO2 is a refractory oxide that is relatively inert at high temperature in hostile environments. Three solid equilibrium phases occur: cubic (stable above 2360~ tetragonal (1100-2360~ and monoclinic (25-1100~ The high temperature cubic phase, c-ZrO2, has the fluorite structure and can be stabilized to room temperature by dissolving solutes such as MgO, CaO, or Y203 . It is primarily used in thin film form or as a precipitate phase in ceramic composites and thus unfortunately very little is known about its surface structure. The same holds true for tantalum oxide (Ta2Os) and vanadium oxide (V205). Both these oxides have layered structures, and it is difficult to prepare large enough samples with well-defined surfaces that can be analyzed using surface science techniques. 5.5. Carbide surface structures
Carbides generally have high hardness, good thermal and electrical conductivity, and a high melting point. Their crystal structures are most appropriately viewed as
Su~. ace structure of ceramics
213
being close-packed metal lattices with carbon atoms residing in the interstitial sites, as discussed in w 5.2. Both the surface and bulk structures, as well as most other physical properties, are very dependent on the exact stoichiometry. Because most carbides can exist over a wide stoichiometry range, one of the important considerations in studying their surfaces is whether or not the surface stoichiometry is the same as the bulk stoichiometry. Forming surfaces by crystal cleavage in vacuum is the preferred method to avoid this stoichiometry question, but crystals are often expensive and relatively small. Creating polished carbide surfaces is a challenge because of their extreme hardness; extended polishing with diamond paste is required. Heating the polished surfaces to typically above 1400~ in vacuum is effective in removing oxide contamination and annealing out polishing damage, but the question of differences in surface versus bulk stoiochiometry must be carefully ascertained. 5.5.1. Transition metal carbides
The group IV and V transition-metal carbides (TIC, ZrC, HfC, and VC, NbC, TaC) have the defective rocksalt structure, MCx, in which the composition typically can range between 0.5 < x < 1. Although this class of carbides exhibits very similar properties, there are some interesting differences particularly relating to their superconductivity and high temperature deformation behavior (Williams, 1988). The group VI transition-metal carbides (Cr3C 2, Mo2C, and WC) crystallize in structures that are more complicated than the simple rocksalt structure, with carbon occupying trigonal prism interstitial sites in addition to octahedral sites. For example, the hexagonal WC lattice (Fig. 5.4) has carbon atoms exclusively in trigonal prism sites which allows the W atoms to be in closer contact with each other than in the rocksalt structure. A large number of surface studies of transition-metal carbides have focused on TiC, since the surface properties of this material have been used in many technological applications ranging from catalytic reactions to hard coatings to highly stable field emitters. Also, relatively large sized single crystals of TiCx can be grown using the floating-zone technique (Otani et al., 1985). The (100) and (111) surfaces have quite different surface terminations and densities which makes the chemical properties of TiC very face-specific. Figure 5.17 contrasts the relatively 'open' (100) surface, which contains both Ti and C atoms in the top surface layer, to the more densely packed (111) surface, which terminates with a hexagonal array of either all Ti or all C atoms. These models are drawn with ideal TiC stoichiometry; in general, carbon vacancies will be present in proportion to the degree of nonstoichiometry in the TiCx sample. Several surface studies have been carried out on both (111) and (100) faces, but experimental attempts to create (110) surfaces result in faceting (Jansen and Hoffmann, 1988). The structure of the TiC (100) surface has been studied by impact-collision ion-scattering spectroscopy, and it has been shown that the surface plane containing both Ti and C atoms is a bulk lattice termination, free of rumpling or relaxation to within + 0.1 ~ngstrom (Aono et al., 1983). Repeated flash annealing above 1600~ is sufficient to maintain a clean reproducible surface. Carbon vacancies can be
214
R.J. Lad
(a) TiC (111) [121] ----"
~
lb) TiC (100) 121]
C--" '
[001] .-.-~
t'
_[011]
Ti
< Fig. 5.17. Lattice models showing the contrast between the relatively "open" (100) surface and more densely packed (11 1) surface of TiC (Souda et al., 1994).
introduced at the surface via preferential ion bombardment, but since the carbon atoms are much smaller than the Ti atoms, the presence of a surface carbon vacancy induces negligible relaxation of the adjacent Ti atoms (Aono et al., 1983). The clean TiC (111) surface exhibits a ( l x l ) LEED pattern after flashing to 1600~ and ion scattering spectroscopy indicates that the preferred termination is the close-packed Ti layer sitting on top of the closed-packed carbon layer (Oshima et al., 198 ! ). This Ti termination is responsible for the essentially metal-like surface reactivity of TiC (111) towards gases such as O2 and H 2 (Edamoto et al., 1992; Souda et al., 1991). A carbon-terminated (111) surface prepared by exposure to ethylene (CzH 4) at 1200~ takes on the form of a single epitaxially grown (0001) graphite monolayer (Itoh et al., 1991). This layer exhibits commensurate (2x2) and (7x7) superlattices rotated 30 ~ with respect to each other, and can be vaporized by flashing above 1600~ The (111) surface of vanadium carbide exhibits an interesting reconstruction that is not evident on TiC or NbC surfaces. In bulk VCx (0.75 < x < 0.88) specimens, periodic ordering of vacancies into superlattice structures has been observed by TEM (Epicier et al., 1989). LEED shows no evidence of superstructure on the (100) o r ( l l 0 ) faces of VC,,8~ but a three-domain (8x l) pattern is visible on the (111) surface after annealing above 1000~ (Hammar et al., 1992). STM images of this reconstruction indicate that the surface consists of three equivalent domains of a square lattice of vanadium atoms on top of the subsurface hexagonal carbon and vanadium layers. A metastable (1• surface structure can be produced by limiting the annealing temperature to below 900~ and in this case the vanadium-carbon interlayer spacing is contracted by 10% compared to the bulk (Rundgren et al., 1992). It is not theoretically understood why VCx (111) substantially reconstructs and other (111) transition-metal carbide surfaces do not, but it may be related to vacancy ordering effects or differences in cohesive bond energies. The (100) surfaces of the third transition-metal series carbides (HfC, TaC) have ( I x 1) terminations but exhibit a rippled relaxation similar to that discussed for the rocksalt monoxides (w 5.4.1). In these systems, the surface stoichiometry (carbon vacancy concentration) can be varied by using an Ar § ion bombardment process to
Su~. ace structure of ceramics
215
induce carbon deficiency and then controlling the amount of carbon diffusion to the surface via an appropriate annealing temperature/time sequence (Gruzalski and Zehner, 1990). LEED I(V) analyses of nearly stoichiometric HfC and TaC (100) surfaces indicate that the carbon atoms are displaced outward by 0.5%, 2.4% and metal atoms inward by 3%, 4.8%, respectively (Gruzalski et al., 1989). The degree of rippling is believed to be dictated by the amount of added charge that accumulates near the metal atoms in the surface layer. The (110) surfaces of TaC and HfC are unstable and like TiC (110) become faceted. STM and LEED studies by Zuo et al. (1994) reveal a unusual unidirectional faceting behavior. A TaC (100) surface heated to near 2000~ and slowly cooled has a structure made up of alternate (010).and (100) facets which periodically propagate along the [110] direction. Figure 5.18 shows an STM image and lattice model of this grating-like structure which has an average periodicity of approximately six lattice spacings. This faceted surface is the stable equilibrium configuration and contains a distribution of facet sizes. TaC (110) is also discussed in Chapter 12 (w 12.4.1.3). m
5.5.2. Silicon carbide
Silicon carbide has received a great deal of attention in the ceramics community over the last several decades in applications spanning from hard cutting materials to structural components for high-temperature applications such as engine components (Brook, 1991). The surfaces of SiC exhibit bonding and structural properties similar to elemental as well as compound semiconductors, and hence also have been of interest for a variety of high power electronic device applications (Morkoq et al., 1994). There are over 300 crystalline modifications of silicon carbide, known as polytypes. These structures are built from tetrahedrally coordinated silicon-carbon bilayers which can be stacked in one of three possible lattice sites (i.e. sites A, B, or C). If the stacking is ABCABC..., the cubic sphalerite structure is produced (also called zinc-blende; see Fig. 6.1); this form is called 3C-SiC or l-SiC. If the stacking is ABABAB .... the hexagonal wurtzite structure is formed (see Fig. 5.3); this structure is denoted 2H-SiC. The other layered polytypes are collectively called a-SiC and contain mixtures of hexagonal and cubic symmetry, but overall have hexagonal crystal symmetry. 4H-SiC has an equal number of cubic and hexagonal bonded layers, and in 6H-SiC two-thirds of the layers have cubic bonding as shown in Fig. 5.19. The I]-phase is metastable and reverts to the c~-phase upon high temperature annealing. The 6H-SiC a-phase is the most common polytype. 6H-SiC substrates as large as 2 inches in diameter, fabricated using a sublimation growth method, are commercially available (Morkoq et al., 1994). Recently, a two-step CVD process has been developed which produces high quality 13-SIC single crystalline layers when grown on Si and a-SiC substrates (Liaw and Davis, 1985). The availability of this CVD material has made it possible to study the structures of the (001) and (111) surfaces of 13-SIC and the (0001) surface of 6H-SiC. However, one of the experimental difficulties is that SiC surfaces become Si-rich by annealing in vacuum at temperatures required to volatilize surface oxide
216
R.J. Lad
[110]
L
(010)
\oo o
I-~ o o
6a
[11"0]
.._1 "-' o
(100)
,,,,,,,4 o
I
,
o
o
o
9 9 9 9 0~9 0 0 0 0 TO 0 0 0 0 0 0 9 9 9 9 9 9 9 9 9 9 9 @ 0 0 o 0 0 o o o 0 o o o o 0 9
9
9
Fig. 5.18. STM image of a faceted TaC (110) surface and the corresponding lattice model showing an irregular grating formed by alternating (100) and (010) facet planes (Zuo et al., 1993). contamination (Kaplan, 1989). Annealing in the presence of a Si flux aids in removing the surface oxide, and in addition provides a means for adjusting the surface composition.
Su~. ace structure ~ceramics
217
_f
) C
I1
I1 A C B
~c
A C
B
B
A B
B
B
C
A A
B C
A
I1 C -
B ^
~ ~ A
(a)
B
C
A
(b)
B
C
A
A A
B
C
A
B
C
A
(c)
Fig. 5.19. Crystal structures of several polytypes of SiC showing the stacking sequences of the silicon-carbon bilayers (Morkoq et al., 1994). (a) cubic 3C-SiC also known as I3-SiC; (b) 4H-SiC containing half cubic and half hexagonal bonded sites; and (c) the most common polytype, 6H-SiC, containing two-thirds cubic and one-third hexagonal sites.
Five different reconstructed phases have been observed on 13-SIC (001), all of which are related to the relaxation of the 'dangling' bonds at the surface as discussed by Duke in chapter 6 for other tetrahedrally coordinated compound semiconductors. In order of decreasing Si concentration, the observed phases exhibit (3• c(4• (2• c(2• and ( l x l ) LEED patterns (Kaplan, 1989). The first three phases are believed to be associated with different ordering configurations of asymmetric Si-Si dimers. The origin of the c(2• phase is controversial. Proposed models include a zigzag pattern of Si-Si dimers (Dayan, 1985), a halfmonolayer of Si atoms residing in the c(2• hollow sites of the top carbon layer (Kaplan, 1989), a zigzag pattern of C-C dimers (Hara et al., 1990) and top-layer carbon atoms forming bridges in between second-layer Si atoms (Badziag, 1992). The (1• phase appears after extended annealing at 1200~ and is accompanied by graphitic features in the AES and ELS spectra indicating a carbon layer surface termination. The 13-SIC (111) surface is similar to (001) in that different surface phases can exist depending on the Si concentration present in the surface region. In order of decreasing amount of Si, the observed phases are (3• (~f3-x~/3-)R30, and (1• (Kaplan, 1989). The (3• phase is proposed to consist of an adsorbed Si bilayer on top of a Si-terminated (111) bulk 13-SIC lattice, but a detailed structure determination has not been carried out to confirm this. Transition to a (~/3-• phase is associated either with the presence of 1/3 monolayer of Si adatoms, similar to the case of Si (111), or with ordered Si Vacancies in the outer bilayer. The ( l x l ) phase is ill-defined and is accompanied by graphite signatures in the spectroscopy data as on the (001) surface. Studies of this graphitic-like layer by Chang et al. (1991)
218
R.J. Lad
consistently showed a (6"~-x~/3-) LEED pattern while STM revealed a (6x6) surface geometry. This discrepancy can be explained by differences between diffraction versus electronic structure imaging of an incomensurate graphite monolayer residing on top of a ( l x l ) Si-terminated (111) [3-SIC layer. Again, a detailed structure determination is needed. The behavior of the 6H-SiC (0001) surface is essentially identical to the [3-SIC (11 l) surface. This is not surprising since, as shown in Fig. 5.19, the top four Si-C bi-layers are identical in the two cases. Following this same arguments the surfaces of all the different ct-SiC polytypes are difficult to distinguish.
5.6. Nitride surface structures Almost all metals form nitrides. However, because of the instability of nitrides towards oxidation, they do not occur naturally in the earth's crust and are therefore regarded as "synthetic" materials (Brook, 1991). Most nitrides have simple structures with l:l MN stoichiometry. However, MN2 and M2N nitrides also form, many of which have complex bulk structures that are not yet known with certainty. Like carbides, the nitrides are technologically attractive because of their hardness, high melting points, and unique thermal and electrical properties. To date, bulk and thin film nitrides have been used only in a limited number of commercial applications, and the structure of very few nitride surfaces have been characterized in detail. Oxygen contamination during synthesis or processing of nitrides is very difficult to avoid. The propensity to form glassy oxynitride phases is very common making single crystal material, except in a few instances, very scarce. The recent interest in developing III-V nitrides, particularly GaN, for optical devices in the blue wavelength regime (Strite and Morko~:, 1992) has initiated considerable research on thin nitride films. Hopefully, the methods developed to fabricate these well-defined III-V nitride films will be applied in the near future to other classes of nitrides to further our understanding of their surface structures. 5.6.1. Transition metal nitrides
The structures and properties of the transition metal nitrides are very similar to the transition metal carbides. Some of them (TIN, VN, ZrN, HfN) have the defective rocksalt structure consisting of interpenetrating fcc metal and nitrogen sublattices, while others (NbN, TaN) have the anti-NiAs structure consisting of a simple hexagonal nitrogen sublattice interleaved with a hcp metal sublattice. Nitrogen vacancies are easily formed, making large deviations from ideal l:l MN stoichiometry very common. Metal vacancies and metal-deficient stoichiometries are also possible. For example, the TiN x phase exists between 0.6 < x < 1.16, and in HfN X both sublattices have approximately a 12% vacancy concentration at the composition x-1 (Brook, 1991). Thus, as with the carbides, when studying the surfaces of transition metal nitrides, differences in surface versus bulk stoichiometry and defect concentration gradients can be very important and must be carefully examined.
Su~. ace structure of ceramics
219
Very few surface studies of single crystal transition metal nitrides have been reported primarily because of the unavailability of samples. Two notable exceptions are TiN and ZrN which have been grown using a zone-annealing technique (Johansson et al., 1980; Christensen, 1976). Angle-resolved photoemission studies of the (100) surfaces for both these materials have been performed in the context of understanding the electronic band structure of this class of nitrides (Lindstr6m et al., 1987; Lindberg et al., 1987). The surfaces were prepared by mechanical polishing followed by repeated flash heating above 1300~ This procedure produces a clean well ordered ( l x l ) surface as inferred by LEED and AES, but the issues of surface relaxation and gradients in stoichiometry from the surface to the bulk have not yet been addressed in any detail. A recent review of electronic and structural properties of transition metal nitrides has been written by Johansson (1995). 5.6.2. G r o u p I I I - V nitrides
The group III-V nitrides include BN, AIN, GaN, and InN. These materials have received a great deal of attention, particularly in thin film form, but few details are known about their surface structures. Epitaxial growth on single crystal substrates using MBE or CVD methods yields very high quality crystalline films, but they contain a large number of defects such as low-angle grain boundaries, dislocations, and nitrogen vacancies which induce atomic scale roughness on the surface. B N crystallizes in either of two forms, a hexagonal form which is chemically inert and has a very low coefficient of friction, and a cubic form which has the diamond structure and is very hard and wear resistant (McColm, 1990). The hexagonal form known as "white graphite" consists of layers of sp 2 covalently bonded B-N atoms in a planar array as shown in Fig. 5.20. The covalent radius of the boron atoms is about 20% larger than the nitrogen radius, and the boron (nitrogen) atoms have an empty (full) p, orbital oriented perpendicular to the planar B - N layer. Studies of rumpling or relaxation on this surface have not been carried out. 0 "145
0"333
nm
nm
!
I
o
I
'I 1 ' I I
I I
i
I , ,
~ t n IL, ,
, "' '
I, 9
, ,I:
I ! _
_#-,~__
~
~
A"
I I
, A_
_
A
I
Fig. 5.20. The hexagonal form of BN known as "white graphite" containing sheets of covalently bonded boron (solid spheres) and nitrogen (open spheres) atoms (McColm, 1990).
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R.J. Lad
The cubic form of boron nitride, c-BN, has the sphalerite structure (the same structure as 13-SIC; w 5.5.2) in which the B and N atoms are tetrahedrally coordinated with sp 3 hybrid bonding. The surface reconstruction of c-BN (001) has recently been theoretically studied (Osuch and Verwoerd, 1993) and contrasted with reconstruction behavior that occurs on other diamond-like lattices; viz. ]3-SIC (001), Si (001) and GaAs (001). The lowest energy structure that emerges from the calculations consists of slightly buckled B-B dimers forming a (2x2) reconstruction on a boron-terminated surface. On a nitrogen-terminated surface, the lowest energy configuration is postulated to be a (2x2) structure created by strongly bonded N - N pairs forming bridges on top of a second layer of dimerized boron atoms. The energetics of these surface structures are probably very dependent on the surface stoichiometry. Variations in surface composition were not considered by Osuch and Verwoerd in their quantum chemical calculations and no experimental data has been reported for this surface. GaN, AIN, and InN crystallize in the wurtzite structure (see w 5.4.2), although so-called zinc-blende polytypes (sphalerite structure) can be also grown in thin film form. Strite and Morkoq (1992) have reviewed the status of thin film research in these systems including crystal growth techniques, as well as structural, electrical and optical properties. The crystal structures are strongly influenced by the substrate type and orientation. The majority of III-V nitride growth has been on sapphire substrates and the hexagonal symmetry is mimicked in the wurtzite structure of the deposited film. Moreover, the lattice constant can change as a function of growth conditions, impurity concentrations, and film stoichiometry. The zinc-blende phases can be induced by using cubic substrates, and they offer interesting electrical characteristics such as reduced phonon scattering due to higher crystal symmetry. Improvement in material quality are still required before details of surface structure can be understood for this class of materials. 5.6.3. Silicon nitride
Silicon nitride has exceptional strength and creep resistance at high temperatures and is very resistant to thermal shock, making it an important high temperature structural ceramic (Riley, 1983). There is also interest in silicon nitride as an electronic thin film material in the role of an optical coating, insulating layer, or diffusion barrier (Dzioba and Rousina, 1994). In vacuum, applications are generally limited to below 1500~ since above this temperature the silicon nitride surface is unstable and decomposes to Si. In air, a silica layer forms on silicon nitride surfaces which is a protective barrier against further oxidation up to about 1400~ Silicon nitride can have either of two crystalline structures. The o~-Si3N4 phase is hexagonal, with each nitrogen atom bonded to three silicon atoms in a distorted trigonal planar configuration and each silicon atom tetrahedrally bonded to four nitrogen atoms (Wyckoff, 1964). The more common phase, I3-Si3N4, has the phenacite structure consisting of a trigonal arrangement of silicon atoms tetrahedrally bonded to nitrogen atoms as shown in Fig. 5.21. When viewed along the c-axis looking down onto the basal plane, a hexagonal arrangement of void channels is visible.
Su~. ace structure r
221
/4
o I
I
I_
I
I
5~ I
(b) Fig. 5.21. (a) Atomic positions within the 13-Si3N4unit cell and (b) top view onto the basal plane showing a hexagonal arrangement of open channels (Wyckoff, 1964). Silicon nitride is produced by either by reaction-bonding Si powder in the presence of a N2 atmosphere or hot-pressing silicon nitride powder containing a densification aid such as MgO. In both these processes, very thin glassy phases associated with oxygen impurities are often formed at the surfaces and interfaces of the particles (Clarke, 1989). Films have been prepared by various chemical vapor deposition (CVD) methods, but these films contain a variety of defect structures and compositions. No single crystal material is available and hence structural studies of well-defined surfaces have yet to be carried out. 5.7. Defect microstructure on ceramic surfaces
Defects in ceramic materials markedly influence their electronic, mechanical, and optical properties. In bulk ceramic systems, a great deal of attention has been given to point defects and their associated effects including defect-derived electronic states, enhanced diffusion, lattice polarization, and lattice relaxation as described in the text by Hayes and Stoneham (1985). At ceramic surfaces, point defects are just as important as in the bulk, and they are known to govern surface properties
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R.J. Lad
such as conductivity and chemisorption behavior (Henrich and Cox, 1994). Unfortunately, the present understanding of defects on ceramic surfaces is very rudimentary compared to metals and semiconductors where the knowledge is large enough to warrant an entire chapter in this book (Chapter 12). Although point defects (e.g. oxygen vacancies) are conceptually the easiest to understand and model, extended defect structures such as steps, facets, dislocations, and phase boundaries are very abundant on real ceramic surfaces. This defect microstructure arises not only because of the poor quality of available single crystal material but also because many competing low energy defect configurations are possible. Defect microstructure is very dependent on the sample preparation history and undoubtedly influences many experimental measurements.
5.7.1. Point defects and defect clusters
Oxides, carbides, and nitrides can exist over wide stoichiometry ranges and typically have metal-rich compositions. The simplest model to describe the nonstoichiometry involves vacancies in the non-metal sublattice. In the dilute limit, isolated non-interacting vacancies exist. At higher defect concentrations, vacancies can agglomerate into clusters or ordered crystallographic shear structures (SOrensen, 1981). In ionic ceramics, an anion vacancy induces changes in the valency of the surrounding cations leading to defect-derived electronic states. For example, a surface oxygen vacancy on NiO (100) has two additional electrons that must be shared by the neighboring Ni cations reducing them to a lower valence state; this additional electronic charge is often visible within the bulk bandgap as measured by UPS or EELS. Figure 5.22 shows a schematic model of this situation, drawn with Ni cations surrounding the oxygen vacancy having a 25% larger ionic radius compared to the other Ni cations (Henrich, 1986). Although the electronic effects associated with this local increase in covalent bonding are large, lattice relaxation
Fig. 5.22. Schematic model of an oxygen vacancy defect on NiO (100). The Ni cations surrounding the vacancy are drawn with a 25% larger ionic radius to reflect their increased electronic charge (Henrich, 1986).
223
Su~ace structure of ceramics
around these isolated vacancies is not believed to be very important (see LaFemina, Chapter 4, w 4.4). Experimental approaches to creating point defects on ceramic surfaces have included vacuum annealing, electron or photon bombardment, or ion bombardment. While all of these methods produce lattice defects, it is usually not the case that isolated vacancies are formed. Atomically resolved direct imaging of isolated point defects on insulating ceramics by AFM and STM is difficult, and therefore many studies have indirectly probed point defects through surface chemisorption behavior (Henrich and Cox, 1994). At high point defect concentrations, clustering and strong defect-defect interactions occur; one can speculate that these interactions are responsible for many of the surface reconstructions discussed in this chapter.
5. 7.2. Surface steps and extended defect structures All ceramics contain surface steps and terraces and extended defects such as dislocations that intersect the surface. Surface steps are responsible for enhanced surface reactivity, act as vacancy sinks and sources, and are nucleation sites for film growth. The reduced coordination at step edges and kink sites can induce defectderived valence band and bandgap states, similar to those derived from point defects, that influence the surface electronic and optical properties. While steps are easily discernible by experimental direct imaging (Figs. 5.6-5.8), the exact lattice geometry and relaxation around step sites is difficult to determine. LEED spot profile analysis has been successfully used to determine the step height and terrace width distributions on vicinal ZnO surfaces (Kroll et al., 1991), but very little data exists for other ceramic systems. Relatively simple static lattice calculations of the energies and relaxations along steps on rocksalt oxides have been carried out by Tasker and Duffy (1984). The general trend emerging from these calculations is that relaxations on the order of 20% of the interatomic distance are possible. For example, Fig. 5.23 shows the atomic displacements calculated around cation and anion terminated steps on MgO (100). While these simplistic calculations may not accurately reflect the true step 9 Cation o
(oo~)
Anion
~ (~oo)
Fig. 5.23. Atomic relaxations surrounding cation and anion terminated steps on MgO (100) as determined from calculations by Tasker and Duffy (1984). The arrows show the displacements magnified by a factor of ten.
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geometry, they do illustrate that relaxation at step edges can be significant and may affect surface processes on these materials. Surface steps also play a very important role during epitaxial film growth. For example, on c~-A1203 (0001) substrates (basal sapphire), the perfect surface has three-fold rotational symmetry, and hence during film growth, islands should nucleate in either of three variants. However, due to the presence of single atomic height steps (one-sixth the height of the hexagonal unit cell), two types of terraces each rotated 60 ~ with respect to each other are produced and thus six island variants are observed (Guo et al., 1992b). The same general result holds for all ceramic surfaces: the presence of steps can lead to several unequivalent terraces on which different orientations of nuclei can form. 5.7.3. Facets and phase boundaries Faceting of flat ceramic surfaces into hill and valley structures made up of other exposed low index planes is a common occurrence. Examples have already been discussed for the (111 ) surface of MgO (w 5.4.1 ), the (100) surface of TiO2 (w 5.4.2), and (110) surfaces of transition-metal carbides (w 5.5.1). The surface free energies for low index planes can vary considerably, particularly if the crystal lattice gives rise to polar surfaces. The tendency of many ceramic systems to facet means that the Wulff plots of crystal equilibrium shape contain relatively sharp cusps, although accurate Wulff plot data has not been measured. The equilibrium phase diagrams of binary and ternary ceramic compounds containing unintentionally or intentionally doped impurities have numerous phase fields and, more often than not, ceramics are multiphase materials. This chapter has emphasized well-defined single crystal ceramic surfaces, but real ceramic surfaces are more complex. Phenomena such as surface segregation, formation of second phase precipitates and glassy phases, and porosity are all extremely important in influencing surface structure and properties but are beyond the scope of this chapter.
5.8. General conclusions
The common theme that is repeated throughout this chapter is that ceramic surfaces have complex structures which are very dependent on the exact sample treatment. The wide range of accessible stoichiometries, the relative ease of creating metastable and nonequilibrium structures, and the importance of surface defects in this class of materials poses a challenge for experimentalists and theorists alike in understanding their properties. The field of ceramic surface science is rapidly maturing as evidenced by the fact that in this tutorial chapter more than 50% of papers in the reference list have been published since 1990. Progress in the areas of epitaxial growth of high quality ceramic films, scanning probe microscopies and spectroscopies, and an improved database will all markedly improve our understanding of surface structure which will continue to increase the widespread application of ceramics in tribology, catalysis, corrosion, and electronics technologies.
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Lind, D.M., S.D. Berry, G. Chern, H. Mathias and L.R. Testardi, 1992, Phys. Rev. B 45, 1838. Lindberg, P.A.P., L.I. Johansson, J.B. Lindstr6m and D.S.L. Law, 1987, Phys. Rev. B 36, 939. Lindsley, D.H., 1976, The Crystal Chemistry and Structure of Oxide Minerals as Exemplified by the Fe-Ti Oxides, in: Oxide Minerals, ed. D. Rumble (Mineralogical Society of America). Lindstr6m, J., L.I. Johansson, A. Callen5s, D.S.L. Law and A.N. Christensen, 1987, Phys. Rev. B 35, 7891. Lubinsky, A.R., C.B. Duke, S.C. Chang, B.W. Lee and P. Mark, 1976, J. Vac. Sci. Technol. 13, 189. Manassidis, I., A. De Vita and M.J. Gillan, 1993, Surf. Sci. 285, L517. Maschhoff, B.L., J.M. Pan and T.E. Madey, 1991, Surf. Sci. 259, 190. Matsumoto, T., H. Tanaka, T. Kawai and S. Kawai, 1992, Surf. Sci. Lett. 278, L153. McColm, I.J., 1990, Ceramic Hardness (Plenum Press, New York). Mr P.J., S.A Komolov and E.F. Lazneva, 1994, Surf. Sci. 307-309, 1177. Morko~, H., S. Strite, G.B. Gao, M.E. Lin, B. Sverdlov and M. Burns, 1994, J. Appl. Phys. 76, 1363 Murray, P.W., F.M. Leibsle, C.A. Muryn, H.J. Fisher, C.F.J. Flipse and G. Thornton, 1994, Surf. Sci. 321, 217. Murray, P.W., N.G. Condon and G. Thornton, 1995, Phys. Rev. B 51, 10989. Nolder, R. and I. Cadoff, 1965, Trans. Metallurgical Soc. of AIME 233, 549. Novak, D., E. Garfunkel and T. Gustafsson, 1994, Phys. Rev. B 50, 5000. Ohnesorge, F. and G. Binnig, 1993, Science 260, 1451. Oshima, C., M. Aono, S. Zaima, Y. Shibata and S. Kawai, 1981, J. Less-Common Metals 82, 69. Osuch, K. and W.S. Verwoerd, 1993, Surf. Sci. 285, 59. Otani, S., T. Tanaka and Y. Ishizawa, (1985), J. Cryst. Growth 71, 615. Pauling, L., The Nature of the Chemical Bond, 3rd ed. (Cornell University Press, Ithaca, NY) p. 514. Peacor, S.D. and T. Hibma, 1993, Surf. Sci. 287/288, 403. Peacor, S.D. and T. Hibma, 1994, Surf. Sci. 301, 11. Phillips, J.M., 1995, MRS Bulletin 20:4, 35. Poirer, G.E., B.K. Hnace and J.M. White, 1992, J. Vac. Sci. Technol. B 10, 6. Prutton, M., J.A. Walker, M.R. Welton-Cook, R.C. Felton and J.A. Ramsey, 1979, Surf. Sci. 89, 95. Rakova, E.V., 1994, Surf. Sci. 307-309, 1172. Ramamoorthy, M., R.D. King-Smith and D. Vanderbilt, 1994, Phys. Rev. B 49, 7709. Ravikumar, V., D. Wolf and V.P. Dravid, 1995, Phys. Rev. Lett. 74, 960. Riley, F.L., ed., 1983, Progress in Nitrogen Ceramics (NATO Advanced Study Institutes Series, No. 65, Nijhoff, The Hague, Netherlands). Rohrer, G.S., V.E. Henrich and D.A. Bonnell, 1992, Surf. Sci. 278, 146. Rundgren, J., Y. Gauthier, R. Baudoing-Savois, Y. Joly and L.I. Johansson, 1992, Phys. Rev. B 45, 4445. Sander, M. and T. Engel, 1994, Surf. Sci. 302, L263. Smith, D..J., L.A. Bursill and D.A. Jefferson, 1986, Surf. Sci. 175, 673. Smith, R.L., W. Lu and G.S. Rohrer, 1995, Surf. Sci. 322, 293. SCrensen, O.T., ed., 1981, Nonstoichiometeric Oxides (Academic Press, New York). Souda, R., T. Aizawa, S. Otani, Y. lshizawa and C. Oshima, 1991, Surf. Sci. 256, 19. Souda, R., W. Hayami, T. Aizawa, S. Otani and Y. Ishizawa, 1994, Surf. Sci. 303, 179. Strite, S. and H. Morkoq, 1992, J. Vac. Sci. Technol. B 10, 1237. Szabo, A. and T. Engel, 1995, Surf. Sci. 329, 241. Tanaka, H., T. Matsumoto, T. Kawai and S. Kawai, 1994, Surf. Sci. 318, 29. Tarrach, G., D. Btirgler, T. Schaub, R. Wiesendanger and H.J. Giintherodt, (1993), Surf. Sci. 285, 1. Tasker, P.W. and D.M. Duffy, 1984, Surf. Sci. 137, 91. Toth, L.E., 1971, Transition Metal Carbides and Nitrides (Academic Press, New York). Tran, T.T. and S.A. Chambers, 1994, Appl. Surf. Sci. 81, 161. Wang, L.Q., D.R. Baer and M.H. Engelhard, 1994, Surf. Sci. 320, 295. Wang, Z.L., J. Bentley, E.A. Kenik, L.L. Horton and R.A. McKee, 1992, Surf. Sci. 273, 88.
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CHAPTER 6
Surface Structures of Elemental and Compound Semiconductors C.B. D U K E Xerox Webster Research Center Webster, NY, USA
Handbook of Surface Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B.V.
All rights reserved
229
Contents
6.1.
6.2.
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1.
S c o p e and p u r p o s e
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2.
C o n c e p t s in s e m i c o n d u c t o r surface c h e m i s t r y
6.1.3.
R e l a x a t i o n and reconstruction
. . . . . . . . . . . . . . . . . . . . .
231
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233
6.1.4.
M e c h a n i s m s and scaling of s e m i c o n d u c t o r surface r e c o n s t r u c t i o n s
6.1.5.
O r g a n i z a t i o n of the c h a p t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..........
234 235
Elemental semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
236
6.2.1.
236
Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1.1. O v e r v i e w o f Si s u r f a c e s t r u c t u r e s 6.2.1.2.
6.2.2.
Si(l I l)(2xl)
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
236 240
6.2.1.3.
Si(111)(7x7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
241
6.2.1.4.
Si(100)(2xl)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
241
6.2.1.5.
Stepped surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
242
Germanium
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
243
6.2.2.1.
Ge(lll)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
243
6.2.2.2.
Ge(100)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
6.3.
Principles of s e m i c o n d u c t o r surface reconstruction . . . . . . . . . . . . . . . . . . . . . . .
6.4.
Tetrahedrally coordinated compound semiconductors
6.5.
231 231
248
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
253
6.4.1.
Zincblende(ll0)cleavage
6.4.2.
G a A s ( l I 1) and (111)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
6.4.3.
GaAs(100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
260
6.4.4.
W u r t z i t e c l e a v a g e faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
262
Synopsis
faces
245
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
265
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
265
230
6.1. Introduction
6.1.1. Scope and purpose The purpose of this chapter is to provide a survey of the known atomic geometries of semiconductor surfaces and of the broad conceptual constructs which permit the interpretation of these geometries in a coherent framework. Its intended audience consists of nonspecialists with backgrounds in physics and chemistry. Emphasis is placed on extracting reasonable generalizations from the wealth of detailed results for specific systems rather than on compiling these results. Compilations of individual surface structures may be found in reviews by MacLaren et al. (1987), Duke (1988), and van Hove et al. (1989). Our attention is focused on the low-index (i.e., nearly flat or close packed) surfaces of tetrahedrally coordinated elemental (Si, Ge) and compound (e.g., GaAs, CdSe) semiconductors. The structure of most of these semiconductors is illustrated in Fig. 6.1 in which a ball-and-stick model of the zincblende (compound semiconductors) and diamond (elemental semiconductors) atomic geometries is given The tetrahedral nature of the local atomic coordination is indicated by the shaded lines Some tetrahedrally coordinated semiconductors, mostly II-VI compounds like CdS or ZnO, crystallize in the wurtzite (or "zincite") structure which differs from the zincblende structure shown in Fig. 6.1 by virtue of the second-nearest-neighbor coordination (Wyckoff, 1963). Typical low-index surfaces which are of interest to us are illustrated in Fig. 6.2 for the diamond lattice. Vicinal (i.e., stepped) surfaces are considered only cursorily. We examine the atomic geometries of ordered structures which are periodic parallel to the surface. In particular, we do not consider surface defects or the roughening, melting, and decomensuration transitions which occur at high temperatures when the two-dimensional translational symmetry parallel to the surface is lost. These topics are treated in Chapters 12 and 13.
6.1.2. Concepts in semiconductor surface chemistry The semiconductors which we consider are held together by covalent bonds (Si and Ge) or by partially covalent and partially ionic bonds (compound semiconductors) as described, e.g., by Phillips (1973). Roughly speaking, we can visualize tetrahedrally coordinated semiconductors as bound together by tetrahedral nearest-neighbor bonds, each of which contains two spin-paired electrons. These are indicated by the heavy lines around a second-layer anion in Fig. 6.1. At a surface which exhibits the bulk atomic geometry, some of these bonds will be broken and the associated surface charge density will only contain one electron. These "dangling" 231
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C.B. Duke
Fig. 6.1. Ball-and-stick model of the zincblende atomic geometry. Open circles represent cations (e.g., Zn) and closed circles anions (e.g., S). If both species are identical (e.g., C, Si, Ge), then this structure becomes the diamond atomic geometry. Balls represent atomic species and lines (i.e., "sticks") the bonds between them. The heavy shadowed lines around the second-layer atom indicate the tetrahedral coordination of the individual atomic species in these structures.
(111)
(001)
(110)
Fig. 6.2. Plan view of a diamond structure elemental semiconductor normal to a (111) plane. Solid circles indicate atoms, and lines indicate bonds. The lines at the surfaces indicate the "dangling" bonds associated with a truncated bulk surface. bonds are indicated explicitly in Fig. 6.2 for several c o m m o n low-index faces. They are unstable. At a clean surface the atoms in the surface region become displaced from their bulk positions (i.e., they "relax") in order to reduce the electronic surface energy by forming new bonds (see Chapter 3). Alternatively, by reacting with suitable adsorbate atoms, especially H, the half-full "dangling" electronic bonds from the substrate semiconductor can form completed bonds, stabilizing the bulk atomic geometry and passivating the surface. The central consequences of bulk covalent bonding for surface structures are, therefore, that the bulk atomic geometry is likely to be altered significantly at a surface and that clean surfaces tend to be chemically reactive.
Semiconductor su~'ace structures
233
Because of the instability and reactivity of truncated bulk semiconductor surfaces, new types of surface chemical bonding occur which are not characteristic of either bulk solids or small molecules. Two particularly well-developed examples of these new types of bonds are the cleavage faces of tetrahedrally coordinated compound semiconductors, as described, e.g., by Duke (1988, 1992), and the adsorption of column five elements (e.g., Sb, Bi) thereon to form saturated overlayers, as described, e.g., by Mailhiot et al. (1985b) or LaFemina et al. (1990). A third consists of the varied surface structures which form on the low index faces of Si and Ge as a function of surface preparation and treatment as reviewed, e.g., by Schluter (1988). These new types of chemical bonding are identified by combining three types of information. The surface atomic geometries, determined experimentally by, e.g., low-energy electron diffraction (LEED) intensity analysis or ion scattering spectrometry, provide the structural basis for suspecting the occurrence of a new type of chemical bonding. Quantum mechanical calculations of the minimum-energy surface structures are used to interpret trends in these structures with differing atomic sizes and ionicities of the bonds involved. These calculations also predict, however, the eigenvalues and eigenfunctions of localized electronic surface states associated with the surface chemical bonding. These surface states, in turn, can be observed by angle resolved photoemission spectroscopy (ARPES) c~r scanning tunneling spectroscopy (STS). They provide an independent "signature" of the nature of the surface chemical bonding which leads to confirmation of the theoretical predictions of novel types of bonding associated with specific types of surface atomic geometries. A survey of the literature on the calculation of surface atomic geometries and surface states has been given by LaFemina (1992). A comprehensive review of the experimental determination and theoretical prediction of tetrahedrally-coordinated semiconductor surface structures and surface-state excitation spectra has been given by Duke (1996). From the above we may deduce that quantitative determinations of surface atomic geometries are important to achieving an understanding of semiconductor surface chemistry for two reasons. First, the systematics of the dependence of surface structural coordinates on atomic size and chemical energetics yield important clues about occurrence of novel types of surface chemical bonding. Second, the existence of quantitative structural data permits the test and evaluation of competing theoretical models of the nature of surface chemical bonds. 6.1.3. Relaxation and reconstruction
In Chapter I it was shown that all surface atomic geometries which are periodic parallel to the surface may be described by one of five two-dimensional Bravais lattices plus the specification of the structure of the surface unit mesh associated with that lattice. The semiconductors discussed in this chapter exhibit cubic (diamond or zincblende) or hexagonal (wurtzite) bulk structures. If the atomic geometry of a clean semiconductor surface exhibits the same symmetry as the truncated bulk solid, we use an adaptation of the Wood notation (see Chapter 1) and refer to it as a " ( I x 1)" structure. The atomic positions of the surface atoms can differ from those of their bulk counterparts by as much as 1 ~, however, so such surfaces are
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called "relaxed" relative to the truncated bulk surface. Stated in the language of Chapter l, for ( l x l ) structures, the atomic layer(s) parallel to the surface at the surface in general exhibit different bases from those in the bulk although they exhibit the same symmetry properties parallel to the surface. In many cases, the instability of the surface layers associated with the tendency to saturate dangling bonds causes the surface layers to exhibit a lower symmetry parallel to the surface than the truncated bulk solid. In this case the surface is said to be "reconstructed". Its symmetry properties parallel to the surface are described in the Wood notation just as in the case of the overlayers described in Chapter 1. In addition to the specification of the symmetry, determination of the bases of the various atomic layers parallel to the surface is also required to specify the surface atomic geometry. The results and interpretation of such determinations are the topic of this chapter. In general, the atomic geometry of a clean surface depends both on the way in which it was formed (e.g., cleavage, ion-bombardment and annealing, growth by molecular bean epitaxy (MBE) using specified fluxes), and on the temperature. Usually one expects only a single atomic geometry to be associated with a structure of a given symmetry at a given temperature. Thus, for S i ( l l l ) one speaks of a "(2x l)" structure obtained by cleavage at T < 300-400~ a "(7x7)" structure obtained by high temperature cleavage or thermal annealing of a sample cleaved at Ic~w temperature, and a " ( l x l ) " disordered high-temperature structure above T = 850~ as described, e.g., by Olmstead (1987). The reduced symmetry structures typically occur in domains of different equivalent orientations on the surface, so the symmetry of the observed LEED pattern may be lower than that of the surface structure itself, e.g., (2x2) in the case of equivalent (2xl) and (Ix2) domains. It is not necessarily true, however, that a LEED pattern of a given symmetry is associated with a unique surface atomic geometry. For example, scanning tunneling microscopy (STM) has revealed that for a given LEED pattern various regions of the surface may exhibit different surface atomic geometries. Therefore the relationship between the surface atomic geometry and the surface symmetry as measured by LEED or reflection high energy electron diffraction (RHEED) must be established in each case individually.
6.1.4. Mechanisms and scaling of semiconductor surface reconstructions Why do semiconductor surfaces relax and reconstruct? Attempts to answer this question constitute a significant fraction of the literature on the energetics of semiconductor surface structure. Several general principles have been identified as reviewed, e.g., by Chadi (1989, 1991), by Duke (1992, 1993, 1996), and by LaFemina (1992). For cleaved surfaces relaxations and reconstructions are driven by the lowering in energy of bands of electronic surface states associated with the rehybridization of the dangling bonds characteristic of a truncated bulk surface. The features of these energy bands are dominated by the atomic connectivity (i.e., topology) of the surface, so that each surface exhibits its own type of reconstruction. For annealed surfaces of elemental semiconductors, complicated surface structures occur which tend to minimize the number of dangling bonds. A variety of nearly degenerate structures is possible. Which one occurs in a particular case
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seems to depend on the details of the electronic structure of the surface involved as reflected, e.g., in both the dangling bond charge density and the local strain at the surface. For polar surfaces of compound semiconductors, the behavior of the electronic surface states plays a dominant role just as for cleavage surfaces. The surface atomic composition (and hence stoichiometry) of these surfaces can vary, however, since they are typically produced by MBE growth or annealing cycles. Stoichiometries generally occur which lead to autocompensated surfaces, i.e., to completely filled bands of electronic surface states separated by a gap from completely empty bands. Once a surface stoichiometry leading to such an autocompensated surface has been selected, the detailed atomic positions are determined primarily by minimizing the electronic energies of the occupied surface state bands, just as in the case of the cleavage surfaces. Thus, one can think of the criterion of autocompensated surface structures as determining various allowed classes of polar surface atomic topologies, each of which exhibits its own characteristic minimumenergy relaxed atomic geometry associated with rehybridizations of the surface chemical bonding. We discuss these principles in detail in w 6.3 and use them as a guide to our interpretation of the observed surface structures. The principles of relaxation and reconstruction articulated in the preceding paragraph typically are discovered by comparing calculated minimum energy atomic geometries with experimentally measured ones. In order to verify the generality of these principles, however, it is necessary to study the systematic variations in surface structure from one material to another. One would expect simple scaling laws, based on atomic size or measures of the strength of the chemical bonding, to characterize homologous classes of surface structures (i.e., those with the same surface stoichiometry and type of surface chemical bonding). Such scaling laws have been established for the cleavage surfaces of tetrahedrally coordinated compound semiconductors, leading to the concept of "universal" atomic geometries independent of any particular material. An extensive review of this scaling law analysis has been given by Duke (1992). The discovery of such scaling laws is a confirmation that new types of chemical bonding, which do not exist either in molecules or in bulk solids, occur at semiconductor surfaces.
6.1.5. Organization of the chapter The literature on semiconductor surface structure consists primarily of studies of individual structures on specific materials, e.g., the Si(111)(2• cleavage surface or the GaAs(100)c(2• structure grown by MBE. For each structure a variety of experimental measurements associated with features of the surface atomic geometry and electronic structure may be performed, the results of which must be fit together in a consistent fashion to provide a coherent description of the structure in question. The results of such a complete program are rarely available. More typically fragmentary or even inconsistent results are available in the literature, a complete discussion of which is inappropriate in an introductory survey like this chapter, but which may be found in comprehensive reviews, e.g. Duke (1996). Given the fragmentary and often contentious nature of the literature, we focus on the current generally accepted features of semiconductor surface structures.
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Among the elemental semiconductors Si is by far the most extensively studied, so we begin by reviewing the features of its major low-index surface structures. Stepped surfaces based on (100) and (111 ) terraces are considered, but in less detail. Similarly, the comparable results for Ge are noted but are considered in detail only in those situations where they seem to differ significantly from those for Si. Following our review of Si and Ge surface atomic geometries, we extract from these results three general principles of semiconductor surface reconstruction which permit us to predict and interpret the formation of these geometries. We supplement these with two additional principles pertinent to compound semiconductors to construct a conceptual model of the determinants of semiconductor surface structures. This model enables us to discuss the more complex situation for compound semiconductors in an orderly systematic fashion. In the case of tetrahedrally coordinated compound semiconductors, synoptic overviews are available for the cleavage faces, so these are considered as a whole rather than material by material. The situation for the polar surfaces is more like that for the elemental semiconductors: studies of GaAs predominate. Therefore the sections on these surfaces are organized around the results for GaAs with comments on other materials included as appropriate. To the extent possible references are made to the review literature. Particularly contentious topics are noted but not examined in depth. The chapter is intended as an introductory survey, not a textbook or review of a particular surface of a given material. By examination of the references cited, the reader may discover the intricacies of the measurements and their interpretation for specific surface structures as well as the rich dynamics of this field of research as a whole.
6.2. Elemental semiconductors In this section we examine the atomic geometries of surfaces of elemental semiconductors, specifically Si and Ge, crystallizing in the diamond lattice as shown in Fig. 6.1. In Fig. 6.2 we show a plan view normal to a (11 I) plane in the diamond lattices illustrating the bulk termination of the three most studied low-index faces: (111), (100) and (110). At the actual surfaces the atoms relax. We begin with the surfaces of Si because they are the most thoroughly studied, and subsequently extend our consideration to those of Ge. The surface structures of diamond have not yet been determined. 6.2. I. Silicon 6.2. I. 1. Overview of Si surface structures The (111 ) surface of Si is its cleavage face. Two possible cleavage terminations can be envisioned. To obtain the one illustrated in Fig. 6.2, the surface is cleaved so that only one bond per surface atom is broken ("single bond scission"). As noted earlier, the truncated bulk geometry is not stable because the surface atoms relax to saturate the dangling bonds. For S i ( l l l ) this is believed to occur via the formation of two
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Semiconductor su~ace structures
Si (111) - ( 2 x l ) Single-Bond Scission
Fig. 6.3. Ball-and-stick model of the (2xl) n-bonded chain structure resulting from the single-bondscission cleavage of silicon. Adapted from Haneman and Chernov (1989).
surface layers comprised of n-bonded chains of locally sp 2 coordinated Si atoms as shown in Fig. 6.3 (Haneman, 1987; Schluter, 1988). The sp 2 chain is a recurring structural motif in the surface structure of tetrahedrally coordinated semiconductors. It is an example of how the surface atoms relax so that the normally tetravalent, fourfold coordinated Si atoms can satisfy their fourfold valence with threefold coordination. The way in which this can occur is thoroughly understood for carbon containing molecules as described in many introductory chemistry texts, e.g., Gray (1965). The tetrahedrally coordinated analogs of bulk Si are methane (CH4) and ethane (H3CCH:~) which exhibit sp 3 bonding. A threefold coordinated variant of ethane, i.e., ethylene (HzCCH2), occurs in which the C species are threefold coordinated and exhibit sp 2 bonding. In addition, however, the p,~ electrons normal to the plane of the molecule also form a n bond which stabilizes the ethylene molecule. Such n bonds occur in both dimer and chain structural motifs on Si surfaces, thereby permitting the "dangling" sp 3 electrons of the truncated bulk surface to participate in chemical bonds which stabilize the reconstructed surface. Readers not familiar with such chemical bonding concepts and notation are advised to consult an introductory chemistry text like Gray (1965) loc. cit. In recent years it has been proposed by Haneman and coworkers (Haneman, 1987) that the (111) surface cleaves by breaking three bonds per silicon atom instead of one, thereby uncovering a different atomic plane parallel to the (111) surface. In this case the sp 2 surface chain would appear as shown in Fig. 6.4. The sp 2 chain is a common structural motif in both models, and by itself accounts for many features of the experimental surface characterization measurements (Haneman, 1987). Upon annealing above about 300~ the (2x l) cleavage structure converts irreversibly into a (7x7) structure, although the precise nature of this conversion is still being discussed (Haneman et al., 1989; Duke, 1996). A schematic diagram of the dimer-adatom-stacking fault (DAS) model for this structure (Takayanagi et al., 1985; Haneman, 1987; Schluter, 1988; Uhrberg and Hansson, 1991) is given in Fig.
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Si (111) -(2xl)
Triple-Bond Scission
Fig. 6.4. Ball-and-stick model of the (2xl) 7t-bonded chain structure resulting from the triple bond scission cleavage of silicon. Adapted from Haneman and McAlpine (1991 ).
Si (111)-7x7
(a) Top View
(b) Side Plan View
"
Fig. 6.5. Schematic illustration of the top (panel a) and side (panel b) views of the dimer-adatomstacking-fault (DAS) model of the S i ( l l l ) - 7 x 7 structure. The side view is given along the long diagonal of the unit cell. In the top view (panel a) the large shaded circles designate the adatoms in the top layer of the structure. The large solid circles designate "rest atoms" in the second layer which are not bonded to an adatom. Large open circles designate triply bonded atoms in this layer, whereas small open circles designate fourfold coordinated atoms in the bilayer beneath. Smaller solid circles designate atoms in the fourth and fifth bilayers from the surface. The size of all circles is proportional to the proximity to the surface. The side view (panel b) is a plan view of nearest neighbor bonding in a plane normal to the surface containing the long diagonal of the surface unit cell. Smaller circles indicate atoms out of the plane of this diagonal. Adapted from Takayanagi et al. (1985).
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Si (100)- (2 x 1)
[oo~] ,,~[]~0] = [11o] Fig. 6.6. Ball-and-stick model of the buckled dimer structure of Si(100)(2• MacLauren et al. (1987).
Adapted from
6.5. Three new surface motifs occur in this structure. Dimers appear along the outside edges of the unit cell. Adatoms (which are threefold coordinated) appear over some of the atoms in the second layer but not over others (called rest atoms). By the third layer all of the atoms are fourfold coordinated as in the bulk. On one half the unit cell the atoms in the second layer lie directly over those in the fifth layer as in the wurtzite rather than the zincblende packing sequence. Thus, they constitute a stacking fault in the diamond lattice. The (100) surface of Si is also important because it is a common template for MBE and a widely used substrate for microelectronics fabrication. This surface is prepared by ion bombardment and annealing. It exhibits a (2x l) structure. The current consensus about the atomic geometry of this surface is that it consists of tilted dimers as shown in Fig. 6.6 (Schluter, 1988; Duke, 1993, 1996). Thus, we again find a dimer motif as a means to achieve threefold surface coordination of a group IV atom. Vicinal surfaces of silicon have been studied in the vicinity of both Si(111), as reviewed, e.g., by Williams and Bartelt (1991), and Si(100), as reviewed, e.g., by Griffith and Kochanski (1990). Important issues for these surfaces include the thermodynamics of surface morphology (Alerhand et al., 1990; Williams and Bartelt, 1991), the kinetics of the Si(l 11)(2• to Si(111)(7x7) transition (Feenstra and Lutz, 1991), the significance of subsurface elastic distortions in determining surface morphology (Vanderbilt et al., 1989), and the role of surface morphology in epitaxial growth (Lagally et al., 1990). These surfaces may be visualized as terraces of Si(111) or Si(100), respectively, separated by steps of varying height. The step edges are rough because they contain kinks. A scanning tunneling microscope
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Fig. 6.7. Scanning tunneling microscope image of a Si(100) surface misoriented by 0.3 ~ toward the 1110] direction, as measured by Swartzentruber et al. (1990). Step height is 1.4 ~ and mean terrace width is 260 ,A,.Adjacent terraces have alternating directions of dimerization, visible as parallel lines (separation, 7.7 ,~)in the image. From Lagally (1993). (STM) image of a vicinal Si(100) surface is shown in Fig. 6.7. The tilted dimers on the Si(100) terraces are evident in the figure. The dimers are aligned in opposite directions on adjacent terraces, indicating that the steps are one atomic dimension in height. Moreover, the step edges parallel to the dimers are relatively smooth whereas those normal to the dimers are highly kinked. Thus, it is evident that the terrace surface atomic geometry is an important factor in determining the energetics of the kinks, and hence the dynamics and kinetics of crystal growth or epitaxy, on such vicinal surfaces. 6.2.1.2. Si(l 11)(2• Upon low-temperature (T <_ 350~ cleavage, a (2• structure is exhibited by S i ( l l l ) . The structural motif characteristic of this surface is planar sp 2 bonded ("x-bonded") chains, as illustrated in Figs. 6.3 and 6.4 for the single- and triplebond scissions, respectively. The single-bond scission is widely believed to provide the correct termination (Schluter, 1988) although the three-bond scission model provides a comparable description of available structural data (Haneman, 1987). For the single-bond-scission termination the x-bonded structure was first proposed by Pandey (1981). Detailed surface structure determinations have been given by Himpsel et al. (1984) and Sakama et al. (1986), in which specifications of the atomic coordinates of the surface atoms may be found. This structure illustrates an
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important feature of semiconductor reconstructions: the saturation of the dangling bonds characteristic of the truncated bulk surface. In the case of sp 2 chains, the electrons in Pz orbitals normal to the chain form delocalized rt bonds which stabilize the chain geometry by 0.3 eV < AE < 0.4 eV per atom (Schluter, 1988). Thus, the surface reconstruction allows the fourfold valence of the surface Si species to be fully saturated even though these species are only threefold coordinated.
6.2.1.3. Si(111)(7x7) The low-temperature (T < 850~ equilibrium structure of S i ( l l l ) exhibits a (7x7) symmetry (Schluter, 1988; Haneman, 1987). First observed via the symmetry of LEED beams by Schlier and Farnsworth (1959), the (7• nature of the surface unit cell was directly confirmed by scanning tunnelling microscopy (STM) 25 years later (Binnig et al., 1983). The first quantitative surface structure determination was reported two years later via a detailed analysis of transmission electron diffraction (TED) intensities from thinned samples in ultrahigh vacuum (UHV), leading to the DAS model shown in Fig. 6.5 (Takayangi et al., 1985). This model subsequently has been refined using various techniques. A complete set of structural coordinates obtained via LEED intensity analysis has been reported by Tong et al. (1988). Total energy calculations confirm that the DAS model yields the lowest energy S i ( l l l ) ( 7 x 7 ) structure relative to the other structural models proposed thus far (Qian and Chadi, 1987; Badziag and Verwoerd, 1989). Heuristic rationalizations of this result usually focus on counting the number of dangling bonds in the various models, while noting that the (unrelaxed) DAS structure exhibits only 19 (out of a possible 49) dangling bonds (Schluter, 1988; Tong et al., 1990). On the basis of studies of smaller structures (e.g., Si(111)(2• ), Si(100)(2• )), however, we expect relaxations of the DAS structure to rehydridize the unrelaxed dangling bonds, thus tending to saturate the valences of the surface species. Indeed the vertical displacements of the atoms in the top two layers in the LEED Si(111)(7• structure are greatly reduced relative to their bulk sp 3 values leading to a more planar structure which can be stabilized by delocalized ~ bonds, as in the case of analogous structures in metal-free phthalocyanine (Duke, 1984). The most plausible explanation for the low energy of this structure is (Vanderbilt, 1987) that it results from a tradeoff between the energy gain via the elimination of dangling bonds by the dimer-row domain walls and the cost in energy of stacking faults and corner holes. This explanation is consistent with the principles of semiconductor reconstruction described below and describes a host of otherwise puzzling results on the stress dependence of Si(111) and Ge(111) surface structures (Duke, 1996). Each of the various surface structural features (e.g., adatoms, rest atoms, corner atoms) exhibits its own unique chemical reactivity, which can be probed directly by atom resolved tunneling spectroscopy as reported by Avouris and Lyo (1990).
6.2.1.4. Si(lO0)(2xl) The Si(100)(2• structure is prepared by ion-bombardment and anneal cycles of suitably oriented Si crystals. The structural motifs characteristic of this structure are rows of tilted dimers, shown in Fig. 6.6, analogous to the (1010) cleavage faces of wurtzite structure compound semiconductors (Duke, 1992; Duke and Wang, D
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These may be visualized as consisting of a ~ bond formed from a pair of adjoining dangling bonds and a much weaker r~ bond formed from the remaining two (rehybridized) dangling bonds. A major question concerning this structure is whether or not the dimers are tilted. Recent ab initio density function theory calculations (Dabrowski and Scheffler, 1992) have resolved this controversy in favor of tilted dimers which can be regarded (Duke, 1993, 1996) as a consequence of a metal-semiconductor transition in quasi-one-dimensional systems (i.e., rows of dimers) caused by the fact that for untilted dimers the surface is metallic due to the weakness of the r~ bond (Chadi, 1979a). This surface is characterized by a high degree of disorder, with both tilted and untilted dimers being observed by STM (Haneman, 1987). It is currently believed (Dabrowski and Scheffler, 1992; Weakliem et al., 1990) that the observation of symmetric dimers by STM is a dynamic effect, caused by the rapid interconversion of the dimers between nearly degenerate tilted configurations. Higher order structures, especially c(4x2) and (2x2), have been reported which are readily understood in terms of correlations between the local tilts of the dimers and whose existence is required to achieve a quantitative description of the ARPES measurements of the dispersion of the surface-state n and n* bands associated with the tilted dimers (Uhrberg and Hansson, 1991 ).
6.2.1.5. Stepped surfaces In recent years considerable effort has been devoted to the study of stepped surfaces of Si based both on its (100) (Griffin and Kochanski, 1990) and ( 111 ) (Williams and Bartelt, 1991 ; and Chapter 2) surfaces. Interest has been focused on the structure of the steps which occur (Aspnes and Ihm, 1986; Chadi, 1987a; Griffin and Kochanski, 1990), the thermodynamics of vicinal surfaces (Williams and Bartelt, 1990), the role of steps on the kinetics of epitaxial growth (Lagally et al., 1990), and the role of elastic energy, including external stress, in determining step distributions (Webb et al., 1990; Vanderbilt et al., 1989). The free energy of a vicinal surface can be expressed as a function of macroscopic parameters (i.e., the miscut angle, ~, its orientation, and the temperature T) and microscopic phenomenological surface energetic parameters (e.g., the step formation energy, the kink formation energy, step-step interaction energies, and the surface free energy of the associated low index face) (Williams and Bartelt, 1990). Macroscopic elastic constants enter the step creation energy. When the anisotropy of the surface state tensor permits the relief of bulk elastic stress by forming stress domains separated by steps on the surface, steps can form spontaneously even on low index surfaces, as illustrated by Si(100) (Vanderbilt et al., 1989). Such models have been applied to describe the mixture of single and double steps on vicinal surfaces of Si(100) miscut along the <110> direction. At low 9 and high T, single layer steps tend to occur whereas at larger 9 and low T double layer steps are observed. Although a phase transition between these conformations has been proposed, the observations do not seem to support this hypothesis (Aumann et al., 1992). This topic is developed in more detail in Chapter 2. Another extensively examined case consists of vicinal surfaces of Si(111) as a
Semiconductor su.rface structures
243
function of 9 and miscut directions (Williams and Bartelt, 1991). In this case the reversible (7x7) ( l x l ) phase transition on Si(111) leads to regions of unstable stepped surfaces which are characterized by faceting. Thus, the kink and step energetics can lead to a wide variety of thermodynamic behaviors which are subjects of active current research (Williams and Bartelt, 1991; Aumann et al., 1992).
6.2.2. Germanium 6.2.2.1. Ge(111) The surface phase diagram of Ge(111) is roughly analogous to that of Si(111) with several important differences. First, a metastable ( l x l ) structure has been reported for cleavage at cryogenic temperatures (T < 20 K) the atomic geometry of which has not yet been determined (Olmstead, 1987). For cleavage between 40 K _< Tc~ < 380 K, a 2xl metastable structure is obtained whose atomic geometry is believed to be described by the r~-bonded chain model show in Fig. 6.3 (Takeuchi et al., 1991 ; Zhu and Louie, 1991). The low temperature equilibrium structure exhibits a c(2x8) symmetry (Phaneuf and Webb, 1985), and the adatom geometry indicated in Fig. 6.8 (Feidenhans'l et al., 1988; Becker et al., 1989). Detailed atomic coordinates may be found in van Silfhout et al. (1990). At approximately 300~ (i.e., 600 K) the c(2x8) structure experiences a reversible first-order phase transition to a ( l x l ) symmetry (Phaneuf and Webb, 1985) currently regarded as a "premelting" transition (Feenstra et al., 1991). Finally, at 785~ the (1• surface layer "melts" reversibly (McRae and Malic, 1987, 1988; Denier van der Gon et al., 1991) before the bulk crystal melts at 937~ Although they differ in detail from those for Si( I 11 ), the Ge(l I 1) surface atomic geometries satisfy the same general "principles" of clean surface reconstruction (Duke, 1993, 1996) (see also w 6.3): surface bonds tend to be saturated; the surface tends to be insulating (as opposed to metallic); and the cleavage surface structures can exhibit non-equilibrium geometries because the relaxation pathway from the truncated bulk geometry to the equilibrium geometry is activated. The n-bonded chain (2x1) structures occur due to their accessibility via cleavage from the bulk geometry (Northrup and Cohen, 1982) but are metastable. They are semiconducting due to their quasi-one-dimensional character (Duke, 1993, 1996). They saturate the fourfold valence of the threefold coordinated surface Si or Ge species because of the delocalized rt bonding. The adatom c(2• equilibrium geometry also satisfies the saturated-bond and insulating criteria. The surface states associated predominately with the adatoms are full whereas those associated predominately with the rest atoms are empty, although the charge transfer is incomplete due to the rehydridization that accompanies relaxations of the surface atoms (Hirschorn et al., 1991). Thus, the surface valences are saturated, with the adatom relaxing to a distorted p3 local conformation and the rest atom to a distorted sp 2 conformation. In addition, since the number of adatoms equals the number of rest atoms, the surface is non-metallic, i.e., the distorted p3 surface states are full and the predominately Pz character nonbonding sp 2 surface states are empty. Other geometries based on the Si(111) DAS structure have been proposed for Ge(111)(2x8) (Takayanagi and Tanishiro, 1986) but have not been found to be compatible with available structural
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G e (111 ) - c ( 2 x 8 )
Adatom 9 Second-Layer Atom
9 Rest-Atom O Backbond Atom
Fig. 6.8. Schematic illustration of the top view of the adatom model of the Ge(l I 1)c(2• low-temperature equilibrium structure. The adatoms, top-layer atoms bonded to the adatoms ("backbond atoms"), top-layer atoms not bonded to an adatom ("rest atoms") and second-layer atoms are indicated. Adapted from Klitsner and Nelson (1991). data (Feidenhans'l et al., 1988; Maree et al., 1988; Becker et al., 1989). This result, plus the site-specific surface chemistry of Ge( 111 )c(2• (Klitsner and Nelson, 1991 ), indicate that these three general principles of clean surface reconstruction describe the behavior of Ge(111) quite well. We develop these principles more fully in w 6.3.
6.2.2.2. Ge(100) As in the case of Si(100), Ge(100) surfaces are prepared by ion-bombarding and annealing suitably cut bulk Ge crystals. The fully ordered surface of Ge(100) is believed to be a c(4x2) structure which exhibits a reversible transition to a ( 2 x l ) geometry at Tc = 200 K (Kevan, 1985; Lambert et al., 1987). A variety of studies favor the tilted dimer model of the (2xl) structure as shown in Fig. 6.6 for Si(100) (Lambert et al., 1987; Kubby et al., 1987; Grey et al., 1988; Landemark et al., 1990). The low temperature c(4x2) phase is interpreted as the suitably ordered array of the tilted dimers as shown in Fig. 6.9 (Kevan, 1985) in accordance with the
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Semiconductor su~. "ace structures
Ge (100)- c (4x2)
Fig. 6.9. Ball-and-stick model of the low temperature c(4x2) structure of Ge(001). Adapted from Needels et al., 1988). predictions of total energy calculations (Needels et al., 1988). On the basis of high-resolution ARPES data, the disordered (2x l) structure is believed to be metallic (Kevan, 1985), although the general features of its surface state spectrum are well described by an insulating tilted dimer model (Landemark et al., 1990). A metallic peak in the photoemission spectrum is attributed to defects arising in the disordered (2x l) structure (Kevan, 1985).
6.3. Principles of semiconductor surface reconstruction Having described the plethora of reconstructions observed for elemental semiconductors, it is helpful to abstract some generalizations from these structures as well as from the corresponding results for compound semiconductors. We refer to these abstractions as "principles" of semiconductor surface reconstruction. They are useful in predicting new surface structures as well as in identifying the unique characteristics of surface chemical bonding. They also assist us in the organization of our discussion of the more complicated case of compound semiconductors given in w 6.4. Our presentation follows that given by Duke (1993,1996). A useful conceptual model of semiconductor surface reconstruction is afforded by the recognition that the atomic geometry of the uppermost few atomic layers is driven by chemical forces which tend to saturate the valences of the atomic species in these layers. If chemical bonds are formed in this process, the energy gain per bond per atom is a substantial 1 eV (Chadi, 1989). These bonds form a new surface compound which places the substrate under elastic stress. The substrate atoms therefore relax to new equilibrium positions. The energy gain in this relaxation is about 0.01 eV/surface atom (Duke, 1993, 1996; Chadi, 1979b). Thus, we envisage semiconductor surface reconstructions as occurring via the formation of a new
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"epitaxially constrained" chemical compound on the surface (AE ~ 1 eV/atom) together with the local atomic elastic relaxation of the substrate (AE ~ 0.01 eV/ atom) on which it is "grown" epitaxially (Duke, 1987, 1993, 1996). Tetrahedrally coordinated semiconductor reconstructions are driven primarily by the need to saturate the "dangling" unsaturated tetrahedrally chemical bonds which would occur if the surface retained its truncated bulk geometry. This concept is abstracted into our first "principle" of semiconductor surface reconstructions: (1) Reconstructions tend to either saturate surface "dangling " bonds via rehydridization or to convert them into non-bonding electronic states. On the (100) faces of both elemental and compound semiconductors, dimers form to saturate the valences of the surface atoms. The simplest examples are the (100) surfaces of Si and Ge, for which ( 2 x l ) and c(4x2) surface structures consist of rows of dimers. These structures are illustrated in Figs. 6.6 and 6.9, respectively. Another example is the (2xl) cleavage structure of the (111) surfaces of Si and Ge indicated in Figs. 6.3 and 6.4. In these cases delocalized x bonding in the surface sp 2 chains saturates the fourfold valence of the threefold coordinated surface species. In general, we expect the atoms at the surface to relax in such a fashion that their local chemical valence tends to be saturated. This conceptual model requires further refinement via the incorporation of solid-state effects associated with the periodicity of the surface. Consideration of these effects leads to the second principle of semiconductor surface reconstruction: (2) In many cases (and all quasi-one-dimensional ones) surfaces can lower their energies by atomic relaxations leading to semiconducting (as opposed to metallic) surface state eigenvalue spectra. An example of this principle is given by the tilted dimer (2• structures of the (100) surfaces of Si and Ge shown in Fig. 6.6. From a chemical perspective, we can regard the surface dimer in Fig. 6.6 as being bound by a c~ bond emanating from the two dangling bond orbitals in between the dimerized surface atoms and a weaker x-like bond emanating from the two dangling bond orbitals pointed away from the dimer. The associated rt* bonding orbital is empty but is separated from the bonding orbital by only a small energy gap (Eg -- 0.5 eV (Chadi, 1979a; Ihm et al., 1980). On the surface the molecular rt and re* orbitals broaden into bands associated with the k vectors in the surface Brillouin zone. These bands overlap for a symmetric (i.e., untilted) dimer, so that the surface becomes metallic. Since these bonds are nearly one-dimensional (along the rows of dimers), however, it is energetically favorable to lower the energy by an atomic relaxation (Duke, 1993, 1996), and hence the dimers tilt opening up a gap between filled electronic states originating primarily from the "up" atom and empty states originating primarily from the "down" atom. The resulting "asymmetric" or tilted dimer model is in good quantitative accord both with experimental determinations of the surface excitation spectra (Uhrberg and Hansson, 1991) and with modern (i.e., converged) total energy calculations (Dabrowski and Scheffler, 1992). The third general principle illustrated by Si and Ge surfaces is the role of the distinction between activationless and activated reconstruction in determining what structure is observed in a given experimental situation:
Semiconductor surface structures
247
(3) The surface structure observed will be the lowest energy structure kinetically accessible under the preparation conditions. The cleavage face of Si and Ge is the (111) surface. When cleaved, a reconstruction of the uppermost four atomic layers occurs leading to a (2xl) chain-like top-two-layer structure as shown in Fig. 6.3. This structure yields a semiconducting surface state spectrum characterized by a gap between the ~ and r~* states of the surface chains. Because of the r~ bonding along the chains, all the surface bonds are saturated, with the new surface epitaxially-constrained compound consisting of the uppermost two layers of "r~-bonded-chains" on an elastically distorted Si substrate (Pandey, 1981). But the Si(111)(2xl) and Ge(111)(2xl) r~-bonded-chains are not the lowest-energy structures. Rather, these are the DAS structures shown in Fig. 6.5 for Si(111)(7x7) or the adatom structure shown in Fig. 6.8 for Ge(111)c(2x8). The reason that low temperature cleavage yields the higher-energy (2x l) structure is believed to be that this geometry can be reached from the truncated bulk geometry via a nearly activationless (E a < 0.03 eV) process (Northrup and Cohen, 1982). The DAS and c(2x8) structures require large-scale atomic motions which can be accessed only at high temperatures. Compound semiconductors introduce an additional ingredient into the conceptual model: charge transfer between the anion and cation. Typically cation "dangling bond" orbitals at surfaces occur at higher energies than the corresponding anion orbitals, so charge is transferred from the former to the latter. If the surface is to remain uncharged, the anion derived orbitals must be completely occupied and the cation derived orbitals completely empty (Harrison, 1979). Such a surface is said to be "autocompensated". This observation leads to the fourth principle of tetrahedrally-coordinated semiconductor surface reconstruction: (4) Surfaces tend to be autocompensated. This principle describes a remarkable variety of structures on the polar surfaces of compound semiconductors (Pashley, 1989; Biegelsen et al., 1990a,b). It determines the stoichiometry of these surfaces and is satisfied trivially for the 1:1 non-polar cleavage faces. While this principle does not predict the detailed atomic geometry, it does identify the candidate structures from which a possible geometry must be selected. Thus, for example, it predicts the 3:1 ratio of anion dimers to missing anion dimers on Ill-V(100) (Pashley, 1989; Biegelsen et al., 1990a), the change to a uniform (2xl) dimer structure on II-VI(100) (Pashley, 1989), and a further change to a c(2x2) adatom (or vacancy) structure for I-VII(100). It also is pertinent to the (111) surfaces, for example, the cation vacancy structure characteristic of III-V(111)-(2• (Chadi, 1989) and the anion trimer structure exhibited by III-V(i 11)-(2x2) (Biegelsen et al., 1990b). A more detailed discussion of this topic is given in w 6.4.2 and 6.4.3. The autocompensation principle yields allowed candidates for a surface atomic geometry, but not to a geometry itself. To obtain the detailed structure, we require the fifth surface reconstruction principle: (5) For a given surface stoichiometry, the surface atomic geometry is determined primarily by a rehydridization-induced lowering of the surface-state bands associated with the (filled) anion dangling bond orbitals.
248
C.B. Duke
This principle has been validated in detail for the non-polar cleavage faces of both zincblende and wurtzite structure compound semiconductors (Duke, 1988; Duke and Wang, 1988b) and is believed to be pertinent to the polar surfaces as well (Chadi, 1989). It underlies the concept that for a given surface, the structures may be "universal" for tetrahedrally coordinated compound semiconductors in that all individual materials exhibit the same structure as measured in suitably scaled dimensionless units (Duke, 1993, 1996). As articulated, these principles apply only to the surfaces of clean, tetrahedrally coordinated semiconductors. Additional principles must be invoked in cases of adsorbed overlayers or heteroepitaxial growth (Duke, 1993, 1996; LaFemina, 1992). Nevertheless, the five principles noted above afford a convenient context for our discussion in w 6.4 of the surface atomic geometries of tetrahedrally coordinated (binary) compound semiconductors.
6.4. Tetrahedrally coordinated compound semiconductors Tetrahedrally coordinated compound semiconductors occur in two crystallographic allotropes: zincblende (shown in Fig. 6.1) and wurtzite. Zincblende materials exhibit a single cleavage face: the (110) surface. This surface, whose truncated bulk structure is shown in Fig. 6.10, consists of equal numbers of anions and cations
Fig. 6.10. Schematic illustration of the truncated-bulk (i.e., unrelaxed) (110) cleavage lhce of zincblende structure compound semiconductors. Shaded circles typically represent anions and open circles cations. From Duke (1977).
249
Semiconductor su~. ace structures
m
Fig. 6.11. Schematic illustration of the truncated-bulk (i.e., unrelaxed) (1010) cleavage surface of wurtzite structurecompound semiconductors. Shaded circles typically represent anions and open circle cations. From Duke (1988). which form zig-zag chains directed along (110) directions in the surfaces. Wurtzite materials exhibit two cleavage faces, both consisting of equal numbers of anion and cation species. The (1010) cleavage surface, whose truncated bulk structure is shown in Fig. 6.11, consists of isolated anion-cation dimers back bonded to the layer beneath whereas the (1120) surface, characterized by the truncated bulk structure shown in Fig. 6.12, consists of anion-cation chains, analogous to those on zincblende(110) but with four rather than two inequivalent atoms per surface unit cell. All three surfaces exhibit relaxations which do not alter the symmetry of the surface unit cell but which lead to large (~1 ]k) deviations of the positions of the atomic species in the uppermost layer(s) from those in the truncated bulk solid. These relaxed surface geometries have been determined quantitatively for the (110) surfaces of zincblende structure AlP, AlAs, GaP, GaAs, GaSb, InP, InAs, InSb, ZnS, ZnSe, ZnTe, CdTe, and CuCI; the (1010) surfaces of wurtzite structure ZnO and CdSe; and the (1120) surfaces of CdS and CdSe (Duke, 1988, 1992; Kahn et al., 1992). Theoretical predictions of these relaxed geometries have been given which are in either quantitative or semiquantitative correspondence with the experimentally determined structures. Analysis of the trends exhibited by the members of each class of cleavage surface and the comparison thereof with theoretical predictions m
250
C.B. Duke
TOP VIEW
Fig. 6.12. Schematic illustration of the truncated-bulk (i.e., unrelaxed) (1120) cleavage surface of wurtzite structure compound semiconductors. Shaded circles typically represent anions and open circle cations. From Duke (1988). permit the extraction from these results of generalizations characteristic of novel types of surface chemical bonding (Duke, 1992). The most important of these is the notion that, for each class of surface, the atomic geometries are approximately "universal" when their coordinates are properly scaled with the bulk lattice constant. Extensions of the concepts of inorganic molecular coordination chemistry are required to predict the cleavage-surface atomic geometries and electronic structures of binary tetrahedrally coordinated compound semiconductors (Duke, 1987, 1996). A comprehensive review of the structures of the cleavage faces of tetrahedrally coordinated compound semiconductors has been given by Duke (1994). Other surfaces of tetrahedrally coordinated compound semiconductors can be fabricated either by ion bombardment and annealing of suitably cut bulk crystals or by MBE. The most commonly studied of these are the (111) and (100) surfaces of zincblende structure materials, the truncated bulk geometries of which are indicated schematically in Figs. 6.13 and 6.14, respectively. These ideal surfaces consist
Semiconductor su~. "ace structures
251
Fig. 6.13. Schematic illustration of the truncated-bulk (i.e., unreconstructed) zincblende ( 111 ) surface. The shaded circles representing surface atoms can be either cations ("A" termination, usually designated (111)) or anions ("B" termination, usually designated (1 I1)). From Duke (1988).
Fig. 6.14. Schematic illustration of the truncated-bulk (i.e., un,reconstructed) zincblende (100). The shaded circles representing surface atoms can be either cations (cation-terminated surface) or anions (anion-terminated surface). From Duke(1988).
252
C.B. Duke
Fig. 6.15. Schematic illustration of the truncated-bulk (i.e., unreconstructed) wurtzite (0001). Open circles indicate cations and shaded circles anions. The corresponding anion-terminated surface is labeled (0001). From Duke (1988).
entirely of either cations ("A" termination) or anions ("B" termination), and hence are called "polar" surfaces. Such (1 1 1) and (100) surfaces prepared experimentally do not exhibit these ideal geometries. Both their composition and structure depend on the conditions used to fabricate them. Wurtzite structure compound semiconductors possess analogous polar surfaces, such as the unrelaxed (0001) surface for which the truncated bulk geometry is indicated schematically in Fig. 6.15. All of these polar surfaces exhibit complicated symmetry lowering reconstructions just like the (111) surfaces of Si and Ge. The structures of the polar surfaces of zincblende materials have been studied extensively in special cases (e.g., GaAs) and are discussed in more detail in w 6.4.2 and 6.4.3. The atomic geometries of the polar surfaces of wurtzite structure materials are essentially unknown except for a few fragmentary results (Duke, 1988, 1966; LaFemina, 1992). They are not considered further.
Semiconductor su~ace structures
253
6.4.1. Zincblende (110) cleavage faces The cleavage surfaces of tetrahedrally-coordinated compound semiconductors occupy an important niche in the universe of semiconductor surfaces. First, they are charge neutral with a fixed, stable 1:1 stoichiometry of anions to cations. At these surfaces, charge has been transferred from the cation "dangling bond" derived surface states, leaving them empty, to the corresponding anion derived surfaces states, leaving them full. Thus, the cleavage surfaces automatically satisfy the autocompensation rule which is the fourth principle of semiconductor surface reconstruction articulated in w 6.3. Second, because of their intrinsic charge neutrality, the cleavage surface atomic geometries exhibit the same symmetry parallel to the surface as a truncated bulk solid (Kahn, 1983; Duke, 1988). This fact is often referred to as the cleavage surfaces exhibiting "relaxed" but not "reconstructed" structures. Third, the mechanism of the surface relaxation on all the cleavage surfaces is the same: the rehybridization of the bonds associated with atoms in the surface layer due to the lowering in energy of the (filled) anion derived danglingbond band of surface states as these states acquire back and surface bonding character (Duke, 1988; Duke and Wang, 1988a; LaFemina, 1992). This mechanism encompasses the different atomic topologies of the three zincblende and wurtzite cleavage surfaces as well as the differing molecular coordination chemistries of III-V versus II-VI compounds. Its recognition and documentation forms the basis for Principle 5 in w 6.3. Fourth and finally, the resulting surface structures are found to be "universal" in that, to within current experimental uncertainties, all materials except CuCI (Kahn et al., 1992) exhibit the same surface structure when dimensions are measured in units of the bulk lattice constant (Duke, 1983, 1988, 1992). Although this is not expected to be an exact result, its approximate validity affords important insight into the relative magnitudes of the various driving forces for surface reconstructions. Specifically, it reveals that surface topology exerts a far more significant influence on the surface atomic geometry than the ionicity of the surface chemical bonds (Duke, 1992, 1993, 1996). As noted irr the discussion of Fig. 6.10, the cleavage face of zincblende structure compound semiconductors is the (110) surface consisting of zig-zag chains of equal numbers of anions and cations. Each surface anion (cation) exhibits two surface bonds to other surface cations (anions) and one backbond to a cation (anion) in the layer beneath. These surfaces relax via approximately bond-length-conserving rotations with the anion moving outward and the cation inward, as illustrated in Fig. 6.16. The independent surface structural variables are defined in Fig. 6.17, including the tilt-angle, to, of the top-layer chain relative to the substrate. The (110) surface relaxations of zincblende structure binary compound-semiconductors consist of bond-length-conserving top layer rotations characterized by to = 29 + 3 ~ sometimes accompanied by a small (< 5%) relaxation of the top layer toward the substrate, a second-layer relaxation in the opposite sense to that in the top layer, and small lateral displacements of the top layer atoms away from their bond-rotation positions (Duke, 1988, 1992). Comprehensive tables of both the best bond-lengthconserving structures and the overall best-fit structures obtained from low-energy
254
C.B. Duke
Zincblende (11 O)
[110]
/ ~ [OO1l
9 Anion 0 Cation
[71o] Fig. 6. ! 6. Atomic geometry of the relaxed non-polar (110) cleavage faces of zincblende structure binary compound semiconductors. Adapted from Duke and Wang (1988b).
electron diffraction (LEED) may be found in the literature (Duke, 1988; L a F e m i n a , 1992). Of all the s e m i c o n d u c t o r surface s t r u c t u r e s , that of GaAs(110) is both the first one to be determined (Lubinsky et al., 1976) and the most extensively studied by a wide variety of surface science techniques. A detailed account of the interesting history of its determination has been given by Duke and Paton (1985). Of equal importance as the surface relaxations themselves is their systematic dependence on parameters characteristic of the nature of the chemical bonding in the various compound semiconductors. The two parameters of historical interest are the bulk lattice constant, ao, and the ionicity - - both spectroscopic (Phillips, 1973) and Pauling (Pauling, 1960). A brief history of research on the dependence of the surface structures on these parameters has been recounted by Duke (1992). The central results of that research are the linear dependence of the perpendicular displacement of the uppermost anion-cation sublattices (i.e., A~,+ from Fig. 6.17) on ao and the approximate equality of surface and bulk bond lengths (Duke, 1988, 1992). The former initially was demonstrated experimentally (Duke, 1983) as shown in panel (a) of Fig. 6.18. It soon was established theoretically as well (Mailhiot et al., 1985a; Godin et al., 1992), as indicated in panel (b) of Fig. 6.18. For bond-length-conserving rotations A~,I is related to the top-layer tilt angle, o3, via a o
Al,• -~- sin (o3)
(6.1)
Semiconductor su~. ace structures
255
Zincblende (110) Side View .~,,.x
~
y
I
I I
f
I
z
(a)
Al,y A1 _L /'
- -- ?i~-
I
I
I
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I
- - -
(b) Top View
-,k ~.
x
ao
i
.---------~-I
. -
/ --, ~'N ~Top
~ -w,, 2ndLayer/ Layer /
~ I AniOn O Cation
Fig. 6.17. Panel (a): Schematic indication of the independent structural variables which define surface atomic geometry for the (] 10) surface of zincblende structure compound semiconductors. The top-layer tilt angle is labeled co in the figure. Panel (b): Zincblende(110) surface unit cell. Adapted from Duke (1983).
Therefore the linear scaling of A~,• with ao shown in Fig. 6.18 implies that co = 29 +_ 3 ~ independent of material, as noted above, provided that the surface reconstruction is a bond-length-conserving rotation. While inexplicable from the perspective of molecular coordination chemistry (Duke, 1987; Duke and Wang, 1988a) these results faithfully reflect the principles of surface reconstruction articulated in the preceding section. The surface bonds are saturated via charge transfer from the cation dangling-bond surface-state orbitals to the corresponding states on the anion (Principle 1). Because a bond-length-conserving rotation is activationless (Principle 3), the electronic energy associated with the occupied anion-derived surface states is lowered by such a relaxation until the cation exhibits sp 2 coordination (Principle 5). Principle 4 (autocompensation) is satisfied automatically due to the 1: 1 stoichiometry of the non-polar (110) cleavage
256
C.B. Duke
,
/
I
I
o< 0.7 ~'~ I-
I
I
I
AIAsc~:y~ ~ l n P
/ <1 0.6r/ I ~
znTe
A , P ~ . f GaAs
"-
1 CdTe~ ~,~,
~,,~oGaP
0.5 5.0
ZnS
i 5.2
1 5.4
I 5.6
I 5.8
I 6.0
i 6.2
6.4
6.6
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(a)
I
0.8
"-~0.7
I
I
I
I
1
I
I
I
.~
.j,,~"
O
I
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_
_
<1 GaAs
oGaP 0.6 ~,.4
I
i
I
I
1
i
6.0
i
I
I
1
I
6.6
ao(A) (b) Fig. 6.18. Panel (a): Experimental correlation between values of AI,• for the (I 10) cleavage surface found by LEED intensity analyses and the bulk lattice parameter, a,,, of the substratc. Adapted from Duke (1983). Panel(b): Theoretical prediction of the linear dependence of A l.l on a,; obtained from tight-binding total energy calculations. Adapted from Godin et al. (1992).
faces. From the perspective of molecular coordination chemistry, the commonality of the Ill-V, II-VI, and most recently I-VII (Kahn et al., 1992) atomic geometries is a puzzle, since the surface structure reflects the molecular coordination of III-V molecules but not that of either II-VI or I-VII molecular analogs. Thus, the five principles of tetrahedrally-coordinated semiconductor surface structure constitute additions to analogous principles of molecular and bulk coordination chemistry. Moreover, they predict novel types of surface chemical bonding. For tetrahedrally coordinated compound semiconductor cleavage surfaces, even more powerful general results may exist. It has been proposed (Duke, 1992; Godin et al., 1992) that the potential energy surfaces for zincblende(l I0) surfaces exhibit a common form for all materials, depending only on bulk materials parameters like lattice constant, ao, and spectroscopic ionicity, f. In such a situation, both the structures and vibrational force constants exhibit universal scaling laws with ao and f. Scaling with ao has been established (Duke, 1992; LaFemina, 1992) but that with f has not. A recent study (Kahn et al., 1992) of the structure of CuCI(110) relative
257
Semiconductor su~. "ace structures
0.14 0.12
~'
v'
-
I '
'
'
'
I E) ' '
o
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I '
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GaAs
'
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'
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ZnS
0.08
0.06 0.04
ao
't
ZnTe O~3 CdTe o
0.10
A1,_L
'
!
0.02
0012
0.3
0.4
0.5
0.6
0.7
0.8
Spectroscopic Ionicity (a) c o .-4--, O tl:l .,...,
0.10
c0 0.08
0 r
c c
0.06
o
0.04
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0.02
133
InAs J
O9 c 0
u_
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000 9G O s b
-0.02
0.2
,
t
0.3
,
I
0.4
,
Spectroscopic
I
0.5
(b)
,
I
0.6
~
0.7
,
0.8
Ionicity
Fig. 6.19. Panel (a): Dependence of the top-layer anion-cation vertical displacements for zincblendc(110) cleavage faces on the spectroscopic ionicity, f. Adapted from Kahn et al. (1992). Pancl (b): Dependence of the top-layer anion-cation bond length for zincblende(110) cleavage faces on spectroscopic ionicity,f Adapted from Lessor et al. (1993).
to that of other zincblende (110) surfaces established that A,,• is approximately independent off, as shown in panel (a) of Fig. 6.19. The surface anion-cation bond length, on the other hand, seems to exhibit a small but measurable contraction from its bulk value which is approximately linear in f, as shown in panel ( b ) o f Fig. 6.19 (Lessor et al., 1993). Thus, to within the accuracy of current experimental results, the structures of the zincblende (110) cleavage faces may be regarded as universal for all materials with the values of the independent structural parameters for a given material being determined from scaling laws like those shown in Fig. 6.19. Furthermore, a similar result is believed to be characteristic of the non-polar (1010) and m (1120) cleavage faces of wurtzite-structure compound-semiconductor cleavage faces (Duke and Wang, 1988a, 1989). Therefore surface-structure scaling laws, m
258
C.B. Duke
even more general and powerful than the five principles of surface structure determination, may characterize the cleavage-surface atomic geometries of tetrahedrally coordinated compound semiconductors (Duke, 1992). 6.4.2. G a A s ( l l l ) and (11~
Zincblende structure compound semiconductors exhibit two distinct (111) oriented polar surfaces as indicated in Fig. 6.13: a cation terminated surface (e.g., GaAs(111)) and an anion terminated one (e.g., G a A s ( l l l ) ) . Neither surface is found in its truncated bulk form. In the case of GaAs, both may be prepared by MBE and by ion bombardment and annealing. The resulting structures depend on the preparation conditions (Ranke and Jacobi, 1977; Bringans and Bachrach, 1984). Recent summaries of the literature on the preparation dependence of the surface structures have been given by LaFemina (1992), and by Kaxiras et al. (1987a,b). The most thoroughly characterized of the various observed structures is G a A s ( l l 1)-(2x2) formed by ion-bombardment and annealing of a suitably cut crystal (Tong et al., 1984). Its formation may be visualized by generating a Ga vacancy, as shown in Fig. 6.20, to produce an autocompensated surface as required by Principle 4 in w 6.3. Then, as expected from Principle 5 in w 6.3, the top two layers relax extensively to form 12-membered ring analogs to the sp 2 chain found on GaAs(110) but with one As in each ring exhibiting four rather than three bonds. The final structure, found for GAP(111)-(2x2) as well as for GaAs(l 11)-(2x2), is illustrated in Fig. 6.21 (Tong et al., 1985). It also is believed to be characteristic of InSb(l 11)-(2x2) (Bohr et al., 1985). This structure was predicted by tight-binding total energy calculations at the time of its determination by LEED (Chadi, 1984). A table of the atomic coordinates of both the predicted and experimental structures for GaAs(111)-(2x2) is given by Chadi (1984).
Fig. 6.20. Schematic indication of the ideal (unrelaxed) GaAs(l I 1)-p(2x2)-Ga vacancy structure. From Duke (1988).
Semiconductor su~. ace structures
259
Fig. 6.21. Atomic geometry of the reconstructed GaAs(l 1l)-p(2x2)-Ga vacancy structure. Adapted from Tong et al. Puga (1985).
Another way to generate an autocompensated GaAs(l 11)-(2x2) surface is to adsorb As trimers to bond to three surface Ga species and to complete the occupancy of the As trimer surface states by transfer of charge from one remaining Ga in accordance with Principle 4 in w 6.3. This structure is indicated in Fig. 6.22. Atoms in the top three layers relax substantially in order to reduce the energy of the occupied surface state as required by Principle 5 in w 6.3. A suitably relaxed structure is predicted to provide the lowest energy geometry in an As rich environment (Kaxiras et al., 1987a). Kaxiras et al. (1987a) argue that the transition between the Ga ~acancy structure and the As trimer structure was observed (but not quantitatively confirmed) in MBE experiments by Cho and Arthur (1975). Thus, Figs. 6.21 and 6.22 both illustrate minimum free energy structures which are believed to describe G a A s ( l l l ) - ( 2 x 2 ) structures which are grown by MBE under differing conditions. Figure 6.21 depicts the structure obtained by ion bombardment and annealing of suitably cut bulk crystals. Analogous arguments to those given above for G a A s ~ 11) have been advanced for G a A s ( l l l ) by Kaxiras et al. (1987b). The situatiorr',is predicted to be more complex than that for GaAs(l l l) with three (2x2) structures (Ga trimer, substitutional Ga or As vacancy plus Ga adatom pair, As trimer) predicted to have the lowest free energy for various growth conditions. Experimentally, only the As trimer structure for G a A s ( l l 1)-(2x2) has been found by STM (Biegelsen et al., 1990b). Its structure is indicated in Fig. 6.22. No quantitative determination of the atomic positions for this structure has yet been reported. Heating this GaAs(11 I)(2x2) structure leads to a (fi--9-x'4-~) geometry which is expected to be metallic as
260
C.B. Duke
GaAs (111 )/(111 )-(2x2)
~~ .
9
.~ ~
0
0
.
.
"
9
0
"
"Q
O"
0
9
"'"".... ~......'"""
~ I Top Layer Anion 0 Second Layer Cation/Anion 9Third Layer Anion/Cation Fig. 6.22. Schematic illustration of the As trimer model of the GaAs( 111 )-(2x2) and GaAs(l ! 1)-(2x2) reconstructions. Large solid circles denote top layer As atoms in the "trimer." For GaAs( 111 )-(2x2) small open circles designate top-layer Ga atoms, and small solid circles denote second layer As atoms. For GaAs(l I 1)-(2x2) small open circles designate top-layer As atoms, and small solid circles denote second-layer 6a atoms. Adapted from Biegelsen et al. (1990b).
described by Biegelsen et al. (1990b), who also suggests an atomic geometry for this structure. The metallic nature of this surface does not violate Principle 4 of w 6.3 because the atomic geometry is not one-dimensional in character. Moreover, the Ge(100)-c(4x2) structure discussed in w 6.2.2.2 reveals another possibility: the occurrence of defects in an intrinsically lower symmetry structure. The determination of the atomic geometries of zincblende structure compound semiconductor (! 1 I) surfaces clearly constitutes a frontier which remains to be explored. 6.4.3. G a A s ( l O 0 )
The (100) surfaces of tetrahedrally coordinated c o m p o u n d semiconductors are prepared almost exclusively by MBE (Joyce et al., 1988). As in the case of elemental semiconductors, the structural motif for these surfaces is dimers, but Principle 4 in w 6.3 predicts differing arrays of dimers and missing dimers depending
S e m i c o n d u c t o r su .rface structures
261
GaAs ( 1 0 0 ) - c (4x4) Top View 0
0
o"
0
0
0
0
0
0
9
o" 9
0
0
"
9
9 ~q
"
9
9
~
, ~
~
~
o
o
~ o ~ o ~_o2 o ~ . 0 ~ o
o
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,to,,, o ,Lo_~ o 'LO..," o
o
o" 9
Side
9
qleg
9
9oql
9
~oql
o" 9
View As
As Ga As Ga Fig. 6.23. Schematicillustrationof the GaAs(100)-c(4• reconstruction. Dashedlines indicate the surface unit cell. Adapted from Biegelsen et al. (1990a). upon the surface stoichiometry (Pashley, 1989). Moreover, many different such stoichiometries are possible, depending upon MBE growth conditions (Farrell et al., 1987; Biegelsen et al., 1990a). GaAs(100) constitutes an extensively studied example, exhibiting a complex array of ordered structures as growth conditions are altered from As rich to Ga rich. Three examples of proposed structures with increasing Ga surface composition are shown in Figs. 6.23-6.25. The structures shown in Figs. 6.23-6.25 illustrate the application of the principles of semiconductor surface reconstruction articulated in w 6.3. Dimers form to saturate the surface dangling bonds as required by Principle 1. The importance of kinetic accessibility, captured as Principle 3, is immediately evident by virtue of the wide range of stable surface structures each generated by its own unique growth conditions. Five distinct stable reconstructed structures, each of a different symmetry, have been observed for GaAs(100) corresponding to different preparation conditions as discussed, e.g., by Bachrach et al. (1981) and by Biegelsen et al. (1990a). Three of these are shown in Figs. 6.23-6.25. The requirement of surface charge neutrality (Principle 4) guarantees semiconducting surface states (Principle 2). The charge neutrality principle dominates the features of the surface structures by requiring arrays of dimers which leave the anion-derived surface states full and cation derived states empty (Farrell et al., 1987; Pashley, 1989). Indeed this principle is so dominating that for doped GaAs charged defects in the (100) dimer structure (i.e., kinks) form during growth in order to compensate the space charge
262
C.B. Duke
G a A s (100) - c ( 2 x 8 ) Top View 9
9
,'
o
9
~'" - "~
~
~'-" ~
o ~ ~ 1 7 6 1 7 6
"Jli" ~
.... .---,---.---..-.-...-.---,
~
*
:
9
o
~
~
9
o
Side View As
As Ga
Fig. 6.24. Schematic illustration of the GaAs(100)-c(2x8) (or (2x4)) reconstruction. Dashed lines indicate the surface unit cell. Adapted from Biegelsen et al. (1990a).
due to the charged dopants in the surface depletion region (Pashley and Haberen, 1991 ; Pashley et al., 1992). Although the detailed atomic geometries of the surface structures shown in Figs. 6.23-6.25 have not yet been determined, they are expected to be relaxed in accordance with Principle 5. Currently available theoretical calculations (Chadi, 1987b) indicate that the expected relaxations occur. Another way to prepare GaAs(100) surfaces is MBE growth followed by As capping in the MBE chamber. The samples can then be removed and decapped in another chamber where they are subjected to thermal treatments and to other structural and electrical studies. Experiments performed in this manner have confirmed that the As dimers associated with the c(2x8) structure shown in Fig. 6.24 are untilted (Chambers, 1992). They have also verified that the work functions of the structures shown in Figs. 6.23-6.25 behave as expected from the surface stoichiometries (Chen et al., 1992). Thus, the qualitative features of these three GaAs(100) surface structures seem reasonably well-established, although their precise atomic geometries have yet to be determined.
6.4.4. Wurtzite cleavage faces As noted at the beginning of w 6.4, tetrahedrally coordinated compound semiconductors occur in wurtzite (or "zincite") as well as zincblende structures. In particular,
263
Semiconductor su~. ace structures
GaAs (100) - c (8x2) Top View . . . o o . .
9
~
9 O--,,O
9 ~-.-O
W
. o , . o o . o o o
Side View
Fig. 6.25. Schematic illustration of the GaAs(100)-c(8x2)reconstruction. Dashed lines indicate surface
unit cell. Adapted from Biegelsen et al. (1990a). common III-V semiconductors crystallize in the zincblende structure (Figs. 6.1, 6.10, 6.13, and 6.14) whereas common II-VI compounds prefer the wurtzite structure (Figs. 6.11, 6.12, and 6.15). Wurtzite structure crystals exhibit two cleavage faces. The ( 1010) surface, whose truncated bulk geometry is shown in Fig. 6.11, exhibits an anion-cation dimer structural motif. The (1120)surface, whose truncated bulk geometry is shown in Fig. 6.12, exhibits a sp 2 chain motif but in the cis conformation rather than the t r a n s conformation characteristic of zincblende(110) (Fig. 6.10). As in the case of zincblende(110), both of these surfaces exhibit structural relaxations which do not lower the symmetry of the crystal parallel to its surface. A detailed review of the structures of these surfaces has been given by Duke (1994). For the ( 1010)surfacesthedimerstiltinan approximatelybond-length-conserving rotation in which thecation almost exhibits aplanarsp 2 local conformation (Duke and Wang, 1988a). A schematic drawing of the relaxed surface is shown in Fig. 6.26. Quantitative surface structure determinations have been reported for ZnO (Duke et al., 1978) and CdSe (Duke et al., 1988; Horsky et al., 1992). For the (1120) surface the chains pucker in such a way that the cation again almost exhibits a planar sp 2 local conformation (Duke and Wang (1988a). A schematic drawing of the relaxed surface is shown in Fig. 6.27. Quantitative surface structure determinations have been reported for CdS (Kahn et al., 1991) and CdSe (Horsky et al., 1992). m
264
C.B. Duke
Wurtzite (1010)
O Anion Q Cation Fig. 6.26. Atomicgeometry of the reconstructed ("relaxed") non-polar ( 1010) cleavage laces of wurtzite structure binary compound semiconductors. From Duke and Wang (1988b).
Wurtzite (11~.0)
Anion 0 Cation Fig. 6.27. Atomic geometry of the reconstructed ("relaxed") non-polar (1120) cleavage faces of wurtzite structure binary compound semiconductors. From Duke and Wang (1988b). The remarkable feature of both surface structures is the local atomic conformation of the anion and cation. The cations, which are group II species, exhibit the approximate sp 2 conformation characteristic of the molecular conformation of group III species. Similarly, the anions, which are group VI species, exhibit the distorted p3 local conformations characteristic of the molecular conformation of group V species. Thus, unlike the case for III-V compounds, the surface structures
Semiconductor su~. ace structures
265
of these I I - V I c o m p o u n d s cannot be regarded as small-molecule adaptations of the bulk tetrahedral structure. Instead, a new concept is needed to interpret these structures. The requisite concept is that of surface states whose features are characteristic of the crystal surface topology (Duke and Wang, 1988a). These surface states, whose effect is described by Principle 5 in w 6.3, dominate the surface bonding and give rise to surface chemical bonds which have neither bulk nor small-molecule analogs (Duke, 1987). Principle 4 (autocompensation) is satisfied by the truncated-bulk structure, so that Principles 1 and 2 are satisfied as well. Hence, Principles 3 and 5 control the details of the relaxed structure. For approximately bond-length-conserving relaxations, no kinetic barriers occur (Duke and Wang, 1988a) so that Principle 5 dominates the determination of the observed surface geometry. Therefore for these II-VI cleavage faces the relaxed atomic geometries reflect directly the consequences of surface-state mediated chemical bonding (Duke and Wang, 1988a).
6.5. Synopsis In this chapter we have reviewed the known structures of the surfaces of tetrahedrally coordinated semiconductors. These surfaces exhibit a variety of structural motifs, some of them associated with metastable rather than equilibrium geometries. We introduced a set of five principles of semiconductor surface reconstruction on the basis of which we could identify which motifs could occur and rationalize which ones do occur in particular cases. These principles permitted the classification and interpretation of the diversity of observed structures so that the physical p h e n o m e n a leading to these structures could be identified, and hence the origin of the structures could be comprehended. Even more powerful scaling laws were found to lead to the concept of universal semiconductor surface structures for the cleavage faces of c o m p o u n d semiconductors. Thus, while they are complicated in detail, the geometries of semiconductor surfaces exhibit regularities which lead to their description and interpretation by a rather small number of physical principles and scaling constructs.
References Alcrhand, O.L., A.N. Berker, J.D. Joannopoulos, D. Vanderbilt, R.J. Hamers and J.E. Demuth, 1990, Phys. Rcv. Lett. 64, 2406. Aspnes, D.E. and J. lhm, 1986, Phys. Rev. Lett. 57, 3054. Aumann, C.E., J.J. de Miguel, R. Kariotis and M.G. Lagally, 1992, Surf. Sci. 275, 1. Avouris, P. and S.-W. Lyo, 1990, Studying Surface Chemistry Atom-by-Atom Using the Scanning Tunneling Microscope, in: Chemistry and Physics of Solid Surfaces VIII, eds. R. Vanselow and R. Howe. Springer Series in Surface Sciences 22. Springer, Berlin, p. 371. Bachrach, R.Z., R.S. Bauer, P. Chiaradia and G.V. Hansson, 1981, J. Vac. Sci. Technol. 19, 335. Badziag, P. and W.S. Verwoerd, 1989, Phys. Rev. B 40, 1023. Beckcr, R.S., R.S. Swartzentruber, J.S. Vickers and T. Klitsner, 1989, Phys. Rev. B 39, 1633.
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Biegelsen, D.K., R.D. Bringans, J.E. Northrup and L.E. Swartz, 1990a, Phys. Rev. B 41, 5701. Biegelsen, D.K., R.D. Bringans, J.E. Northrup and L.E. Swartz, 1990b, Phys. Rev. Lett. 65, 452. Binnig, G., H. Rohrer, C. Gerber and E. Weibel, 1983, Phys. Rev. Lett. 50, 120. Bohr, J., R. Feidenhans'l, M. Nielsen, M. Toney, R.L. Johnson and I.K. Robinson, 1985, Phys. Rev. Lett. 54, 1275. Bringans, R.D. and R.Z. Bachrach, 1984, Phys. Rev. Lett. 53, 1954. Chadi, D.J., 1979a, Phys. Rev. Lett. 43, 43. Chadi, D.J., 1979b, Phys. Rev. B 19, 2074. Chadi, D.J., 1984, Phys. Rev. Lett. 52, 1911. Chadi, D.J., 1987a, Phys. Rev. Lett. 59, 1691. Chadi, D.J., 1987b, J. Vac. Sci. Technol. A 5, 834. Chadi, D.J., 1989, Ultramicroscopy 31, 1. Chadi, D.J., 1991, Electron-Hole Counting Rule at III-V Surfaces: Applications to Surface Structure and Passivation, in: The Structure of Surfaces III, eds. S.Y. Tong, M.A. van Hove, K. Takayanagi and S.D. Xie, Springer Series in Surfaces Sciences, 24. Springer, Berlin, p. 532. Chambers, S.A., 1991, Surf. Sci. 261, 48. Chen, W., M. Dumas, D. Mao and A. Kahn, 1992, J. Vac. Sci. Technol. B 10, 1886. Cho, A.Y. and J.R. Arthur, 1975, Prog. Solid State Chem. 10, 157. Dabrowski, J. and M. Scheffler, 1992, Appl. Surf. Sci. 56-58, 15. Denier van der Gon, A.W., J.M. Gay, J.W.M. Frenken and J.F. van der Veen, 1991, Surf. Sci. 241,335. Duke, C.B., 1977, J. Vac. Sci. Technol. 14, 870. Duke, C.B., 1983, J. Vac. Sci. Technol. B 1,732. Duke, C.B., 1984, Canad. J. Chem. 63, 236. Duke, C.B., 1987, Surface Structural Chemistry for Microelectronics, in: Atomic and Molecular Processing o1" Electronic and Ceramic Materials, eds. I.A. Aksay, G.L. McVay, J.T. Stroebe and J.F. Wagner. Materials Research Society, Pittsburgh, PA, p. 3. Duke, C.B., 1988, Atomic geometry and electronic structure of tetrahedrally coordinated compound semiconductors, in: Surface Properties of Electronic Materials, eds. D.A. King and D.P. Woodruff, The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Vol. 5. Elsevier, Amsterdam, p. 69. Duke, C.B., 1992, J. Vac. Sci. Technol. A 10, 2032. Duke, C.B., 1993, Appl. Surf. Sci. 65/66, 543. Duke, C.B., 1994, FestkOrperprobleme. Advances in Solid State Physics, ed. R. Helbig, Vol. 33. Vieweg, Braunschweig/Wiesbaden, p. 1. Duke, C.B, 1996, Chem. Rev., 96, 1237. Duke, C.B and A. Paton, 1985, Surf. Sci. 164, L797. Duke, C.B, A. Paton, Y.R. Wang, K. Stiles and A. Kahn, 1988, Surf. Sci. 197, 11; (E) Surf. Sci. 214, 334. Duke, C.B, R.J. Meyer, A. Paton and P. Mark, 1978, Phys. Rev. B 18, 4225. Duke, C.B and Y.R. Wang, 1988a, J. Vac. Sci. Technol. A 6, 692. Duke, C.B and Y.R. Wang, 1988b, J. Vac. Sci. Technol. B 6, 1440. Duke, C.B and Y.R. Wang, 1989, J. Vac. Sci. Technol. A 7, 2035. Farrell, H.H., J.P. Harbison and L.D. Peterson, 1987, J. Vac. Sci. Technol. B 5, 1482. Feenstra, R.M. and M.A. Lutz, 1991, Deposition and annealing of silicon on cleaved silicon surfaces studied by scanning tunneling microscopy, in: The Structure of Surfaces III, eds. S.Y. Tong, M.A. van Hove, K. Takayanagi and S.D. Xie, Springer Series in Surface Sciences, 24. Springer, Berlin, p. 480. Feenstra, R.M., A.J. Slavin, G.A. Held and M.A. Lutz, 1991, Phys. Rev. Lett. 66, 3257. Feidenhans'l, R., J.S. Pederson, J. Bohr, M. Nielson, F. Greg and R.L. Johnson, 1988, Phys. Rev. B 38, 9715. Godin, T.J., J.P. LaFemina and C.B. Duke, 1992, J. Vac. Sci. Technol. A 10, 2059. Gray, H.B., 1965, Electrons and Chemical Bonding. Benjamin, New York, pp. 155-175.
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Grey, F., R.L. Johnson, J.S. Pederson, R. Feidenhans'l and M. Nielsen, 1988, Surface X-ray diffraction: the Ge(001)-(2• reconstruction and surface relaxation, in: The Structure of Surfaces II, eds. J.F. van der Veen and M.A. van Hove. Springer Series in Surface Sciences, 11. Springer, Berlin, pp. 292. Griffith J.E. and G.P. Kochanski, 1990, Crit. Rev. Solid State Mater. Sci. 16, 255. Haneman, D., 1987, Rep. Prog. Phys. 50, 1045. Haneman, D. and A.A. Chernov, 1989, Surf. Sci. 215, 135. Haneman, D. and N. McAlpine, 1991, Appl. Surf. Sci. 48/49, 111. Haneman, D., J.J. Rownd and M.G. Lagally, 1989, Surf. Sci. 224, L965. Harrison, W.A., 1979, J. Vac. Sci. Technol. 16, 1492. Himpsel, F.J., P.M. Marcus, R. Tromp, I.P. Batra, M.R. Cook, F. Jona and H. Liu, 1984, Phys. Rev. B. 30, 2257. Hirschorn, E.S., D.S. Lin, F.M. Leibsle, A. Samsavar and T.C. Chiang, 1991, Phys. Rev. B 44, 1403. Horsky, T.N., G.R. Brandes, K.F. Canter, C.B. Duke, A. Paton, D.L. Lessor, A. Kahn, S.F. Horng, K. Stevens, K. Stiles and A.P. Mills, Jr., 1992, Phys. Rev. B 46, 7011 lhm, J., M.L. Cohen and D.J. Chadi, 1980, Phys. Rev. B 21, 4592. Joyce, B.A., P.J. Dobson and P.K. Larsen, 1988, Molecular beam epitaxy of III-V compounds: aspects of growth kinetics and dynamics, in: Surface Properties of Electronic Materials, eds. D.A. King and D.P. Woodruff, The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Vol. 5. Elsevier, Amsterdam, p. 271. Kevan, S. D., 1985, Phys. Rev. B 32, 2344. Kahn, A., 1983, Surf. Sci. Rep. 3, 193. Kahn, A., C.B. Duke and Y.R. Wang, 1991, Phys. Rev. B 44, 5606. Kahn, A., S. Ahsan, W. Chen, M. Dumas, C.B. Duke and A. Paton, 1992, Phys. Rev. Lett. 68, 3200. Kaxiras, E., Y. Bar-Yam, J.D. Joannopoulos and K.C. Pandey, 1987a, Phys. Rev. B 35, 9625. Kaxiras, E., Y. Bar-Yam, J.D. Joannopoulos and K.C. Pandey, 1987b, Phys. Rev. B 35, 9636. Klitsner, T. and J.S. Nelson, 1991, Phys. Rev. Left. 27, 3800. Kubby, J.A., J.E. Griffith, R.S. Becker and J.S. Vickers, 1987, Phys. Rev. B 36, 6079. LaFemina, J.P., 1992, Surf. Sci. Rep. 16, 133. LaFemina, J.P., C.B. Duke, and C. Mailhiot, 1990, J. Vac. Sci. Technol. B 8, 888. Lagally, M.G., 1993, private communication. Lagally, M.G., Y.-W. Mo, R. Kariotis, B.S. Swartzentruber and M.B, Webb, 1990, Microscopic aspects of the initial stages of epitaxial grovvth: a scanning tunneling microscopy study of Si on Si(lO0), in: Kinetics of Ordering and Growth at Surfaces, ed. M.G. Lagally. Plenum, New York, p. 145. Lambert, W.R., P.R. Trevor, M.J. Cardillo, A. Sakai and D.R. Hamann, 1987, Phys. Rev. B 35, 8055. Landemark, E., R.I.G. Uhrberg, P. Kruger and J. Pollmann, 1990, Surf. Sci. 236, L359. Lessor, D.L., C.B. Duke, A. Kahn and W.K. Ford, 1993, J. Vac. Sci. Technol., A 11, 2205. Lubinsky, A.R., C.B. Duke, B.W. Lee and P. Mark, 1976, Phys. Rev. Lett. 36, 1058. MacLaren, J.M., J.B. Pendry, P.J. Rous, D.K. Saldin, G.A. Somorjai, M.A. van Hove and D.D. Vvedensky, 1987, Surface Crystallographic Information Service: A Handbook of Surface Structures. D. Reidei, Dordrecht, 352 pp. Mailhiot, C., C.B. Duke and D.J. Chadi, 1985a, Surf. Sci. 149, 366. Mailhiot, C., C.B. Duke and D.J. Chadi, 1985b, Phys. Rev. B 31, 2213. Maree, P.M.J., K. Nakagawa, J.F. van der Veen, and R.M. Tromp, 1988, Phys. Rev. B 38, 1585. McRae, E.G. and R.A. Malic, 1987, Phys. Rev. Lett. 58, 1437. McRae, E.G. and R.A. Malic, 1988, Phys. Rev. B 38, 1363. Needeis, M., M.C. Payne and J.D. Joannopoulos, 1988, Phys. Rev. B 38, 5543. Northrup, J.E. and M.L. Cohen, 1982, Phys. Rev. Lett. 49, 1349. Olmstead, M.A., 1987, Surf. Sci. Rep. 6, 159. Pandy, K.C., 1981, Phys. Rev. Lett. 47, 1913.
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Pashley, M.D., 1989, Phys. Rev. B 40, 10481. Pashley, M.D. and K.W. Haberen, 1991, Phys. Rev. Lett. 67, 2697. Pashley, M.D., K.W. Haberen and R.M. Feenstra, 1992, J. Vac. Sci. Technol. B 10, 1874. Pauling, L., 1960, The Nature of the Chemical Bond, 3rd edn. Cornell University Press, Ithaca, NY, pp. 221-264. Phaneuf, R.J. and M.B. Webb, 1985, Surf. Sci. 164, 167. Phillips, J.C., 1973, Bands and Bonds in Semiconductors. Academic, New York, 279 pp. Qian, G.-X. and D.J. Chadi, 1987, Phys. Rev. B 35, 1288. Ranke, W. and K. Jacobi, 1977, Surf. Sci. 63, 33. Sakama, H., A. Kawazu and K. Ueda, 1986, Phys. Rev. B 34, 1367. Schlier, R.E. and H.E. Farnsworth, 1959, J. Chem. Phys. 30, 917. Schluter, M., 1988, Structural and electronic properties of elemental semiconductors and surfaces, in: Surface Properties of Electronic Materials, eds. D.A. King and D.P. Woodruff, The Chemical Physics of Solid Surface and Heterogeneous Catalysis, Vol. 5. Elsevier, Amsterdam, p. 37. Swartzentruber, B.S., Y.-W. Mo, R. Kariotis, M.G. Lagally and M.B. Webb, 1990, Phys. Rev. Lett. 65, 1913. Takayanagi, K. and Y. Tanishiro, 1986, Phys. Rev. B 34, 1034. Takayanagi, K., Y. Tanishiro, M. Takahashi and S. Takahashi, 1985, Surf. Sci. 164, 367. Takcuchi, N., A. Selloni, A.I. Shkrebtii and E. Tosatti, 1991, Phys. Rev. B 44, 13611. Tong, S.Y., G. Xu and M.N. Mei, 1984, Phys. Rev. Lett. 52, 1693. Tong, S.Y., G. Xu, W.Y. Hu and M.W. Puga, 1985, J. Vac. Sci. Technoi. B 3, 1076. Tong, S.Y., H. Huang and C.M. Wci, 1990, Bonding and structure on semiconductor surfaces, in: Chemistry and Physics of Solid Surfaces VIII, eds. R. Vanselow and R. Howe. Springer Series in Surface Sciences 22. Springer, Berlin, p. 395. Tong, S.Y., H. Huanq, C.M. Wei, W.E. Packard, F.K. Men, G. Glander and M.B. Webb, 1988, J. Vac. Sci. Tcchnol. A 6, 615. Uhrbcrg, R.I.G. and G.V. Hansson, 1991, Crit. Rev. Solid State Mater. Sci. 17, 133. Van Hovc, M.A., S.W. Wang, D.F. Ogletree and G.A. Somorjai, 1989, Adv. Quant. Chem. 20, I. Van Silfhout, R.G., J.F. van der Veen, C. Norris and J.E. MacDonald, 1990, Farad. Disc. Chem. Soc. 89, 169. Vanderbilt, D., 1987, Phys. Rev. B 36, 6209. Vandcrbilt, D., O.L. Alerhand, R.D. Meade and J.D. Joanopoulos, 1989, J. Vac. Sci. Technol. B 7, 1013. Wcaklicm, P.C., G.W. Smith and E.A. Carter, 1990, Surf. Sci. Lett. 232, L219. Wcbb, M.B., F.K. Men, B.S. Swartzentruber and M.G. Lagally, 1990, J. Vac. Sci. Technol. A 8, 2658. Williams, E.D. and N.C. Bartelt, 1991, Science 251,393. Wyckoff, W.G., 1963, Crystal Structures, Voi. 1, 2nd edn. Wiley Interscience, New York, pp. 108-113. Zhu, X. and S.G. Louie, 1991, Phys. Rev. B 43, 12146.
Part II
Experimental Methods to Study Surface Structure
This Page Intentionally Left Blank
CHAPTER 7
Diffraction Methods
E. C O N R A D School of Physics Georgia Institute of Technology Atlanta, GA 30332-0430, USA
Handbook ~" Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B.V. All rights reserved
271
Contents 7.1.
7.2.
Surface diffraction basics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
273
7.1.1.
Kinematic approximation in 3-d
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
274
7.1.2.
Diffraction from surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
7.1.3.
Diffraction from a reconstructed surface . . . . . . . . . . . . . . . . . . . . . . . .
281
7.1.4.
Thermal effects
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
284
Diffraction from disordered structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
289
7.2.1.
7.2.2.
Qualitative description of diffraction from defects . . . . . . . . . . . . . . . . . . .
290
7.2.1.1.
290
Sample mosaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1.2.
M i s c u t s a m p l e with steps . . . . . . . . . . . . . . . . . . . . . . . . . .
291
7.2 1.3.
Flat surface with local r o u g h n e s s . . . . . . . . . . . . . . . . . . . . . .
295
General description of diffraction from defects . . . . . . . . . . . . . . . . . . . . .
298
7.3.
Instrument resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
303
7.4.
Surface diffraction techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
305
7.4.1.
Low energy electron diffraction 7.4.1.1.
7.4.2.
7.4.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
305
Experimental LEED systems ........................
308
7.4.1.2.
High resolution L E E D
313
7.4.1.3.
Qualitative LEED
7.4.1.4.
Quantitative LEED
7.4.1.5.
L E E D resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
320
7.4.1.6.
D y n a m i c L E E D and multiple scattering . . . . . . . . . . . . . . . . . .
322
7.4.1.7.
S u m m a r y of a d v a n t a g e s and d i s a d v a n t a g e s o f L E E D . . . . . . . . . . .
325
Surface X-ray diffraction
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316
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318
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2.1.
S u r f a c e sensitivity in X - r a y scattering . . . . . . . . . . . . . . . . . . .
7.4.2.2.
E x p e r i m e n t a l c o n s i d e r a t i o n s and g e o m e t r y
7.4.2.3.
Integrated intensities
326 327
................
331
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
335
7.4.2.4.
R e s o l u t i o n function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
338
7.4.2.5.
Surface crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . .
339
7.4.2.6.
Reflectivity
7.4.2.7.
S u m m a r y of advantages and disadvantages of surface X-ray s c a t t e r i n g . .
Atom diffraction 7.4.3.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eikonal a p p r o x i m a t i o n to the hard wall
..................
342 343 344 345
7.4.3.2.
M u l t i p l e scattering in the hard wall m o d e l
7.4.3.3.
C o u p l e d channel t e c h n i q u e s
................
347
7.4.3.4.
T h e inversion p r o b l e m in the hard wall a p p r o x i m a t i o n . . . . . . . . . .
349
7.4.3.5.
R e s o n a n t scattering
352
........................
348
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.3.6.
H e l i u m diffraction e q u i p m e n t
7.4.3.7.
S u m m a r y of a d v a n t a g e s and d i s a d v a n t a g e s o f atom scattering
.......................
355 .....
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
272
357 358
7.1. Surface diffraction basics
The atomic structure of surfaces is of such fundamental importance in the understanding of surface electronic and chemical properties that a great variety of experimental techniques have been developed to determine atomic positions. These techniques can be generally grouped into two categories; direct space and reciprocal space imaging methods. In the former category, a real space image of the distribution of atoms, or more precisely, the electron distribution around the atom, is measured. Examples of direct space techniques are scanning electron microscopy (SEM), transmission electron microscopy (TEM), and scanning tunneling microscopy (STM). These techniques will be discussed in Chapter 8. Reciprocal space methods do not give the positions of the atoms but instead measure the Fourier transform of the atom-atom correlation function. All diffraction techniques such as Low Energy Electron Diffraction (LEED), High Energy Electron Diffraction (HEED), Surface X-ray Diffraction, and Atom Scattering are examples of reciprocal space imaging and are the topic of this chapter. The complexity of surface structural analysis as it is carried out today requires a variety of different measurable parameters. It is a moot point to argue which diffraction technique is "better" since each one is sensitive to different properties of the surface. Rather than quibble over measurement preferences, it is more appropriate to exploit the advantages of several different techniques so that the structural data can be thoroughly understood. This chapter will discuss the basics of using diffraction methods for studying surfaces. There are two solutions for achieving sensitivity to the surface while minimizing bulk scattering contributions. The first is to use a high intensity weakly interacting probe (X-rays) in an appropriate geometry to reduce bulk diffraction effects. The second is to use a strongly interacting probe (electrons or neutral atoms) to reduce transmission to deeper layers. The two approaches have their trade offs. For a weakly interacting probe, a simple kinematic analysis (discussed below) makes data analysis simple. This is balanced by the cost and complexity of the intense X-ray source. Strongly interacting probes are generally less expensive and easier to operate. On the other hand, the strong interaction that gives them their surface sensitivity means that multiple scattering effects will complicate the structural analysis. With this proviso, surface scattering will be discussed as a perturbation to weakly interacting radiation in order to shed light on the physical information contained in surface diffraction data. In later sections when individual diffraction techniques are reviewed, the applicability of these weakly interacting approximations will be discussed.
273
274
E. Conrad
Q
kf
k,/, ki
~
ki ~
~
2 ki
.
!" 2
o
(a)
(b)
Fig. 7.1. (a) A plane wave, ki, incident on a scattering medium with an outgoing spherical wave, kf. The momentum transfer vector Q is also defined. (b) Scattering from two sources.
7.1.1. Kinematic approximation in 3-d Consider the geometry shown in Fig. 7. l(a). A plane wave, exp[iki.r], is incident on a scattering object and produces an outgoing spherical wave. k~ and kf are the incident and diffracted wave vectors, respectively; Ikl = 2rt/L. At a distance D from the scatterer the scattered wave has an amplitude (see Schiff, 1968); J(k,,k,) ~-
D
e ik'.
(7.1)
The functionf(kf,k~) is called the form factor (or in neutron scattering it is called the scattering length) for the object. If the scatterers are the bound electrons of an atom, ./"is the atomic form factor. In the general case f(kf,k~) is proportional to
f(k,, k~) ~ I exp [-ik i . r] V(r) ~(kf,r)dr,
(7.2)
where V(r) is the potential due to all scatterers in the system. If the interaction is weak, two assumptions can be make. First, the scattered wave is assumed not to make any collisions with any other scatterers in the system before reaching the detector. That is, ~/(kf, r) is the final state wavefunction after scattering from an isolated scatterer. This is referred to as the single scattering limit, andf(kf, k~) can be calculated from the local potential of a single scatterer. The second assumption is that the outgoing wavefunction is also a plane wave, = exp[ikcr]. This is simply the Born approximation (see Schiff, 1968). In this case the form factor only depends on the vector difference Q = k f - ki , (7.3) IQI = Ik(E)l sin [(20)/2]. Q is the momentum transfer vector and plays a fundamental role in diffraction physics. This notation is somewhat incorrect since the momentum transfer vector is
Diflractionmethods
275
actually hQ. In order to parallel the current notation in the literature the momentum transfer vector will simply be written as Q. Within these approximations, f(kf, ki) is a function of the incident radiation's energy and the scattering angle alone; f(kf, k~) = f(E,20). These two assumptions comprise the kinematic approximation. For X-ray scattering f(E,20) can be calculated assuming that the electromagnetic field of the X-rays weakly interacts with the electrons in the atoms. This is true as long as the incident angle with the surface is far from the angle of total reflection (see w 7.4.2.1 ). However, in lieu of the discussion in w 7.4.1.6, the application of the Born approximation to low energy electrons is somewhat tenuous. Accurate calculations of the scattered intensity for electrons must be made from a full dynamic calculation. For Helium atom scattering, the problem is further compounded. Helium atoms interact with the surface through two types of interactions. At short distance from the surface (d < 3 A,), the overlap of surface electron wavefunctions with the filled helium s level electrons produces an extremely repulsive potential, V(z), so that V(z) >> EHe. Far from the surface the interaction can be approximated by a long-range van der Waals potential. Because of the long-range nature of this potential, the interaction time of the helium atom with the surface electrons is very long and perturbation approaches to calculatingf(E,20) are unreasonable. Complete quantum mechanical treatments such as coupled channel techniques have been developed for calculating the scattered amplitude (Liebsch and Harris, 1982" Garcia et al., 1983). In spite of these difficulties, a single scattering approach to the problem of surface scattering serves as a convenient approximation for understanding the physics of the scattering process. Consider two scatterers at positions r~ and r 2 from the origin. The total scattered amplitude is (7.4)
A
where ~i is the phase shift of the wave scattered from point 2 associated with the path difference in going from point 1 to point 2. From Fig. 7. l(b), 8 - (rl - r 2 ) 9k~. The intensity at a distance D from the scatterers is the scattered flux per unit solid angle and is given by 2
I(Q)-AA'-I~
Z
eliQ ~r'-5)l"
(7.5)
ij=i
The intensity has been normalized to the incident beam flux Io which for further discussions will be considered equal to 1.0. This treatment can be generalized to a collection of N atoms where the atomic scattering factor of the ith atom is f~(E,20). In the kinematic approximation the amplitude scattered by the solid is given by (James, 1962; Guinier, 1963) N-I
a = ao 2 i=0
f(E,20) e iQr,.
(7.6)
E. Conrad
276
The corresponding intensity is N-I
I(Q) = A A * = I o Z
f/(E,2O)g(E,20) e i Q ( r ' - r j )
(7.7)
ij=0
If the atoms are arranged in a crystalline structure, the position vectors of each atom can be written as (see Chapter 1) r i - h,a + kib + l,c + urea + vmb + WmC,
(7.8)
where (hkl) are integers. The reader is reminded that the first three terms give the position, Ri(hkl), of the ith unit cell and the last three terms are the positions, [3rn, of the mth basis atom in the unit cell. Substituting this expression into Eq. (7.6) and separating the sums over i and m gives
A(Q) -
f m ( E , 2 0 ) e (iQO") m=()
e (iQlr
(7.9)
i=11
The term in brackets contains the scattering amplitude from all the atoms within the unit cell and is called the c r y s t a l s t r u c t u r e f a c t o r , F ( E , 2 0 , Q ) . The scattered intensity is therefore N-I
I(Q) - IF(E,20,Q)I 2 ~_~ eC'~
(7.10)
i j=()
The last term in Eq. (7.10) is called the i n t e r f e r e n c e f u n c t i o n , 3(Q), and contains all of the information on the ordered arrangement of the N unit cells. Note that the interference function only depends on the diffraction geometry through the momentum transfer vector. Since the sum in Eq. (7.10) is over all atoms in the solid, the product I ( Q ) cannot depend on either the order of summation or the choice of a basis. The non-unique choice of lattice vectors often offers convenient choices for a set of basis vectors that can substantially simplify the summation in Eq. (7.10) for either F ( E , 2 0 , Q ) or 3(Q). Suppose that the 3-d crystal is composed of N~, N2, and N 3 unit cells in the three perpendicular directions (for a general treatment for non-orthogonal Bravais lattices, see Jackson, 1991 ). The sums in Eq. (7.10)can be readily evaluated using the identity N-I
1
-
y_x"n=()
x N
.
(7.11)
1-x
Using this result, the interference function from a finite number of scatterers is
Diffraction methods
277
30 25 9~
-
--
N=5
20
10
="
-3
-2
-1
0
1
2
3
Q.a Fig. 7.2. The 1-d sin2 function in Eq. (7.12) for N - 5 (solid line). The diffraction peak has an intensity equal to N2 and a full width at half maximum (FWHM) given by Qa = 5.66/N. Open circles are a Gaussian approximation. N 2- I
Nt-!
3(Q)-
E
eiO,b'k,-k,, y--' eiQ, c(/,-/j )
eiO'"(h'-h" Z
h hi={}
N-!
kik={}
/,/=0
sin" l-~N~ Q . a ] sin2 (2N2 Q .B sin~(2 Q . a ) j
sin2"(1Q . b)
)2, sin
I-~N3Q . c
)
(7.12)
sinZ'(1Q . c ) "
The sin z functions are sharply peaked when Q.a~ = 2rcn (see Fig. 7.2). They have a peak intensity proportional to N 2 and a full width at half maximum ( F W H M ) proportional to I/N (QaFw.M = 2rt 0.901/N). Note that this function also has small oscillations away from the peak. The oscillations are due to the loss of the high Q components of the diffracted wave because the solid has been truncated into finite coherent domains. In real crystals the finite coherent domains are due to crystalline defects (grain boundaries, etc.) and occur with some distribution of sizes. In general, the incident radiation illuminates an area of the surface consisting of many of these domains. Since each domain has a different phase relationship with respect to the others, the scattered intensity can be assumed to be an incoherent average of the scattered intensity from each domain. The effect is to average out the small oscillations. The shape of the diffracted spot due to this average is given by the Warren approximation (Warren, 1941). In one dimension the intensity is sin
3 (Q~) =
N ! Q. a
~ t"1 sin [-~ Q 9a
)
1>
,.
:
,v = (N,) 2 el- ~Q,,)/4/t]
(7.13)
where (N~) is the average number of unit cells per domain. Equation (7.13) is a
278
E. Conrad
G a u s s i a n with a F W H M of QaFwHM= 2n: 0.939/(N,), w h i c h is slightly b r o a d e r than the sin 2 function. If the crystal is infinite, the interference function reduces to a series o f delta functions, ~3(Q) = 8 ( Q ~ a - 2rch) 8(Q J , - 2n:k) 8 ( Q z c - 2rtl) Thus the total scattered intensity, is only a p p r e c i a b l e w h e n the m o m e n t u m transfer vector is equal to (7.14)
Q = k f - k i = Ghkl
w h e r e Gj, kt is the r e c i p r o c a l l a t t i c e v e c t o r defined in C h a p t e r 1. (7.15a)
Ghk z -- ha* + kb* + lc*
and
a*-
2rt(b • )
2 rc(c xa) , - b * = ~ ,
a . (bxc)
2rt(axb )
c * = ~
a . (bxc)
(7.15b)
a . (b•
It is easy to s h o w n that a . a * , b . b * , and c.c* - 2ft.
9
9
9
9
9
9
9
9
9
9
9
i" 9
i l l i i i i '/ i J" i
9
9
9
9
9
9
9
9
9
" 9
9
Fig. 7.3. Ewald construction. For an incident wave vector, ki, ending at an arbitrary reciprocal lattice point, the Ewald sphere is constructed with radius Ikl centered on the origin ofk~. Since the momentum transfer, Q, must be equal to a reciprocal lattice vector, non-zero diffraction peaks will only be observed if the final momentum, kf, ends on a reciprocal lattice point that lies on the sphere.
Diffraction methods
279
The condition that the momentum transfer must equal G in order to give a measurable scattered intensity is known as the Laue condition for diffraction. Since Ghk I a r e a set of translation vectors in reciprocal space corresponding to the real space crystal structure given by R(hkl) (see Chapter 1), the Laue condition simply states that the maximum scattered intensity occurs at the reciprocal lattice points of the real space crystal structure! The Laue condition can be conveniently visualized by the Ewald construction. The first step in the construction is to draw the 3-d reciprocal lattice. The incident wave vector is drawn with its tip at any reciprocal lattice point. Because the scattered wave vector for elastic scattering has the same magnitude as ki, its tip must lie on a sphere of radius Ikel with its origin at the tail of k~ (see Fig. 7.3). This sphere is called the Ewald sphere. The Laue condition is satisfied whenever the Ewald sphere intersects a reciprocal lattice point; since by construction Q = G. Of course, the magnitude of the scattered intensity still depends on the crystal structure factor IF(E,20,Q)I 2 that modulates the interference function.
7.1.2. Diffraction from surfaces To extend the previous section to include diffraction from surfaces requires detailed consideration of the solid-probe radiation interaction. How a truncated infinite lattice influences the diffraction depends on whether the incident radiation is penetrating (X-rays and neutrons), attenuated (electrons), or perfectly reflecting (He atoms). Each of these cases will be treated separately in following sections but for now a general approach to surface diffraction, common to all of these techniques, will be discussed. To begin with, assume that the radiation is attenuated as it passes through the solid and that the kinematic approximation is still valid. Consider two cases; a surface derived from an ideally terminated bulk crystal and a surface whose top layer has a (2• reconstruction. The elastic mean free path, A, is first defined in the conventional way 1 = 1oe-~A),
(7.16)
where x is the path traveled through the solid. A takes into account all losses, elastic and inelastic, that remove particles from the beam. This will be discussed in more detail in w 7.4.1. Because of these losses, radiation penetrating deeper into the surface will be more attenuated and have a smaller contribution to the radiation collected at the detector. If the inelastic mean free path for the radiation is assumed uniform in the solid, the attenuation coefficient for the relative amplitude scattered from deeper planes can be defined in the following way. Consider a particle incident on the surface at an angle 0~ to the normal that scatters off the nth layer of atoms and emerges into an angle Of. It will travel a total distance through the solid of n(c/cosO~ + c/cos0f). Then the ratio of the amplitude scattered from the nth and (n+l)th layer is
An
-e
-c~.
(7.17)
E. Conrad
280
For simplicity, the scattered intensity from the (001) surface of a simple cubic lattice that has an ideal bulk termination is calculated first. Once again c is parallel to the surface normal and the basis vectors (a~,a2) lie in the plane of the surface. Starting from Eq. (7.6), the total amplitude scattered from the crystal is N I ,N 2-1
N z- 1
eiQ'(a'h+a2k) Z
~
A(Q)=
hk=O
1
O~I eiQ3cl--e i-~(Q2a~N2+Q'aN')
I=0
•
sin(l/2NiQlal) sin(l/2Qlal)
sin(l/2N2Q2a2) 1 - o~u~ e i NzQzc sin(l/2Q2a2)
1 - o~eiQzc
(7.18)
The crystal structure factor is assumed to be 1.0. The scattered intensity is
I(Q) =
sin 2 (1/2 NiQial) sin 2 (1/2 N2Q2a2) 1 + o~zu` - 2o~N~cos (NzQzc) sin2 (1/2 Qlal) sin 2 (1/2 Q2a2) 1 + 0:2 - 2cz cos (Qzc) "
(7.19)
For the case of perfect absorption, o~ = 0, only the top layer contributes to the scattered intensity, and I(Q) does not depend on Q~. If the crystal is again allowed to be infinite parallel to the surface (i.e., N~ and N2 ---) co). the condition for a maximum in the scattered intensity is relaxed from the 3-d Laue conditions. Instead the maximum diffraction intensity occurs for all values of Qz as long as the momentum transfer parallel to the surface, Qal, is equal to the surface reciprocal lattice v e c t o r Ghk Qj; -
Ghk = ha~ + ka'2 .
(7.20)
The reciprocal space picture with attenuation now consists of a series of surface diffraction rods perpendicular to the surface instead of points (see Fig. 7.4), with the position of these rods given by Eq. (7.20). The diffraction rods are also referred to as crystal truncation rods (CTR). This is because the interference function for a surface can alternately be derived by considering the surface as a product of an infinite solid truncated by a step function | (Andrews, 1985). The form of I(Q) derived from the step function is the same as Eq. (7.18) in the limit that N2 ---) oo and that cz < 1.0 (Robinson, 1986). The CTR terminology is most commonly found in surface X-ray scattering literature. The interpretation of Eq. (7.19) is that the periodicity perpendicular to the surface has been lost and, therefore, the third Laue condition, Qz c = 2rtl, is no longer strictly valid,.The normally sharp 3-d diffraction peaks remains sharp in directions parallel to the surface but are broadened normal to the surface because the attenuation only allows the probe radiation to see a finite distance into the bulk. This has essentially the same effect as a finite bulk crystal has on the 3-d peak width. The bulk diffraction points are elongated perpendicular to the surface so that all values of Qz are allowed, subject only to the condition given by Eq. (7.20). Using the Ewald construction, the diffraction condition for surface scattering occurs when the sphere intersect a rod.
D~ffraction methods
281
[0011 (20)
(00)
(10)
(io)
(20)
/ Or'/
.
_
~Qzl
0~ /Q
\ k
QII=Q1
[lOOI
Fig. 7.4. Surface diffraction rods normal to the (QI,Q2) plane. For the parallel-piped geometry discussed in the text, the rods are separated by 2trial in theal direction. The rods are indexed according to the notation in Chapter 1. Diffraction occurs when the Ewald sphere intersects a rod at Oil = 2r~h/a~.
The picture of diffraction rods does not change if the attenuation is allowed to be non-zero. The intensity along a rod, however, is modulated. In the simple case of a bulk truncated surface the modulation is given by the last term in Eq. (7.19). Relaxation, reconstruction, and surface defects will also affect l(Qz) and will be discussed later in this chapter. Assuming that the attenuation length is much smaller than the finite domains in the crystal, then N2 in Eq. (7.19) can be taken to infinity with little error. The result is that the intensity along the rod is modulated by [1 + 122_ 212cosQzc]-~ (see Fig. 7.5). As in the 3-d cases, the intensity is still peaked when Qzc is equal to 2rtl, i.e. at the Laue condition. This is a general result: the intensity modulation along the rod gives information on the vertical spacing of atoms in the surface region. Note that the width of the diffraction peaks along the rod (i.e., measured in the Q~ direction) are broader for smaller 12. This is one way in which the attenuation can be measured. From Eq. (7.19) the peak to background ratio, l(Qzc - O)/l(Qzc = nrt), is [(1+12)/(1-12)] 2. Remember that 12 depends on Qz as well through Eq. (7.17) so that calculations of 12 from the peak to background ratio must be used with some caution.
7.1.3. Diffraction from a reconstructed surface Consider the reconstructed p(lx2) surface of a parallel-piped crystal shown in Fig. 7.6. Every other row of atoms in the y direction has been removed. The scattered amplitude can be written by breaking the sum in Eq. (7.9) into terms that contain only bulk and only surface atoms.
E. Conrad
282
1.2 =0.0
1.0 0.8 .~
0.6
=
0.4
0.5
0.2 0.0 0
2
6 2r~
4
8
10
Qzc Fig. 7.5 The intensity distribution along the (00) surface diffraction rod for the (001) surface of a simple cubic crystal with finite attenuation. The diffraction intensity is still peaked at the momentum transfer corresponding to the third Laue condition.
Fig. 7.6. A p( I •
reconstruction of the (001 ) surface of a simple cubic lattice. The top layer has also been relaxed outward by Ac.
N t -
1
N z -
1
A(Q) = f ~_~ e' e~,,h ~ h=O
N I -
e i Q,a2k Z
k=O
1
~t eiQc,
I=(1
(7.21) N I -
1
N2/2-
I
+ f ~_~ e iQ~a'h ~_a e iQr2azk [ e i q ~ A c ] h=()
k=O
The s e c o n d term is a sum o v e r only the surface atoms. S q u a r i n g Eq. (7.21) and letting N 3 ~ oo gives the scattered intensity
D~ffractionmethods
283
1.8 A i l x 2 ) ' " (10) ro~ /,. \
f . . . . . 1.6 Bulk Terminated
", (lO)rod
N .,-,4
O
. ,...q
/,; ",,\~
1.4
(b)
1.2 1.0 0.8 (0 89
0.6 0.4
i
. . . . . . 2 4
o
6
8
10
Qz c Fig. 7.7. (a) The reciprocal space picture of the p( 1x2) surface of Fig. 7.6. (b) The intensity distribution along the (0,0) (solid curve) and ( 1/2,0) rods (heavy solid line) are shown. For comparison the intensity profile for a bulk terminated surface is also shown (dotted curve). The effect of the relaxation is to add a modulation to the intensity profile that slightly shifts the peaks. Data is for c = 2.5 ~, o~= 0.2 and Ac = 0.8 A.
sin ( 1 Q2a221
I(Q) =/Bulk
+ Isurf,~ + 41S,,-.,:~
sinl2Q2a21 (7.22a)
Qza2- Qz Ac -
cos
l+ot"
otcos
~-Qla,
2o~co
(Qzc)
- Qz[Ac c]
where 9 2
1
.2
1
/Bulk
s,n
12Q2a2J
1 + c~2 - 2o~ cos
(Qzc)
(7.22b)
284
E. Conrad
and
sin212N,Q,a,i sin2 l 1 - ~ Q22a2/ Isu~-~,cc=If 12 sin2(1Q,a,!. sin~(1Q22a2),
(7.22c)
Equations (7.22a-c) show that the total scattered intensity is the incoherent sum of the intensity scattered from the ideal bulk crystal (with a nonzero attenuation coefficient) plus the intensity scattered from the surface layer and an additional interference term between the two. The reciprocal space picture has twice as many rods in the a2 direction, which from Eq. (7.22c), are given by instead of 2nrt (see Fig. 7.7). The extra rods come purely from the surface term, l,,rf. This could have been anticipated since the surface reciprocal lattice vector is twice as small in this direction compared to the underlying substrate. These additional rods, associated with the reconstruction, are called The relative importance of these three terms in Eq. (7.22a) depends on the attenuation. For o~ near 1.0 (penetrating radiation) the ratio of the bulk to surface contribution (Eqs. (7.22b) and (7.22c)) is about equal to N 3 with the bulk rod widths determined by the size of the bulk crystallites (at least for large N3). Also, the interference term (third term in Eq. (7.22a)) only contributes at the bulk truncated rod positions, = 2nrt, with its intensity determined in part by o~. Since the interference term is zero everywhere along the superlattice rods, the intensity of the these rods as a function of is constant. This is a general result" within the single scattering model, the superlattice rods only contain information about the structure of the surface layer. Had two atoms per surface cell been included, Eq. (7.22c) would contain a term (proportional to cos[Q,Az], where A. is the vertical height difference between the two surface atoms). This term would modulate the intensity along the superlattice rods. In terms of analyzing diffraction data from a reconstructed surface one need only consider the atoms in the surface cell when comparing calculated diffraction spectra to the experimental intensity distribution measured along the superlattice rods. This interpretation is no longer correct if multiple scattering is included.
Q2a2=nrt
superlatticerods.
Q2a2
Qz
7.1.4.Thermaleffects So far the scattering intensity has been calculated assuming that the lattice is static and that even zero point motion is neglected. At elevated temperatures the atoms vibrate, so that at any instant in time an atom will be displaced from its T = 0 K position by an amount ui(T) (see Fig. 7.8). The position of the ith atom will be ~; = r, + u;, where r i is the static lattice position of the ith atom, Eq. (7.8). To calculate the scattered intensity remember that the ui's are changing in time and that the incident wave sees a time dependent distribution of atomic positions. To handle the time dependent structure it is common to assume that the collision time between the incident radiation and the lattice is short compared to a typical vibration frequency.
Diffraction methods
285
Fig. 7.8. Instantaneous position of an atom (filled circles) from their T= 0 K positions (r~) (open circles).
Then the lattice can be considered "frozen" during a single scattering event and the scattered intensity can be found by averaging I(Q) over a large number of such scattering events for different possible ui's. This is referred to as a thermal average (James, 1962). This assumption is valid for electrons and photons, but the longrange helium-substrate interaction means that it is questionable for atom scattering. Nevertheless, it is found that many of the results that will be derived below are applicable to atoms as well as electrons and photons. Assuming f = 1.0, the average scattered intensity can be written using Eq. (7.7) by inserting the instantaneous atomic position ~i = ri + u~ and taking the average N
l~ve(a) = ~
etiQCr'-r)l
(eliQ'Cu'-u)l)
(7.23)
ij
The first part of Eq. (7.23) is the T = 0 K intensity distribution lo(Q). The average can be written in a convenient form by expanding the exponential 1 )2 i (e iQ ~u,-u,i) = (l + i Q . (u i - uj) . 2! (Q " ( u i - ui) --~. (Q . ( u , - uj)) 3 + . . . -l = exp -2- ((Q" ( u i - uj))2)
(7.24)
Equation (7.24) follows from the fact that all terms to odd powers of Au must average to zero by symmetry. To proceed, the displacement of each atom is written as a sum over normal vibrational modes. The displacements have contributions from both surface and bulk phonons. Because the amplitude of bulk modes decays rapidly towards the surface, only penetrating radiation will scatter from atoms with significant amplitudes from these modes. Atom scattering will see vibrations almost entirely from surface modes. Regardless, the effect of both surface and bulk modes leads to qualitatively the same effect on the scattered intensity as well as on the diffraction lineshapes (McKinney et al., 1967). A derivation of the effect of finite temperatures on diffraction is outlined below assuming only surface phonons (Conrad, 1987). A general discussion of phonon diffraction effects can be found in James (1962) and a similar derivation including both bulk and surface phonons is given by McKinney (McKinney et al., 1967).
E. C o n r a d
286
In terms of surface phonons the instantaneous displacement of the ith atom can be written as (Maladudin, 1971)
Uq~i= E Uq~eq~ sin(q 9ri, + Oaq~t + ~qr,) e-e tqt,, c.
(7.25)
q~
Here K: is one of the two polarization directions in the surface plane that have a polarization vector eq~. UqK is the amplitude of the qrcth normal mode. The terms COq,, and 8q,~ are the frequency and phase of the qx:th normal mode, respectively. The momentum of a surface phonon, q, only has a component in the plane of the surface (i.e., q = q,). The spacing between layers is c, and ni is the number of the layer in which the atom i is located. The quantity c is a measure of the damping of the phonon vibrational amplitude into the bulk. The scattered intensity is calculated by substituting Eq. (7.25) into Eqs. (7.24) and combining with Eq. (7.23). While the calculation is straightforward, it is rather tedious and will not be presented here (see Conrad, 1987). The result is found to be 2 I(Q) = e -2M I,,(Q) + e -2M y_, Uq~ (Q . eq~)2 I,,(Q +_q) +...
(7.26)
qtr
where N
/,,(Q)- ~
e lio'r'-r')l.
ia
The Debye-Waller exponent, 2M(T), is defined as l
2M--}- ~
2
2
Uq,~ ( Q . e q O .
(7.27)
qw,
The diffracted intensity is the sum of two terms. The first term in Eq. (7.26) is the normal Debye-Waller effect. To first order the thermal vibration of each atom is independent of the others. In that case ( ( u i - uj)) in Eq. (7.24) can be replaced by ~2, the mean squared vibrational amplitude. The first term in Eq. (7.26) is also referred to as the zero phonon contribution since it is derived assuming that no correlations exist between vibrating atoms (i.e., no phonons). From equipartition theory ~2 must be proportional to T for T > | where OD is the bulk Debye temperature. The Debye-Waller exponent in Eq. (7.27) for a bulk solid is usually written as (Warren, 1990):
2M -
2B(T) sin20 ~2 ,
where
12h2T 2 B ( T ) - mkBO------~
(7.28)
In general B(T) depends on the crystal geometry through the different phonon propagation vectors. Also, the expression for B(T) in Eq. (7.28) will be modified
D~ffraction methods
287
near the surface, since the surface Debye temperature will be substantially different from the bulk. To first order then, the effect of atomic vibrations is to decrease the scattered intensity by a term proportional to exp(-T) without changing the shape of the diffraction peaks. Another way of viewing this result is that the uncorrelated vibrations act only as point defects so that the total number of scatterers remains the same. Since the diffraction widths are proportional to 1/N, the lineshapes remain constant. The second term in Eq. (7.26) is the contribution to the scattered intensity due to the creation or annihilation of a single phonon (two and three phonon effects are included in the higher order terms that are not shown in Eq. (7.26)). From equipartition theory the mean squared amplitude of a surface phonon is kT2~c
UqK 2 =~
t 2
1
-
(7.29)
mNcq~ q
where N' is the number of atoms per surface plane (each plane separated by c). It has been assumed that the dispersion relationship for the surface phonons is C0q,:= qc,i ~, where cue is the speed of sound of the qK:th mode. For simplicity let the scattered intensity from the static lattice, Io, be represent by a delta function rod (i.e. 1o- 8 ( Q , - G,) where G, is a surface reciprocal lattice vector). Substituting Eq. (7.29) into Eq. (7.26) and assuming an isotropic surface ()f area A gives I(Q) = e -2M 6(Q - G,.) + e -2M
kTQ 2 A v. c m n 2 N pc, 2q,, IG, + Q,I
(7.30)
Equation (7.30) shows that the phonon contribution to the scattered intensity adds a term proportional to Q2Te-ZM. More importantly this term broadens the diffraction lineshapes by contributing a tail that decays as I / Q , (see Fig. 7.9). The form of the decay is independent of temperature. Note that the ratio of the zero and one-ohonon intensities is proportional to T. The equation for the one-phonon contribution is only valid near a diffraction rod. As can be seen, it diverges at Q, = Gll because the phonon lifetime has not been included in the calculation. A similar expression for the phonon broadened lineshape is also obtained using bulk phonons as long as the intensity variations along the rod (i.e. l(Qz)) are not too dramatic (McKinney et al., 1967). Two important points should be made about one-phonon scattering. First, the calculation of the one-phonon scattered intensity was made independent of the incident type of wave. In other words the inelastic scattering contribution in the kinematic approximation is the same for electrons, X-rays, or helium atoms and is only a function of temperature and total momentum transfer. The phonon contribution can be minimized by taking diffraction measurements in a scattering geometry that minimizes Q. This will be discussed in w 7.4.2.
288
E. Conrad
I(Q u)
-- Qll Fig. 7.9. The zero (dotted line) and one-phonon (solid line) contributions to a diffraction lineshape due to surface phonon scattering.
The second important consideration is that at high temperatures the one-phonon scattering gives a significant contribution to the diffraction lineshape. This must be kept in mind when analyzing any particular diffraction profile. It will play an important role in the ability to distinguish various type of surface disorder as discussed in the next section. The relative importance of inelastic scattering can be estimated by simply knowing the value of 2M. Barnes et al. have estimated the total intensity scattered into one Brillouin zone for each contribution to the inelastic scattering; i.e., zerophonon, one-phonon, and multi-phonon terms (Barnes et al., 1968). The integrated intensity over the Brillouin zone, normalized to the total scattered intensity is found to be 1.0 "~~ 0.8 "~
0.6
'~
0.4
0.0
0
1
2
3
4
5
2M Fig. 7.10. The integrated intensity over the Brillouin zone of the zero-, one-, and multi-phonon contribution to the total scattered intensity.
D~ffraction methods
289
I z e r o ( 2 M ) = e -2M
lone(2M) = 2Me -2M
(7.31 )
lmum(2M) = e-2~e Mw- 1 - 2M) The three contributions are drawn in Fig. 7.10. Note that the one-phonon term maximizes at 2M = 1.0. Also note that all three components are nearly equal when 2M = 1.0. Even though the multi-phonon component is larger than either the zero or one-phonon term above 2M = 1.0, the zero and one-phonon terms are peaked near G,. The multi-phonon term on the other hand is roughly uniform throughout the zone. Therefore, near a diffraction peak the intensity measured in the detector is heavily weighted towards the elastic and one-phonon contributions.
7.2. Diffraction from disordered structures So far the discussion of diffraction has focused on a perfectly ordered array of two-dimensional unit cells. Perfect surfaces are, however, the exception rather than the rule in surface physics and defects such as vacancies, adatoms, atomic steps, and mosaics are usually present on real surfaces. If the unit cell structure is to be determined, it is at least necessary to be able to separate or account for changes in the scattered intensity due to the loss of long-range order. On the other hand, many interesting physical phenomena such as enhanced catalytic activity, the instability of surfaces at high temperatures, disorder transitions, and others (see further volumes in this series) are directly related to the defect distribution on an otherwise perfectly ordered surface. For both reasons one would like to have a method of analyzing both the concentration and distribution of these defects on surfaces. From w 7.1.1 there are two contributions to the scattered intensity: the crystal structure factor and the interference function. Each is sensitive to different types of defects. The interference function is sensitive to partially ordered defects that reduce the periodicity of the surface. These will be discussed in the next section. Defects which change the structure of an individual unit cell affect I(Q) through either the atomic scattering factor, fiE,| or through the crystal structure factor, F(E,| This type of disorder is referred to as a point defect (e.g., vacancies, adatom, or substitutional atoms). If such defects are assumed to be distributed randomly throughout the surface and are therefore not correlated with one another, the scattered intensity can be calculated from the average of I(Q). For the case of one atom per unit cell, the scattered intensity (from Eq (7.9)) is equal to N-I
N-I
N-I
i4
i=j
i~j
(7.32)
where the intensity has been broken into two sums, one over identical atoms with i j and the other with i c j .
290
E. C o n r a d
Consider the case of a surface with n random vacancies and N - n undisturbed crystalline sites. The vacancy form factor will be assumed to be zero while all other atoms have a form f a c t o r f 2. For the first sum, i = j , there are N-n pairs of identical atoms each contributing an intensity f2. F o r i ~: j there is a probability (N-n)/N of having an atom at site i and the same probability of having another atom at a site j. The total intensity becomes (Henzler, 1979; Cowley, 1984)
I(Q) = ( N - n)f 2 +
f2 Z
eiQ(R'-~}
(7.33)
i,~j
Equation (7.33) can be written as an unrestricted sum by subtracting the contribution from the N pairs of surface atoms, i.e., N[(N-n)/N]Zf"z. N-I
I(Q) = n(N - n) f2 + Uun 12 ~_~ e iQ, . , - ~) N
(7.34)
ij
The last term in Eq. (7.34) is the diffraction from a perfect sample but with a reduced atomic scattering factor. Because the interference function itself remains unchanged, the width of the diffraction peaks for the disordered surface are the same as for the perfect surface. The other effect of random point defects is the addition of a uniform background (the first term in Eq. (7.34) which does not depend on Q). As an example, consider the case of a cubic primitive lattice with f = 1.0. If a concentration of 10% vacancies is present in an ordered surface containing 1000 atoms, the peak intensity would be reduced by 20%. The background on the other hand, which is the true indicator for the presence of the vacancies, would only amount to <0.01% of the peak intensity. Typical point defect concentrations are usually much smaller (typically a few percent) and are therefore almost impossible to measure in a diffraction experiment. The exception to this is atom scattering where the cross section for impurity scattering, which is proportional to f, is very large. When the concentrations of point defects does approach 5-10%, the situation is significantly different. At that point correlations between impurities will become important and the random impurity model leading to Eq. (7.34) will break down. Even though diffraction is insensitive to random point defects, its ability to detect correlated defects is much better. As will be shown below, correlated defect concentration substantially smaller than 1% can easily be measured.
7.2.1. Qualitative description of diffraction from defects 7.2. I. 1. Sample mosaic Bulk crystals contain a large number of small crystallites. Because of either the growth process, impurities, or mechanical stress, these crystallites are not all oriented in the same direction. Furthermore, each of these domains has a random phase relationship between the others because of the stress and lattice mismatch at
Diffraction methods
291
(a)
(]0)
(00)
(10)
zA0
(b) I_. I
2n --~ a
Qll
Fig. 7.11. (a) A surface prepared from a bulk sample containing a mosaic spread A0. (b) The corresponding reciprocal space picture showing the broadening of the diffraction rods. their grain boundaries. In semiconductors these crystallites may be many microns in diameter, while in metals they are usually no larger than a few thousand angstroms. If a solid is cut to expose a particular crystallographic direction, the surface will therefore contain a distribution of surface normals as a result of the angular misorientation of each of the domains. The angular width, A0, of the distribution of surface normals about the average direction is called the mosaic spread of the sample. The effect of a continuous distribution of surface normals on the diffraction is shown in Fig. 7.11. Because there is no phase relationship between the domains, the diffraction is an incoherent sum of the diffraction from each domain. The reciprocal space picture of a surface with a mosaic A0 contains a distribution of rods that are rotated about the point Q = (000)a*. This causes the rods to be broadened with a width that increases linearly with Qz; AQll = Qz A0
(7.35)
In some cases the mosaic is peaked at certain 0 around the average direction. The reciprocal space picture will then contain several tilted rods whose separation in Qtl also increases linearly as in Eq. (7.35). In that case A0 is the angular separation between these domains.
7.2.1.2. Miscut sample with steps Disorder influences I(Q) not only because of the concentration of a particular defect but because of the defect distribution as well. If the distribution is rather uniform and thus ordered to some degree, the defected surface can be thought of as a new superlattice, giving rise to a higher order diffraction pattern. Consider the case of a surface with an equal number of up and down steps. For a random distribution of
292
E. Conrad
]c al [N-1]a 1
~-,
.I
Fig. 7.12. (a) A surface with random steps of height c. (b) An ordered arrangement of up-down steps with a periodicity 6 times the bulk terminated surface.
al
Fig. 7.13. An ordered staircase of up steps. these steps the surface is quite rough (Fig. 7.12a). If, on the other hand, every up step is adjacent to a down step (Fig. 7.12b), the surface appears ordered and would give rise to a diffraction pattern with sharp rods separated by 2 n / L , where L = 2(N-1)al. Even though the distribution of steps must be considered in any serious analysis of diffraction line shapes, an intuitive understanding of diffraction from disordered surfaces can be obtained by studying a few characteristic ordered defect arrangements. In the subsequent discussion these examples will be analyzed. The general case will be reviewed in w 7.2.2. Consider the ordered one dimensional staircase of identical atoms shown in Fig. 7.13. Physically this would correspond to a surface that was cut at a slight angle away from a low index face such as the (001) surface. If a strong step-step repulsion existed, each terrace width would be the same. Let each terrace contains N atoms with an inter-atomic spacing a~. Adjacent terraces are separated by mono-atomic steps of height c. The position of each surface atom is given by the position vector r..~ = n, a I + m ( N a , + c ~
(7.36)
That is, r,,, is the position of the nlth atom on the mth terrace. Using rnm from Eq. (7.36), the amplitude scattered from this arrangement of surface atoms is given by Eq (7.6)
Diffraction methods
293 N-l
A(Q) = f (E,|
M-I
~_~ ein'(Q"a') Z n=O
e im(NQ"a'+cQ~)
(7.37)
m=()
The scattered intensity is easily found to be
I(Q) = f 2
sin 2 (Q,a,
N/2) sin2(MQ. {Na, +c}/2)
sin 2 (Q,,a,/2)
sin2(Q 9{Ua, +c}/2 )
(7.38)
The first sin 2 term in Eq. (7.38) is a set of diffraction rods perpendicular to the flat terraces broadened to --2r~/Na~ by the finite terrace size. For M --) oo, the second term in Eq. (7.38) is a set of delta functions rods tilted by an angle a (c~ = tan-~[c/Na~]) with respect to the flat terraces (see Fig. 7.14). The spacing between these rods is 2rt/L where L is the spacing between terraces edges measured in the plane of the miscut surface, L = 2rtcosc~/Na~ (see Fig. 7.13). For finite M these tilted rods are also broadened parallel to the miscut plane by --2rt/ML. The measured diffraction intensity (Eq. (7.38)) is the product of these two functions so that I(Q) only has appreciable intensity where the two terms overlap (see Fig. 7.14). Note that the intensity along the tilted rods is a m a x i m u m at (Q,,Qz) = (n2rt/a~, m2rt/c), where n and m are integers. This point is precisely the third Laue condition, which is a general result imposed by the discrete lattice structure: i.e, all superlattice rods must intersect the Bragg points of the 3-d reciprocal lattice. The condition for a m a x i m u m in the intensity is summarized by
Fig. 7.14. Reciprocal space picture of the ordered staircase structure of Fig. 7.13. The average surface is tilted by an angle ~ from the flat terraces. The staircase produces a set of superlattice rods perpendicular to the miscut plane with a spacing 2rdL, where L = Na~/cosot. The light shade indicates the intensity of the first sine function in Eq. (7.38) due to the finite size terraces. The dark shade is the region of overlap between the terrace rods and superlattice rods where the intensity is a maximum.
E. Conrad
294
0i .
.
.
.
.
.
.
.
.
.
|
ii
F
c
Fig. 7.15. Scattering from a two-level surface where the height difference between planes is c. The path difference, d =dl + d2, for scattering from the bottom layer is shown.
Qllal = 2nrc
(7.39a)
QlialN + Qzc = 2mrt
(7.39b)
The origin of this diffraction pattern could have been derived from a more intuitive point of view. Instead of beginning with a lattice vector a~ in the plane of the terrace, a primitive lattice vector L could have been chosen that lies in the plane of the tilted surface having a magnitude ILl = Na~/cosot (see Fig. 7.13). The diffraction would then be a set of diffraction rods perpendicular to the tilted surface but modulated by a new crystal structure factor F'(| given by N-I
F'(E,O) = i(E,O) ~., e 'c)'"'''
(7.40)
n=()
The condition for maximum intensity, Eq. (7.39), could also have been easily derived from elementary arguments. Consider the isolated step of height c in Fig. 7.15. A plane wave incident on this surface can scatter from either the top layer or from a plane c below the top layer. As a result there is a path difference d between a wave scattered from the upper and lower layers. This path difference is d = c(cos0i + cos0f)
(7.41)
When this path difference is an integral number of wavelengths, the scattered waves from the two levels interfere constructively and the intensity is a maximum. Conversely, when the path difference is an odd multiple of half wavelengths, the waves interfere destructively and the intensity must be a minimum. The condition for m a x i m u m intensity is then d-c(cos0i+cos0f)=n)~-
2nn
k -c
Qz k
(7.42)
where Qz = k(cos0~ + cos0f) has been taken from the construction shown in Fig. 7.4. The condition for m a x i m u m intensity, Qzc = 2nrt, is identical to Eq. (7.39b) if we substitute Qlla~N = 2n'rt, where n' is an integer equal to nN from Eq. (7.39a).
Diffraction
methods
295
The in-phase scattering conditions take on a special importance in studying defects for any surface diffraction technique. Note first that the in-phase condition is equivalent to the Bragg condition for the bulk. At the in-phase condition all crystal planes parallel to the terraces scatter constructively. From the diffraction point of view it is impossible to tell whether there are two or more surface levels or just one level since each level has the same phase relation. Therefore, the diffraction is insensitive to the presence of steps at this scattering condition. When d = (2n+ 1)~ or equivalently when Qzc = (2n+l)rt, levels separated by a single step of height c scatter out-of-phase (sometimes known as the anti-Bragg condition) and the intensity is a minimum. The lost intensity is redistributed in k-space according to the distribution of steps. At the out-of-phase condition the diffraction is maximally sensitive to steps. By analyzing the shape of diffraction rods as a function of Qz, not only can a qualitative description of the disorder be made, but a quantitative defect distribution can be calculated. This will be made clear in the examples below. The extension of the uniform stepped surface model to the case where there is a distribution of terrace widths can be qualitatively described without complicated algebra. Consider a stepped surface consisting of local regions where the terrace length is approximately constant. Let c~; be the local tilt of the surface plane from the terraces in this region. If each of the terraces contain Ni atoms, the length of the locally uniform stepped region will be MiN~a~/cos~, where Mi < M. The entire surface can be though of as a collection of these locally uniform regions that give rise to a distribution of angles Ac~ about the average normal, , of the stepped surface. If the M~'s are sufficiently large to neglect interference between adjacent tilted surfaces, each surface can be thought to contribute incoherently to the total intensity, so that the diffracted intensity is the sum of the scattered intensity from each surface. In this case, the diffraction is constructed by rocking the tilted rods about the (000) point by +Ac~. Because the average stepped region for each Mi is small, the tilted rods are considerably broadened from the perfect stepped surface. If the broadening is large enough, two tilted rods can overlap near the out-of-phase condition and instead of observing two distinct tilted rods the diffraction will appear as a waving rod normal to the terraces (see Fig. 7.16). The difference between the two reciprocal space pictures shown in Figs. 7.14 and 7.16 is the distribution of terrace sizes. If (N~) - (N;) 2 is small, the staircase structure has considerable long-range order and the reciprocal space picture will resemble Fig. 7.14. On the other hand when (N~) - {Ni)2 is large, there will be a broad distribution of terrace sizes and the reciprocal space picture will look more like Fig. 7.16. A quantitative solution for the general case has been derived in terms of two parameters: the probability to make a step, and the probability that the step is either up or down (Presicci and Lu, 1984).
7.2.1.3. Flat surface with local roughness The case where a step up is equally as likely as a step down can also be easily understood in the same manner as the staircase structure. Consider the surface illustrated in Fig. 7.12b, with a finite length Ma~. The surface consists of a periodic
296
E. Conrad
Fig. 7.16 The reciprocal space picture of a staircase structure with a distribution of terrace widths about the mean, a~, and an average tilt from the terrace plane of . up-down structure with each flat terrace having N cells (i.e. [N-1]a~ wide). Each atom position is given by the vector r ..... = n l a ~ + m
Nal-(-l)
m-~
(7.43)
where again r,,,, is the position of the nzth unit cell on the mth terrace. Instead of going through the algebra leading to an equation similar to Eq. (7.38), it is more instructive to qualitatively derive the reciprocal space picture of the diffraction. From Eq. (7.10), the total scattered intensity is the product of the intensity scattered from a single unit cell (the crystal structure factor) times the intensity scattered from the collection of u~lit cells (interference function). The calculation of the scattered intensity is made simply by choosing the unit cell to be a single up-down step (Fig. 7.17a). By inspecting Fig. 7.12b it is obvious that the periodicity of the stepped surface is larger than the flat surface by a factor of 2(N-I). The reciprocal space picture from the steps alone will therefore be a series of superlattice rods with a spacing 2r~/2(N-l)a~ modulated by the form factor for a single up-down step (Fig. 7.17a). The width of these rods will be 2rt/M'a~, where M' = M/(2N-2), assuming that the sample consists of M total scatterers. From the discussion in the last section, the intensity along the rods will be zero between two Bragg points and a maximum at the Bragg condition Q z c - 2nrt. ~ In addition to the steps, there are N unit cells (separated by a~) on each terrace. The diffraction from this collection of scatterers gives a set of diffraction rods spaced 2rt/a~ apart with a width 2rt/(N-1)a~ (Fig. 7.17b). 1 The form factor for an up-down step is calculated by summing over the two lattice points at (0,0) and ([N-I ]al,-c~, which after squaring gives I
Fstep = Z e't2 re.n=()
-r,,)= 4cos2 Qz ~ -
al Qll
Diffraction methods
297
Fig. 7.17 Diffraction from one dimensional up-down steps. The reciprocal space picture is the product of the diffraction from: (a) a collection of steps each of length 2(N-I )a~; and (b) the diffraction from a finite terrace of width (N-l)a~. The total diffracted intensity (c) consists of a single peak at the in-phase conditions and two split peaks (the splitting in QII will be equal to 2n:/(N- 1)az) at the out-of-phase conditions.
The total scattered intensity is the product of these two terms and is shown schematically in Fig. 7.17c. Note that the maximum intensity again occurs at the Bragg condition Qzc = 2nrt. Note also that at the out-of-phase condition, Qzc -(2n+l)rt, the rods appear split with a separation inversely proportional to the size of a terrace. If the distribution of terrace size is allowed to broaden so the width of a step is characterized by an average terrace size, ((N-l)>a~, the superlattice rods will broaden (Henzler, 1979; Presicci and Lu, 1984). For a sufficiently broad distribution of step sizes, the splitting at the out-of-phase condition will not be observed, but
E. Conrad
298
Fig. 7.18. Diffraction from a surface with up~own steps where terraces have an average width (N)a~. The shaded area represents the FWHM of the CTR. The diffraction is characterized by delta function peaks at the in-phase conditions (filled circles) and broad peaks at the out-of-phase conditions.
would be replaced by a very broad peak centered around the flat surface's crystal truncation rods, Qita~ - 27rn (see Fig. 7.18).
7.2.2. General description of diffraction from defects Even though the calculation of the diffraction from a general distribution of steps is more involved than for the idealized structures of the last section, the basic physics remains unchanged. It will, however, be more convenient to approach the calculation of the diffracted intensity from a slightly different formulation. Consider the case of a solid with a single type of atom. The normalized scattered amplitude, Eq. (7.6), can be rewritten as A(Q)
-.[(E,O) ~
n(R,) e ~~
,
(7.44)
i
where n(Ri) is the occupation probability of an atom at Ri" n(Ri) - 1 if an atom is at R, and zero otherwise. Note that the atoms are still confined to lattice points. The scattered intensity is
I(Q ) - f2
~_. ~., n(R~) n(Rm) m
e ~Q "' e -~ o. R,,,
(7.45)
i
Using the substitution R m = R + R i, and breaking R into components parallel and perpendicular to the, surface (R = Rit + Z), it is easy to show that the intensity scattered from a distribution of atoms is I(Q)
_ f2 Z
P(R,,.Z) e ; Q R = FIP(R.,.Z)~f 2 .
(7.46)
R, ,Z
F{ P} stands for the Fourier transform of the pair distribution function, P(RIt,Z), that is given by
Doyractionmethods
299
P{R,.,Z} = ~_~ n(r,, z,) n(r, + R,,, z, + Z)
(7.47)
i
P(RII,Z) is the probability that any pair of atoms will be separated by a vector R. ~ Equation (7.46) states that the scattered intensity is the Fourier transform of the pair distribution function. Therefore the maximum information in a diffraction experiment is P(R). Absolute positions cannot be determined; only relative separations can be extracted from the diffraction data. Since there is no phase information in a diffraction experiment, a unique distribution of defects cannot be specified and the problem of determining the surface disorder becomes an iterative one. First a distribution is chosen. This is done by guessing both the probability of having a terrace of width L and a probability that a step will be either up or down. A good initial guess can be made from the data by using the qualitative arguments of the last section. It is usually assumed that steps are mono-atomic so that double steps can be ignored. Once the distribution is known, P(R) can in principle be generated and the diffraction intensity found from Eq. (7.46). The calculated intensity is compared to the data and the distribution of steps is modified to increase the agreement with the experimental data (Henzler, 1978). The difficult part of this process is transforming the step distribution into P(R). There are several techniques for doing this. The first method is based on the solid-on-solid model (SOS). It assumes that an expression for the mean squared height fluctuation, ([h(r)-h(O)]2), is known. Consider the semi-infinite solid shown in Fig. 7.19. The height of the surface is given by the function h(x,y). If the surface is ideally truncated, h(x,y) will be a constant, which tk)r convenience will be set to zero. In the kinematic approximation the amplitude scattered (Eq. (7.6)) from a collection of atoms is (Robinson et al., 1990) N
h(x,y)
XV.
.~ = -
A-Z
1)
N
Z
ei~"'"+~176
ei~C'"'"+~
~
.0,.'
Z
eiQ::'~
(7.48)
Z ' = -- oo
where the attenuation has been included. The spacing between planes normal to the surface is c (fhas been assumed to be equal to 1.0). The scattered intensity is found by taking an ensemble average over all possible height configurations of the surface, i.e., / - (AA*). Using Eq. (7.48). I(Q) becomes (Robinson et al., 1990) N
N
I ( Q ) - 1 + c~2 + 2otcosQzc ~" eiO"r"
(eiQjhc,,,) _
hC,,)t).
(7.49)
rll
The term outside the summation is the crystal truncation rod defined after Eq. 1
In X-ray scattering n(r) will be proportional to the electron density and the sum in Eq. (7.44) will be replaced by an integral. P(RII,Z) is more commonly referred to as the Pattersonfunction in the X-ray literature. It is also called the auto correlationfunction.
300
E. Conrad
Z ,~. \ ,
./r
h(x,y)
~r
Fig. 7.19. The SOS model of a rough surface. The surface boundary is represented by a single valued discrete function h(x,y) (i.e., no overhangs).
(7.19). Comparison with Eq. (7.46) reveals that the term in average brackets is the pair correlation function parallel to the surface, P(Rla), for the surface atoms. Assuming h(r) to be a discrete Gaussian random variable, P(r) can be expanded similar to Eq. (7.24) to give
P(r) = (e i Q
[h(r)-h((,)]~ ~_ e-O~
(7.50)
Here the periodicity along a rod has been taken to be [Qzc] modulo 2ft. Once ([h(r)-h(O)] 2) is specified, the diffracted intensity is found by carrying out the Fourier transform in Eq. (7.49). A number of important examples are discussed below and shown in Table 7.1. For a surface consisting of random steps, the mean squared height fluctuation is independent of R. This gives rise to a Debye-Waller-like term in the diffraction peak intensities that changes the peak height but not the peak shape. For ([h(r)h(0)] 2) = u 2, and combining Eqs. (7.49 and (7.50) the diffraction intensity becomes
N iQcl2u2/2 I(Q) = 1 + ct2 + 2otcosQzc e 8(Q,, a x - 2nn:) 8(Qy a r - 2nrt) _
(7.51)
For a finite crystal, the delta functions in Eq. (7.51) can be replaced by the Gaussian lineshape in Eq. (7.13). Another common situation arises when steps occur with a probability p but when each step is statistically independent of any other step (i.e. no step-step interactions). The mean squared height fluctuation in this case is linear in R up to a distance R, after which it becomes a constant (Lent, 1984). The resulting diffraction line shape is the sum of a delta function peak corresponding to the long-range order (the R independent part of the ([h(r)-h(O)]2)) and a broad Lorentzian with a width proportional to p. The ratio of the two line shape components depends on both the coverage of atoms in the top layer and the perpendicular momentum transfer. This example will be discussed in more detail in a moment.
Diffraction methods
301
Table 7.1
Diffraction lineshapesfor different ([h(r) P(r) a
([h(r) - h(0)] 2) a
- h(0)] 2)
I(Q) (in-phase)
I(Q) (out-of-phase)
all\
I
all\
rll
Q
rll
J
Q II
a Solid lines correspond to Qzc = (2n+ 1)re. Dashed lines correspond to Qzc= 2nn.
Another example is the case of a surface above its roughening transition. Above the roughening temperature, TR ([h(r)-h(O)] 2) diverges logarithmically (Kosterlitz and Thouless, 1973) ([h(r)-
h ( 0 ) ] 2) =
C2 2X(T) ln(R ~
(7.52a)
R*2 =/-~112/~x 32 +/a-~z32/-~~./ 2
(7.52b)
Again a~ and a 2 are the surface unit cell vectors in the x and y direction, respectively. c~ and c2 are constants related to the surface tension in the x and y direction, and X(T) is the roughness parameter related to the degree of surface disorder above TR (Villain et al., 1985). Substituting these two expressions for ([h(r)-h(O)] 2) into Eqs. (7.49) and (7.50) gives a power-law line shape (Villain et al., 1985; Robinson et al., 1990) N
F(2 - rl)
I(Q) = 1 + c~2+ 2o~cosQzc F(rl/2)F(3/2- rl/'2)
a~Ql + cl
(7.53) ~, c2
where the exponent rl(T, Qz) is defined as
q(T, Qz) = X(T)[Qzc]2;
[Qzc] modulo 2r~
(7.54)
302
E. Conrad
Note that at the in-phase condition (corresponding to [Qz c] - 0), Eq. (7.54) gives 3"1 = 0 so that I(Q) becomes a delta function peak (or a Gaussian if the surface finite). At the out-of-phase condition, [Qzc] - ~, the peaks broaden and have tails that decay like [a.Q] n-2. Another method has been developed to calculate the diffraction from a stepped surface starting from a known distribution of step lengths M(L), and a known distribution of step heights H(s) (here s is the step height in units of c). Pukite et al. begin by deriving a general expression for P(RII,Z) for a system of one-dimensional steps and calculate I(Q) from Eq. (7.46) (Pukite et al., 1985). The general case is broken into two regimes" (1) a surface with an infinite number of levels, each level having the same M(L), and (2) a finite number of levels, each with a different terrace length distribution M;(L). As an example, only the latter case is considered for the special case when the number of levels equals 2. For this type of disorder I(Q) becomes
I(Q)
-
[02 "f" (]
-
-
0) 2 q- 20(1 -- O) Cos(Qzc)I 2~6(Q,,) + (7.55a)
40
[ 1 - cos(Qzc)] R e ( [1 - M I ( Q I I ) ] [ I - M2(Q")]]
Q: r
M;(Q,,) - f M,(L) e -'c)''; dL.
(7.55b)
11
M;(QI I) is the Fourier transform of the terrace length distribution of the ith level M,(L), 0 is the coverage of the top layer, and (L~) is the average terrace length in the top level. For non-interacting steps, the probability of a terrace of length L is given by a Boltzmann distribution M,(L) - ere -'~'',
and
M2(L) - 13e-lsL
(7.56)
where 0 - ~3/(o~ + 13). Substituting Eq. 7.56) into Eq. (7.55) gives a diffraction line shape
I ( Q ) - (I - 20(! - 0)11 - cos(Q.c)])2rcS(Q,,- G,,) + (7.57) 20(1 - 0)[1 - cos(Q,c)]
2(oc + [3) (O~ + ~)2 + Q _
GI,)2
For a 2-level system the line shape is the sum of a broad Lorentzian and a delta function peak (or a Gaussian for a finite crystal). At the in-phase condition, the intensity of the Lorentzian term is zero. Note that the F W H M of the Lorentzian is inversely proportional to (L) as discussed in w 7.2.1.3. At the out-of-phase condition, the ratio of the Lorentzian to delta function peak depends on 0. For 0 = 0.5 the delta function term is exactly zero.
D~ffractionmethods
303
7.3. Instrument resolution Any analysis of diffraction data, regardless of the probe radiation, requires a detailed understanding of the momentum or Q-resolution of the diffractometer. The diffracted beams will in general be broadened and distorted due to many factors dependent on both the particular instrument and the scattering geometry. The instrument response also plays an important part in measuring integrated intensities, and a correction for finite resolution (Lorentz factor, see w 7.4.2.3) must be included before data can be compared to model calculation. In this section some general considerations about the finite instrument resolution will be discussed. Later, when individual diffraction techniques are reviewed, the specific factors that limit their Q-resolution will be pointed out. It will be necessary to begin by defining a coordinate system that will be common to all of the specific diffraction techniques discussed in this chapter. The coordinate system is shown in Fig. 7.20. The plane containing the incident and diffracted beam is defined as the scattering plane. The detector angle (20) and the crystal angle (0) are measured relative to the incident beam direction in the scattering plane. These two rotations have a co-linear axis (instrument axis) that is perpendicular to the primary beam direction. The sample is fixed on the ~ axis and thus rotates with both Z and ~. The crystal can be tilted by an angle ~ that lies in the scattering plane, which is rotated 0 from the incident direction. Finally the sample can be rotated about the ~ axis as shown. A real diffraction instrument will detect a range of momentum transfer vectors Q +_ AQ. For instance, a detector with a finite aperture will accept waves that have been scattered into a range of angles (20) _+A(20). The measured diffraction signal, J(Q), will therefore be the sum over all possible beams with momentum transfers that reach the detector
~strtaneot Axis
t
.
.
.
.
.
.
ztee/den,,;,
Fig. 7.20. The standard convention for a four-circle diffraction geometry.
E. Conrad
304
J(Q) = ~
I(Q - Q')T(Q') d Q ' = T(Q) 9I(Q).
(7.58)
AQ
T(Q) is the instrument response function that determines which momentum transfers are accepted by the diffractometer. If the transfer function, t(R), is defined as the Fourier transform of T(Q), the pair correlation function can be determined by taking the Fourier transform of Eq. (7.58). Using the convolution theorem, P(R) is P(R) = F{ J(Q) }/t(R)
(7.59)
The broader the function t(R) is, the more the Fourier transform of J(Q) resembles P(R). The F W H M of t(R) (which is a direct space distance) is known as the transfer width of the instrument. It is a measure of the largest distance that can be resolved by a diffraction experiment. If t(R) is known for all R, then P(R) could bc determined exactly. In practice this is never the case so the calculated P(R) from the data always contains an uncertainty. The Q-resolution, of a diffraction instrument is determined by three terms: the energy uncertainty in either the incident beam or the measured signal, the uncertainty in the direction of the incident k vector, and the uncertainty in the detected k vector direction. The uncertainty in Q due to the energy spread of the incident beam is found by differentiating Eq. (7.3); 5Q = 5ksin[(20)/2]. If the incident beam has mass, the particle has a deBroglie wavelength such that k is proportional to E ~/2 so ~Sk or 1/2)E-1/25E. Therefore, the uncertainty in Q due to the energy spread will be AE AQE--~- Q
for radiation with mass.
(7.60a)
For photons E is proportional to k and
I~i
ki
Fig. 7.21. The in-plane resolution function taken from Robinson (1991).
Diffraction methods
305
AE AQE = Q --~ for massless radiation
(7.60b)
For simplicity it is best to define the resolution function in terms of its components both in the scattering plane and normal to the scattering plane. Uncertainties in k~ and kf are a result of the angular uncertainty in the incident beam (Aoq) and the acceptance angle of the detector (Ao~f), respectively. Here AO~i and A~f are measured in the scattering plane. These angular uncertainties lead to an in-plane resolution function that is trapezoidal, as shown in Fig. 7.21. The in-plane resolution can be specified by components AQT and AQR, which are the resolution widths transverse and parallel to the momentum transfer vector, respectively. From the geometry of Fig. 7.21 they are (Robinson, 1991)
(7.61)
AQT- k(Ao~i + Aotf) sin [2(20)]= (Aff, i + Ao~f)lQ I The out-of-plane, resolution, AQN, is determined by both the out-of-plane angular width of the incident beam (Aa) and detector out-of-plane acceptance (A[3). The out-of-plane resolution width is AQN = k(Aot + A[3)
(7.62)
The physical origin of Ac~, AI3, Aoq and Ac~fwill be described in more detail in w 7.4. The contribution from each of these terms (Eqs. (7.60), (7.61), and (7.62)) on the broadening of J(Q) will depend in detail on both the instrument and the way the data is taken. (i.e, how Q is scanned across the rod). For instance, suppose that the diffraction rod is normal to the scattering plane and Q is scanned transverse to the crystal truncation rod. In that case only the transverse component of the instrument function and the energy dispersion will contribute to the rod broadening. Assuming that the instrument response function for each term has a Gaussian form (Park et al., 1971), the rod will be broadened parallel to the surface by an amount AQII 2
AQ~= A/~ - QII] + (A~i+A~f) z IQI 2
(7.63)
7.4. Surface diffraction techniques
7.4.1. Low energy electron diffraction Low energy electron diffraction (LEED) is the oldest diffraction technique applied to the study of surfaces dating back to Davisson and Germer's experiments of electron scattering from nickel (Davisson and Germer, 1927). Today LEED is used in one of two basic ways. Its most common use is as a tool to study the structural
306
E. Conrad
!I
,<
.~ 10 3
g ~ ~.
1 02
=
10 i
10
I
I
10 2
I
j
I
!.
I
10 3
10 4
10 5
Electron Energy (eV) Fig. 7.22. Plot of the electron mean free path length, Ain(E), vs electron energy. Most materials have A~,(E) that lie within the shaded region (Quinn, 1962).
arrangement of atoms within the surface unit cell (Ertl and Kuppers, 1985). With the introduction of higher quality electron sources and electron detectors, LEED is now also used as a quantitative probe of surface long-range order and 2-d phase transitions. The surface sensitivity of LEED is derived from the strong interaction of medium to low energy electrons with the conduction electrons in the solid. To see this consider an electron of energy E as it passes through a solid. The attenuation of the elastic electron flux can be described by an inelastic mean free path length A~,, where Ain is the mean distance an electron travels before being inelastically scattered. The elastically scattered electron flux after traversing a distance x through the solid is given by Eq. (7.16) using Ai, I - I,, e -(~/A,,,)
A schematic plot of A~,(E) is shown in Fig. 7.22. For most materials Ain reaches a minimum of about 20 A near 100 eV (Quinn, 1962). This means that an electron beam with this energy will be severely attenuated by the time it has passed through only a few crystal planes (the spacing between planes is 3 ~). The surface sensitivity of the electrons is now obvious. An electron of energy E that passes through the solid and is subsequently detected at the same energy E must have been scattered from the top few layers of the solid. The probability that this electron came from deeper in the bulk is very low since it would most likely have suffered an inelastic scattering event before reaching the detector. Experimentally, the detector energy resolution need only be a few eV to successfully discriminate electrons originating in the surface layers from those originating in the bulk. This is because energy losses of 10 to 100 eV are typical after an electron traverses a single mean free path. An electron beam incident on a solid, with primary energy Ep, will produce a distribution of electrons near the surface due to elastic and inelastic collisions with electrons in the solid. Some of these electrons will be backscattered from the
D~ff'raction methods
307
I
I
I
(d)
(a)
Z
I
50
150
200
250
Energy (eV) Fig. 7.23. The energy distribution of emitted electrons from a target bombarded by electrons with primary energy E o. Several important regions are shown: (a) the true secondary electron peak; (b) Auger electrons shown in an exaggerated scale; (c) Plasmon and interband losses; (d) Elastic scattered electrons. Taken from Rudburg (1930).
surface. A typical energy spectrum for these backscattered electrons is given in Fig. 7.23. The broad peak below 50-100 eV is due to electrons that have suffered many losses due to single particle interactions with electrons in the bulk as well as those ejected from valance states. These electrons are referred to as the true s e c o n d a r i e s . At higher energies, small fluctuations in the secondary electron spectrum appear. These electrons are the A u g e r electrons produced during the de-excitation of ionized core holes caused by the incident electron beam (for a review see Ertl and Kuppers, 1985). The narrow peak in N ( E ) at the incident energy, Ep, contains elastic and quasielastic electrons. These electrons have either been scattered elastically or have lost (or gained) energy due to the creation (or annihilation) of phonons. Slightly below the primary electron peak are a series of smaller peaks. These are a result of losses due to bulk and surface plasmons or valance band transitions. The total elastic yield depends on the incident energy (see Fig. 7.24). For E o below 100 eV the elastic and quasi-elastic contribution to the scattering yield can be more than 60%. The ratio of the total number of electrons scattered from the surface per incident electrons is known as the secondary emission ratio 6 s. It is a function of energy and can be both greater than or less than 1.0. This means that unless the sample is properly grounded, the surface can charge up positively or negatively, making the study of insulators particularly difficult with LEED. One way around this is to use a flood electron gun to discharge the sample. The secondary emission ratio is generally greater than 1.0 at higher energies so that the incident electron beam will cause the sample to charge positively. 5~ is less than 1.0 for lower energies and will cause the sample to charge negatively. If a low energy beam of electrons from the flood gun is directed at the sample (8~ < 1.0) it will charge the sample negatively. A proper choice of flood gun electron energy and current can prevent the sample
E. Conrad
308
25
20
t"q
15
.< ~o
0
,
0
t
,
100
t
200
,
I
300
,
t
,
400
t
500
,
600
E l e c t r o n E n e r g y (eV) Fig. 7.24. The electron total collision cross section (OT)and total elastic-scattering cross section (oe~) for mercury. At 500 eV o~l/C~-r---4/9. From Webb and Lagally (1973).
from developing a net charge even in the presence of the primary electron beam. Because the flood gfan energy is lower than the primary energy, the electrons from the flood gun will not get into the detector and interfere with the diffraction data. In some cases the surface may still charge up and it should be kept in mind that under this circumstance the electron energy measured relative to the sample (and thus the relevant electron wavelength) is not the same as its energy relative to ground.
7.4.1.1. Experimental LEED systems One general requirement of a LEED system is to produce a nearly mono-energetic, collimated electron beam at the sample. Some type of detector is then used to measure the scattered electron current as a function of angle and thus determine J(Q). Exposed insulators must be avoided because they may become charged by the incident or scattered electrons and produce strong electric fields that alter trajectories of the low energy electrons. Similarly, small magnetic fields of the order of > 100 mG (approximately the Earth's field) must be reduced by either using magnetic shielding or Helmholtz coils. The most common type of LEED system to do this is the full view LEED system that comes in three variations. When only qualitative information about the surface structure is required, a basic three grid optics system with phosphor screen is used (in general most LEED systems have four grids; the fourth grid is used when the LEED optics are set up to do Auger spectroscopy). A basic 4-grid LEED optics is shown in Fig. 7.25. The first grid (from the sample) is grounded and provides a field free region in the scattering volume. The electrons pass through the retarding grids that act as a high pass energy filter to remove the inelastic secondaries. Two retarding grids are used to improve the energy resolution, not so much for LEED, but because the same four grid system can double as an Auger electron analyzer. The holes in the mesh material that make
309
D ~f f rac t i o n m e t h o d s
Electron Gun
....
'''
Fig. 7.25. Standard 4-grid LEED/Auger system. In a rear viewing mode (looking into the gun) the phosphor screen is made of glass with a conducting coating (SnO2) to prevent charging. In a back view LEED system (looking into the incident beam direction) the phosphor is deposited onto a metal hemisphere. In some cases the grids are flat and the distortion caused by non-radial fields is taken out in the computer software. up the retarding grids each act like electron lens. This causes a broadening of the diffracted beams especially when the retarding voltage is greater than 80% of the beam voltage (Lagally and Martin, 1983). This is not an important consideration while viewing a LEED pattern, but when lineshape measurements are important, as in long-range order measurements, this distortion must be considered. The electrons are finally accelerated through a 1-5 kV potential into a phosphor screen for viewing. Since the elastic electrons only scatter into directions corresponding to a surface reciprocal lattice vector (determined by the Ewald construction), the pattern on the screen is the intersection of the diffraction rods with the screen (see Fig. 7.26). Thus the observed spot pattern is the reciprocal lattice of the surface at a Qz determined by the electron energy. Information about surface symmetry using LEED is discussed in w 7.4.1.3. The number of beams seen in the pattern depends of course on the electron energy. LEED uses electrons with energies typically in the range of 2 0 - 5 0 0 eV. An electron with energy E has a wavelength given by the deBroglie relation
~,v.~(]k)=~2n-4 E-(-(-~iI50"4
(7.64)
k The wavelength given by Eq. (7.64) refers to the wavelength measured in vacuum. For larger k values (larger energies) more diffraction rods intersect a larger Ewald
310
E. Conrad
(00)
(oo)
L",
i i
e
i
ki k"x\\X'x"q ~ \ \ \ x , x l
E] (a)
Qil (b)
E2< E l
QIt
Fig. 7.26. Spot profile geometry in direct space (a) and corresponding reciprocal space picture (b) for two different energies. Both pictures are for normal incidence geometry.
sphere (see Fig. 7.26b). In real space, as the energy is raised, more diffraction spots emerge from the boundary of the screen and move radially inward towards the (00) or specular reflection. Note that the position of the specular reflection is independent of electron ~.nergy. By measuring the intensity of each spot as a function of energy, l(Qz) can be determined for each beam. The measurement of l(Qz) is known as a LEED IV profile. Simply viewing a LEED pattern on a phosphor screen is not enough in some experiments. Relative diffraction intensities, intensities as a function of temperature, monitoring changes in the diffraction pattern as a function of adsorbate coverage or electron energy are some examples of the types of data that might be collected. There are three methods to obtain quantitative intensities of one or more diffraction beams simultaneously. A popular system is a video technique shown in Fig. 7.27. The standard four-grid optics is still used. The diffraction pattern on the phosphor screen is then photographed in real time with a high resolution Vidicon system outside the vacuum (Lang et al., 1979). The image is digitized and stored by computer. In this fashion many diffraction beams can be measured. Beam intensity can be automatically integrated over the beam profile, background can be subtracted, etc. Because data is collected for many beams at once, measurement times are reduced by N -~, where N is the number of diffraction beams on the screen. The measured intensities must be normalized by the sensitivity of the phosphor screen (which is not spatially uniform), grid transmission (which is also spatially varying), and stray light that increases the background. In many cases either the scattered intensity is weak or a low incident current must be used to prevent electron stimulated desorption or decomposition of the overlayer material. In such experiments the phosphor screen is not efficient enough to produce a measurable image. To get around this problem an image intensifier is used as shown in Fig. 7.28 (Chinn, 1977). The image intensifier is basically a collection of parallel electron multipliers (chevron). Usually two chevrons are placed in series to get electron multiplications of 106. This effectively increases the
D(ff'raction methods
311
Fig. 7.27. Four-grid LEED system with Vidicon camera viewing system.
Fig. 7.28. LEED system using a channel electron multiplier array. A fifth grid is added between the LEED optics and the chevron plates to minimize distortions due to non-radial fields.
S/N ratio for a given collection time by 106 over the Vidicon system. The output of the last chevron is accelerated to the phosphor screen and data collection is processed in the same way as in the Vidicon method. While the S/N ratio is increased, there is considerable spatial broadening of the diffraction beams in this method. The finite channel width of a single chevron plus the spreading of the output pulse from the first chevron to several channels in the second chevron give a beam broadening of--80 ktm F W H M . This is about double the instrument broadening of the Vidicon system. In addition the non-radial fields between the fifth grid and the chevrons cause a distortion of the diffraction pattern.
312
E. Conrad
Fig. 7.29. LEED system with resistive anode detector.
Vidicon systems are often large because of the camera and optics. An alternative is to replace the phosphor screen with a two-dimensional electron detector. One type of 2-d detector is a position sensitive resistive anode. The basic device is shown in Fig. 7.29 (Stair, 1980). As before, an image intensifier is used to amplify the electron signal passed by the LEED optics. When a pulse of electrons hits the anode at a specific point, the current is collected at the four corners of the anode. Because the resistance is linear with the distance from the collector to the electron burst, each corner will measure a current spike with a different magnitude. The farther the burst is from the collector, the smaller the signal. A position computer measures the current spike to each corner, and through a known algorithm, triangulates the position of the input pulse. The number and position of pulses are collected over a specific integration time and kept in a 2-d array; thus a map of the diffracted intensity as a function of position on the anode is generated. The resistive anode has the additional advantage that its sensitivity makes it possible to resolve individual current pulses. In comparison a florescent screen has been estimated to resolve no fewer than 1000 pulses/s. The resistive anode method of data collection has three significant problems. First, its spatial resolution is poor due to thermal noise (Ax-- 300--400 l.tm). This problem can be partially surmounted by going to a more grazing incidence scattering geometry that improves Q-resolution for a fixed angular resolution (see w 7.4.1.5). The second problem is their small size. This limits the corresponding size
Diffraction methods
313
of reciprocal space that can be measured. Finally, data collection is slow because new pulses can arrive in the detector before the triangulation procedure is completed. The maximum total count rate at the anode that can be tolerated before pulse overlap occurs is <105 counts/s. The resistive anode is therefore not used when kinetics data, which require rapid collection times, are needed.
7.4.1.2. High resolution LEED The systems discussed above are generally used either to obtain qualitative information about the surface symmetry or for analysis of total beam intensity for unit cell structural determination. As will be discussed, the reciprocal space resolution of the above systems is not very good. When information on the shape of a diffrac:ted beam is necessary for determining the long-range order of the surface (i.e., steps), both the electron source and the detection system must be modified. To improve resolution a movable collector is used (see Fig. 7.30). In the simplest case the collector is a small pinhole in front of a metal cup (Faraday Cup) similar to the one originally used by Davisson and Germer (Davisson and Germer, 1927). The cup is biased about 1 eV positive with respect to the electron energy to screen out secondary electrons. The collected current to the cup is measured with a pico-ammeter. Because the typical surface resistance of insulators is about 1014 f~, leakage currents from the detector to ground limit the minimum measurable diffraction current to about 10-14 amps. Slit apertures are also used. They have the advantage of decoupling the resolution function in two orthogonal directions. On the other hand they increase the background current since they integrate the diffracted intensity along the long axis of the slit. To improve S/N and to allow for the use of lower current incident beams, an electron multiplier is substituted for the Faraday cup. A small aperture is again used on a movable arm. To screen out secondaries a small retarding grid is placed after the aperture (Fig. 7.30). This design allows diffracted currents as low as 10 -~7 amps to be easily measured. While the dynamic range of an electron multiplier is less than the Faraday cup, it is still large (--104). The retarding potential behind the aperture has the additional advantage of making the aperture act like a divergent lens. This
e" Ep Aperture Grid
---
i ~ . ~O~2rOr _k_
Fig. 7.30. Faraday cup/electron multiplier design.
314
E. Conrad
Table 7.2 Operational parameters of various LEED systems
Incident beam current Beam diameter at sample Angular divergence of incident beam Minimum detectable current: Fluorescent screen Electron multiplier Energy spread of incident beam: W cathode Indirectly heated cathode Field emitter (energy dependent) Angular resolution of detector: Fluorescent screen Area detector Faraday cup Energy resolution of detector: 4-grid screen Aperture detector Spatial coherence of system
Normal LEED system
High resolution systems
1-10 I.tA 0.3-1 mm 0.25 ~
1-50 nA 20--100 I.tm 0.02-0.15 ~
10-12 A 10-17 A
10-17 A
0.25 eV 0.1-0.25 eV >0.15 eV 2~ 0.25 ~ 0.002~ ~ 1% 0.25 eV 150-500 ,~
0.25 eV 500-20,000
can reduce the apparent acceptance angle, df~, of the detector over the geometric acceptance angle by a factor of 4. Because of the small entrance aperture in this type of detector ( 2 0 - 4 0 ktm), magnetic field requirements are much more stringent. Fields as low as 10 m G can move the diffracted beam out of the detector even when the electron energy changes by as little as 10 eV. Standard electron guns suitable for L E E D are commercially available with typical operating parameters given in Table 7.2. They can produce very large incident currents but have poor spatial properties, i.e., large beam diameters and large angular divergence. When high Q-resolution performance is required a more elaborate system is required. Three types of electron guns meet the higher standards of spatial coherence. Henzler has designed a gun that uses a BaO2 cathode and a magnetic lens to produce a small well collimated electron beam. This gun is c o m m e r c i a l l y available and is an integral part of a commercial Spatially Analyzed LEED system ( S P A L E E D ) (Gronwald and Henzler, 1982). The S P A L E E D system has a fixed sample and uses four pairs of electrostatic deflection plates to change the incident angle. A sample manipulator provides course positioning of a diffracted beam into the detector. Instead of mechanically scanning the detector, the S P A L E E D sweeps the diffracted beam across a fixed detector aperture by two sets of deflection plates as shown in Fig. 7.31. This arrangement allows a rapid 2-d peak profile to be collected.
D~ffraction methods
315
Sample
Deflection Plates
Electron Gun Channeltron I
Computer Fig. 7.31. SPALEED system of Gronwald and Henzler (1982).
Several investigators have developed low energy electron guns using field emission tip cathodes (Martin and Lagally, 1983; Williams et al., 1984). The object of the field emission design is to have a small electron source (~1 ~m). Starting from such a small source the beam can be demagnified with an electrostatic lens to improve the angular divergence and still keep the spot size at the sample less than 100 ~tm. Field emission sources require an extraction voltage greater than 1 kV meaning that the emitted electrons must be decelerated to typical LEED energies. This requires that the gun be refocused when the energy is changed. Martin and Lagally have found that periodic Cs coating of the field emission tip not only stabilizes the beam current but allows extraction voltages as low as a few 100 eV. Although the brightness of these sources was originally thought to be able to produce intense electron beams even for small beam diameters, the total current from the tip is not really a factor. The maximum current the gun can produce is limited by the space charge in the beam, which is only a function of the electron energy and the beam divergence half angle, 13 (Klemperer and Barnett, 1971) ]max(~lA) -- V3/21~2
(7.65)
Since it is the divergence and not the beam diameter that is usually the important parameter in Q-resolution (see w 7.4.1.5), /max is usually determined by [3. The constraint on 13for Q-resolution and the typical brightness of LaB6 and BaO2 makes these cathode materials equivalent to field emission sources for most LEED applications. Cao and Conrad have designed a high resolution unipotential electron gun using an indirectly heated LaB 6 cathode (Cao and Conrad, 1989). The cathode has a 15 ~tm diameter tip. The source image is demagnified in the same way used by field
316
E. Conrad
emitter designs. The advantage of this gun is that the unipotential design means that electron energy can be changed without refocusing. It is also inexpensive, relatively small, and does not require constant Cs recoating. 7.4.1.3. Qualitative L E E D
The most common use of LEED is to determine the symmetry of the surface structure. As already discussed in w 7.4.1.1, the spot pattern on the LEED screen is an image of a portion of 2-d reciprocal space (the size of this image is determined by the Ewald sphere diameter, see Fig. 7.26). Without any detailed analysis, this picture gives an immediate description of the surface symmetry. As an example consider the LEED pattern from clean Ni(001) and the same surface covered with a partial coverage of oxygen, giving a p ( 2 x l ) structure (i.e., a Ni(001)-p(2xl)-O in Wood notation). For the clean unreconstructed surface the lattice vectors are shown in Fig. 7.32. The reciprocal lattice vectors are a~ =
2n ^ a~
and
2n ^ a2 = ,-S-Ta2 lu21
(7.66)
The reciprocal lattice for the clean surface is shown in Fig. 7.33a (which is also the pattern observed on the LEED screen when the incident beam is at normal incidence). If oxygen is adsorbed on Ni(001) to form a p(lx2) pattern as in Fig. 7.32, the new lattice vector, b2, is twice as long in the a2 direction. The corresponding reciprocal lattice vector, b~, is therefore half as small as a2. This means that the diffraction pattern will have twice as many spots in the a~ direction (see Fig. 7.33b). In general when the surface has a lower symmetry than the bulk, more than one type of domain can exist. In the example of oxygen, both p ( 2 x l ) and p ( l x 2 ) overlayers are equally likely (see Fig. 7.32). The actual LEED pattern is just the incoherent sum of the individual LEED patterns from the different domains (see Fig. 7.33c).
Fig. 7.32. The Ni(001 ) surface (open circles)'with adsorbed oxygen (shaded circles). Two 90 ~rotated domains are shown. The (lx2) domain are shown adsorbed on 4-fold hollow sites, and the (2xl) domain on 2-fold bridge sites.
Diffraction methods
317
(01)
b~ ~ _ ~ (01)
9
'-'1
(oo) 9 (1o) .
9
9
9
(01) 9
9
p(lx2)
(oo) 9
9
9
9
9
9
9
(60) 9
9
9
(Ol) 9
9
9
9
9
9
9
9
9
9
a~ (lxl)
(0o)
9
9
9
9
9
9
p(2xl) (a)
(b)
(c)
Fig. 7.33. (a) the LEED pattern from an unreconstructed Ni(001) surface. (b) The LEED pattern for both a p(2xl ) and p(1 x2) structure. (c) The LEED pattern for a surface with both p(2xl) and p(1 x2) domains. Note also that the type of adsorption site does not change the symmetry of the reciprocal lattice. The IV profiles, however, will be different. To analyze a LEED pattern the first step is to identify the possibility of rotated domains. Once this is done, possible choices for the new surface reciprocal lattice vectors ( b ] , b 2 ) are identified. From these vectors the surface symmetry relative to the unreconstructed bulk surface can be determined. While this is simple in the system already described, it is sometimes extremely difficult with higher order periodicities. The transfer from reciprocal to direct space can be made more tractable using a matrix method (Ertl and Kuppers, 1985). Let a~ and a2 be the bulk terminated pr!mitive lattice vector. The primitive lattice vectors of the reconstructed surface, b l and b2, can be written as a linear combination of the a ' s b~ = m] la~ + m~a (7.67) b ~= m 2 la ! + m22a Likewise the reciprocal lattice vectors of the reconstructed surface can be written as a linear combination of the bulk terminated reciprocal lattice vectors. = m~a~ + ml2a 2
(7.68) b2
= m'z~a~* + m22a * 2*
It can be shown that the coefficients of the reciprocal and real space vectors are related:
318
E. Conrad
liml'm'21_ l (m*z2-m:'i det M" ~-m;2 m~l
where M* =
(m! "/ l m~2
(7.69)
Since the mij s are known from the diffraction pattern, the mij s can be determined. From the diffraction pattern for the p ( 2 • surface in Fig. 7.33, m~l = 1/2, mlz = 0, m21 = 0, and m~2 = 1.
7.4.1.4. Quantitative LEED Quantitative information about either the structure of the surface unit cell or the ordered arrangement of the unit cells requires a more detailed understanding of LEED. As discussed in w 7.1.2, information about the unit cell structure is contained in the intensity variations of the reciprocal lattice rods as a function of the perpendicular momentum. In order to use the experimental data, the relationship between experimental geometry and reciprocal space must be understood. The standard LEED geometry with a movable detector is shown in Fig. 7.34. With reference to Fig. 7.20, this geometry corresponds to X = - 9 0 ~ so that the ~ axis is in the scattering plane (it is assumed that the surface normal is co-linear with the axis). In this geometry the components of Q perpendicular and Q parallel to the surface are
Sample Normal
ilo
i
~
kf ,fkf
ki
Sample
(0o)
(0o)
(0o)
k2
"',
k
k2>kl
(a)
~
(b)
v
(c)
Fig. 7.34. (a) LEED IV profile, (b) thermal diffuse scan (or rocking curve), and (c) a detector scan that gives I vs. 20.
D~ffmction methods
319
Q• = k(cos0f + cos0i) (7.70) Q, = k(sin0e- sin0 i) where k = 2~/~, is the magnitude of the electron wave vector. In all diffraction experiments the quantity of interest is the intensity scattered as a function of the momentum transfer vector (or a component of Q). In addition to the IV profile already discussed in w 7.4.1.1 (see Figs. 7.26 and 7.34a), there are several other important types of scans through reciprocal space that are mentioned below. The first type of scan is called the thermal diffuse scan (TDS). In this geometry the crystal angle is tilted while the gun energy and detector position are kept constant (see Fig. 7.34a). The scattering angle 20 is fixed meaning that f(E,20) is held constant and does not affect the shape of the measured diffraction peak. Note also that the detector path is symmetric in Qll for the specular rod (0i = Of) and nearly symmetric for the off-specular rod. In some cases the contribution off(E,20) must be known to correct the actual diffraction line shape. This is done by scanning the detector angle (scanning 20) with 0~ and electron energy constant (see Fig. 7.34c). As the detector scans through 20, the momentum transfer vector is changing so that both f(E,20) and I(Q) contribute to the intensity variations. The effect of I(Q) can be eliminated by preparing the surface in a completely disordered state (either by sputtering, quenched deposition, or heating to high temperatures). The electron wavelength in Eq. (7.64) was calculated in vacuum. As the electron moves into the solid, however, it experiences the periodic potential due to the lattice. The spatial average of this potential is called the inner potential U. The wavelength of the electron in the solid is decreased 150.4 Z,(h) =
(7.71)
E(eV) + U
While the inner potential is a function of E, it is typically of the order of 5-15 eV. The inner potential has an important effect on LEED analysis. The component of electric field in the plane of the surface of the incoming radiation must be continuous across the surface. On the other hand the electrons are accelerated normal to the surface by the inner potential thus causing the beam to be refracted. This means that, while the momentum transfer parallel to the surface is the same inside and outside the surface, the momentum transfer perpendicular to the surface changes. This is illustrated in Fig. 7.35. Inside the surface the perpendicular momentum transfer is (Webb and Lagally, 1973) Q~, =
2____~_~{Ecos2 0~ + U] '/z + [Ecos 2 0f + U] '~} (7.72) 4150.4 At low energies or grazing angles (i.e., low Qz) the contribution to the momentum transfer from the inner potential is very large. This means that instead of in collecting J(Qz) in an IV profile, the measured signal is j (Qz). As an estimate of AQ z = Qzi,-Qz, consider a solid with U = 15 eV. A 100 eV electron incident on this solid
320
E. Conrad
(10)
(oo)
(I0)
Of
ki-in
y
v
QJi Fig. 7.35. The effect of the inner potential on the electron momentum measured inside (superscript "in") the sample. Solid arcs are portions of the Ewald sphere. Note that QII is the same with or without the inner potential.
at an angle, 0~ = 45 ~ will scatter into the specular rod with a Qz shifted by AQz = 1 ~-J. This is half of the reciprocal unit cell size (i.e., Ghk "- 2-3 ,/k-~) and cannot be ignored! 7.4.1.5. LEED resolution
The resolution function in the surface plane (AQll) and perpendicular to the surface (AQz) is found by using the standard scanning geometry in Fig. 7.4. Differentiating Eqs. (7.70) with respect to the incident and scattered angle gives AQI*i =
k(cos0f)Aot,
f ~
(7.73a)
AQII = k(cos0i)Act~
(7.73b)
AQ~ = k(sin0f)A~f
(7.73c)
AQ~z= k(sin0i)At~
(7.73d)
where the At~'s are those defined in Eq. (7.61). The contributions to At~ and Actf in a LEED apparatus are dominated by three terms; the angular dispersion in the incident beam, the angular acceptance of the detector, the finite size of the beam on the sample. The largest contribution to LEED's resolution function is Act i, and it is dominated by the divergence of the incident electron beam, Ate. The minimum electron divergence angle is usually determined by the space charge in the beam and is sometimes referred to as the source extension. As a rule of thumb, the more
D!ffraction methods
321
Detector
Fig. 7.36. Beam diameter elongation on the sample. collimated the incident beam is, the larger the spot size on the sample. Too large a diameter beam on the sample will increase Acxf as discussed below. The best L E E D systems have A~v-- 0.04 ~ although typical values are an order of magnitude worse (see Table 7.2). The AQ, resolution of the instrument can be improved by going to a grazing angle geometry to reduce the cos0f term in Eqs. (7.73a and b). Because the electron beam has a finite diameter, its projected size on the sample will depend on the incident angle and give rise to an error in the scattered angle (see Fig. 7.36). Collimating slits, such as solar slits used in X-ray scattering, in principle could remove this problem. However, strong electron scattering from the slit material and space limitations in the vacuum make them unusable in LEED. For a zero divergence incidence beam of diameter d and a point detector, Ao~fwill have a contribution from the spot size of cos (Or) d AOtBD= ~ --, COS(0i) D
(7.74)
where D is the distance between the sample and the detector. Equation (7.74) shows that AC~nDcan be minimized at normal incidence as expected. The detector aperture also contributes to 8orr. The acceptance angle of the detector is given by AcxD= s/D, where s is the effective aperture diameter discussed in w 7.4.1.2. Typical values are listed in Table 7.2. Assuming that the spot size and detector acceptance angle contribute to the response function as Gaussian variables, the total contribution to the instrument function will be the square root of the sum of the squares of each term (i.e., Ac~ = A~ZD + ACX~D). The total resolution function, including the finite energy resolution (Eq. 7.60a), is found by adding up the contributions to the resolution in quadrature. The combined resolution width of the system is 2
=
2+
LI,DJ LkD ;
2
+
kcos(0,)
l+ (~--~-j Q~
(7.75a)
J + (,-2-E-J Qz2
(7.75b)
J
r + ' ~'-~(0i) D
322
E. Conrad
7.4.1.6. Dynamic LEED and multiple scattering While LEED has historically been the most often used diffraction method for surface structural analysis, it suffers from one serious drawback: LEED is not accurately described by the kinematic approximation. The strong electron interaction with the solid necessary to insure surface sensitivity also leads to strong multiple scattering. Simply put, the plane wave description of the outgoing scattered wave is incomplete. Spherical waves diffracted near the surface in turn diffract from other scatters as they pass through the solid. Each multiple scattered beam contributes to the collected intensity. Therefore, structural determination of the unit cell using LEED must include a full quantum mechanical treatment of the problem. Complex computational techniques have been developed for this purpose. In general they require an assumed structure for the atoms in the unit cell. As a first guess, the diffraction data can be treated as kinematic. Using this first guess structure the diffracted intensity is calculated and compared to the experimental data. A slightly new structure is guessed from this comparison and the intensity recalculated. This procedure is iterated to a desired degree of accuracy. The process becomes extremely complicated for large unit cells. Surface cells containing more than a 10 atom basis set are rarely calculable. There has been a good deal of advancement in full multiple scattering treatments of the LEED scattering problem (Van Hove et al., 1986). Fast algorithms for handling the scattering matrix problem as well as perturbation techniques to optimize structural parameter searches allow larger unit cell structures to be determined. A complete treatment of dynamic LEED calculations is beyond the scope of this book. Instead, a common approach to these calculations is outlined below. It includes a discussion of the accuracy of these calculations both from the standpoint of critical assumptions in the calculations and on the errors associated with experimental data collection. The problem of calculating the scattered LEED intensity from a known atomic structure can be broken into two parts. First, a self consistent scattering potential including band effects, thermal vibrations, inner potential, and adsorption must be known. Second, given this potential, the scattering matrix for a single atom can be calculated. The transition matrix takes the incident plane wave and scatters it into a given diffraction beam. The greatest improvement in calculating LEED intensities has been in determining the scattering matrix. The basic problem is to solve the wave equation
i-~mV2+k21~(r)=O,
k2 = --~ 2m (E- V(R))
(7.76)
The difficulty in obtaining a solution to Eq. (7.76) is that the amplitude scattered from the ith atom depends not only on the incident plane wave but also on the amplitude of the outgoing scattered wave from every other atom in the unit cell. At the same time the amplitude scattered from all the other atoms in the surface depends on the amplitude scattered from the ith atom. This requires the solution of a set of self consistent equations.
323
Diffraction methods
There are several methods for solving for the scattered amplitude. Most of these are based on a two level solution. First the surface is divided into vertical planes (with their normals co-linear with the surface normal); each plane contains one or more atoms. The scattering matrix for a plane wave incident on each of these planes is then calculated. In the second step, the inter-planer scattering matrix is then calculated for the collection of planes, including a set of bulk planes. A powerful example of this approach is the Combined Space Method of Van Hove and Tong (1980). In this method a spherical wave representation for the scattering is used within the plane and a plane wave representation for the scattering between planes. In general the layer scattering matrix is the difficult part of the calculation. Several methods are commonly employed for its solution: Renormalized Forward Scattering (RFS) (Pendry, 1971), Reverse Scattering Perturbation (RSP) (Zimmer and Holland, 1975), or the Beeby matrix inversion (Kinniburgh, 1975). After the intensity is calculated from the model structure, the results must be compared with the experimental LEED I-V profiles. This means that the scattering matrix must be calculated for a set of energies between 10 and 200 eV. For accurate comparison the matrix must be calculated for a large set of diffraction rods. The comparison is quantified by use of reliability or R factors. In X-ray scattering the Rx factor is defined as (Robinson, 1991) Z [IFi'call 2 -IFi.exol 2 I
Rx = i
(7.77) Z IFi.~xpI 2 i
where F is the crystal structure factor defined below Eq. (7.9). In LEED the intensity modulations along a rod are much richer than in X-ray diffraction (in large part due to the multiple scattering). Information about the structure is contained in both the shape of a peak in the IV profile as well as in the position and intensities of their minima and maxima. To take this into account a LEED RL factor was developed by Zanazzi and Jona (1977) RL=A
' - l~pI " I~I " ,:,i - l"e,,pI I~I~,,, dE
II'r
Ima,, + I l'~xo I
(7.78)
where A = 37.04/j" le,,pdE and ' and " represent first and second derivatives, respectively. The weighting factor in this analysis is purely for normalization
I] = :b
lexo dE (7.79)
J Ic.,l dE Zanazzi and Jona have concluded that R L > 0.5 is a bad fit and R L < 0.2 is a good fit.
324
E. C o n r a d
Once the calculation and comparison is done for a trial structure, a new guess for the structure must be made, the calculation performed again, and a new comparison with the data is made to see if the new structure improves the fit. Since the computing time goes as the cube of the number of experimental CTRs, and because derivatives of the calculated IV profile must be made for the R factor determination, this task is extremely time consuming. This is especially true because no algorithm is available for iterating the guess for the next trial structure. Recently Pendry has developed a method called Tensor LEED analysis that provides a means to speed up the iteration process (Rous et al., 1986). Starting from a base structure, the full dynamic calculations are performed. Using the transfer matrix for the base structure, the scattering amplitude can be calculated for small perturbations of the atomic positions without recalculating the transfer matrix. The perturbation technique is good as long as atomic coordinates are not change by more than about 0.4 A,. The change in amplitude for N displaced atoms is N
~A (k,,,,. k~,,) = Y_, ~ i
Tiv (k,,,, ki,,)tz, &~t~L,(Sty)
(7.80)
LL"
where T~.v is the transfer matrix for the base structure. T,.,, (k,,,,, k~,,) = (k,,,,IVvV ( r - r,) I k~,~) (7.81) 8ZLL. (Sri) = e -iO" ~".
and
Tensor LEED also has the advantage that derivatives of the IV profile are also easily calculated. This makes it ideally suited as the basis for optimization routines using RI.
In spite of the advances in computational solutions to the scattering process, there are inherent limits to the accuracy of dynamic LEED analysis. A fundamental limit is imposed by diffraction theory. Unertl and McKay have shown that both the finite extent of the data (Qma~Qmin.) and the instrument resolution (AQ) lead to a maximum accuracy in any structural length, d, of (Unertl and McKay, 1984) Ad d
> -
2rt AQ
n,
(7.82)
C ( O m a x - Omin)
where C is a constant near 4 and n is the highest distinguishable order of multiple scattering. In a typical LEED experiment the energy range is limited to 30 < E < 300 eV. The lower limit is set by the inter-atomic spacing because below 30 eV (measured inside the crystal) the Ewald sphere will not intersect any rods. The higher energy limit is due to the Debye-Waller factor that significantly reduces the intensity at higher Q. For all practical purposes Qmax--Qmin < 12 ,/k-~. Using a typical LEED system with AQ -- 0.1 ]k-1 and assuming that n < 3, gives A d / d > 0.5%.
Diffractionmethods
325
Higher system resolution may push this limit to 0.05% but because of both experimental uncertainties and because of certain assumptions in the dynamic calculations even the conservative limit of 0.5% is never achieved. Errors in electron trajectories due to either stray magnetic fields or absolute angle determination can lead to a degradation of the ideal structural resolution. Averaging IV profiles of equivalent reflections can reduce these errors. Sample quality can also degrade the IV profiles because beam width modulation from the defect structure causes the intensity along a rod to fluctuate. More important, however, are assumptions in the model calculations themselves. Vibrational amplitudes, especially when systems have large anisotropy, are not well known for surface atoms or interfaces. Usually simple models assuming a slightly enhanced bulk amplitude are used in the dynamic calculations and in many cases these assumptions are very bad. Data collection at lower temperatures can reduce these effects. More fundamental limits to increasing the uncertainty in Ad/d come from other model assumptions. The vacuum-solid or solid-overlayer interface potentials are not known. Even the inner potential itself is not a priori known. The assumed energy dependence, or lack of any at all, of the spatially averaged inner pote__ntial can lead to much larger errors because of the introduced uncertainty in Qz. U = V(R) is chosen as a free parameter in most calculations to minimize RL. The energy dependence of the attenuation coefficient is also a source of error. Usually a simple smoothly varying A(E) is used even near resonant adsorption energies. All of these effects reduce Ad/d > 2%.
7.4.1.7. Summary of advantages and disadvantages of LEED LEED's main advantage as a surface analysis tool is its ability to determine a wide variety of structural information quickly. Because it can sample a large portion of k-space (using a phosphor screen or area detector), it can determine the surface symmetry in a single measurement. Computer controlled IV profiles can be taken on 4-10 diffraction beams at once to provide detailed structural data. Even longrange order information contained in diffraction profiles can be taken quickly. Rapid determination of diffraction data is a nontrivial advantage especially on surfaces that are highly reactive. In a large number of experiments this type of information is required as a function of adsorbate coverage, substrate temperature, or time when the kinetics of surface processes are being studied. Because of its high incident beam currents and large elastic scattering cross section, LEED is sensitive to very low Z adsorbates, which are difficult to detect in X-ray measurements. This allows a wide variety of adsorbate systems to be studied. Structural measurements of adsorbed hydrogen, however, are very limited because the atomic form factor decreases as Z 2. Unless the hydrogen induces a substrate reconstruction, little change in the electron diffraction can be observed with even a monolayer of adsorbed hydrogen. The high surface sensitivity of LEED also has an advantage in determining vertical spacing information, l(Qz) data contains information primarily from the top 3-4 layers. This means that LEED intensities are sensitive to interlayer spacings free from bulk intensity contribution. This makes LEED ideal for measuring layer
326
E. Conrad
spacings and normal thermal expansion coefficients. To a smaller extent LEED can measure some average dynamic properties of the surface. Surface vertical and lateral thermal vibration amplitudes can be measured through Debye-Waller factors. Experimentally, LEED is confined to operate in vacuum environments (p < 10-5 torr). Measurements in chemical vapor deposition (CVD) growth environments or in high pressure gas dosing experiments are difficult or impossible and are better handled by X-ray diffraction. The main disadvantage of LEED is dynamical scattering caused by the strong electron-solid interaction. While the surface symmetry is easily determined by LEED, detailed structural information is difficult to obtain from the data and requires an intense theoretical input. Even then, the results may not be unambiguous. It should be noted that line shape measurements are much less effected by multiple scattering. This is because multiple scattering modulates the intensity over a length scale 2n/a and primarily affects the crystal structure factor term in Eq. (7.9). Dynamic scattering over length scales of 3-4 unit cell lengths is significantly reduced by electron attenuation. Line shape measurements on the other hand are sensitive to structural changes on length scales of 10-1000 /~, where multiple scattering terms in the interference function are small. 7.4.2. Surface X-ray diffraction
The problems associated with strong dynamic scattering in electron diffraction are overcome by using a weakly interacting probe such as X-rays. The method of X-ray diffraction for structural analysis is old and well developed (Warren, 1990). Absolute intensities can be measured and compared to predictions based on the kinematic model in order to determine the unit cell structure. Its main problem as a surface experimental probe has been the low diffracted intensity at surface sensitive reciprocal space positions (i.e., Qz= (2n+l)rr,/c). With the advent of synchrotron radiation sources capable of delivering fluxes of 101~-10 ~2 photons/mm2/s at the sample, its use as a surface probe has developed rapidly. All of the discussion in w 7.1.2 is applicable to X-ray scattering. In this section a more careful and quantitative approach is outlined for the derivation of the absolute scattered intensity. From this derivation the relationship between experimental measurements and the determination of the surface atomic structure can be made. Several reviews are available for a more complete description of the analysis of surface X-ray scattering (Robinson, 1991 ; Feidenhans'l, 1989). The amplitude scattered from a single electron (at a point r relative to an arbitrary origin) measured at a distance D from the electron, is given by the Thomson scattering formula ( e2 )
A = Eo mc2D P ~ e iO'"
(7.83)
where Eo is the incident electric field intensity of the photon. The polarization factor P is 1.0 if the incident polarization is normal to the scattering plane defined by ki
D~ffractionmethods
327
and kf (see Fig. 7.20). If the polarization is in the scattering plane, then P = cos220. For a distribution of electrons, p(r), about an atom, the amplitude scattered by the atom is the sum over the amplitudes scattered from each electron times an appropriate phase factor exp[iQ.r~]. Using Eq. (7.83), the amplitude scattered from an atom is
A, - Eo
m c2D
pl/2
atom
p(r)eiQ.rdr 3 = Eo
m c2D
pl/2 f(Q)
(7.84)
This defines the atomic form factorflQ) for X-ray scattering as the Fourier transform of the atomic charge distribution. Assume that the surface contains N~xN2 surface unit cells. Using the amplitude given by Eq. (7.84) and following the derivation leading to Eq. (7.10), the scattered intensity is
sin211N, Q~a,I sin2flN2Q2a2)
e4
l(Q)=l"lm2cnD21 PIF(Q)I2
sin~(l Qla~)
sin2/1 Q2a2)
(7.85a)
,:m
F(Q) = ~ fm ei Q . ( u , , , a ,
+ v, a 2)
ei Q,w,,
(7.86b)
m=O
where Io = E~ and F(Q) is the same crystal structure factor defined in Eq. (7.9) (with m atoms per unit cell). Note that in Eq. (7.85b) the unit cell has been defined as a column of atoms extended into the entire bulk.
7.4.2.1. Surface sensitivity in X-ray scattering Equation (7.85) is completely general and includes scattering from atoms in the bulk and at the surface. The question arises as to how X-rays can be made surface sensitive. The answer to this question is threefold. As in w 7.1.3, a reconstructed surface will always have superlattice reflections where the intensity scattered with Q, = G.,uoerhas no contribution from the bulk. The intensity along the rod, l(Qz), of these diffraction peaks therefore depends only on the surface structure. Recall from w 7.1.3 that some of the superlattice rods fall on integer order rods as well, where the bulk scattering is important. The integer order rods could be ignored, but that would arbitrarily limit the number of Fourier coefficients in the analysis, and significantly restrict the amount of obtainable surface structural information. The problem of the bulk contribution in the integer order rods is also related to the case of a surface with only relaxations (i.e., no superlattice peaks). To estimate the contribution from the surface and the bulk, the crystal structure factor in Eq. (7.85b) can be broken into two parts: a sum over the Ms basis atoms in the surface unit cell and a sum over the remaining bulk atoms.
E. Conrad
328
M
F ( Q ) : ~_,fm e i e "("-", + v,. as) eiQ, w,c m=0
(7.86) Mb
-I- Z
oo
fi e i Q
(ulal "1"v~2"F
Wic) ~
i=0
ei Qzn3c
n3=O
is the number of basis atoms in the bulk unit cell. The first term in Eq. (7.86) is due to the surface. If Qz is allowed to have an imaginary part corresponding to a finite attenuation of the X-ray beam, the second term will gives rise to bulk CTRs (see the derivation leading to Eq. (7.18). For simplicity assume that there is only one atom per unit cell in both the bulk and the surface. Also assume for the moment that the amplitude scattered from the surface is incoherent with the scattered amplitude from the bulk. Under these conditions, the surface and bulk contribution to the integer order rods when Q = 2rt( 1~a l, 1]a 2, = 1/2c) are m b
(7.87a)
l,(Qz) = I,, m2c4D2 P N~ N~
e4)
Ib(Qz) = 1,, m2caD2 P N~ N~
si. / oc/
(7.87b)
The bulk contribution to the intensity as a function of Qz, Eq. (7.87b), has the same form as the CTR derived in Eq. (7.22b). It is peaked at the bulk allowed point Qzc = 2nr~ corresponding to the third Laue condition (in-phase condition). The surface term Eq. (7.87) is independent of Qz and therefore contributes a background to the CTR. Near a bulk allowed reflection the bulk term will clearly dominate. At the out of phase condition (Qzc = (2n+l)r~), however, the surface term is approximately twice as big as the bulk contribution. An example of the surface contribution to the CTR is shown in Fig. 7.37. The data shows two rod from W(001). The (1,0,k) rod was scanned between the (100) and (101) points, and the (2,0,k) rod was scanned between the (200) and (202) points (the shaded area in the insert of Fig. 7.37 shows the range of Q~ scanned). The dashed line is a fit using only the bulk contribution, Eq. (7.87b), to the CTR. The surface contribution (in this example it is the surface roughness) has the effect of lowering the diffracted intensity at the out-of-phase point (i.e., the (101) point on the (1,0,k) rod and the (201) point on the (2,0,k) rod). The surface contribution to the CTR is more dramatically shown in Fig. 7.38. In this example a rod scan along the (1,0,1) rod is shown from the Si(111 ) 7x7 surface. The dashed lines are again the bulk contribution from an ideally truncated Si(111) surface, which clearly do not fit the data. This reconstruction has many atoms in the surface unit cell so that the sum over the amplitude scattered from the surface atoms
329
D~ffraction meth()ds
10"/ _-='-r ' -
_
'
I
i
,
,
,
I
I
o
'
l
i
w' I
'
i
i
w I
Wlt0Ol
O0
11,0,K)
/3=0.46
'
i
--
(0(~2)
I
'
t
i
,
~
O/.
o
~_
-_ -
~ Oo-
(lOft
o
I
oo
(z6o)
(2oz)
oO
i o.
o
t0 3
~
i
~
__
03
'
--
12,O,K)
/3=0.54
tO6 -
(...) uJ
'
WltOO1
-
,.-...
'
SURFACE
d"/
o Io
t0z
11011
?::~12001
- o
t
~-
~_ (~
i
i
t
0
I
I
i
i
i
i
0.5
I
....--.
SURFACE
- O
o
~
o
o_~
2)
~
_-
O
q::
l 0
t.0
PERPENDICULAR
-~vb
MOMENTUM
i
,
i
i
I
l
I
t
_
,
0.5 TRANSFER,
l
i
i
,
t.0 K (RECIPROCAL
,
/
i
i
~ L
|
t.5
-
2.0
L A T T I C E UNITS)
Fig. 7.37. Crystal truncation rods from W(001 ). Dashed line is an attempt to fit the ideal bulk truncation rod profile of Eq. (7.87b). The solid lines are independent fits including surface roughness (Robinson, 1986).
"i'' i'I' ' '' I,
104
z
10 3
.......,"t
'
_
j
I0 2
z
I0
-5
!
-4
I
-3
I
-2
I
-I
I
0
I
I
I
2
I
3
!,.
4
!
5
PERPENDICULAR MOMENTUM TRANSFER (HEXAGONAL UNITS) Fig. 7.38. The (1,0,l) CTR from Si(111) 7x7. The dashed curve is a fit to the data using only the bulk CTR intensity, Eq. (7.87b). The solid curve is a fit to a 4-layer model of the interface (Robinson et al., 1986).
6
330
E. Conrad
(the first term in Eq. (7.86)) is a strong function of Qz. The surface contribution to the scattered intensity produce strong oscillations in the rod intensity between the bulk allowed reflections (l = - 4 , - 1 , and 5). The fact that the bulk and surface components of the CTRs are of the same order of magnitude at an out-of-phase point means separation of the two components requires a high signal to background ratio. One significant contribution to the background comes from bulk phonons. To reduce the phonon component of the background, data must be collected at as low a Q (and temperature) as possible, i.e., data is collected in a grazing angle geometry. If the surface is well ordered, the background can be further suppressed by using a crystal analyzer to improve the q-resolution of the detector. In this way the volume of Q space that is detected is reduced and the contribution from the bulk thermal diffuse scattering (TDS) is lowered without sacrificing the signal. In fact the large asymmetry of the resolution function in a standard synchrotron X-ray experiment can be exploited in a grazing angle geometry to increase the signal-to-background ratio (see w 7.4.2.4). The third way of increasing surface sensitivity is again a result of using a grazing angle geometry. Near grazing, the X-ray reflectivity of the surface, 9~, and the electric field strength at the surface, E, are given by (Born and Wolf, 1975) (e.g., Fresnel's equations) as sin~ - ~/n2 - cos2~ sin[3 + Ntn2 - cos2[~
(7.88a)
and 2sin[3 E
__
sinl3 + ~tn2 - cos213
(7.88b)
where n is the index of refraction n = l-
K2e2F(Q=0) 2rcmc2V
(7.89)
where F is the crystal structure factor. For X-rays n is less than 1.0 by about 1 ppm (James, 1962). From Eq. (7.88a) ~R - 1.0 when n - cos]3c, where 13c is the critical angle. At ]3~ the electric field strength, and thus the scattered intensity from the surface, are enhanced by a factor of 4. A similar additional enhancement occurs if the exit angle is near 13c. For most materials the critical angle for X-rays is between 0.1 ~ < [3c < 0.6 ~ In general the incident beam is not set at the critical angle for two reasons. First, the beam divergence of the incident radiation is larger than 13c. This means that only a fraction of the incident radiation could be used and the total signal strength would be arbitrarily reduced. A similar problem occurs if the exit angle is restricted to be ~c. The second problem operating at the critical angle is the finite penetration near ]3c (which is responsible for the total reflection). The finite penetration is an
Diffraction methods
331
indicator of an increasing dynamic scattering component that further complicates the data analysis. In practice the choice of incident and exit angle is optimized to collect the largest portion of the beam and still remain near [~c. Typically enhancements of 1.5-2.0 in the reflected intensities are obtained within a few tenths of a degree from I]c (Robinson, 1991). 7.4.2.2. Experimental considerations and geometry The design of a UHV surface X-ray scattering system must meet several stringent requirements. First, because the samples must be cleaned and kept so, the diffraction must take place in UHV. Not only does the design have to allow for precise 3-axis rotations of the sample within this environment (angular resolutions of <0.001 ~ but the X-rays must be able to penetrate the vacuum system walls before and after scattering from the sample. Almost all UHV X-ray scattering systems incorporate a Be entrance and exit window for this purpose. A 0.5 mm thick Be plate is strong enough to withstand the vacuum load, while attenuating the incident flux by only about 10% (for 8 kV X-rays). The larger the area of this window, the greater the range of scattering angles that can be achieved. The methods used to obtain sample motion while simultaneously maintaining UHV conditions are diverse. A few examples are given below. The simplest solution is to mount the entire vacuum chamber on a standard 4-circle diffractometer. Weight and size restrictions on the diffractometer, however, limit the size of the chamber and therefore the number of surface preparation and analysis techniques that can be installed in it. The most versatile X-ray scattering systems keep the vacuum chamber and all of the surface analysis equipment fixed and rotate only the sample. An example of such a system is shown in Fig. 7.39. The sample holder is mounted on a translation manipulator so that the sample can be moved into the larger vacuum chamber for cleaning and other surface analysis studies, or it can be fixed to the diffractometer. The main vacuum chamber is attached to the diffractometer through a bellows (to allow for a ~Z rotation) and a differentially pumped rotary seal (/'or the ~ rotation). The design allows full 0, 20 and ~ rotation and +15 ~ of ~ rotation. The 4-circle geometry is redundant in that any position in k-space can be specified with only three degrees of freedom. The fourth degree of freedom adds an additional constraint that can be used to control the angle of incidence or outgoing angle with respect to the sample surface. Several scan modes have been developed for the 4-circle geometry. The first type is the symmetric or "60 = 0" mode (Bushing, 1967; Fleming, 1985; Brauslau, 1987). The sample surface is adjusted on the diffractometer so that its normal coincides with the ~-axis. In this type of scan the constraint is 0 = 1/2(20), and the )(;-circle is half way between the incident and diffracted beam directions. The incident and exit angle (relative to the sample surface) are equal, and are given by. sin 13'= sin 13= sin~ sin0
(7.90)
In this mode, rod-scans are essentially ~ scans (equivalent to LEED IV profiles Fig 7.34a).
332
E. Conrad
Fig. 7.39. Top view of AT&T Bell Laboratory's feedthrough-based surface diffractometer built by Fuoss and Robinson at the National Synchrotron Light Source, Brookhaven (Fuoss and Robinson, 1984). Alignment of the sample normal with the ~-axis is often difficult, particularly when the sample is slightly miscut from a nominal crystallographic plane. Instead, it is preferable to scan in such a way as to decouple the bulk crystallographic coordinate system from the surface coordinate frame. This is done by removing the 0 = 1/2(20) constraint and replacing it with either a constraint on the incident angle (Mochrie, 1988), exit angle (Mochrie, 1988), or to constrain 13 = 13' (Robinson, 1989). The additional advantage of the last approach is that it allows 13 and ~' to be kept far from the critical angle. In some designs a fifth degree of freedom is added. The 5-circle geometry has two advantages. It extends the range of incident and scattered angles that can be reached, which would normally be limited by the maximum g rotation, and allows 13 and 13' to be specified independently. One 5-circle design adds a detector rotation with an axis perpendicular to the 20 axis (Liang et al., 1990). Another 5-circle design rotates the entire diffractometer about an axis perpendicular to the scattering plane as shown in Fig. 7.40 (Vlieg et al., 1987). This design also allows the diffractometer to be run in a reflectivity measurement mode as discussed in w 7.4.2.6. In Eq. (7.87) the scattered intensity is proportional to the number of scatters (N~ and N2). This in turn is proportional to the illuminated area of the sample, A,, which is determined by the entrance and exit slits L~ and L2 as shown in Fig. 7.41. When the sample surface is in the scattering plane (i.e., Z = 0). the illuminated area only depends on 20. For out of plane measurements As, as seen from the detector, is longer and is given by
D~ffraction methods
333
Fig. 7.40. FOM design with rotary table to provide the fifth rotational degree of freedom (Vlieg, 1988).
Fig. 7.41. Illuminated area of sample for the case of the sample surface in the scattering plane.
A~=
L~L2
cosg sin20
(7.91)
For small scattering angles or large Qz, it is possible that the illuminated area A~ is larger than the sample size. Thus, care must be taken to include the finite sample size, when calculating absolute intensities as discussed in w 7.4.2.3. While a few X-ray experiments can achieve sufficient count rates on a rotating anode X-ray generator (when the samples have a sufficiently high Z, such as Au), most experiments are performed at a synchrotron light source. A typical experimen-
334
E. Conrad
Top View
[~
:==::zz::(~~i
n D Detector Parallel Crystal Monochromator
DoubleFocusing Mirror
Slit
Slit
Sample
Lt
Side View Fig. 7.42. Typical X-ray beam line at a synchrotron source. tal a r r a n g e m e n t is shown in Fig. 7.42. Because the electrons in the storage ring orbit in a plane, the light is polarized in the plane of the ring. The advantage of a synchrotron is that the vertical divergence of the light from the source is extremely small (<0.2 mrad), determined by the ring emittance (see Table 7.3). The horizontal divergence is much larger, and for bending magnets can be 20 mrad. For wiggler insertion devices the horizontal divergence is lower, about 2 mrad. Undulator insertion devices have an even smaller horizontal divergence (<1 mrad). The large path from the source to the sample requires the beam to be focused to c o m p e n s a t e for both the vertical and horizontal divergence of the beam. This is done by a toroidal focusing mirror. Beam sizes at the sample are typically greater than 1 mm, and are currently limited by the production quality of the toroidal mirrors. The high heat load on the mirror (>900 W / m m ) requires a large water cooling capacity. Table 7.3 Typical beam line parameters for a surface X-ray scattering experiment 0.5•
Incident Ilux (/o) Monochromator Rocking curve width Ar Energy resolution AE/E Beam parameters Horizontal divergence (from mirror): AI3 Vertical divergence Size at sample Slits Sample-detector distance Horizontal width Vertical width
for Gc( ! 1I) E - 15kV E=7.5 kV
photons/s 8 mm
0.008~ 10-3 5•
-4
horizontal vertical
5 mrad 0.1 mrad -3 mm -1 mm
D Llor L2 L
500 mm 5 mm 2 mm
D(ff'raction methods
335
After the mirror, the beam is monochromated with a parallel crystal monochromator. The double crystal design allows the wavelength to be tuned without displacing the beam on the sample. The monochromator usually consists of two parallel Si(1 1 1) or Ge(1 1 1) crystals. From Bragg's law a specific wavelength will be diffracted into an angle 0. In Fig. 7.42 the angle is set by the location of the entrance and exit slits. Rotating the crystals selects different wavelengths with a resolution A)~/)~-- 10-3. Also from Bragg's law, the monochromator will pass L/3, )~/4 etc. These wavelengths are filtered out in two ways. The higher harmonics can be removed by pulse height discrimination of the detector output, since the pulse height is proportional to the photon energy. In some beam line designs the focusing mirror is placed after the monochromator. In this geometry most of the higher harmonics can be reduced by scattering from the Au coated focusing mirror. This is done by taking advantage of the effect of the index of refraction on the photon wavelength. The critical angle, below which the incident wave is totally reflected is given by Eq. (7.89), and is approximately
13 = cos-'(n) =
)'2e2F(-------o---0) ~: X, 21r,mc2V
(7.92)
If the incident angle at the Au mirror is set slightly below the critical angle for the longest wavelength passed by the monochromator, the shorter wavelength harmonics will be significantly attenuated. The problem of having the monochromator as the first optical element in the beam is the high heat load on the monochromator crystals. The thermal expansion of the top layers of the crystal can cause a distortion in the crystal planes that will significantly worsen the Darwin rocking curve widths (the Darwin width of a crystal is the broadening due to the finite attenuation of the beam, which limits the number of scatters contributing to the diffracted beam (Warren, 1990). Furthermore, the large monochromator to sample distance means that small drifts in the beam exiting the monochromator (due to thermal or mechanical drifts) will produce a large change in the position of the beam at the sample. For these reasons it is best to have the monochromator as close to the sample as possible. After the monochromator, the beam enters the experimental area. All of the flight paths from the monochromator to the chamber and from the chamber to the detector are evacuated to reduce attenuation. Because the incident flux is a function of both the current in the synchrotron ring and mechanical drifts in the beam line, it must be monitored. This is accomplished by placing a thin Kapton sheet in the incident beam. A small fraction of the X-rays in the incident beam are scattered by the plastic perpendicular to the beam and are detected in a scintillation counter. By counting the X-rays diffracted from the sample for a fixed number of monitor counts, instead of a fixed time, an absolute scattering cross section can be measured. 7.4.2.3. Integrated intensities Surface X-ray diffraction's power comes from its ability to make accurate compari-
336
E. Conrad
sons of experimental data to the kinematic scattering model. To achieve this end, peak intensity measurements are not enough to extract structural information. This is because the finite resolution of the instrument and the finite extent of the crystal, due to sample mosaics, cause the CTRs to be broadened. Accurate structure determination requires measurements of the integrated peak intensities, which account for these effects. The integrated intensities are determined by scanning the detector slit across the CTR. The common mode of data collection is by an o-scan (see Fig. 7.43). In this scan mode the crystal is rotated about the surface normal through an angle co at a constant scan rate of f2 (deg/s). The angle 0 = 1/2(20) so that the Z axis is along the Q direction and the incident and exit angles are equal. Defining A~ as the acceptance angle of the detector parallel to the surface plane and Ay as its acceptance angle normal to the surface, the integrated intensity in units of total number of photons is
li.t- J"p,~,k dtr .~, l(Q)dA=f o.~,k--~ do3 ~ ~ l(Q)OZ d v d y
(7.93)
The second integral is over the entrance aperture of the detector. The acceptance angles of the detector, A~ and Ay, are shown in Fig. 7.44
diffraction rod I
Qz
kf
(b)
Qx
Fig. 7.43. Real (a) and reciprocal (b) space picture of the scattering geometry for an to scan. The incident and outgoing angles are labeled 13and 13'and remain constant throughoutthe scan (Vlieg, 1988).
337
Diffraction methods
9
,
." A
!
kd~
Qudo~
/ (a)
.
.
.
.
.
.
(b)
.
(c) Fig. 7.44. (a) Projected top view of the scattering plane onto the crystal surface. (b) A detailed drawing of the acceptance angle of the detector in the surface plane. (c) Cross-sectional view along the line AA' (Vlieg, 1988). The intensity collected over the projected detector area is given by an integration over the area element in reciprocal space of dA that can be written as dA = kQi I cos(a;l)do~dt ~
(7.94)
The differential area in reciprocal space can also be written as dA = a l a 2 d Q l d Q 2 / ( a I • a2). Equating this with Eq. (7.94) gives (Vlieg, 1988) do~d~ --
a I a2 FL,,r k 2 A 9 dQ, dQ,
(7.95a)
where k FL'" = Qll c~
(7.95b)
FLo,. is the L o r e n t z f a c t o r and A* = (a~ • While FLor is a complicated function of the scattering angles, it can be calculated in the symmetric 03-scan mode where the incident and scattered angles are equal; Eq. (7.90). In this mode FLor = cosl3/[coszsin20]. Using this result for the Lorentz factor and substituting Eq. (7.85) for I(Q)
338
E. Conrad
into Eq. (7.93) gives e ) lint -- I o m2c4 P
COS[~
~2
~N~N2
cos% sin20 ~A*
~ IF(Q)I z dT. det
(7.96)
Assuming a large sample and small illuminated area, the number of unit cells exposed is N~N2 = AJA*. If F(Q) does not vary appreciably over the detector slit, Eq. (7.96) becomes (Robinson, 1991 ) e ) COS~ ~2 lint-" lo m2c4 P cos% sin202 ~A .2 L~L2 IF(Q)I2 Ay.
(7.97)
Remember that Ay is the acceptance angle of the detector measured normal to the
surface plane. For an in-plane scattering geometry it is simply the vertical acceptance of the detector, A'/= L/D. For out-of-plane measurements A7 is a function of the 0, 20, and )(;, and must be carefully evaluated before intensity measurements can be accurately interpreted. The counting rate in a typical synchrotron experiment can now be estimated. Assuming that the angular width of the diffraction rod is determined by the sample mosaic, Am, and not the resolution function, the peak counting rate would be
o
1,.~,,~ - li,,, --
A03
= 1,,
/e 4 ) m 2C4
P
cos[~
~
COSX2 sin202 A0~A"2 LIL2 IF(Q)I2 A T
(7.98)
Using typical values for a monolayer of gold (Robinson, 1991): IFI 2 ,:,,: Z 2 = 6,241, LIL 2 = 4 m m 2
A* = (2.9) 2 ~
= 7 . 3 , 2 , Am = 9 mrad
and taking typical values for the beam line constants from Table 7.3 with X = 1.7 A, the count rate would be >7x105 photons/s. For a Si (Z = 14) monolayer the signal would be lower by a factor of 32.
7.4.2.4. Resolution function As mentioned in w 7.1.4, the background in a surface scattering experiment has a major contribution from the bulk thermal diffuse scattering. This background can be reduced by reducing the volume of reciprocal space sampled by the detector. As long as the detector volume is large enough to accept the entire surface peak, the measured peak signal will be constant; only the background will go down. It is for this reason that single crystal analyzers are used in front of the detector. The diffracted beam from the sample is incident on the analyzer crystal and diffracted into the detector through the entrance slit, which defines the Bragg angle. Because
Diffraction method~
339
the vertical divergence of the synchrotron is less than the Darwin width of the analyzer crystal, the acceptance angle of the analyzer-detector pair will be limited by the Darwin width. Of course the overall count rate goes down because of scattering from the analyzer crystal, even though the signal to background rate goes up. If the surface perfection is poor, the defect surface scattering will broaden the diffracted beam to the extent that the analyzer crystal will no longer give an advantage. The angular uncertainty in the incident and detected beam are limited by the rocking curve width of the monochromator and analyzer crystal, respectively. Typical values are given in Table 7.3, Ac~i = Ac~f = 0.008". From Eqs. (7.61) and (7.62) a typical X-ray resolution function has values of (Robinson, 1991)
AQT= 10-3 sin [1 (20)1 ~-1
(7.99)
AQN = 0.02 ~k-' Using these values, the transfer width is larger than 6,000 ~. AQN is often purposefully degraded by increasing the horizontal detector slit width. This increases the detected signal and does not significantly effect the in-plane resolution as long as Z is small. For high-resolution diffraction data the CTRs are scanned by sweeping the momentum transfer vector perpendicular to Q, because the resolution function is narrowest in the AQT direction (see Fig. 7.12). This type of scan is known as a transverse scan. Lower resolution data is taken in a radial scan mode. This is done by increasing the magnitude of Q.
7.4.2.5. Surface cr~'stallography At this point the structure of the surface unit cell can be determined from the integrated intensities of an o3 scan (Buerger, 1960; Lipson and Cochran, 1966; Stout and Jensen, 1968). Equation (7.98) gives a direct relationship between the crystal structure factor and the more easily measurable integrated intensity. For now the analysis is restricted to superlattice rods to exclude bulk effects. First, the scattered amplitude is written as (Robinson, 1983; Feidenhans'l, 1989) A(Q)- ~
p(r)e iQ' dr.
(7.100)
crystal
It can also be rewritten as (see Eq. (7.9)) oo
A(Q) = F(Q) Z ei~
(2rt)2 ,~_., 8(Q - Ghk) F(Q) ata2 h,k
(7.101)
340
E. Conrad
where Ghk is again the surface reciprocal lattice vector and R i is the position vector parallel to the surface. Using this definition for A(Q) in Eq. (7.100) and the atomic position vector (Eq. (7.8)), the electron density is found from the Fourier transformation. p ( r ) - _______~_l ~ e_;2~,hx+k:,, I F~k(Qz)e-~e~z dz (2rt)ala2 hk
(7.102)
where Fhk(Oz) is
Fhk(Qz)- ~__,fm ei2'~(%h+~"k)eiQ~w"'z
(7.103)
m=(I
To determine the structure it is necessary to extract Fhk(Qz) from the diffraction data. Once this is done, the problem of inverting the surface structure from the diffraction is broken into several steps. Since the size of the unit cell is not known (either in the number of atoms in a plane or the number of surface planes involved), it is much easier to determine the electron density projected onto a 2-d plane,
p,,(x,y). In grazing incidents X-ray scattering the momentum transfer perpendicular to the surface is almost zero, so to a very good approximation Fhk = Fhk(Q~ = 0). It is more convenient to write Fhk as a product of a real and imaginary term: IFhklexp[iO~hk]. Then the projection of the electron density onto the surface plane is given by p,,(x,y) ~ Z hk
F',k e-; z,~,,1.,-+~-,., = Z
IF,,kl cosl2rt(hx + ky) - o~,,k l.
(7.104)
hk
If the structure factor were known, the projected electron density could be calculated. Unfortunately, only the amplitude of Ft,~ is measured (see Eq. (7.97) for the integrated intensity). One solution to the phase problem is to make a self consistent guess for c~,k using a technique called a difference map. The start of the iteration is to assume a model structure for the surface. IFhklm~ and t~hk .... dc~ for the model are then calculated from Eq. (7.103). IFhkl is determined from the integrated intensity given by Eq. (7.97). If the model is sufficiently close to the real structure, the approximation that O~m~)del .exp ~,k ---t~j,~ can be used. The difference between the real charge density and the model charge density is then approximately given by _ model p,,(u v) -- Z (IF~h~p[ -I"m"aCll) [cos (2rt(hu + kv)) + ~Xhk ]9 ~J hk
(7.105)
hk
This difference map is plotted as a function of u and v. A positive peak in the difference map indicates that the assumed model did not put an atom at that
341
Diffraction methods
position. From this map a new model structure is refined accordingly and a new guess for the phase is calculated. This process is repeated until the X;2 value,
~Z2= N----7-Py_~
,, ~k
hk
,
(7.106)
(YAk
between the model structure factor and the experimental value is minimized. N is the number of rods measured, and Ohk is the error in the measurement of Fhk. P is the number of free parameters in the model (e.g., atomic coordinates, D e b y e - W a l l e r factors, etc). At this point only the 2-d projected structure of the surface is known (at least to the uncertainty in the measurement). The perpendicular positions of the atoms still remain to be found. To complete the analysis, a comparison with Fhk(Qz) must be made. For the superlattice rods the measured rod profiles can be compared directly to Fhk(Qz)m~ and out-of-plane corrections can be made to refine the model structure. Integer order rod profiles can also be added to the analysis to form a complete data set. The complete structure factor, Eq. (7.86), is used. Since the bulk structure is known, the second half of Eq. (7.86) is already determined (including its phase) and Fhk(Q~) will be M
Fhk(Q,,)- ~,fm ei 2={,,,,h+ ~,k,)eiOW,z m = ()
(7.107) m h
l l _ e i C ) : , -. Z
fi
e i 2rt~,k + v,k) e i Q w.z ~ '
An example of a fit to a rod scan is shown in Fig. 7.38. Because the bulk atomic positions are known, rod intensities lower than the bulk CTR imply that the phase of the surface structure factor must be 180 ~ with respect to the bulk lattice. This provides a starting point to adjust the vertical layer spacing that can be further refined by a least squares method. Refinements to the surface structural model can be made by including thermal vibration as discussed in w 7.1.4. Another important consideration in determining the structure is partial occupancy or substitutional occupancy of an adatom site (Robinson et al., 1986; Pedersen, 1988). For a more complete review of surface crystallographic methods see Robinson ( 1991 ). Unlike in LEED, the accuracy with which a structural model can be determined in X-ray scattering is set by the quality of the diffraction data. For this reason R factors are less often presented in the literature. Instead error bars on individual inter-atomic spacings are derived from ~2 values used in the minimization routine discussed above (to first order, Ad -- {~d/~xZ}Ag2). The error in the integrated intensity measurement of an individual reflection, CYhk,comes from several sources.
E. Conrad
342
First, the standard counting noise is given by the normal square-root noise in the integrated peak plus background counts. Systematic errors such as sample misalignment, surface contamination, surface preparation, etc. are more difficult to evaluate. An analysis method based on symmetry equivalent rods has been devised by Robinson to handle these errors (Robinson, 1991). In the best of cases absolute errors of Ad/d-- 0.5% have been reported. In most systems, however, this level of accuracy does not apply to all of the free atomic coordinates used in the model (e.g., horizontal positions are usually more accurate than vertical spacing).
7.4.2.6. Reflectivity Another important X-ray technique for determining vertical structural information at a surface or interface is X-ray reflectivity. The geometry for X-ray reflectivity is identical to most LEED geometries shown in Fig. 7.34b. The sample normal is in the scattering plane, and the incident angle is scanned The magnitude of the reflectivity is related to the electron density profile 9(z), and is given by (Braslau et al., 1985)
9~(Q) 9~v(Q)
(--~SI0(p(z))0z
ei o z d z
(7.108)
where (9) is the average bulk electron density and 9~r: is the Fresnel reflectivity (Born and Wolf, 1975). For 13 >> 13~(where [3c is the angle tbr total reflection given by Eq. (7.92)), the Fresnel reflectivity is 4
(7.109a) with Qr = 2k sinl3c.
(7.109b)
Equations (7.108a and b) state that the reflectivity is the Fourier transform of the electron density gradient. Because of this, the technique has several applications. First, it can be used as an accurate check or refinement to structural models. After a crystallographic analysis of a surface structure (outlined in the last section), reflectivity data can be used to refine the vertical positions of the surface atoms. This is done by calculating an average electron density due to the proposed model structure and comparing it to the reflectivity data. A least squared fit is then used to modify the vertical positions of the surface atoms. X-ray reflectivity can also be used to measure the sharpness of the solid-vacuum or solid-solid interface. An example of this type of analysis is the work done on the surface melting transition of Pb(110) (Pluis et al., 1989). Because the molten layer density is less than the solid surface density, a gradient exists that substantially changes the reflectivity from the Fresnel value as shown in Fig. 7.45.
D~ff'raction methods
343
"1 ....
~.o
I ....
-,-*
I ....
I ....
I ....
I'"~
p-~ ~- --*- L ~
1.0
(Q) 0.5
0.5
1.0
"-,."
(
rY o
1.0
0.5
0.5
" ~ +
o.s 0
0
,,, .... , .... , .... , .... , .... ,..., 1
2
3
4
Q/Q~
5
6
0.5 7
Fig. 7.45. ~(Q)/9~F(Q)vs Q/Q~for Pb(110) at different temperatures: (a) 300 K, (b) 581.2 K, (c) 592.8 K, (d) 599.8 K, (e) 600.50 K. The bulk melting point is 600.7 K. Solid curves are best fit to the data using Eq. (7.108) assuming a thin liquid layer. Dashed curves are fits to Eq. (7.108) assuming the surface is dry and rough (Pluis et al., 1989).
7.4.2.7. Summary of advantages and disadvantages of surface X-ray scattering With the advent of bright synchrotron sources, surface X-ray scattering has become an important tool in surface physics. Its main advantage is the direct interpretation of diffraction data through the kinematic scattering model. The inversion problem is only limited by the quality of the data, and is not subject to further errors due to the treatment of the theoretical scattering model. The other important advantage of X-rays is the high Q-resolution of a synchrotron source. This makes X-ray scattering ideal for studying problems in 2-d phase transitions, defects, and other systems where long range order or finite size effects are important. A disadvantage of surface X-ray scattering is its low scattering cross section. This has currently limited studies to adsorbate systems that have a relatively high Z. On the other hand, its low sensitivity to low Z adsorbates can also be an advantage. X-ray diffraction intensities are much less sensitive to small amounts of
344
E. Conrad
contaminates, which can plague LEED or helium atom scattering data. Therefore, the quality of integrated intensity data is better for structural analysis. The large penetration depth of X-rays means that surfaces can be studied in very high background pressures. Surface X-ray scattering can study 2-d adsorbate phases in equilibrium with their 3-d gas to pressures well above atmosphere. Structural studies of growing interfaces can also be carried out under high pressure growth conditions where other diffraction methods simply cannot operate. X-ray's large penetration also allows structural studies of buried interfaces. This is especially true if the overlayer has a different periodicity than the interface. In that case the interface superlattice rods can be studied without interference from the overlayer diffraction. Solid-liquid interfaces such as those found in electrochemical cells can also be studied. The relatively low count rates compared to LEED or atom scattering makes certain experiments more difficult using X-rays, for instance, in studies of highly reactive surfaces where contamination times are short. Kinetics studies using X-rays have been limited to the bulk or special high Z adsorbate systems. The planned X-ray sources now under construction, however, should allow surface kinetics studies to open up as a new area of research using X-ray scattering. 7. 4.3. Atom diffraction
The inert properties of the noble gases have been used to make a purely surface sensitive diffraction probe. Noble gas atoms interact with the surface through a van der Waals type potential. Far from the surface there is an attractive interaction, while near the surface their closed shell electronic configuration produces a large repulsive potential between their 2s electrons and the electrons in the sample surface. Typically the repulsive interaction causes the atoms to be turned around about 3 A from the ion cores of the surface atoms. For this reason noble gas atom scattering is only sensitive to the distribution of electrons at the surface. Since noble gas atoms interact with the surface electron charge distribution, the scattering cross section is independent of atomic number. Atom scattering, therefore, has the advantage of being able to detect adsorbed hydrogen. As a consequence of their strong interaction with the surface electron distribution, atom scattering data is difficult to interpret. The first reason for this is because the Born approximation is strictly not valid. Not only is the scattering potential comparable to the incident energy of the atom (-63 meV at room temperature), the amount of time the atom spends in the potential is comparable with a surface atom vibrational period (this is because of the long range nature of the van der Waals potential). A complete description of the scattering problem requires a full time dependent solution of the Schr6dinger equation. These full scattering calculations are referred to as coupled channel techniques (Liebsch and Harris, 1982; Garcia et al., 1983). The second problem in interpreting the data is to extract the ion core positions. As will be shown below, the diffraction data can be inverted to give the electron charge corrugation 2-3 IX above the surface. In order to relate the measured charge
345
Diffraction methods
corrugation to the ion core position a structural model must be proposed. Once this is done, a full self-consistent electronic calculation of the electron density for this model is made to compare with the measured charge corrugation. This last step is extremely difficult especially when it is noted that the surface charge density at the point where a helium atom turns around after a collision with the surface is extremely low, -- 104 au -3 (Haneman and Haydock, 1982). Nevertheless, the information in an atom diffraction measurement can be very valuable. To understand what information is obtained in a helium scattering measurement, the scattering problem must be discussed. Rather than begin with a full QM treatment, a more intuitive and instructive approach is to present an approximation more in line with the discussion in w 7.1.1. This model is known as the hard wall approximation (Engel and Rieder, 1982). 7. 4.3.1. The Eikonal approximation to the hard wall model The electron charge distribution at the surface is a smoothly varying function. Since the repulsive part of the potential develops rapidly over a short distance perpendicular to the surface, the potential can be approximated by an infinite hard wall as shown in Fig. 7.46. The position of this wall relative to the surface ion cores defines the classical turning point and is described by the corrugation function z = ~(R), where R is a position vector in the surface plane. The hard wall potential is oo z < ; ( R )
v(R,z) =
t
(7.1 lO)
0 z > ;(R)
For the time being it will be assumed that the kinematic scattering approximation is valid. Then, if the charge surface at the helium turn around point is thought of as an infinite collection of point scatters, the amplitude scattered from a single surface unit cell can be written using Eq. (7.6)
V(z)
o0
Rt ~ l ~ - z 9
Ion c~ ~ Classical Turning Point
9
9
p(R) o~ ~(z)
@ @
/ vl
!
09 I9 Qi~ @ Q
(a)
(b)
Fig. 7.46. Hard wall potential for helium-surface scattering.
346
E. Conrad
AEik(Q) = Z ei~Q"6+Q~;~RJ))~ A*-' [____. j
e i~~176
dR.
(7.111)
Unit cell
where A* is the area of the surface unit cell and f(E,0) has been set equal to 1.0. The scattering amplitude written this way represents a single-scattering picture of the helium-surface interaction and is referred to as the Eikonal a p p r o x i m a t i o n (Garibaldi et al., 1976). As an example, consider a 1-d surface described by the corrugation function ~(R) = (c/2)cos(2rcx/a). Substituting this expression into Eq. (7.1 1 1) and changing variables to 0 = 2rcx/a gives 2~
a A zik(2~n/a, Qz) - 2~:A* " e-; ,0 +7 Q~r
dO '
(7.112)
0
where the amplitude has been calculated at a surface reciprocal lattice vector, Qll = G = 2nrta. This integral is a Bessel function of order n
,Eik2n,a
/
/
2
(7.113)
Unlike LEED or X-ray scattering, the scattered intensity is not a periodic function of Q~. This simply reflects the fact that only a single layer is being probed. The amplitude of the corrugation, c/2, is found from the minimums in I(Q~), which are given by the zeros of the Bessel function. The Eikonal approximation has serious drawbacks, the most important coming from neglecting multiple scattering. The reason multiple scattering plays such an important role in atom scattering is obvious. At more grazing angles the hard wall potential shadows part of the incoming or outgoing beams. Simple ray tracing of a trajectory as in Fig. 7.47 shows that double scattering is important, especially for large corrugation amplitudes. Helium \ Atom B e a m ~
O
Ion Core
Fig. 7.47. Multiple scattering in the hard wall approximation.
Diffraction methods
347
While multiple scattering is difficult to handle in LEED, it is in fact much less complicated in atom scattering. The main reason is that the number of free parameters necessary to model the smooth corrugation is small (2-8 Fourier terms). It, therefore, does not take much time to search through all parameter space using the iterative method described below.
7.4.3.2. Multiple scattering in the hard wall model Because the hard wall potential in Eq. (7.110) is infinite at ~(R), the solution to the Schr6dinger equation (Eq. (7.76)) becomes a boundary value problem. In this limit the wave function of the atom is given by the Rayleigh ansatz (Swendsen and Rieder, 1982) ~ ( R , z ) = exp[i(k~zz +
kiji- R)]
+
~_~A a exp[i(kczz
+ (ki.** + G ) .
R)]
(7.114)
G
where R is measured parallel to the surface and the incoming and outgoing wave vector are k i = (kiz,ki,ll) and kf = (kGz, k~,ll + G), respectively. The G's are again the surface reciprocal lattice vectors, and AG is the amplitude scattered with a momentum transfer G. The hard wall boundary condition is that ~ is zero at ~(R), leading to the solution for the AG'S
-1 = Z AG exp[iG.
R + i(kGz - k~z)~(R )]
(7.115)
G
As a limit to this ansatz, the corrugation amplitude must be less than 14.3% of the unit cell length (Hill and Celli, 1978). If the corrugation exceeds this limit, a consistent set of Ac's cannot be found that satisfy the normalization condition given be Eq. (7.115). The scattering amplitude, AG'S, are found by multiplying both sides of Eq. (7.115) by the complex conjugate of the exponential and integrating over the surface unit cell. This gives A Eik Aa = "'a + Z Ma'~ Aa"
(7.116a)
G'
MG.G,=..._A.V[.Unicet, dR exp [i(G' . G). R]{I . .exp [i(kG,z kGz) ~(R)I}
(7.116b)
Equation (7.116) identifies immediately the multiple scattering component of the hard wall approximation. The scattered amplitude is the single scattering amplitude, 'cAEik,plus the sum over all other scattering channels with amplitude, At,,' s. The contribution from each multiple scattering channel is determined by the scattering matrix, Mc.~.
E. Conrad
348
7.4.3.3. Coupled channel techniques When the corrugation is large or when the scattering takes place at low perpendicular momentum transfers, the hard wall model is no longer adequate and a full quantum mechanical treatment must be used. Real gas-surface potentials are not infinite. Higher incident helium energies will sample the charge distribution slightly closer to the ion cores than lower incident energies. For this reason the classical turning point, and thus ~(R), are influenced by the perpendicular component of the incident helium kinetic energy. This means that a data set for I(Q) taken at one incident angle will yield a different ~(R) compared to a set taken at a different incident angle, if the data is analyzed within a hard wall model. Because the real potential is soft, a hard wall model artificially limits the number of multiple scattering channels available. In the hard wall model only diffraction channels with k 2cz > 0 are allowed (referred to as open channels). For a soft wall potential the atom wave function can exist in the classically forbidden region of the potential. This gives rise to diffraction channels with k~ < 0 (closed channels). These additional channels can have a significant contribution to the total multiple scattering component of I(Q). The other drawback to the hard wall potential is that it does not take into account the attractive part of the well. In particular, under the correct energy and momentum conditions it is possible for the incoming helium atom to be temporarily trapped in a bound state of the helium-surface potential well. The coupling of the incoming wave to the scattered wave through these bound state resonances is very large especially at low momentum transfers, and gives rise to large dips and spikes in I(Q) that cannot be accounted for in a hard wall model. A full treatment of a coupled channel calculation is beyond the scope of this chapter. The method, however will be outlined. A 2-d periodic potential can be written as
(7.117)
V(R,z) = ~ vc(z) e i ~ "R G
If V(R) is known, the Fourier coefficients, vc, can be found from the inverse Fourier transform of the potential. Likewise the wave function is written as ~13(R,z) = ~ ~ol~(z) exp[i(Q + G). R ]
(7.118)
G
Substituting Eqs. (7.117) and (7.118) into the Schr6dinger equation (Eq. (7.76)) and integrating over the unit cell gives the set of coupled second order differential equations 2m
[k2 -- (Q + G)2] +
t~t:;l~(Z) = E VG'6'(Z) t~g'fI(z) G'
The boundary conditions for the soft potential are now
(7.119)
349
Diffraction methods
0,
when z --->+ oo and
v(R,z) =
k2 <
0
(7.120) exp[i(ki z z
+ ki,ll " R ) ]
+ ~ Ac exp [i(kczz + (ki., + G) 9R)], when z ~ oo and k 2 > 0 G
Here ~ ( R , z ) is a linear combination of the ~ ( R , z ) ' s . Equations (7.119) and (7.120) give a set of equations to solve for the scattered amplitudes, Ac's. Numerical methods have been developed to solve these equations and invert the diffraction data to give the charge corrugation (Wolken, 1973; Liebsch and Harris, 1982). The inversion problem, using the coupled channel method, is long and tedious. Usually the structure is first determined by using a hard wall potential as is outlined in the next section. 7.4.3.4. The inversion problem in the hard wall approximation To determine the corrugation amplitude, the scattered intensities must be related to the A~;'s given by the hard wall model, Eq. (7.116). Once again the phase problem prevents this from being done uniquely. Several methods have been proposed to invert the data (Rieder et al., 1981; Rieder, 1982; James et al., 1983: Kaufman et al., 1984). Before discussing these, however, it is important to point out that the measured scattered intensities must be adjusted for a number of experimental reasons. First, the experimental intensities must be normalized in order to conserve the number of helium atoms incident and scattered from the surface. This is clone by multiplying IAcl2 by the ratio of the normal component of the outgoing and incoming momentum. The scattered intensity is then related to the amplitudes by l~xP _
IAcl2
kGz ki z
(7.121)
A slightly Joetter approximation to both the Eikonal and hard wall approximations is to include the attractive part of the helium-surface potential. In general the well depth in Fig. 7.46 is about 4-10 meV. This is a sizable fraction of the incident helium beam kinetic energy (-63 meV for a room temperature source). This means that, like electrons, helium atoms refract just before the surface. The attractive well is included by assuming a spatially uniform square well of depth U. In the helium scattering literature this correction is know as the Beeby approximation (Beeby, 1971). The perpendicular momentum at the classical turning point is then given by Eq. (7.72). The depth of the well can be fairly accurately estimated by using helium's resonant scattering phenomena described in w 7.4.3.5. The inversion problem is made difficult because Eqs. (7.116) are a set of transcendental coupled equations. Swendsen and Rieder use an iterative method assuming that multiple scattering is small (Swendsen and Rieder, 1982). They start by making an initial guess for the corrugation. If single scattering dominates, the second term in Eq. (7.116a) should be small. In that case the Ac,'s can, to a good
E. Conrad
350
approximation, be written as ,zlEik , c ' . Using this as the first guess, a new set ofA6~ s are calculated and used as a next guess. After n iterations of this procedure the amplitude scattered for the (n+ 1)th guess is A c("§~) = A cEik + ~_, Ma.a" Aa(n)
(7.122)
G'
This procedure converges quite rapidly since the scattering matrix needs only to be calculated once for each trial structure. A least squares minimization is used to modify the structure so that the experimental and calculated Ac's agree. Kaufman et al. have developed a technique similar to that described in w 7.4.2.5 for X-rays (Kaufman et al., 1984). Basically, they begin by assuming a model structure and solve for the A c' s in the same way as used in Eq. (7.122). Rather than do a least squares fit to refine the model structure, a new model structure is calculated in the following way. A new structure is guessed using Eq. (7.115). This is done by substituting the magnitudes of the experimental Ac's along with the phases from the previously guessed model into Eq. (7.115) to generate ~(R). Using this model, a new set of Ac's are generated from Eqs. (7.122) and (7.116b). The procedure is repeated until the experimental Ac,' s and the calculated ones converge. The inversion of the diffraction data by both techniques is facilitated by the fact that only a small number of parameters in the Fourier expansion of the charge corrugation are necessary to adequately describe the surface. This means that almost all of parameter space can easily be searched to minimize the error between the calculated intensities and the experimental data.
!
Fig. 7.48. Classical rainbow scattering from a sinusoidal corrugation, ~(x) = (~m/2)cos(2rtx/a). Dashed lines are the rainbow angle + 50.4~for the values shown. Data is plotted relative to the specular direction.
Diffraction methods
351
The iteration procedure can be substantially speeded up by an appropriate first guess for the corrugation amplitude. This is done by introducing the concept of rainbow scattering. Consider the corrugation shown in Fig. 7.48. For an incident beam of helium atoms consisting of parallel trajectories, the classical scattered intensity into a given angle dO is proportional to the range of impact parameters db that can scatter into that direction, i.e., I(0) ,,~ db/dO. It is simple to show that I(0) is bounded by two angles 0 = 0i + 2 tan-' (/)~(xX)~.,x
(7.123)
where 0~ is the incident angle relative to the average surface normal and a r e points of maximum slope in the surface corrugation. The maximum scattered angles are called the rainbow angles. Furthermore I(0) is a maximum at these two angles. A simple example of rainbow scattering would be light scattering from a periodic triangular wave surface, where only two scattered directions are allowed. For a sinusoidal corrugation ~(x) = (~m/2)cos(2rtx/a), the rainbow angle measured from the specular scattering angle is given by Eq. (7.123) (Engel and Rieder, 1982) (O~(X)/~X)max ,
0: tan
/
l 4,
An example of rainbow scattering can be seen in Fig. 7.49. The measured rainbow angles provide a useful first guess for the amplitude of the first Fourier component of the corrugation function. Note that this classical approach predicts a minimum intensity at the specular direction. For helium atom scattering the wavelength of the atom is comparable to the corrugation repeat length. Therefore, specular scattering from a corrugated surface is an interference effect! While it is possible to calculate the corrugation amplitude from the diffraction data, the position of the atoms in the top layer is by no means solved. A complete solution requires a method to calculate the scattering potential (and thus the position of the repulsive wall) due to the electron charge density p(r) from the surface atoms. This in turn requires a self consistent solution of the Schr6dinger equation for the surface atoms, which is far from trivial (Haneman, 1981). Even though the electron densities are low (~ 10-4/au 3) near the helium turning point, a simple superposition of atomic orbitals can give a reasonable estimate of the corrugation amplitude. These estimates, however, have error bars of 10% in metals and are significantly higher in semiconductors (Haneman and Haydock, 1982). Another limit to the inversion problem is the effect of thermal motion. Because the helium atom is large, the incoming atom interacts with more than one surface atom. Also, the long-range He-surface potential means that the collision time can be comparable to a typical period for a thermal vibration. These effects mean that the frozen lattice approximation used to produce the thermal average of Eq. (7.24) is much less valid for helium-surface scattering (Levi and Suhl, 1979). Theoretically,
352
E. Conrad (0 0)
He-Li F (001) (110)
(2,0)
(-2,0)
al
(-1,0) .001 o
>. I-.
m
(3,01
A. I Jt i
,-4.o, I
0o
r
30 ~
60 ~
m z
Z.
=.
,-2,-2)
( 0,0 )
(100)
I
(2.2)
.001 (-3;3)
1
~ - 60 o
l
.i,
_1 •
- 30 ~
'~ 0o
A 30 ~
60 o
S C A T T E R I N G A N G L E Of
Fig. 7.49. Example of rainbow scattering of helium from LiF(001). The incident beam direction is in the: (a) [ 110l and (b) [ 100] azimuths. The incident beam is normal to the surface in both cases. In (a) the rainbow angle is + 22~ in (b) it is + 35 ~ The hatched areas represent the instrument resolution. Taken from Boato et al. (1976). a simple D e b y e - W a l l e r analysis is t e n u o u s at best. E x p e r i m e n t a l l y , h o w e v e r , it has been f o u n d that at low t e m p e r a t u r e s and small m o m e n t u m transfers (TQ 2 < 1x10 -5 K]k-2), a simple D e b y e - W a l l e r factor can be used to a c c o u n t for thermal m o t i o n in atom scattering ( C o n r a d , 1987).
7. 4.3.5. Resonant scattering The low kinetic energies o f thermal helium atoms also m a k e possible m e a s u r e m e n t s o f the shape o f the h e l i u m - s u r f a c e interaction potential. W h e n the normal c o m p o nent o f the helium atom energy is c o m p a r a b l e to the attractive well depth, it is possible for the atom to be trapped in a b o u n d state o f the h e l i u m - s u r f a c e potential. This process is k n o w n as selective adsorption. F r o m c o n s e r v a t i o n of e n e r g y k~ :
kff :
(kil I -t- O ) 2 -t" k Gz 2
(7.125)
Diffraction methods
353
The normal component of the scattered helium atom energy is defined as h2 h2 Ec" - 2m k~z = -~m [k~ k
-
(7.126)
(ki, , + G)21
J
Ecz is always greater than zero for scattering into one of the diffraction rods. Near the surface it is possible for the helium atom to pick up kinetic energy at the expense of potential energy and make a transition to a bound state level, v, with energy ev = Ec, z, where ev is less than zero. If the bound state energy satisfies Eqs. (7.125) and (7.126) for some particular G (this G must lie outside the Ewald sphere), the incoming helium atom is in resonance with the bound state making it likely that the transition will occur. The helium atom no longer has a vertical m o m e n t u m in this state but instead travels along the surface with a m o m e n t u m k i II § G and has a total energy h2 E,,,, - ~ k~ +
ISvl
(7.127)
The resonance condition can be seen graphically by rewriting Eq. (7.126) as (ki,' + G) 2 - -2m ~ [E~ - Is~l]
(7.128)
where E~ is the energy of the incident helium atom. Equation (7.128) shows that the resonance condition is a circle of radius k~e, = [ (2m/h 2) ( E l - [13vl)]1/2. When this circle intersects a reciprocal lattice rod, a resonance scattering condition occurs (see Fig. 7.50). The resonance condition means that intensity can be removed (or added) from a particular diffraction channel to a new diffraction channel by coupling through a reciprocal lattice vector G that satisfies Eq. (7.128).
9 (o~)! G
k\O
(olO)........o
]'
9
Fig. 7.50. Graphic representation of the resonance condition for selective adsorption. The sample has been rotated about the surface normal until the (11) diffraction beam is in resonance with the (00) beam. From Hoinkes (1980).
354
E. Conrad !
!
I
I
!
I
I
D~-..-- NoF (001)
=- ~ V
E, : 53.2meV
En : -t6meV
TSF: 2&OK
[1,1) 14,-I) Glrl)(lol)
(-1,-2!10,-4)
,d~,,~-o ~.41 q,j)
io,4!,(4,.a..
'~9 70 4
11 11.
~ 70- - 14.-II
(I,l
I1.11 (0,41
.
.
~--
(O,rll(.l, 2)
I1.11
10-11!0,11 (0,4!10,11
tAI ;'
10,41
10.-11
( I1
~ 10.-11
0
,
l
H,~.(o,~)...,-
(.I,2110,1 14
10.11
11.-I1
,o.,I~,,,~,,o,, J
'
........
9 .~
-20*
!
:55"
~,. .j
I
-t,5 ~ - t,0*
0,11
10.11
t,~-
.-..
~ - ~ .... ~
"
h:'
10.r
,o,.,,,,.,, ,o:,
90- -
10,1 )(.I,2) _
(o,41~" 1o,.~)!o,41 (o,~)
~,~,.
'~176
I
F.~ = .O,3meV
-I0"
a
-
10" azimuthal angle u
Fig. 7.51. Example of resonance structure in atomic hydrogen scattering from NaF(001 ). Plots show the specular intensity vs azimuthal angle for different incident energies. Bound states resonances with G = (0 + la*) are shown. From Finzel et al. (1975).
Diffraction methods
355
In the diffraction data the resonance conditions are seen as either dips or peaks in the rod scans as shown in Fig. 7.51. The bound state energies are found by changing the crystal azimuth, which in Fig. 7.50 corresponds to a rotation of the reciprocal lattice about the (00) rod. As this is done different G vectors will come into resonance. The dips in I vs ~ for fixed E i and 0i can be correlated to the bound state energies through Eq. (7.126). This is done by plotting I(~) for either different incident energies or different incident angles. The A~ shift of the resonance peak as E~ or 0~ is changed allows the G vector responsible for the resonance to be identified. The set of bound state energies can then be used to construct the scattering potential. Starting from an assumed form for the atom-surface interaction (van der Waals, Morse, etc.) the potential parameters are calculated by comparing the predicted energy levels for the model potential to those derived from the experiment (Hoinkes, 1980).
7.4.3.6. Helium diffraction equipment A helium diffractometer only differs from diffractometers used in other diffraction techniques in the source and the detector. The source must produce a well collimated and highly monochromic beam. The detector must be able to measure the scattered beam without being too sensitive to the substantial background helium gas. The helium source is a supersonic nozzle shown in Fig. 7.52. Helium gas at a pressure p,, and temperature To is allowed to expand into vacuum through a pinhole of diameter d. The shock wave produced by the supersonic flow is "skimmed" off by a sharp skimmer nozzle placed in front of the source, preventing turbulence in the beam. The rapid expansion produces a collision volume that extends several nozzle diameters out past the source. Slow atoms are hit by faster atoms causing them to pick up energy in the direction of flow or otherwise be scattered out of the beam direction. Faster atoms also lose energy in the direction of flow or are scattered out of the beam. This supersonic flow process produces a monochromatic beam. The energy spread in the beam and the helium beam effective temperature are determined by the speed ratio, S (Toennies and Winkelmann, 1977):
1 t PoTo r To Helium d_~_ Bottle T
~ ~
~
-
2nd Aperture
z Beam Helium
rture i / ~ Ape To Pump
Fig. 7.52. Expanded view of a helium nozzle source.
E. Conrad
356
Table 7.4 Typical helium atom scattering parameters for a surface diffraction experiment Source Incident flux (I o) Energy resolution AE/E
1.0• atoms/sr s 0.5-2.0%
Beam Parameters Divergence A[3 Size at sample (diameter)
0.5-3.0 mrad 0.5-1.0 mm
Detector Sensitivity Sample-detector distance Signal-to-noise ratio
10-16 A cm3/atom 20-25 cm 105 • Io
Spatial coherence of system
200-1000
Av 1.65 _-- ~ v S
(7.129)
T 2.5 T,, - S 2
(7.130)
The speed ratio is related to the initial pressure and nozzle diameter; S o~ (pod)O where 13 -- 0.5. Speed ratios approaching 350 have been reported using d = 0.03 mm and Po = 1400 atm ( C a m p a r g u e et al., 1977). Velocity resolutions of less than 0.3% have been achieved (see Table 7.4). The large gas load in the s k i m m e r c h a m b e r requires a high t h r o u g h p u t p u m p i n g system (10,000 to 50,000 L/s, 50 tort L/s). The p u m p i n g of this stage is done with large diffusion pumps and high throughput mechanical pumps to keep the pressure below 10 -5 tort. Higher pressures in this c h a m b e r causes more scattering of the helium beam and therefore degrades the energy resolution. The wavelength of the helium atoms is controlled by heating or cooling the nozzle. At room temperature the helium atoms leaves the nozzle with an energy of about 63 meV. The wave vector of a helium atom is related to its energy by the deBroglie relation, 1/2
\
)
A liquid nitrogen cooled nozzle gives a kinetic energy of about 21 m e V for helium. After the s k i m m e r nozzle, the beam passes through a second aperture and into a second vacuum chamber. A diffusion p u m p or a large turbo p u m p is used to bring the pressure in this stage down to 10 -9 torr. Because 99% of the helium from the source is p u m p e d in the nozzle chamber, the pumps after the second aperture need
Diffractionmethods
357
Pumps Differential
Ill -
[---] Skimmer Helium
I ~[f~~
~') /
]
~J0 J ']
Iil1
@
i Quadrapole
Fig. 7.53. Helium scattering system.
only have a small throughput. The beam then passes through a third aperture into the main UHV scattering chamber (see Fig. 7.53). The second and third aperture define and collimate the helium beam. The scattering chamber is pumped with a turbo pump since other pumps have very low helium pumping speeds. After scattering from the sample, the diffracted beam intensity is measured in a quadrupole mass filter. Because of the diffuse helium scattering, a large helium background builds up in the chamber. To increase the signal to background ratio the quadrupole is doubly differentially pumped. This keeps the helium background in the quadrupole one to two orders of magnitude below the main chamber' s helium partial pressure (5• -9 torr). The turbo pumps on the detector volume are usually backed by a diffusion pump before the mechanical pump. This is because helium saturation of the mechanical pumping fluid can cause backstreaming through the turbo and thus increase the helium background pressure in the detector. To further increase the signal to background ratio the quadrupole uses a crossed beam ionizer instead of a Bayard-Alpert type. This design is used to define the ionization volume to less than a millimeter diameter, which is about the diameter of the diffracted beam. Typical operating parameters for a helium diffraction system are given in Table 7.4.
7.4.3.7. Summary of advantages and disadvantages of atom scattering Helium scattering's main advantage is its contribution to studying the structure of low Z adsorbates. Unlike X-ray and electron diffraction, helium's scattered amplitude does not depend on Z, but instead on the charge density at the surface. This means that helium diffraction is just as sensitive to adsorbed hydrogen overlayers as to any other adsorbate structure. Structural determination of adsorbed hydrogen
358
E. Conrad
are extremely difficult or impossible with either electrons or X-rays. Besides the study of low Z structures, helium scattering is just as easily done on insulating surfaces. L E E D is normally unable to study insulators because of surface charging. Furthermore, problems with stimulated desorption that occur in electron scattering do not exist. A n o t h e r result of h e l i u m ' s interaction with the surface charge density is its high sensitive to adsorbed impurities. The charge from a single adsorbed atom can spread out to many neighboring surface unit cells. This means that the effective scattering cross section can be 1 0 0 - 2 0 0 ~2 ( P o e l s e m a and Cornsa, 1991). Both L E E D and X-ray scattering have significantly lower sensitivity to point defects. Because helium atom diffraction is completely surface sensitive, diffraction from i n c o m m e n s u r a t e overlayers have no contribution from the substrate structure. This has definite advantages in following certain 2-d phase transitions into incommensurate fluid phases. H e l i u m ' s main disadvantage as a diffraction technique is the p r o b l e m of inverting the measured corrugation amplitudes back to the ion core positions. This has kept atom scattering from b e c o m i n g a general quantitative structural tool. This criticism, however, does not imply that atom scattering is without merit. It can be used as a method of estimating structures. These estimates can then be used and refined in other diffraction experiments. Finally, the role of helium scattering as a probe of dynamic surface properties has purposely been ignored in this chapter. Inelastic helium scattering is a powerful tool in measuring, surface phonon spectrums, surface self diffusion coefficients, etc. These types of m e a s u r e m e n t s are discussed in a subsequent volume of this handbook.
References Andres, S.R. and R.A. Cowley, 1985, J. Phys. C 18, 6427. Barnes, R.F., M.G. Lagally and M.B. Webb, 1968, Phys. Rev. 171,627. Bceby, J., 1971, J. Phys. C 4, L359. Boato, G., P. Cantini and L. Mattera, 1976, Surf. Sci. 55, 141. Born, M. and E. Wolf, 1975, Principles of Optics, 5. Pergamon, London. Braslau, A., M. Deutsch, P.S. Pershan, A.H. Weiss, J. Als-Melsen and J. Bohr, 1985, Phys. Rev. Lett. 54, 114. Braslau, A., 1987, SPEC program, PhD dissertation Harvard University. Available from G. Swislow (Certified Scientific Software, Cambridge, MA). Buerger, M.J., 1960, Crystal Structure Analysis. Wiley, New York. Busing, W.R. and H.A. Levy, 1967, Acta Crystallogr. 22, 457. Campargue, R., A. Lebehot and J.C. Lemonnier, 1977, in: Rarefied Gas Dynamics, ed. J.L. Potter. AIAA Publ., New York. Cao, Y. and E.H. Conrad, 1989, Rev. Sci. Instrum. 60, 2642. Chinn, M.D. and S.C. Fain, 1977, J. Vac. Sci. Technol. 14, 314. Conrad, E.H., L.R. Allen, D.L. Blanchard and T. Engel, 1987, Surf. Sci. 184, 227. Cowley, J.M., 1984, Diffraction Physics. Elsevier, Amsterdam.
Diffraction methods
359
Davisson, C.J. and L.H. Germer, 1927, Phys. Rev. 30, 705. Engel, T. and K.H. Rieder, 1982, in: Springer Tracts in Modern Physics, Vol. 91, Structural Studies of Surfaces, ed. G. Hohler. Springer, Berlin. Ertl, G. and J. Kuppers, 1985, Low Energy Electrons and Surface Chemistry. VCH, Weiheim. Feidenhans'l, R., 1989, Surf. Sci. Rep. 10, 105. Finzel, H.-U., H. Frank, H. Hoinkes, M. Luschka, H. Nahr, H. Wilsch and U. Wonka, 1975, Surf. Sci. 49, 577. Fleming, R.M., 1985, Super Diffractometer Control Program, unpublished. Fuoss, P.H. and I.K. Robinson, 1984, Nucl. Inst. Meth. 222, 171. Garcia, N., J.A. Baker and I.M. Batra, 1983, J. Electron Spectros. 30, 137. Garibaldi, V., A.C. Levi, R. Spadicini and G.E. Tommei, 1976, Surf. Sci. 48, 649. Gronwald, K.D., 1982, M. Henzler, Surf. Sci. 117, 180. Guinier, A., 1963, X-ray Diffraction. Freeman, San Francisco, CA. Haneman, D.R., 1981, Phys. Rev. Lett. 46, 1227. Haneman, D. and R. Haydock, 1982, J. Vac. Sci. Technol. 21, 330. Henzler, M., 1978, Surf. Sci. 73, 240. Henzler, M., 1979, in: Electron Spectroscopy for Surface Analysis, ed. H. Ibach. Springer, Berlin. Hill, N.R. and V. Celli, 1978, Phys. Rev. B 17, 2478. Hoinkes, H., 1980, Rev. Mod. Phys. 52, 933. James, R.W., 1962, The Optical Principles of the Diffraction of X-Rays. Bell, London. James, R., R. Kaufman and T. Engel, 1983, Surf. Sci. 133, 305. Jackson, A.G., 199 I, Handbook of Crystallography. Springer-Verlag, New York. Kaufman, R., R. James and T. Engel, 1984, Surf. Sci. 148, 72. Kinniburgh, C.G., 1975, J. Phys. C 8, 2382. Klemperer, O. and M.E. Barnett, 1971, Electron Optics, 3rd edn. Cambridge University Press, Cambridge. Kosterlitz, J.M. and D.J. Thouless, 1973, J. Phys. C 6, 1181; J.M. Kosterlitz, ibid. 7, 1064 (1974). Lagally, M.G. and J.A. Martin, 1983, Rev. Sci. lnstrum. 54, 1273. Lang, E., P. Heilman, G. Hanke, K. Heinz and K. Muller, 1979, Appl. Phys. 19, 287. Lent, C.S. and P.I. Cohen, 1984, Surf. Sci. 139, 121. Levi, A.C. and H. Suhl, 1979, Surf. Sci. 88, 221. Liang, K.S., K.L. D'Amico and J. Russo, 1990, unpublished. Liebsch, A. and J. Harris, 1982, Surf. Sci. 123, 355. Lipson, H. and W. Cochran, 1966, The Determination of Crystal Structure, Cornell Univ. Press, Ithaca, NY. Martin, J.A. and M.G. Lagally, 1983, J. Vac. Sci. Technol. AI, 1210. Maradudin, A.A., E.W. Montrol, G.H. Weiss and I.P. lpatova, 1971, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Suppl. 3. Academic Press, New York). McKinney, J.T., E.R. Jones and M.B. Webb, 1967, Phys. Rev. 160, Suppl. 3, 523. Mochrie, S.G.J., 1988, J. Appl. Crystali. 21, 1. Park. R.L., J.E. Houston and D.G. Schreiner, 1971, Rev. Sci. Instr. 42, 60. Pedersen, J. Skov, 1988, PhD dissertation, University of Copenhagen. Pendry, J.B., 1971, Phys. Rev. Lett. 27, 856. Pluis, B., J.M. Gay, J.W.M. Frenken, S. Gierlotka, J.F. van der Veen, J.E. MacDonald, A.A. Williams, N. Piggins and J. Als-Nielsen, 1989, Surf. Sci. Lett. 222, L845. Poelsema, B. and G. Cornsa, 1991, in: Scattering of Thermal Energy Atoms from Disordered Surfaces. Springer Tracts in Modern Physics, Vol. 115. Springer-Verlag, Berlin. Presicci, M. and T.-M. Lu, 1984, Surf. Sci. 141,233. Pukite, P.R., C.S. Lent and P.I. Cohen, 1985, Surf. Sci. 161, 39. Quinn, J.J., 1962, Phys. Rev. 126, 1453. Rieder, K.H., N. Garcia and V. Celli, 1981, Surf. Sci. 108, 169.
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Rieder, K.H., 1982, in: Dynamics of Gas-Surface Interaction. Springer Series in Chemical Physics, Vol. 21. Springer, New York. Robinson, I.K., 1983, Phys. Rev. Lett. 50, 1145 Robinson, I.K., 1986, Phys. Rev. B 33, 3830. Robinson, I.K., W.K. Waskiewicz, R.T. Tung and J. Bohr, 1986, Phys. Rev. Lett. 57.2714. Robinson, I.K., E.H. Conrad and D.S. Reed, 1990, J. Phys. France 51, 103. Robinson, I.K., 1991, in: Handbook on Synchrotron Radiation, Vol. 3, eds. G.S. Brown and D.E. Moncton. North-Holland, Amsterdam, p. 221. Rudberg, E., 1930, Proc. Roy. Soc. (London) A127, 111. Rous, P.J., J.B. Pendry, D.K. Saldin, K. Heinz, K. Muller and N. Bickel, 1986, Phys. Rev. Lett. 57, 2951. Schiff, L.I., 1968, Quantum Mechanics. McGraw-Hill, New York, pp. 318-324. Stair, P.C., 1980, Rev. Sci. Instrum. 51, 132. Stout, G.H. and L.H. Jensen, 1968, X-ray Structure Determination. Macmillan. Swendsen, R.H. and K.H. Rieder, 1982, Surf. Sci. 114, 405. Toennies, J.P. and K. Winkelmann, 1977, J. Chem. Phys. 66, 3965. Unertl, W.N. and S.R. McKay, 1984, in: Determination of Surface Structure by LEED, eds. P.M. Marcus and F. Jona. Plenum, New York, p. 261. Van Hove, M.A. and S.Y. Tong 1980, in: Determination of Surface Structure by LEED, eds. P.M. Marcus and F. Jona. Plenum, New York, p. 43. Van Hove, M., W.H. Weinberg, C.-M. Chan, 1986, Low Energy Electron Diffraction, Springer Series in Surface Science, Vol. 6. Springer, Berlin, Heidelberg, p. 237. Villain, J., D.R. Grempel and J. Lapujoulade, 1985, J. Phys. F 15, 809. Viieg, E., 1988, Ph.D. thesis, Fundamenteel Onderzoek der Materie (FOM), unpublished. Vlieg, E., J.F. van der Veen, J.E. Macdonald and M. Miller, 1987, J. Appl. Cryst. 20, 330. Warren, B.E., 1941, Phys. Rev. 59, 693. Warren, B.E., 1990, X-Ray Diffraction. Dover, New York. Webb, M.B. and M.G. Lagally, 1973, Solid State Physics 28. Academic Press, New York, p. 301. Williams, E.D., R.Q. Hwang and R.L. Park, 1984, J. Vac. Sci. Technol. A2, 1004. Wolken, G. Jr., 1973, J. Chem. Phys. 58. 3047. Zanazzi, E. and F. Jona, 1977, Surf. Sci. 62, 61. Zimmer, R.B. and B.W. Holland, 1975, J. Phys. C 8, 2395.
CHAPTER 8
Direct Imaging and Geometrical Methods W.N. U N E R T L Laboratory for Surface Science and Technology Sawyer Research Center University of Maine Orono, ME 04469, USA
M.E. K O R D E S C H Department of Physics and Astronomy Ohio University Athens, OH 45701, USA
Handbook of Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
361
Contents
8.1.
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
363
8.2.
Scanned probe microscopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
363
8.2.1.
8.2.2.
Scanning tunneling microscope (STM) . . . . . . . . . . . . . . . . . . . . . . . . .
364
8.2.1.1.
Properties of the t u n n e l i n g current . . . . . . . . . . . . . . . . . . . . .
364
8.2.1.2.
R e s o l u t i o n o f the S T M
8.2.1.3.
I n s t r u m e n t a t i o n for S T M
Scanning force microscope (SFM) 8.2.2.1.
8.3.
366
. . . . . . . . . . . . . . . . . . . . . . . . . .
369
. . . . . . . . . . . . . . . . . . . . . . . . . . .
372
T i p - s u b s t r a t e interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
374
8.2.2.2.
R e s o l u t i o n o f the S F M
8.2.2.3.
I n s t r u m e n t a t i o n for force m i c r o s c o p y
. . . . . . . . . . . . . . . . . . . . . . . . . . . ...................
Ion based techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1.
8.3.2.
8.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
376 378 380
Field ion microscopy (FIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
380
8.3.1.1.
Physical basis of FIM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
380 384
8.3.1.2.
R e s o l u t i o n o f FIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1.3.
I n s t r u m e n t a t i o n for FIM
. . . . . . . . . . . . . . . . . . . . . . . . . .
385
8.3.1.4.
S a m p l e s for FIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
386
Ion backscattering techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
387
8.3.2.1.
387
Basic physics of ion scattering . . . . . . . . . . . . . . . . . . . . . . .
8.3.3.2.
H i g h e n e r g y ion scattering
. . . . . . . . . . . . . . . . . . . . . . . . .
392
8.3.3.3.
Low e n e r g y ion scattering
. . . . . . . . . . . . . . . . . . . . . . . . .
296
Electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
398
8.4.1.
Image formation in the electron microscope
400
8.4.2.
Transmission electron microscopy
......................
. . . . . . . . . . . . . . . . . . . . . . . . . . . ......................
401
8.4.2.1.
Physical p r i n c i p l e s of o p e r a t i o n
8.4.2.2.
Resolution
8.4.2.3.
Instrumentation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
402
8.4.2.4.
S a m p l e s for surface T E M . . . . . . . . . . . . . . . . . . . . . . . . . .
402
8.4.3.
Reflection electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
404
8.4.4.
Emission electron microscopy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4.1.
Principles of o p e r a t i o n
8.4.4.2.
R e s o l u t i o n in L E E M
8.4.4.3.
Instrumentation
8.4.4.4.
S a m p l e s for L E E M
. . . . . . . . . . . . . . . . . . . . . . . . . . .
401 402
408 409
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
409
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
410
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
Related methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
416
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418
8.4.5.
362
8.1. Introduction This chapter provides a brief survey of scanned probe microscopies, field ion microscopy, ion backscattering, and electron microscopies. These are the most important experimental methods that provide "direct" information about the structure of surfaces. This information is usually in the form of an image, except in the case of ion scattering where atomic positions are obtained by triangulation methods. The choice of techniques to be covered in this chapter is limited to those that have demonstrated resolution of better than a few nanometers with emission microscopies being the exception. Thus, for example, optical microscopy and X-ray tomography are not considered. A major strength of most direct methods over diffraction or spectroscopic techniques is that they provide real space images of non periodic structures. Thus, direct methods are ideal for characterization of surface imperfections such as point defects, steps, or grain boundaries. Numerous examples are given in Chapters 2 and 13. In contrast, most diffraction methods, as described in Chapter 7, have little sensitivity to point defects and yield only statistical information about the atomic structure of extended defects. Additionally, the extraction of information from diffraction data must often rely on specific models. The main goal of this chapter is to provide the reader with a good understanding of the physical principles of each technique including the fundamental limitations to resolution. Also included are examples of the application to the study of surface structure. In addition, the characteristics of typical instrumentation are given for the subset of methods that are most widely used or require fairly small scale instrumentation. This chapter should be viewed as an introduction. For those readers desiring to progress to a more advanced level of understanding, references are given to review articles and books. 8.2. Scanned probe microscopies Scanned probe microscopies are the newest and most widely applied of the direct methods for determination of crystal structure. Their introduction in the 1980s revolutionized the way surface structures are visualized. In all scanned probe microscopies, a sharp tip is brought to within about a nanometer of the surface. The two most common of the scanned probe techniques that have atomic or near atomic resolution are the Scanning Tunneling Microscope (STM) and the Scanning Force Microscope (SFM, also called the atomic force microscope or AFM). In both cases, an image is obtained by measuring the interaction between the tip and substrate as the tip is scanned over the surface. In the case of the STM, this interaction is the tunneling current, while for the AFM it is the force between the tip and substrate.
363
364
W.N. Unertl and M.E. Kordesch
G v
Fig. 8.1. The basic STM is shown schematically. The tunneling current I between the sharp tip and the sample is measured as the bias voltage V or tip-sample distance d are varied.
8.2.1. Scanning tunneling microscope (STM) The first scanning tunneling microscope was built by Binnig and Rohrer (1982) following earlier work by Young, Ward and Scire (1972) a decade earlier. The STM is based on electron tunneling through vacuum as shown schematically in Fig. 8.1. A small bias voltage is applied between the tip and sample and the tip is brought close enough to the sample for electron tunneling to occur, typically, d = 1 nm. In the most commonly employed imaging mode, the tip is scanned over the surface while a feedback system adjusts the sample to keep the tunneling current constant. The image is constructed by plotting the changes in sample height vs. tip position over the sample. Figure 8.2 is an example for various coverage-dependent benzoic acid structures induced on Cu(ll0). Other examples are presented elsewhere in this handbook. Figures 2.1 and 2.3 show facet formation induced by adsorption. Figure 6.7 shows a reconstructed Si(100) surface with monoatomic height steps. A defect and phase transition on CO covered Pt(l 10) are shown in Figs 9.10 and 9.11. Small islands of (2• 1)-O on Cu(110) are shown in Fig. 9.21. Finally, Chapter 12 contains ten images that illustrate various types of defects that occur at surfaces and during adsorption.
8.2.1.1. Properties of the tunneling current The tunneling current between a tip t and sample s separated by a spacing d is determined by three major components. One is the probability T(E,V,d) that an electron incident from electrode t will tunnel across the barrier. T depends on the electron's energy E, the width and shape of potential barrier, and the bias voltage between the electrodes V. In this section, we assume that E is measured with respect to the Fermi energy EF. The probability that an electron with this energy is incident on the barrier from material t is given by the occupied density of states p,(z,E). Finally, there must be an unoccupied state in s at energy E into which the electron can tunnel. The probability of such an unoccupied state is given by the unoccupied density of states p.,(z,E) of electrode s. The way these components influence the tunneling current can be most simply understood for the case of a one-dimensional potential barrier (Fig. 8.3). More realistic models have been developed (e.g., Tersoff and Hamann, 1986; Chen, 1993) but a numerically accurate quantitative theory is not yet available.
Direct ima~in~ and fceometrical methods
365
Fig. 8.2. A series of three 200 x 200 ~ STM images showing various coverage-dependent benzoic acid induced structures on Cu(ll 0). The left image at low coverage shows the benzoic acid dimers lying flat on the surface; the vertical rows of dots are images of substrate Cu atoms. The middle image at intermediate coverage shows the ordered structure of upright benzoate species and flat-lying molecules. The right image shows areas of the intermediate coverage benzoate structures coexisting with the saturation coverage c(8x2) benzoate structure in which all the benzoate molecules are now upright. (Images courtesy of N.V. Richardson.) TIP
SAMPLE
~
~
d
~
Fig. 8.3. A one-dimensional trapezoidal potential barrier.
W h e n the tip and s a m p l e are electrically c o n n e c t e d but separated by a large v a c u u m gap, the Fermi levels EF are identical and electronic states are o c c u p i e d up to EF and e m p t y b e l o w it. The work function is e@, for the tip and e@., for the substrate. In general, e~, and eq~, will not be equal. Inside each solid, the w a v e functions of states with e i g e n e n e r g i e s b e l o w the v a c u u m level are periodic. Outside, they decay e x p o n e n t i a l l y with distance into the v a c u u m with a decay length on the order o f K-~ = [2m(e~ - E)/h2] -1/2. If a bias voltage V is applied b e t w e e n the tip and sample, the e n e r g i e s of the two electrodes will be rigidly shifted by e V with respect to each ~ther.
W.N. Unertl and M.E. Kordesch
366
As the sample and tip are brought close together (Fig. 8.3), the trapezoidal shaped potential barrier becomes narrow and there is an increase in the interaction between the exponentially decaying wave functions of the sample and tip as d --) ~c-l. If the shape of the potential barrier U(z) is known and is not too narrow or too low, the tunneling transmission probability, as estimated using the semiclassical WKB approximation (Davidov, 1965), is
T = To
e -2~ = exp
- ~-
~/2m[ g(z) - E] dz
(8.1)
0
where the coefficient T o is independent of d and 7. For the case of the trapezoidal barrier in Fig. 8.3 with an applied bias V, T(E,V,d) -- exp { 2'~r~3he(A~ 4d - V) I(e~),- eV - E)3~
- (e~,-
E ) 3''21}
(8.2)
where E is measured with respect to EF. This simple model reveals the main functional characteristics of T including its exponential variation with d as well as its dependence on potential barrier and the applied bias V, which can be either positive or negative. The total tunneling current can then be calculated from eV
l(d,V) - f p,(O,E) p,(l,E- eV) T(E,V,I ) dE
(8.3)
Of course, the WKB approximation provides only a semi-quantitative analysis of the tunneling problem. In particular, it seriously over estimates the tunneling probability when the barrier is very low or very narrow. More precise methods to estimate 1 are presented in Chapter 2 of the book by Chen (1993). 8.2.1.2 Resolution of the STM The dependence of the current on the width of the barrier, Eq. (8.3), comes entirely from T, so that
1 ~ exp(-2yd)
(8.4)
This exponential dependence of I on the barrier thickness is responsible for the very high z-sensitivity of STM. For example, typical values are ~ = 5 V and d -- 0.5 rim, so that 2yd = 11.4 and 1 decreases by nearly one order of magnitude for every 0.1 nm increase in d. Equation (8.4) shows that STM has the sensitivity to detect changes in the vertical component of surface topography as small as a fraction of an angstrom. We consider the lateral resolution below. Equation (8.3) makes it clear that the tip-substrate spacing d is not the only important parameter that affects the structure in an STM image. This is particularly true for materials like semiconductors that have complicated densities of state as
Direct imaging and geometrical methods
367
Fig. 8.4. STM images of GaAs (110) at positive and negative biases. (a) Tunneling from the tip to the substrate; bias = +1.9 V. (b) Tunneling from the substrate to the tip; bias =-1.9 V. (c) A plan view of the GaAs (110) surface with As atoms shown as filled circles and Ga atoms as filled circles. The rectangle has the same position in (a), (b), and (c). (From Feenstra et al., 1987).
illustrated in Fig. 8.4 which shows images of GaAs (1 10) obtained for two biases (Feenstra et al., 1987). In Fig. 8.4a, electrons tunnel from filled states in the tip into empty states of the sample. In Fig. 8.4b, the tunneling is from filled states in the sample into empty states of the tip. The image is interpreted as follows: The conduction band (occupied) states are primarily concentrated on the Ga atoms while the valence band (unoccupied) states are primarily located on the As atoms. Thus, bright features in the left-hand image (Fig. 8.4a) mark the locations of Ga atoms whereas, in Fig. 8.4b, the bright features are predominantly due to As atoms. Analysis of these images reveals that the intensity maxima in the two images are actually separated by about 0.05 nm more than the true spacing, as determined from diffraction studies. Thus, in this case, the influence of the density of states on the tunneling current complicates extraction of structural data from STM images. These effects are most important for non-metallic materials or for adsorbate covered surfaces, particularly when the adsorbate molecule has a complex electronic structure. The lateral resolution of the STM can be estimated using a simple model of a point tip a distance d from the surface as shown in Fig. 8.5. The tunneling current, Eq. (8.4), will fall to one-half of its m a x i m u m value when
xl/z -
0.693 7
~4~a N/1 + 0.693
(8.5)
368
W.N. Unertl and M.E. Kordesch
~------Tip
d
,
,
Fig. 8.5. A point tip above a flat substrate.
--
n"
I
1
1
2
'-r"
o
"=
1
"--/ -1
-20
-
1
-10
1
He 1
0 10 x (BOHR)
20
Fig. 8.6. Calculations of the change in tip distance Ad versus lateral separation x between the tip and substrate adatom required to keep the total tunneling current constant. The 0-level of Ad corresponds to a tip-substrate spacing of 0.85 nm. The tip atom is Na and adatoms are Na, S, and He. (1 bohr = 0.0529 nm.) After Lang (1986). Since T = 10 nm -z and d = 0.5 nm, we find xi/2 - ~ 0.4 nm. This e s t i m a t e is in g o o d a g r e e m e n t with the results of a m o r e detailed theory of S T M r e s o l u t i o n d e v e l o p e d by L a n g (1986), w h o used a m o d e l s y s t e m consisting of two planar metal e l e c t r o d e s each with a single atom a d s o r b e d on it. The t u n n e l i n g c u r r e n t d e n s i t y was c a l c u l a t e d as a f u n c t i o n o f e l e c t r o d e s p a c i n g as the two surfaces are translated o v e r o n e another. F i g u r e 8.6 s h o w s results for the case of a N a atom on o n e e l e c t r o d e and a Na, S, or He atom on the other. T h e resolution as m e a s u r e d by the h a l f - w i d t h at h a l f - m a x i m u m ranges from about 0.25 nm to about 0.5 nm. T h e s e results s h o w that S T M can r e s o l v e individual atoms. H o w e v e r , S T M is not able to a c h i e v e as g o o d a p o i n t - t o - p o i n t resolution as can be o b t a i n e d with diffraction or ion scattering techniques.
Direct imaging and geometrical methods
369
The calculations of profiles shown in Fig. 8.6 provide another illustration of the effects of electronic structure on STM images. In the case of He, the image would actually show a dip at the location of the atom!
8.2.1.3. Instrumentation for STM Typical STM instruments consist of a scanner assembly, an electronic control unit, and vibration isolation. In addition, the environment is often controlled by placing the STM in an ultra high vacuum chamber or a fluid cell. Because of the exponential dependence of the tunneling current on tip-sample separation, Eq. (8.4), the STM instrument must not only be capable of bringing the tip very close to the sample in order to get a significant tunneling current but must also be able to keep that spacing very constant to avoid large fluctuations in the current. I is usually in the range of nanoamperes, d is typically 0.5-1.0 nm, and bias voltages are not more than a few volts. The tip used in STM is normally fabricated from wire using electrochemical methods. The article by Melmed (1991) gives particularly detailed information. Mechanical forming, field desorption, and ion milling are also occasionally used (Chen, 1993; Rohrer, 1993). There has been no published demonstration of the effect of tip material on resolution although it has been suggested that metals with predominantly d-electrons at the Fermi level might have better resolution than those with s-electrons (Tersoff, 1993). Because of the strong dependence of the tunneling current on distance, only the very closest atom (or atoms) on the tip participate. Such tips can be fragile and accidental contact with the sample, exposure to high fields, or interaction with contaminants can alter the performance of a tip. Indeed, it is not unusual for the tip characteristics to change during imaging resulting in either improved or degraded performance. Some tips have multiple protrusions which can result in ghost images, in sections of the surface not being imaged, and other artifacts (Rohrer, 1993). The tip (or equivalently the sample) is usually mounted on a piezoelectric scanning device (Jaffe and Berlincourt, 1965). This allows the tip position to be controlled by changing the voltages applied to the scanner. Scanners are usually fabricated from various lead zirconate titinate polycrystalline ceramics because they are the most efficient of all known piezoelectric materials (Chen, 1993). Two configurations are common: the tube scanner and the tripod scanner. Both are illustrated schematically in Fig. 8.7. The tube scanner is usually provided with metallic electrodes and poled in the radial direction. The change in length Az for a voltage difference V between the inner and outer electrodes is given approximately by (Vernitron) Az --- 2d31LV/t
(8.6)
where L is the length of the tube, t its wall thickness, and d31 a materials constant of the PZT (typically 100-300 • 10-12 V/m). For a tube with L = 2.5 cm and t = 0.5 mm, Az(nm) --- 10-30 V (Volts). The lateral sensitivity, for the case where the inner electrode and one pair of opposing outer electrodes are grounded and a voltage + V is applied to the other electrodes, is given by Ax -_-0.9d31L2V/Dt
(8.7)
370
W.N. Unertl and M.E. Kordesch
t x
z
Tube Scanner Y
L
Tripod Scanner
J Fig. 8.7. Tube and tripod piezoelectric scanners.
where D is the inner diameter of the tube (Chen, 1993). Typically, Ax(nm) -_--20-60 V (Volts). Tripod scanners use the changes in length L of each leg to move the tip according to AL = d33 V
(8.8)
where V is applied between the ends of one leg of the tripod and d33 is typically in the range 150-500 x 10-12 V/m. Unfortunately, Eqs. (8.6), (8.7) and (8.8) are not strictly equalities because the piezoelectric materials have a significant amount of hysteresis, do not respond instantly to the applied voltage, and d31 and d33 usually have a significant temperature dependence (Vernitron). Additionally, there can be strong coupling between the z- and x - y motions of the piezo-tube (Chen, 1992). These factors preclude using piezoelectric scanners for metrological applications in open loop operation. Closed loop control is required in which the displacement is measured using an optical interferometer (Schneir et al., 1994) or capacitative
Direct imaging and geometrical methods
371
displacement sensors or strain gauges (Smith and Chetwynd, 1992). Resonance frequencies of tube scanners limit data acquisition rates; typical values are about 8 kHz in the radial direction and about 40 kHz in the axial direction. The Curie points of the most commonly used piezoelectric materials are in the range 195-350~ so they cannot be used to image surfaces at high temperatures. Recently, new materials and improved methods have been developed that allow imaging at temperatures up to 700 K (Bott et al., 1995). Typical scanners can displace the tip by only 1-100 ktm so a coarse positioning mechanism must be used to initially bring the tip and sample close together. Many devices have been employed including simple mechanical levers, differential springs, electrostrictive "inchworms" or "louses", and electromagnetic "walkers" (Chen, 1993; Smith and Chetwynd, 1992). Because of the very high sensitivity to changes in the tip-sample spacing, vibration isolation and temperature stability are important considerations in the reduction of noise in any STM design (Smith and Chetwynd, 1992; Chen, 1993). Vibrations with frequencies between about 1 Hz and a few kHz are the most deleterious. Rigidity and compactness in STM design and use of materials with compatible thermal expansion properties are features of the best instruments. Mechanical vibration isolation using springs, magnetic damping, or air tables are also usually necessary. The electronic control circuitry typically consists of a bias voltage supply, current amplifier, feedback system, high voltage power supplies, and a computer interface as shown schematically in Fig. 8.8. The bias voltage supply sets the potential difference between the tip and sample. The current amplifier usually serves two purposes. One is to measure the tunneling current I which is in the range 0.01 to 50 nA. The other is to provide a signal that is proportional to the tunneling gap d. Since the I and d are related exponentially by Eq. (8.4), a logarithmic amplifier is used. The feedback system is used to adjust the z-voltage on the piezo scanner to keep the tunneling current constant as the tip is scanned over the surface. The changes in z-voltage are assumed to be directly related to changes in surface height through Eq. (8.6) as long as density of states contributions are not changing. Feedback systems can be either analog or digital. The high voltage power supplies provide the voltages used to control the x-, y-, and z-displacements of the scanner and are usually controlled by the computer. Most STM images are presented as plots of the z-voltage as a function of x- and y-voltages. Since the actual motion of the scanner is not linear, as discussed above, the images may be distorted unless a closed loop feedback system is used to control each axis. The computer interface consists of a set of A/D and D/A converters; 16 bit resolution is typical. Detailed discussions of design considerations of each component of the STM control system are presented by Chen (1993). In order to study well characterized surfaces, STM measurements must usually be carried out either in ultra-high vacuum or in liquid environments. In uhv, standard surface cleaning techniques such as ion sputtering, gas reactions, and annealing are possible. In liquids, electrochemical techniques can be used to clean the surface or to form layers of other materials. In uhv special sample manipulation
372
W.N. Unertl and M.E. Kordesch
I y-axis voltage
II v~ x-a•
II
I feedback ___• electronics ~
F-'L--_.~__ [ z-piezo voltage ~
.
""
.,[
tunnelin ~
t
sample
~ ~
error
logarithmic~ ~]am pl,fier ~ ' )
signal
current
set point
computerand interface
I electronics -
Fig. 8.8. Principal components of the control electronics for a typical STM. capabilities are usually needed as is the ability to replace the tip without breaking the vacuum. Special sample stages and tip fabrication methods are required for measurements in liquids (Bard and Fan, 1993).
8.2.2. Scanningforce microscope (SFM) The scanning force microscope (SFM) was invented by Binnig, Quate and Gerber (1986) to remedy the major limitation of the STM that both the sample and the tip must be conducting. Indeed, the first SFM image was obtained from an AI203 surface using a diamond tip. The scanning force microscope is also often referred to as an atomic force microscope, or AFM. However, unlike STM, it has not proven routine to obtain images with true atomic resolution with the SFM. True atomic resolution requires that individual atoms be resolved in the vicinity of a surface defect and only a few experiments have approached that goal (Ueyama et al., 1995). In general, the principles of the SFM are less well understood than those of the STM, but several good reviews are available (Burnham and Colton, 1993; Quate, 1994; DiNardo, 1994). As illustrated in Fig. 8.9(a), a typical SFM consists of a probe tip, force sensor, sample, and scanner all mounted in a rigid frame. The force acting between the sample and the tip is measured as their relative lateral and vertical positions are changed by the scanning device. Figure 8.9(b) shows a sketch of a typical commercial SFM in which the force sensor and probe tip are an integral assembly made by
Direct imaging and geometrical methods
373
+ Scanner
(a) Position
Cantilever
Laser
Tip
~
~ Sample PiezscTdneCer tric
(b) Fig. 8.9. (a) Basic components of a scanning force microscope. (b) Schematic drawing of an AFM. standard photolithography techniques (Albrecht and Quate, 1988). The force sensor is a cantilever beam with a stiffness ~c that is typically in the range 0.01 to 20 N/m. Bending of the cantilever is proportional to the normal force and twisting of the cantilever is proportional to the lateral force. For small deflections, the normal force is determined using Hooke's Law
374
W.N. Unertl and M.E. Kordesch
FH = ~Az
(8.9)
where Az is the deflection of the end of the cantilever. In most SFMs, the deflection is measured with an optical l e v e r - the angular deflection of a laser beam reflected off the end of the cantilever is measured with a position sensitive detector. The bending and twisting modes can be measured independently since they cause deflections of the light beam that are nearly orthogonal to each other. The sample is mounted on a piezoelectric scanner. Both tubes and tripods are used (see Fig. 8.7). 8.2.2.1.
Tip-substrate
interactions
There are several possible components of the interaction force between the tip and substrate. The van der Waals dispersion force will always be present. Coulomb forces can be important if at least one of the surfaces is charged or a potential is applied between the surface and tip. If the tip is made from a magnetic material, magnetic forces can also be measured. Imaging using Coulomb or magnetic interactions has not played an important role in surface structure studies and will not be considered further in this chapter. If the contact occurs in a liquid, a variety of other interaction mechanisms can also be important (Israelachvili, 1992). Figure 8.10 illustrates the important properties of the interaction force F~, for the case when only van der Waals forces are important and the environment is vacuum or air. The detailed shape of the attractive portion of the curve (F negative) is determined by the shapes of the tip and substrate, by their densities, and their dielectric properties. The book by Israelachvili (1992), particularly Chapters 10 and 11, gives an excellent treatment of this part of the force curve. Most attempts to model this portion of the curve assume that the substrate is flat and that the tip can be approximated as a sphere of radius R. In this case, (8.10)
F~. = - A R / 6 D 2
where D is the distance between the tip and substrate and A is the Hamaker constant with values n e a r 10 -19 J. A depends on the dielectric properties of the tip, substrate
s
NoLoadPoint
J
J
J D ,,,.__
Fig. 8.10. Interaction force between the tip and sample.
Direct imaging and geometrical methods
375
and environment and on the temperature. It can be calculated using the theory developed by Lifshitz (see Israelachvili, 1992) if the frequency dependence of the dielectric constants are known. A is always positive (i.e., F is attractive) in air or vacuum but in other media it can be negative. Once D becomes small enough that there is overlap between the wavefunctions of the substrate and the tip, chemical interactions can occur and adhesive bonds form. As D is decreased further, the Pauli exclusion principle comes into play and a repulsive component starts to act. At the "no load" point, the repulsive and attractive forces are equal; i.e., there is no net force on the tip (Fin = 0). Any additional decrease in the tip-sample separation requires elastic and/or plastic deformation of both the tip and sample and measurements in this force regime are used to study surface mechanical properties (Hues et al., 1994; Agrai't et al., 1995). The total force acting on the tip is (Fin + FH). The dashed straight line in Fig. 8.10 is -FH, Eq. (8.9), for the force sensor; i.e., the slope of the dashed line is K:. Simple analysis of the force balance between F~n and FH on the tip as a function of D shows that when the slopes of the two force curves become equal, the tip becomes unstable to displacements towards the surface and suddenly jumps into contact. This point is labeled J in the figure. Jump to contact can be avoided only if a force sensor is used that has ~ larger than the slope of F~,for all D. This typically requires K: greater than about 30 N/m, a value that is substantially larger than typical SFM sensors. In most SFMs, jump to contact occurs when D is about 4-5 nm. The most common imaging mode is to scan with the tip in contact under a load F~,, near the no load point. Most typically, a feedback system is used to adjust the z-piezo voltage to maintain constant deflection of the cantilever. The image is then constructed by plotting changes in the z-voltage vs. the x - y scan voltages much as in the case of the STM. Images obtained in this way are actually surfaces of constant tip-substrate interaction force assuming that the piezo scanner response is adequately described by Eqs. (8.6)-(8.8). Figure 8.11 shows an image of the surface structure of a fullerene film obtained in this mode.
Fig. 8.11. Force microscope image of the (311 ) surface of a face-centered-cubic film of C60. Reprinted from Snyder et al. (1991). Copyright 1991 American Association for the Advancement of Science.
376
W.N. Unertl and M.E. Kordesch
Non-contact imaging is attractive because there is no possibility of damaging either the tip or surface and the net force between the tip and sample is very small compared to contact imaging. In the non-contact mode, the tip is oscillated at a frequency just above its natural resonance frequency (Ueyama et al., 1995). The vibrating tip is brought close enough (1-5 nm) to the surface so that the influence of the surface force shifts the resonance frequency by an amount proportional to the force gradient. A feedback system adjusts the tip-sample distance to keep this frequency shift constant at =10-20 Hz. High K: cantilevers must be used to avoid jump-to-contact. The highest resolution images have been reported for non-contact imaging under ultra-high vacuum conditions and true atomic resolution may be possible in some cases. Other imaging modes are used but the imaging mechanisms are less well-understood. Some of the more important of these are the tapping mode, the lateral force mode, and adhesion maps. In the tapping mode, the tip is oscillated just below its resonance frequency with a small amplitude vibration (-20 nm). The tip-sample distance is adjusted so that the tip just contacts the surface for only a short time during each cycle (Spatz et al., 1995; Bustamante and Keller, 1995; Burnham et al., 1995). The contact force can be kept low but a detailed understanding of the intermittent contact process is not available. In the lateral force mode, the lateral force on the tip, which is determined in part by friction, is monitored by measuring the twisting of the cantilever (Overney, 1995). Adhesion images are plots of the pull-off force required to separate the tip and sample. This force is determined by the contact area and the work of adhesion both of which can vary from point-topoint over the surface (Burnham and Colton, 1993; Overney, 1995). 8.2.2.2. Resolution of the SFM Two criteria must be satisfied to demonstrate true atomic resolution with the SFM. These are: (i) Images must reproduce the structure of surfaces with known atomic structure. (ii) Atomic structure must be observed at defects. Criterion (i) has been demonstrated many times in the literature. However, such images are not sufficient to prove atomic resolution. Images of graphite, Fig. 8.12, provide a good example. During imaging, a small flake of graphite is believed to be sheared off the surface and attached to the tip. The contact area between the flake and substrate involves multiple atoms, yet features in the image show the expected C-C spacing ingraphite. Images of this type are said to have atomic scale resolution rather than true atomic resolution. Because of this complication, criterion (ii) must also be satisfied before true atomic resolution is proven (Ueyama et al., 1995). None the less, the SFM is a valuable surface structural tool because of its ability to study non-conducting materials and provide information about meso-scale structures on the surface including step distributions and crystal growth. A simple continuum mechanics analysis suggests why true atomic resolution should be difficult to obtain with the SFM. Molecular dynamics simulations show that the predictions of such continuum models are surprisingly accurate even at length scales near atomic dimensions (Landman et al., 1990). Johnson et al. (1971)
Direct imaging and geometrical methods
377
Fig. 8.12. Examples of force microscope images of graphite obtained in the contact mode. From Albrecht and Quate (1988). showed that when a sphere of radius R is brought into contact with a plane, the radius of the contact area is given by a = [ 9 r t W R 2 / E * ] 1/3
(8.11)
where W is the work of adhesion between the two surfaces and E* is an effective modulus calculated from the Young's moduli and Poisson ratios of the sphere and substrate. Equation (8.11) provides an estimate of the size of the contact region between the tip and thus also of the resolution to be expected. We see that hard materials, which have E* of several hundred GPa (Anderson, 1981), will have the smallest a. The smallest value of W will typically occur for the case where the only interaction between the two surfaces is the van der Waals interaction and can be calculated from Eq. (8.10) since Fin = - d W / d D . Using these values yields a -- 0.3R. Thus, only tips with R on the order of an atomic diameter should be expected to produce images with atomic resolution. However, in most cases, the stress produced on such a small area contact by the van der Waals attraction between the tip and sample will substantially exceed the yield stress and the junction will collapse by elastic or plastic deformation of the tip and/or substrate (Pashley, 1984). Thus, true atomic resolution should be difficult to obtain because of the mechanical instability
378
W.N. Unertl and M.E. Kordesch
of the small tips required. This seems to be borne out by the fact that true atomic resolution of defect structures, including steps is rarely achieved with the SFM. This is to be contrasted with STM where such resolution was attained even in the earliest measurements. The situation may be somewhat improved for imaging in liquids since it is possible in some cases for the van der Waals interaction to be repulsive (Israelachvili, 1992).
8.2.2.3. Instrumentation for force microscopy Instrumentation for scanning force microscopes has many similarities to that for STM since piezoelectric materials are used to position the tip and sample and vibration isolation is important. The discussion of these aspects in w 8.2.1.3 is also valid for force microscopy instrumentation. The primary features of force microscopes that are different from the STM are the tip and the sensor used to monitor tip deflection. General considerations concerning the optimization of SFM for specific types of measurements are given by Burnham and Colton (1993). Although tips for AFM use have been manufactured from a large variety of materials, the vast majority of AFM experiments have been carried out using commercially available tips made from Si3N 4 or SiO2. These tips are manufactured as an integral part of a cantilever beam using photolithographic techniques (Albrecht and Quate, 1988). Figure 8.13 shows a typical example. Both the cantilevers and tips are made from Si3N 4. Cantilevers are both rectangular and triangular in shape with force constants ranging from 0.01 to 0.5 N/m. The tips are available as simple pyramids about 3 ktm in height with a tip radius of about 50 nm or as sharpened pyramids with a smaller tip radius of about 20 nm. Figure 8.14 illustrates the methods that have been most widely used to measure the displacement Az of the tip. Many methods, including several not described here, have been developed and are discussed in detail in the book by Sarid (1991). The simplest, and most popular method is the optical lever, Figs. 8.9(b) and 8.14(a). A light beam, usually from a solid state laser, is focused onto the tip of the cantilever with the reflected beam incident on a split photodiode detector consisting of two closely spaced diodes A and B. With the cantilever undeflected, the diodes are positioned so that each element A and B is equally illuminated resulting in a difference signal A-B = 0. For small cantilever deflections, A-B varies linearly with Az. Sensors of this design can detect deflections as small as a few hundredths of a nanometer. The major sources of noise are shot noise in the photodiodes and thermally driven vibrations of the cantilever. Optical interferometry, Fig. 8.14(b), detects tip displacements using interference between a reference beam and a beam reflected off the tip of the cantilever. The reference bean can be reflected from the stationary base of the cantilever, from the end surface of an optical fiber, or from some other stationary surface. Both homodyne and heterodyne interferometer systems are in use. The best interferometer systems are capable of detecting displacements of less than 10-~ m. The first AFM used a tunneling tip placed above the cantilever, as shown in Fig. 8.14(c). Because of the exponential dependence of the tunneling current on the gap width, Eq. (8.4), this devise can easily detect displacements of less than 0.01 nm.
Direct imaging and geometrical methods
379
Fig. 8.13. Views of an integrated tip-cantilever assembly at various magnifications. The longest triangular cantilever is 320 l.tm long with a force constant of about 0.01 N/m. The rectangular cantilever has a torsional force constant of about 0.008 N/rad. (Courtesy of Park Scientific Instruments.) S u r f a c e r o u g h n e s s of the cantilever p r o v i d e s a limitation to the sensitivity since, due to thermal drift or deflection o f the cantilever, tunneling will o c c u r to d i f f e r e n t points on the cantilever. O n e of the m o r e recently d e v e l o p e d s e n s o r designs, uses the p i e z o r e s i s t i v e r e s p o n s e o f the cantilever to the strain that occurs w h e n it is bent. Sensitivity is
380
W.N. Unertl and M.E. Kordesch
a
b
c
Fig. 8.14. Methods used to detect displacement of the cantilever. (a) The optical lever: angular deflection of the reflected laser beam is measured by a position sensitive detector. Sub-Angstrtim resolution is possible and both bending and twisting of the cantilever can be observed. (b) Interferometry: interference between a light beam reflected from the end of the cantilever and a reference surface is used. Only bending is detected. (c) Tunneling: changes in the tunneling current are used to measure cantilever deflections. Only bending is detected. about 0.025 nm (Tortonese et al., 1993). This device is effectively a miniaturized strain gauge and its simplicity makes it attractive for use in ultra-high vacuum where other methods can be difficult to implement. The vast majority of AFMs in operation today are designed for operation in an ambient air environment so that both surface and tip contamination (including condensation of water) severely limit their capability for structural studies on well defined surfaces. Some instruments can also operate in liquid environments and this allows electrochemical methods to be used to clean the surface and to deposit other species on it in a controlled manner. Finally a few instruments have been designed to operated in ultra-high vacuum for the study of well characterized surfaces and surface processes.
8.3. Ion based techniques
8.3.1. Field ion microscopy (FIM) The field ion microscope (FIM) obtained the first atomic resolution images of the atomic structure of surfaces (MUller, 1951). It was invented and developed by E.W. Mtiller and his students starting in the 1950s. Several excellent monographs are available (Tsong, 1990; Gomer, 1961 ; MUller and Tsong, 1973; Wagner, 1985). FIM has made important contributions to understanding the near surface structure of alloys (Tsong, 1990), evaluating the interaction potential between pairs and triplets of atoms adsorbed on surfaces (see Einstein in Chapter 11), and unraveling atomic scale diffusion mechanisms (see Roelofs in Chapter 13).
8.3.1.1 Physical basis of FIM The basic principle of the FIM is simple as illustrated in Fig. 8.15. The sample is in
Direct imaging and geometrical methods
381
Screen
Y
"
Tip
R
T Fig. 8.15. Schematic of the field ion microscope.
the form of a sharp tip of conductive material with end radius r of 10-100 nm. It is placed in an evacuated chamber at the center of a fluorescent viewing screen with radius R = 10 cm. The chamber is then filled to a pressure of about 10-~ Torr with He or Ne and the sample is placed at a high positive voltage V o with respect to the viewing screen. When the field strength near the surface of the sample is a few A/V, gas atoms are polarized and attracted to the sample by a dipole force. As they get near the surface, electrons are able to tunnel from the gas atom to the tip leaving a positively charged ion. This ion is repelled radially away from the surface, strikes the viewing screen and excites the phosphor. Ionization occurs with higher probability near protruding atoms or at the edges of steps because the locally higher radius of curvature produces the highest field strengths. Figure 8.16 is an image obtained from a PtCo alloy tip with about 30 nm radius. Bright spots on the image correspond to atoms that are in more protruding positions on the surface. The magnification of the image is given by
M=R/~r
(8.12)
where the factor [3 corrects for the fact that the ion trajectories are not precisely radial; 13 typically lies between 1.5 and 2. Thus, M = 10+6; i.e., a distance of 1 mm on the imaging screen corresponds to a few/~ngstr6ms on the surface. The properties of the FIM images are determined by several factors: the ionization probability, the rate at which gas atoms reach the surface, and the thermal accommodation of the atom to the tip temperature.
382
W.N. Unertl and M.E. Kordesch
Fig. 8.16. Field ion microscope image of a Pt Co alloy emitter with end radius of about 30 nm. (Tsong, 1990, reprinted with the permission of Cambridge University Press.)
The field ionization of a gas particle (atom or molecule) near a metal surface is due to electron tunneling from the atom to the metal. We illustrate the principle by considering a hydrogen atom near the surface of a free electron metal. Figure 8.17(a) shows the potential energy and ground state for the electron in an isolated H atom. The ionization energy of the isolated atom is I (13.6 eV for H) and the potential is the Coulomb potential of the nucleus. The metal is described by its work function e~, Fermi energy E v , and energy of the bottom of the band W. If the atom is placed in a uniform electric field E, the potential energy becomes asymmetric as shown in Fig. 8.17(b). Once the width of the barrier becomes comparable to the de Broglie wavelength of the electron, the tunneling probability becomes significant. The larger the field, the higher the tunneling probability. The potential energy is also distorted outside the metal, but inside the metal, it remains constant. Figure 8.17(c) shows the potential when the H atom is distance zi from the metal surface. The total potential V ( z ) is e2 V(z) =
4rt~lz~- zl
e2 + eEz - ~
4rtco(4z)
e2 +
16rt~(z~ + z)
(8.13)
where the first term is the potential b~tween the H nucleus and the electron at z, the second term is the applied electric field, the third term is the image potential of the electron, and the fourth term is the interaction of the electron with the image
Direct imaging and geometrical methods
383
(b)
v(C)
e* El.
-
i
Fig. 8.17. Potential energy diagram of the field emission process. (a) Potential for an electron in a state with ionization potential I in an isolated atom. (b) Effect of a constant electric field on the potential of (a). The dotted line indicates the width of the tunneling barrier. (c) Effect of bringing a metal surface near the atom of (b). (Tsong, 1990, reprinted with the permission of Cambridge University Press.)
potential of the nucleus. As the atom is brought close to the surface, the electron can tunnel directly into unoccupied states of the metal as long as the ionization level remains above EF. I will coincide with EF at some critical distance z,.. For closer distances, there are no empty states in the metal for the electron to tunnel to and the tunneling probability drops to zero. Using the geometry of Fig. 8.17(c), z,. can be estimated from (8.14)
e E z c --- I - e ~
A more accurate estimate involves the image potential of the electron and the polarizabilities of the atom and ion. For He, I = 24.6 eV and the best FIM images are obtained at a field strength of about 0.44 V/nm. Since e~ is in the range 2 - 5 eV for most metals, z,. is typically 0.4-0.5 nm. T[E, U(z)], the probability that an electron will tunnel through the barrier when the atom is located at z can be estimated from the potential of Eq. (8.12) using the W K B method.
T [ E , U ( z ) ] --- exp
-
8m
~[V(z) - E] dz o
(8.~5)
384
W.N. Unertl and M.E. Kordesch
where m is the electron mass, E is the energy of the electron state in the atom, and the integral is over the width of the barrier. Equation (8.15) also assumes that the electron is in a planewave state on both sides of the barrier. In practice, T is significant only for values of z within a few tenths of an ,~ngstrtim of z,.. Thus, it is sufficient to evaluate Eq. (8.15) at z =zc. At distance z; from the surface, the lifetime "~(z) of an electron on a gas particle, is determined from Eq. (8.15) and the rate v at which electrons strike the potential barrier is given by x = [vP(z,E)] -~
(8.16)
The Bohr model of the atom gives an estimate for v of about 4 x 1016 Hz. The strong electric field enhances the gas density near the tip because the gas particles are polarized by the strong electric field of the tip. The induced dipole then interacts with the field and an attractive force pulls the particle toward the tip. Thus, the flux of gas particles that strikes the tip is enhanced over the flux dv/dt =
p/~12rcMkT
(8.17)
predicted by kinetic theory by a factor ~ . 9 depends on the detailed shape of the tip and shank and on the work function. Various estimates of 9 have been made, but none of them are quantitatively correct because of the complex shape of a real tip and its supporting shank and because a significant amount of gas reaches the tip by diffusion along the shank. Gas particles are usually not ionized during their first collision with the surface but are reflected and then re-attracted. This results in a hopping motion over the surface. Typically, the particle loses a few percent of its energy during each collision. Up to several hundred collisions occur before ionization takes place. This number of collisions is large enough that the gas particle is thermally equilibrated to the sample temperature. A detailed description of the hopping process is given by Gomer ( 1961 ). 8.3.1.2. Resolution o f FIM Three main factors determine the resolution ~i of the FIM: the Heisenberg uncertainty principle, the lateral velocity of the particle, and the lateral overlap of the particle and substrate wavefunctions (Tsong, 1990). If the tip is approximated as a sphere, a v spread in lateral velocity components causes a spread of A = M~ = 4vR~/(m/2eVo~
(8.18)
in the location of the image point on the screen where m is the mass of the particle. If the particle is confined to a region of lateral dimension x, the indeterminacy in its speed is given by the Heisenberg uncertainty principle as Av = h/2m Ax This results in a contribution to the optimal resolution of
(8.19)
Direct imaging and geometrical methods
385
5n = 2(132ti2r2/2meV,,) TM
(8.20)
The contribution of the lateral component of the particle's velocity to the resolution is estimated assuming a Maxwell-Boltzmann velocity distribution with characteristic temperature T very close to the substrate temperature and yields (8.21)
= 4(kTl]2r2/eVo) I/2
This factor can be decreased substantially by lowering the substrate temperature and is the reason that many FIM measurements are made with the sample at liquid nitrogen or liquid helium temperatures. Finally, although there are no quantitative estimates of the lateral overlap of the substrate and tip wavefunctions, the size of this disk 5o should be no larger than the radius of the image gas particle. The best overall resolution is typically about 0.3 nm and is obtained by working at low substrate temperature and using an imaging gas that has a high ionization potential so that the highest imaging voltage Vo can be used. Helium best satisfies these criteria. 8.3.1.3. Instrumentation f o r FIM A modern field ion microscope, such as that illustrated in Fig. 8.18, consists of a sample holder and imaging screen mounted in an ultra-high vacuum chamber. It also has provision for changing the tip through a vacuum interlock. The majority of FIM instrumentation is custom built.
I n
,i i
~~:~ ~ T
"
~
I3 ~ ' ~ ~ L ~---i C~PPdr
T
f
I
- i~I
Sapphire,..
1
x L.- I
I
han plate -
tip an eal I To Displex.He refrigerator
Fig. 8.18. Modern ultra-high vacuum Field Ion Microscope with interlock system for changing the tip. (Tsong, 1990, reprinted with the permission of Cambridge University Press.)
386
W.N. Unertl and M.E. Kordesch
The sample holder must be able to heat and cool the tip. Heating is often used as part of the tip cleaning procedure and to allow the surface temperature to be controlled by the experimenter. In most FIMs, the tip is heated by conduction from the hot filament to which it is attached. Heating by other means such as a laser beam are also possible (Tsong, 1990). Cooling is not only required to obtain the optimal resolution, as discussed above, but may also be necessary to control surface properties such as diffusion or adsorption of gases. The sample holder is mounted in a manipulator so that the tip's orientation can be adjusted with respect to the screen. The sample holder is usually constructed so that the sample is placed at a high positive voltage with respect to the screen and the walls of the vacuum chamber, which are grounded. Voltages of up to about 20 kV are required. In the first FIMs, imaging was achieved with a phosphor coated screen. The images were of very low intensity and photographs of the image required long times. High image gas pressures in the 10-3 Torr range were required to maximize the intensity. Modern instruments use image i n t e n s i f i e r s - usually a single microchannel plate with a gain of about 1000. This allows gas pressures to be in the range of 10-5 Torr and images to be recorded in a few seconds or less.
8.3.1.4. Samples for FIM Samples suitable for study by FIM must have several properties. Their electrical conductivity has to be high enough to remove electrons as they tunnel to the sample during ionization of the image gas. Otherwise the sample would charge to a potential at which the ionization would cease. The samples must have sufficient tensile strength to avoid fracture in the high fields. Finally, it must be possible to prepare a sharp tip with radius less than 200 nm. Similar methods are used to prepare tips for the STM and AFM. Samples usually start in the form of wires, bars cut or etched from macroscopic samples, or slivers obtained by fracture of brittle materials. It is often necessary to sharpen the tip further. This is usually done with electrochemical polishing and procedures have been developed to handle nearly all types of materials including metals, alloys, pure and compound semiconductors, and ceramics (Melmed, 1991). After inserting the sample into the vacuum chamber, additional processing is often necessary to modify the tip shape or to remove damaged or contaminated material. Processes available for this include field evaporation, heating with or without an electric field present, sputtering with inert gas ions, and deposition of atoms. Field evaporation occurs whenever the electric field applied to the tip is large enough that the atoms that make up the tip are emitted as ions. Typical fields required are in the 10-100 V/nm range. If the evaporating atoms are adsorbed species the process is called field desorption. A good review of the mechanisms involved in these processes is given by Tsong (1990). Since the electric field is largest at asperities and edges, atoms will be preferentially removed from these sites and the tip shape will become smoother. Thus, field evaporation is a self-regulating process to produce a smooth tip. Since imaging can be carried out during field evaporation, the atomic structure of the tip can be revealed layer by layer. This provides a powerful method to study defects. Point defects, stacking faults, twin
Direct imaging and geometrical methods
387
boundaries and grain boundaries have all been studied using field evaporation. Finally, if the evaporating species are passed through an appropriately designed spectrometer, their chemical identity can be determined. An instrument with this capability is called an atom probe (Tsong, 1990). If the field is reversed to operate in the field electron emission (FEM) mode in the presence of the imaging gas, the emitted electrons can ionize gas atoms which will then be accelerated into the surface. The tip is sputtered by these ions. Tips with only three atoms on the (111) tip have been prepared in this way (Fink, 1986).
8.3.2. lon backscattering techniques Ions with energies from a few keV up to an MeV or more are used as direct probes of surface structure. However, unlike the other techniques described in this chapter, ion scattering does not produce a real space image of the surface structure. Instead, the ion trajectories contain information about projections of the structure along various crystallographic directions and structures are determined using triangulation methods in combination with computer simulations of the data.
8.3.2.1. Basic physics of ion scattering We begin this section with a summary of the basic phenomena that determine the properties of the collisions of ions with atoms. To a very good approximation, in most cases, the dynamics of the scattering process are well described in terms of binary collisions between the incident ion and atoms in the lattice. This is because the energies transferred in the collisions are large compared to the binding energies. Figure 8.19 illustrates the geometry of such a collision. Eo is the kinetic energy of an incident ion of mass M~. Conservation of energy and momentum require that the kinetic energy of the scattered ion is E., = E,,[cos 0 + (A 2 +_sin 2 0)1/2/(1 + A)] 2
(8.21)
where 0 is the scattering angle measured from the forward direction and A = M2/M~ with M2 the mass of the surface atom. If the target atom is lighter than the incident ion (M2 < M~), there is a critical angle 0, = sin -! A
(8.22)
above which no ions can be scattered. The energy ER and direction ~ of the recoiling surface atom (or ion) are related by
E R = Eo[4A/( 1 - A)2]cos2~
(8.23)
The actual intensity distribution of scattered and recoiled particles depends on the interaction potential between the ion and atom. Most calculations assume a screened Coulomb potential of the form
V( r) = ( ZnZzeZ/4 rt~or)cp(r/r,)
(8.24)
where Z~ and Z2 are the atomic numbers of the colliding particles and cp(r/r,) is a
388
W.N. Unertl and M.E. Kordesch Eo A W
M
y
1
O
M 2
Before Collision
6)
/ /
J ER After Collision Fig. 8.19. Geometry of ion-atom binary collision.
screening function with r., the characteristic screening length. Various approximations for q)(r/r,) are used (Mashkova and Molchanov, 1985). Two of the most c o m m o n ones are due to Moliere (1947) based on the T h o m a s - F e r m i model and to Biersack and Ziegler (1982) based on a free electron model and including exchange and correlation effects. In all cases, V(r) falls rapidly with r so that, during any particular collision, the contribution due to interactions of the ion with neighboring atoms in the surface is negligible. One of the basic characteristics of the scattering is the existence of a region behind the target atom, called the s h a d o w cone, which is not penetrated by the incident ions. The formation of the shadow cone was first discussed in a classic paper by Lindhard (1965) and is illustrated in Fig. 8.20. The position of the surface atom is shown by the solid dot and the lines show trajectories for ions incident from the left. Those ions with small impact parameters are scattered through large angles while those with large impact parameters are only slightly deflected. The radius R of the shadow cone a distance d behind the target ion can be calculated analytically for the case of pure Coulomb scattering and is given by ~ R(d) = (ZlZ2e2d/rtG E,,) I/2
(8.25)
1 The expression for R given here is in SI units. The equivalent expression most often found in the literature is in cgs units.
389
Direct imaging and geometrical methods
More realistic potentials yield somewhat different values. At high energies ( 0 . 1 - 2 MeV) and for light ions (H or He), the shadow cone radius, measured about one atom spacing behind the target, is about 0.01 nm. For energies of a few keV, R increases by an order of magnitude and is comparable to typical interatomic spacings in the target. Oen (1983) proposed a universal form for R(d). When the incident ion beam direction is aligned to be nearly parallel to a low index crystallographic axis, it is useful to think of the target as composed of s t r i n g s or rows of atoms. Each row starts at a surface atom and all atoms lying behind it are inside the shadow cone and thus not exposed to the incident ions. As the crystal is tilted, deeper atoms will intersect the edge of the shadow cone and the number of back scattered ions will increase. Figure 8.21 a and b illustrate ways in which shadow cones can be used to extract information about surface relaxations and surface reconstructions. For an unrelaxed surface, Fig. 8.21a, only the surface atoms at the ends of low index strings can contribute to the back scattering. If the surface is relaxed outwards by ~5, there will be no change in scattering by ions incident along the surface normal. However, for inclined angles of incidence, the surface atoms are displaced normal to the string and scattering will occur from subsurface atoms. If the crystal is rotated slightly by 0 until the subsurface atoms are again inside the shadow cone, the back scattered intensity will decrease. Measurements are made for incidence along several low index directions and simple geometric triangulation is used to calculate the magnitude of 5. Figure 8.21 b shows the case of a surface that has reconstructed by pairing alternate surface
r
r ,,,..._
w
R
y
y
!
r
r
Fig. 8.20. Formation of the shadow cone.
390
W.N. Unertl and M.E. Kordesch
9
9
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i~
9
UNRELAXED
S
. . . . . . . . . . . .
9
9
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(a)
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ATOM PER STRING
ONE
ATOM
PER
STRING
ONE
A T O M PER
STRING
UNRECONSTRUCTED
TWO
ATOMS PER STRING RECONSTRUCTED
(b)
Fig. 8.21. (a) Measurement of surface relaxation using the shadow cone. (b) Effect of energy dependence of the shadow cone on backscattering before and after a surface reconstruction. atoms. On the right, the incident energy is low and the shadow cone is wide. In this case, reconstruction causes no change in back scattered intensity. As the energy is decreased, shown on the left, there is no change in the number of atoms contributing to the backscattering from the unreconstructed surface; however, the number doubles for the reconstructed surface. In both of these examples the shadow cone is used like a search light. Figure 8.21 shows examples of the single alignment technique for measuring ion backscattering spectra. The backscattering angle 0 is fixed and the crystal rotated
391
Direct imaging and geometrical methods
AO
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0 Q
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(b) Fig. 8.22. The double alignment technique. The displacement of the top layer atoms results in different angular positions of the surface and bulk blocking cones. (b) Variation of the backscattered intensity as a function of detector position showing the blocking dips due to the surface and bulk atoms.
W.N. Unertl and M.E. Kordesch
392
to align the ion beam with a channeling direction of interest. Single alignment is the simplest technique to implement and is widely used. More sophisticated instruments use the double alignment technique developed by Saris and coworkers (Turkenburg et al., 1976; van tier Veen, 1985). Double alignment, also called channeling/blocking, requires a moveable detector and is illustrated in Fig. 8.22. The ion beam is incident along a channeling direction. Some of the incident ions are back scattered from subsurface atoms. In effect, these subsurface atoms become new sources of ions which can be scattered a second time as they leave the crystal. The shadow cones associated with these second scatterings are observed as minima in the detected intensity called blocking dips. Whenever surface atoms are displaced from bulk-like sites, new blocking dips are observed. The displacement Ae of these blocking dips from the bulk blocking dips is related by simple geometry to the displacement of the surface atoms. The above physical picture is actually a little too simple to provide a reliable quantitative analysis of surface structure. Two energy regimes are normally considered with the dividing line between them at about 100 keV.
8.3.3.2. High energy ion scattering High energy ion scattering is called Rutherford backscattering spectroscopy (RBS) if the ions have MeV energies and Medium energy ion scattering (MEIS) if the energy is around 100 keV. Feldman (1994) has recently described the history of high energy ion scattering as applied to studies of surface properties. The ions most often used are He + ions with energies in the range 1.0-2.0 MeV or H § ions with energies above about 100 keV. The basic equation of high energy ion scattering is the Rutherford scattering cross section:
d~ / l d~-
A s'n 0' L~)
2
sin40(l - A-2 sin20) '~
where all of the quantities are measured in the laboratory coordinate system where the surface is at rest. For backscattering (i.e., 0 > 90~ the impact parameter is small and consequently screening corrections to the Rutherford cross section are small (L'Ecuyer et al., 1979). Screening remains important for forward scattering and reduces the shadow cone radius R(d) by as much as 20% from the value calculated using eqn 8.25. Only a small fraction of the incident ions have impact parameters small enough to be back scattered. If the ion beam direction is aligned along a low index direction of the sample, the majority ions have large impact parameters and are not strongly scattered. Instead, they undergo a series of correlated collisions with atoms on neighboring strings (Lindhard, 1965; Tesmer and Nastasi, 1995). These ions penetrate far into the crystal and are said to be channeled. As they travel through the solid, they collide predominantly with loosely bound valence electrons and are gradually degraded in energy. This energy loss is described by the electronic stopping power dE/dx; dE/dx is typically a few hundred eV/nm (Tesmer and Nastasi, 1995). If these channeled ions are eventually backscattered out of the solid, their energies will be smaller than the energy of ions elastically backscattered from
Direct imaging and geometrical methods
,,oo[-- ,k~~
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the surface; i.e., channeling suppresses backscattering from the bulk. Thus, a measured energy spectrum of ions back scattered through the same 0 consists of a "surface peak" due to direct backscattering from atoms at the ends of strings plus a background due to ions scattered from deeper in the crystal. Figure 8.23 shows an example for 2.0 MeV He + ions scattered from a W(001) surface. The open circles are the back scattered spectrum for incidence along the [001 ] direction. The surface peak occurs near 1.8 MeV and its width is limited by the energy resolution of the detector. The area of the surface peak is directly proportional to the number of atoms per string visible to the ion beam. If the ion beam direction is a few degrees from an aligned direction, called a random direction, channeling cannot occur and bulk scattering is not suppressed. The solid dots in Fig. 8.23 show a backscattering spectrum measured along a non-channeling direction. R is comparable in magnitude to the thermal vibrational amplitudes P of atoms in real materials. Thus, more than one atom along a string will be visible to the incoming ions, even for perfect alignment, and the intensity of the surface peak is increased over the value expected for a rigid lattice. Stensgaard et al. (1978) used computer simulations to show that the increase is a universal function of p/R as illustrated in Fig. 8.24.
394
W.N. Unertl and M.E. Kordesch
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The deviation of the scattering potential from the simple Coulomb form and the importance of thermal vibrations require that computer simulations be used for quantitative analysis of the data. This is done using Monte Carlo techniques based on an approach originally developed by Barrett (1971). See Feldman (1989) for additional references. The ion-solid potential is usually taken to be a linear superposition of the individual atomic potentials. Since the backscattering at high energies is dominated by events with very small impact parameters, any potential which accurately describes the inner shell electron distribution is suitable for determination of corrections to the Rutherford cross section. However, the precise flux distribution at the edge of the shadow cone is determined by the small angle scattering and is sensitive to the choice of potential. The Moliere potential is most often used. The probability that an atom on a string will extend beyond the shadow cone is calculated assuming that thermal vibrational amplitudes have a Gaussian distribution. The effects of correlated vibrational motion and variations near the surface must also be included in determination of p (Unertl, 1982). Monte Carlo
Direct imaging and geometrical methods
395
simulations of the double alignment geometry (Tromp and van der Veen, 1983) are additionally complex because each ion trajectory must be followed both on the way into the crystal and on the way out. Equipment for high energy ion scattering studies of surface structure is expens i v e - in the one million dollar r a n g e - and therefore is available at only a few laboratories. The ion accelerator must be capable of producing ions with energies between 0.1 and 2 MeV at beam currents in the 0-100 nA range with an angular divergence less than 0.1 ~ Beams with these properties can cause severe damage to the sample and precautions must be taken to minimize the extent and effects of the damage. Special shielding and licenses are required to use some types of beam because of radiation hazards. The accelerator must be interfaced to an ultra-high vacuum chamber. The crystal sample holder must have at least two rotation axes an azimuthal axis normal to the surface and a tilt axis normal to it. Angular orientations must be reproducible to at least 0.02 ~. In addition the sample must be accessible for cleaning and analysis by other surface sensitive techniques. The capability to vary the temperature over a wide range is essential for many experiments. In the single alignment geometry, the detector can be as simple as a stationary Si surface barrier detector. Although absolute measurements are possible because the scattering process is well understood, calibration standards are also available (L'Ecuyer et al., 1979). Once an apparatus has been calibrated using a standard, only relative measurements are required. In the case of the double alignment technique, the detector must have the capability to detect the spatial distribution of the back scattered ions and, consequently, the apparatus is more complex (Tromp and van der Veen, 1983; van der Veen, 1985). More detailed information about the experimental aspects of high energy ion scattering can be found in the references (Tesmer and Nastasi, 1995; Feldman, 1989). High energy ion scattering measurements have made many significant contributions to the understanding of surface crystal structure of clean and adsorbate covered systems. We briefly mention a few here. Ion scattering can accurately measure deviations from a simple bulk termination of a crystal and a number of important determinations of surface relaxations have been made (van der Veen, 1985; Bohnen and Ho, 1993). In most cases, these measurements are in good agreement with the results of LEED studies which not only gives confidence in the results, but provide a check on the reliabilities of the two techniques. Chapter 3 contains a more detailed discussion of surface relaxations. Very accurate determinations of the number of atoms displaced from bulk positions as viewed along various crystallographic directions is a quantity directly determined by high energy ion scattering. Thus, it has played a very important role in structure determination because knowledge of the number of displaced atoms places severe restrictions on possible structural models. Even though the ion scattering data by itself cannot uniquely determine the structure, it reduces the number of models that need to be tested by other techniques. Some of the most important examples are for surface reconstructions such as those on low index faces of noble metals, as discussed in Chapter 3, and more complex systems (Copel and Gustafsson, 1986; Fenter and Gustafsson, 1988). Adsorbate induced reconstructions
W.N. Unertl and M.E. Kordesch
396
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Z
i
\
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n~ 0
r
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0.6
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;
550 (K)
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Fig. 8.25. Correlation of ion scattering data with CO coverage and work function changes during the hex-to-( I x l ) phase transition on Pt(001 ). (From Jackman et al., 1983.)
such as the H-induced displacive reconstruction on W(001) (see Chapter 13) and the CO induced h e x - t o - ( l x l ) transition on Pt(100) (Jackman et al., 1983) are important examples. Figure 8.25 shows typical data for the latter case. Interfacial structures developed during the initial stages of thin film growth are also accessible to high energy ion scattering as demonstrated by the early work of Feldman on the epitaxial growth of Ag on Au(111) (Feldman, 1989). This is likely to be one of the most important areas of application of ion scattering. 8.3.3.3. Low energy ion scattering Low energy ion scattering most often refers to ions with energies in the 1-10 keV range although both lower and higher energies are used. The technique is also often called ion scattering spectrometry (ISS). A historical review of the development of ISS as a useful tool for surface crystallography has been given by Rablais (1994). The shadow cone is much broader for low energy ions and more sensitive to screening by the valence electrons than is the case for higher energy ions. Thus, simulations of ISS scattering are more sensitive to the choice of scattering potential. Because the radius of the shadow cone is comparable to typical interatomic spacings, the ions are able to penetrate only the outermost few layers of a crystal and
Direct imaging and geometrical methods
397
channeling is usually unimportant. Also, because the scattering cross section and shadow cone radii are large, multiple scattering events are relatively much more important for ISS than for RBS or MEIS. Neutralization and sputtering are additional effects that are important for ISS. Because a low energy ion moves relatively slowly, there is a significant probability that it will pick up an electron and be neutralized during its interaction time with the surface (Woodruff, 1982; Souda et al., 1985; Beckschulte and Taglauer, 1993). Thus only a fraction of the backscattered ions will be charged. Many ISS systems, particularly older ones, use electrostatic analyzers that have high energy resolution but are unable to detect the neutral ions. More modern systems use time-of-flight methods that have lower energy resolution but are able to detect all of the back scattered ions. The neutralization probability is often significantly lower for alkali ions (Hagstrum, 1977) and this has lead to the widespread use of Li § and Na § beams, especially when electrostatic analyzers are used. The impact of a low energy ion with a surface also causes substrate atoms to be ejected (Robinson, 1981; Sigmund, 1981). In fact sputtering by low energy inert gas ions is one of the most widely used methods to clean a surface. Since sputtering can cause significant damage to a surface, special care must be used to insure that the surface structure is not significantly altered during the time required to record a data set. The simulation of ISS experiments has reached a rather sophisticated level in the sense that reasonable approximations to the scattering potentials are available and several standard computer codes, including ones that can be run on personnel computers, are available to carry out the necessary Monte Carlo simulations (Robinson, 1981 ; Teplov et al., 1994). Figure 8.26 gives examples of how the surface structure influences the back scattered intensity for the case of 2 keV Ne ions incident on a reconstructed Pt (110) (Ix2) surface. The data show the backscattered intensity for fixed_scattering angle as a function of the incidence angle for incidence along <001>, <112>, and <1 I0> azimuths (6 = 0, 35.3, and 90 ~ respectively) as shown in the plan view of the surface. The right hand part of the figure shows the scattering events that are responsible for each labeled feature (A, B, C, D) in the backscattered intensity profiles. The ions first scatter from a surface atom and, whenever the edge of the shadow cone intercepts another atom, a second scattering takes place causing the steep rises in backscattered intensity that are indicated by the open circles. Clearly substantial information about the geometrical arrangement of the substrate atoms can be extracted from the locations of the steep rises and a knowledge of the shape of the shadow cone. However, complete quantitative evaluation of structural models, including refinements of atom locations, requires modeling as described in the previous paragraph. Equipment for low energy ion scattering is compact enough to be incorporated as part of an ultra-high vacuum surface analysis system. The primary components are an ion source, a high quality manipulator capable of varying the incidence and azimuthal orientation of the sample, and a detector capable of energy or velocity analysis. Figure 8.27 shows the basic components of an instrument with time of
398
W.N. Unertl and M.E. Kordesch
Fig. 8.26. Low energy ions scattering data from a Pt(110) (lx2) surface for a scattering angle of 149~ and three different azimuths 5. The ball models on the right show the scattering trajectories that cause the major features in the backscattering data. (Reprinted from Rablais, 1994, with kind permission from Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.) flight detector for velocity analysis (Rablais, 1994). The ion source provides a collimated beam of a few keV energy and must usually be differentially pumped to maintain ultra-high vacuum in the sample chamber. If time of flight detection is used, the beam must also be pulsed.
8.4. Electron
microscopy
Electron microscopy and surface science have developed as separate fields due to several significant differences in their goals and methods. Electron microscopes evolved along a course dictated by image resolution; the result was high voltage, high intensity electron beam microscopes, with sample size and access dictated by the focal length of compact, solid lens bodies, and poor vacuum. The latter because commercial microscopes based on non-bakeable construction and materials with diffusion-pumped vacuum were firmly established for thirty years before ultra high vacuum technology became known or semi-practical.
Direct imaging and geometrical methods
399
c..cl,,,,,,!
Ip~. A
B
C
O
[ !
E C
F
/
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l
{ "
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. . . .
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ICONTROLLERJ
Fig. 8.27. Schematic diagram of the major components of a low energy ion scattering system with time of flight detection. A is the ion source, B is a Wien filter to monochromate the beam, C is an electrostatic lens for focusing, plates D are used to pulse the beam, E is an aperture, F are deflection plates to steer the beam, G is the sample, H is an electron multiplier with an energy prefilter, and I is an electrostatic deflector used to remove ions from the detected beam. (Reprinted from Rablais, 1994, with kind permission from Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.) Surface science began with ultra high vacuum and was primarily a spectroscopic science ("non-direct methods") until the advent of scanning Auger spectroscopy and other imaging spectroscopies and the STM and AFM (Duke, 1994). The need for a low penetration depth by the electron probe and the primary goal of surface sensitivity in surface studies has led to low energy electrons as probes or signal in surface spectroscopy and microscopy. Ultra high vacuum microscopes are essential for the comparison of microscopy and surface science results. The goal is for UHV electron microscopy "to show LEED, Auger, electron energy loss, transmission electron diffraction and transmission electron microscopy results all from the same surface" (Marks, 1991a). Today one might include STM or AFM, and substitute LEEM or REM for TEM. The UHV versions of these microscopes are relevant to surface science, and may bridge the gap between direct (imaging) and non-direct methods (spectroscopy and diffraction). The term "direct" imaging, used loosely in this chapter to distinguish methods that produce an image of the surface rather than a spectrum or diffraction pattern alone, is more often applied to "parallel" imaging as opposed to sequential imaging. In parallel imaging methods, the entire image is acquired at one instant: no scanning or sequential image composition occurs. Generally, only "direct" meaning "parallel" imaging methods will be examined here. A comprehensive survey of all types of surface imaging has been made recently by Hubbard (1995).
400
W.N. Unertl and M.E. Kordesch
There are literally hundreds of books on electron microscopy. A personal selection from elementary to extremely specialized is: Wischnitzer (1962), beginners; Watt (1985), broad based introduction; Busek et al. (1988), comprehensive, including surfaces; Spence (1988), practical TEM and related techniques; Zuo and Spence (1992), microdiffraction; Bethge and Heydenreich (1987), application to solid state physics; Septier (1967) and Hawkes and Kasper (1989), electron optics for microscopy. The history of electron microscopy can be found in Ruska (1980) and Hawkes (1985). The journal Ultramicroscopy is a valuable reference, often devoting a single volume to one topic with state of the art discussion by experts. There are Internet newsgroups and list-servers, as well as World Wide Web sites from manufacturers, individual laboratories and professional societies for electron microscopy.
8.4.1. Image formation in the electron microscope Image formation in electron microscopes is analogous to image formation in light optics. Much of the same terminology and concepts are used in both. The detailed discussion of image formation is expertly handled in the book by Spence (1988). An extended discussion of contrast and image interpretation in the many variants of electron microscopy will not be attempted here. The basic ideas necessary to understand how the wealth of accumulated knowledge in electron microscopy can be applied to the imaging of surfaces will be touched upon below, with a guide to further reading. Yagi ( 1988, 1989) has reviewed the progress in surface imaging up to 1989 with several methods. Bonevich and Marks (1992) have made a short and readable progress report up to 1992. The idealized process of image formation in an aberration-free optical system where the optical transfer is linear and the small angle approximation is valid consists of a sequence of two Fourier transforms that combine the Fourier analysis of the object wavefunction into the diffraction pattern and the Fourier synthesis of the diffraction pattern into the image. In the wave optical treatment, the specimenbeam interaction results in an object wavefunction (written in one dimension for simplicity) ~o(X) at the exit or reflection plane of the specimen. The electrons described by ~o have either undergone scattering after transmission through the object (TEM), or reflection at glancing (REM) angles or normal reflection (LEEM, EEM), and have been altered by interaction with the notential of the object or object surface. The action of the objective lens is mathematically equivalent to a Fourier transform of gto that results in a Fraunhofer diffraction pattern at the back focal plane of the objective lens. The wave function in the back focal plane, ~ ( u ) , where u is in reciprocal space, is again transformed to the real space image at the Gaussian image plane: gtd(U) = F[~o(X)] and ~i(x)= F-l[~d(U)]
(8.27)
where F represents the Fourier transformation operation. The degree to which the
Direct imaging and geometrical methods
401
sample characteristics and the optical system allow faithful transformation of the object wavefunction into the image wavefunction defines the essence of electron microscopy. When non-ideal optics and specimen are considered, lens aberrations such as spherical aberration Cs, defocus value Af, finite aperture size, specimen and illumination characteristics must all be taken into account. For coherent, axial illumination, the amplitude ("Scherzer") contrast transfer function describes the phase shift of the electron wave due to spherical aberration and defocus, and is the first step in calculating image parameters such as resolution. For a scattering angle | = u)~, C(|
= exp[(-2xi/~,){ (Af/2)(~ 2 - (C,]4)(~4},]
(8.28)
defines the Scherzer contrast function, with Af the defocus. Thus in a real optical system: ~j.(u) = ~d(u)C(u)A(u) = ~d(u) exp(ix)A(u) = { (C,]2)~,3u 4 - A f L u z }
(8.28) (8.29)
A(u) is the aperture function and includes chromatic aberration, Cc, and beam characteristics and C(u) is the contrast transfer function. The intensity distribution in the image is proportional to [tt/il2, that of the diffraction pattern is proportional to I~d,I2. The contrast transfer function can be oscillatory and have several zeros. The conditions for optimum resolution are derived from the behavior of the real and imaginary parts of C(u), often expressed as the sine and cosine of X. The manipulation of ted' with an aperture in the back focal plane is important for bright and dark field imaging, high resolution TEM, and critical to surface specific imaging in some instances; these effects arise fiom amplitude contrast associated with the real part of the contrast transfer function. Phase contrast is associated with the imaginary part. For a full treatment with extensive mathematical detail and references to several other books, see Spence (1988).
8.4.2. Transmission electron microscopy 8.4.2.1. Physical principles of operation Surfaces are generally imaged in two ways in UHV-TEM: in plan view, using standard TEM methods such as dark field imaging or with surface diffraction beams or HRTEM to achieve sufficient contrast and resolution, and in profile. UHV-TEM instruments are further divided into microscopes where the surface analysis and preparation is made in the microscope, or made in an attached preparation chamber and subsequently transferred through UHV. Some instruments are only UHV in the sample region. Instruments where the surface is prepared and transferred through air are not considered.
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8.4.2.2. Resolution Resolution in TEM is directly related to the contrast transfer function (Eq. 8.28), since the optimum focus and aperture size for optimum resolution are calculated from this function. There are several choices for how, and in what plane, resolution is calculated. Sometimes a disk of confusion is used. A point object is imaged as a disk of diameter d. Here d = Cfc~ 3, w h e r e f i s the focal length of the objective lens, ~ is the aperture and C.~.is the spherical aberration. The necessary use of small apertures and focal lengths to increase resolution is clear from this formulation. For the optimum defocus and aperture values Afopt = -1.22(C,~,)'/2, O~op t "-
1.4(L/C,.)!/4
(8.30) (8.31 )
the often-derived value for point-to-point resolution is 8 = 0.66C1/49~ 3/4 .~.
(8.32)
Tsuno (1993) has reviewed experimental progress in reaching the resolution limits in TEM, 0.1 nm is now relatively common. Sarikaya 1992 reviewed "Resolution in the Microscope", covering TEM, REM and EEM, and several more.
8.4.2.3 Instrumentation UHV TEM instruments are usually commercial instruments that are special order, one-of-a-kind products, built using the combined expertise of the manufacturer on more commonly produced microscopes and the customer. The cost for one of these sophisticated microscopes is in the million dollar range, and can climb much higher as "extras" are added. Skilled operators and maintenance are also a requirement. The "home-made" UHV TEM is not competitive. A recent example of the state-of-the-art UHV TEM is shown in Figure 8.28; a photograph of the Hitachi UHV H-9000 microscope attached to the Sample Preparation Evaluation Analysis and Reaction (SPEAR) system at Northwestern University (Collazo-Davila, 1996). A schematic diagram of this system is shown in Figure 8.29. A list of microscopes from various manufacturers is given in Bonevich and Marks (1992). Usually options such as electron energy loss spectroscopy, Auger spectroscopy and X-ray fluorescence spectrometers can be attached to the TEM. A combination instrument called MIDAS is described by Liu and Cowley (1991).
8.4.2.4. Samples for surface TEM Three types of specimen are common in UHV TEM. Noble metals that can be prepared by heating and have low adsorption coefficients, such as gold. Semiconductor surfaces that can be cleaned by sublimation, and metals deposited on clean semiconductor surfaces. Metals are usually thinned mechanically, dimpled, and ion milled to 10 nm thickness or less for profile imaging.
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Fig. 8.28. A photograph of the Hitachi UHV H-9000 microscope attached to the Sample Preparation Evaluation Analysis and Reaction (SPEAR) system at Northwestern University (Collazo-Davila, 1996).
Fig. 8.29. A schematic diagram of the SPEAR system. Surface studies are usually made at 300 kV or less, at higher voltages, displacement of the atoms by the beam becomes possible. Haga and Takayanagi (1992) have imaged individual Bi atoms on Si in HRTEM. The It(001) surface has been studied in UHV TEM (Dunn et al., 1993). Profile imaging is demonstrated on CdTe by Smith et al. ( 1991).
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Fig. 8.30. UHV Transmission electron diffraction TEM diffraction pattern from a region with predominantly a single domain of the Si(l 11 )(5• structure. This pattern was indexed in terms of a centered (10x2) unit cell, thus the arrow points at a strong surface beam with (h,k)= (13,2). (Marks and Plass, 1995).
Fig. 8.31. Near Scherzer defocus, noise filtered, off-zone HREM image of the Si(11 l)(5x2)-Au surface. Clearly visible are two (arrows) rows of dark features which correspond to gold atoms. (Marks and Plass, 1995).
Figure 8.30 s h o w s a U H V T E M diffraction pattern and Fig. 8.31 a high resolution i m a g e from the Si(1 1 l ) ( 5 x 2 ) - A u structure ( M a r k s and Plass, 1995).
8.4.3. Reflection electron microscopy R e f l e c t i o n electron m i c r o s c o p y ( R E M ) is similar to T E M , so much so that the s a m e m i c r o s c o p e can be used for R E M as for T E M . H i g h voltage b e a m s are used as for T E M . T h e R E M m e t h o d bases its surface sensitivity on very s h a l l o w i n c i d e n c e and reflection angles; it is closely related to R H E E D , much as L E E M is related to
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LEED. The utility of REM is also much like RHEED, surface steps and the morphology of growing surfaces are the ideal specimens for study. Image formation in REM is in principle the same as for TEM, most of the techniques associated with TEM have analogous REM cousins: bright and dark field imaging, interference fringes, SREM, convergent beam-REM, REM holography, and others. Because of the shallow incidence angle, REM images show severe foreshortening effects; only a narrow band of the image perpendicular to the incidence direction of the electron beam is in focus. The imaging of steps is possible mainly through phase contrast. Resolution is typically about 1 nm. Some sample preparation methods are similar to TEM studies, i.e. semiconductors cleaned by sublimation in UHV. An effective technique in REM, not strictly UHV, is to observe the surfaces of very fine melted wires. The cooled wires are terminated by recrystallized, facetted spheres suited to glancing incidence illumination and easy rotation and alignment of the incidence azimuth for diffraction (Lehmpfuhl and Uchida, 1993). A REM example from the study of such gold spheres is shown in Fig. 8.32, which shows the reconstructed Au(l 11) surface with its (23• superstructure. In this particular example, the surface reconstruction was
Fig. 8.32. (a) REM image of the reconstructed Au(ll 1) surface with (23• superstructure recorded with the intensity enhanced (666) rellection near the [ 112] zone axis. Periodicity of the structure is 6.6 nm. The horizontal line of exact locus is shown by the two bars. The dark area is another domain of reconstruction rotated by 60~, where the superstructure is not resolved because of foreshortening. (b) Diffraction pattern with fundamental and superstructure spots (Wang et al., 1992).
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W.N. Unertl and M.E. Kordesch
Fig. 8.33. (a) REM image of the reconstructed Au(100) surface showing in two domains the twisted ccll of reconstruction with 1.4 and 4.0 nm. The image was recorded with the intensity enhanced 10.00 rcflection near the [ 110] zone axis. (b) Diffraction pattern and with the fundamental spots and the superstructure spots of the two lattices (Wang et al., 1992).
maintained after passing the specimen through air by heating the specimen jn the microscope to about 200~ A technique specific to REM due to the b e a m - s p e c i m e n geometry is the excitation of surface resonance scattering parallel to the specimen surface. Intensity enhancement of surface Bragg diffracted beams that meet the resonance conditions (see e.g. Lehmpfuhl and Uchida, 1993; or Wang and Bentley, 1991 ) is observed and is beneficial for surface imaging. The image in Fig. 8.32 and the one in Fig. 8.33 were recorded using surface resonance intensity-enhanced reflections. Buried interfaces have also been imaged in an unusual type of REM (Spence, 1994). The more open specimen geometry in REM allows versatility in additional spectroscopy or measurement in these microscopes. STM and REM have been combined in one instrument (Lo and Spence, 1993). Spectroscopy for chemical analysis can also be added. Such a combination is shown for MgO surfaces in Fig. 8.34, from Crozier and Gajardziska-Josifovska (1993) with EELS of O and Mg included with the REM image. Due to the long working distance and possibilities for
Fig. 8.34. (a) A REM image from the freshly cleaved (100) surface and the accompanying RHEED pattern showing the (800) specular reflection at surface resonance. (b,c) REELS spectra and acquired from the (800) resonance condition from the freshly cleaved (100) surface showing the presence of oxygen (b) and Mg (c) K-edges. (Crozier and Gajdardziska-Josifovska, 1993).
Opposite:
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x 10:~
40o t
'
'
3.20
'
'
I
i
b
O-K
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1.60
0.80
0.00
I
I
ENERGY LOSS [eV] x 104 5.50
I
i
i
i
c'
4.40 Mg-K 3.30 2.20 1.10
0.00
I
I
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ENERGY LOSS leVI Fig. 8.34. C a p t i o n opposite
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W.N. Unertl and M.E. Kordesch
408
elemental analysis some forms of REM may be applied to in situ evaluation of film growth in the future. REM has been comprehensively reviewed by Peng and Kuo (1993) with a bibliography of the technique.
8.4.4. Emission electron microscopy Emission microscopy usually refers to electron microscopy using an immersion objective lens or "cathode lens". An acceleration voltage of several kV is applied between the specimen and first objective lens element, a distance which is commonly several mm. There are many types of emission microscope (see Kordesch, 1995): thermionic emission, photoelectron emission, soft X-ray and X-ray absorption near edge structure microscopes based on synchrotron radiation illumination, and low energy electron microscopy, using low energy electrons and commonly known as LEEM. Because LEEM is an electron probe microscopy, most of the discussion here will concern LEEM and Mirror Electron Microscopy (in the case when the specimen is not a single crystal). Other emission microscopes and variants will be discussed in w 8.4.5.
a. T H E R M I O N I C
d. SECONDARY
g. LEED
b. PEEM
r XPEEM
e. AUGER
f. MIRROR
h. D I F F R A C T I O N
J. D A R K FIELD
"~///)/ff/////~ j. FRESNEL
It. I N T E R F E R E N C E
L SPLEEM
Fig. 8.35. Contrast mechanisms available in a LEEM microscope (Veneklasen, 1992).
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LEEM uses electrons in the energy range of 2-100 eV for specimen illumination; the LEEM geometry also allows observation of solid, thick single crystal specimens. These facts make LEEM images directly comparable to other surface science methods. The history of what is justifiably called the "Bauer LEEM" has been reviewed by Bauer himself (Bauer, 1994a). Results obtained with the several generations of this instrument have also been reviewed by Bauer (1994b, 1995). The design parameters and technical details of LEEM have been presented by Veneklasen (1992). Recent advances in emission microscopy in general are assembled in Griffith and Engel (1991 ).
8.4.4.1. Principles of operation The LEEM is similar in function to TEM, because both a diffraction pattern and real space image are formed by the optical system, but very different in practice. The illumination source and Gaussian image are on the same side of the specimen, making the geometric arrangement of a LEEM very different from the straight columns usual for TEM. Also, normal electron lenses function very poorly at the 2-100 eV used for LEEM. As noted above an immersion objective lens is used. The contrast in LEEM images is also very different from TEM, because it depends on diffraction and electron reflectivity of the specimen surface at very low electron energies compared to TEM. Data known as VLEED, very low energy electron diffraction, phenomena are related to image contrast in LEEM. Veneklasen (1992) has diagrammatically listed the contrast mechanisms available in LEEM; these are shown in Fig. 8.35.
8.4.4.2 Resolution in LEEM The resolution of emission microscopes is commonly calculated (as a rough estimate) with the Recknagel formula, which relates the resolution of a cathode lens to the ratio of the starting voltage of the emitted electrons (Vo) to the electric field strength at the specimen surface (E): ~5 = Vo/E. In this formulation, the resolution limit is due to the limitation of the accelerating field. The field strength is practically limited to about 100 kV/cm. The energy spread of the emitted electrons depends on the illumination method. The Recknagel formula was derived without including an aperture or control of the energy width of the illumination source. An energy distribution of the emitted electrons was assumed from thermodynamic relations of source temperature and a Maxwellian distribution appropriate to thermionic emission and photoelectron emission. In LEEM, contrary to what is expected for thermionic and ultraviolet photoelectron emission microscopy where the Recknagel formula applies, Bauer proposed that a narrow energy spread in the illuminating beam and an optimum aperture could allow LEEM resolution to approach the 5 nm range, and possibly less with further instrument development (Skoczylas et al., 1991). In terms of the discussion of TEM and REM, the resolution in LEEM is calculated differently, due to the inclusion of the electron emission parameters and the action of the homogeneous field between the sample surface and the lens that constitutes the "cathode" lens.
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W.N. Unertl and M.E. Kordesch
The aberrations of the objective lens system is added to these considerations to arrive at the total resolution limit. The principle, however, is the same: a calculation for optimum aperture and energy spread for a final energy of the source is made (See Bauer, 1994b). With the cautionary note that real and theoretical emission microscopes are very different, and that resolution calculations are also very different for different purposes (Rempfer, 1992), the theoretically attainable resolution in an emission microscope can be written as a combination of the aberrations of theelectric field used to accelerate the emitted electrons and the aberrations of the objective lens. The aberrations will depend on both the electron emission energy e Vo and the electron emission energy spread eAVo, and the accelerating field E = V/a, where a is the distance between the specimen and cathode lens. The spherical and chromatic aberration can be written (Engel, 1968):
8x s = [ Vo/E + Cs( VJV)3/2]sin317~,
(8.33)
6Xc = [ V~IE + Cc(V~IV)3/2](AVolVo) sin oq,
(8.34)
where ~0 is the electron emission angle relative to the surface normal, and Cs and Cc are the aberration coefficients of the objective lens (see Engel (1968) for a calculation for a magnetic objective lens in PEEM; Rausenberger (1993) for LEEM with magnetic objective lens; and Veneklasen (1992) and Rempfer and Griffith (1989) for electrostatic objectives). From Eqs. (8.33) and (8.34) the strong dependence of the resolution on emission energy and emission energy spread is clear. Rempfer and Griffith make explicit comparisons of emission microscope resolution with the Scherzer formula, the interested reader is referred to the discussion in Rempfer and Griffith (1989), and Rempfer (1992). The discussion in Rempfer (1992) is a clear and concise introduction to the general question of resolution in the electron microscope. Bauer has commented that resolution without sufficient intensity and contrast for focusing is useless (Bauer and Telieps, 1988). Detailed calculations of electron illumination intensity, expected image intensity and problems of specimen damage, etc., have been made by Veneklasen 1992 for most modes of LEEM operation.
8.4.4.3 Instrumentation There are less than a dozen LEEM instruments in the world today, about half are actively producing LEEM data. Bauer and Tromp offer commercial instruments, some are still "home-made". The commercial instruments are comparable in cost to UHV TEM instruments. It is necessary to separate the illumination and imaging beam paths. The LEEMs in use today are all based on a 60 ~ deflection of the incident and exit beams using a magnetic prism before and after they reach the specimen at normal incidence and exit angles (see Fig. 8.36). The LEEM optics operate at a modest 15-20 kV throughout most of the beam path, so that no special high voltage technology is necessary. At the sample, the electrons are slowed to < 50 eV; in the region of the sample and objective lens,
Direct imaging and geometrical methods
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Fig. 8.36. A photograph of the IBM LEEM. The electron column is on the left, the imaging column on the right-hand side. The sample chamber and objective lens are at the top center, above the magnetic sector deflection magnet in the center of the photograph (Tromp and Reuter, 1992).
compensation of the earth's magnetic field, shielding of non-uniform static fields, and dynamic compensation of AC magnetic fields is often necessary. The magnetic lenses other than the objective are external to the vacuum system. An exception is Rempfer (Skoczylas et al., 1991), who has built a compact, electrostatic LEEM assembled completely on an a electron optical bench. Specimen chambers and sample manipulators can be adapted to LEEM from standard UHV equipment. The original Bauer-Telieps LEEM had a provision for flipping the specimen 90 ~ from imaging to "sputtering" position, other LEEM instruments have sputter guns at glancing incidence. Most LEEMs are equipped with sample heating to 1500 K or more, some with cooling to LN2 temperature, adsorbate dosing arrangements for UHV surface chemistry and a few with fast sample transfer or load lock capability. The IBM LEEM developed by Tromp is shown in Fig. 8.36. The "Y"-type of geometry is characteristic of present-day LEEM instruments.
8.4.4.4. Samples for LEEM The samples for LEEM are the same as for any standard surface science experiment: single crystal disks a few mm thick and 10-20 mm diameter, evaporated films, semiconductors, and some thin insulators; some of the insulators can be imaged by simultaneous illumination with UV light to promote photoconductivity. In order to make use of the diffraction contrast mode, single crystals or epitaxial film sub-
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W.N. Unertl and M.E. Kordesch
Fig. 8.37. (Top) An AFM image of an epitaxial Ag island on Si(l 11). A LEEM image of the circled region is shown (bottom). The field of view in the LEEM image is 4 lam. Arrows: a, interface steps, b, surface steps c, stacking faults. (Tromp et al., 1993). strates are necessary. It is also necessary for LEEM samples to be flat and smooth for best results, because the emission microscope has relatively poor depth of field, and irregularities on the surface cause distortions in the accelerating field. An image recorded with the IBM LEEM by Tromp (Tromp et al., 1993) is shown in Fig. 8.37. In this figure an AFM image is also shown with the LEEM micrograph. Both interfacial atomic steps and surface steps can be identified in the LEEM image of silver islands grown on S i ( l l l ) . This particular example is chosen from an already large LEEM literature because a direct comparison is made between the available information from LEEM, AFM and plan view TEM, including sub-surface features, purely surface phenomena and some features which were observed in real time, in situ with LEEM (features marked c) that resulted from cooling of the Ag film after growth was stopped. The interplay of several imaging techniques is elegantly displayed in this study.
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While the majority of L E E M work to date has concerned silicon or metals adsorbed on other semiconductors or metals, some surface chemical reactions that can be understood only with real time in situ observation of both adsorbate distribution on a surface a n d the dynamics of the underlying surface crystal structure have also been investigated with LEEM. The oxidation of CO to CO2 on platinum surfaces is a problem of surface chemistry that has been investigated by several generations of surface scientists; it may also fall to direct imaging techniques as did the Si(111)(7x7) reconstruction to TEM and later STM. An immense body of literature has been produced with L E E M , P E E M and new optical microscopy techniques (see below) that have solved many of the questions related to the dynamics of the CO oxidation reaction. Rausenberger (1993) has studied the oxidation of CO on the Pt(100) surface in situ and in real time with L E E M and Mirror Electron Microscopy (MEM). Figure 8.38 shows a sequence of LEED patterns recorded during the reactive desorption of CO from the Pt(100)c(4x2)-CO surface with oxygen at 300 K. The
Fig. 8.38. A sequence of LEED patterns observed recorded during the reactive desorption of CO from the Pt(100)c(4x2)-CO surface with oxygen at 300 K. The reaction diffusion front passed through the illuminating electron beam during observation. The succession of LEED structures: (a) c(4x2), (b) c(4x2) and C~-2-x3"~-)R45, (c) (~-x3"~-)R45, (~'~-x'~-)R45 and (5xl), (d) (~-x'~-)R45 and (5xl), (e) (5xl) and (f) (2xl) are clearly visible. The electron energies for (a)-(e) are 27 eV, (f) 54 eV. (Rausenberger, 1993).
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Fig. 8.39. A LEEM image at 23 eV (a,b) of a similar event as is in Fig. 8.38 is shown with a schematic representation of the surface structures (c,d) that give rise to the LEEM image contrast. (Rausenberger, 1993). reaction diffusion front passed through the illuminating electron beam during observation. The succession of c(4x2), c(4x2) and ('#-2-x3~r2-)R45, (~-x3"~r2)R45, (~2-x'(2-)R45 and (5x l), and other complex structures is observed over several seconds as the front passes. In Fig. 8.39, a LEEM image of a similar event is shown with a schematic representation of the surface structures that give rise to the LEEM image contrast. A M E M image of a reaction diffusion front crossing through the field of view is shown in Fig. 8.40. The determination of CO structures involved with the reaction diffusion front allowed the interpretation of previous PEEM results that showed work function contrast in the reaction diffusion front, and the selection between models of the CO-oxidation reaction. In a related measurement on Pt(110) (Rose et al., 1996), the spiral patterns that have become well known from PEEM results (Ertl, 1991; Rotermund, 1993) have been imaged with mirror microscopy in a LEEM-type instrument. Spiral reaction fronts that were pinned to surface d~fects and travelling spirals were observed in real t i m e , in situ, at partial pressures of the reactants of 10 -5 Pa and temperatures of 4 3 0 - 4 5 0 K. An example of this imaging mode is shown in Fig. 8.41.
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Fig. 8.40. A MEM image of a reaction diffusion front crossing through the field of view, similar to that in Figs. 8.38 and 8".39. (Rausenberger, 1993).
Fig. 8.41. A sequence of MEM images showing an elliptically shaped target pattern during the catalytic oxidation of CO on Pt(110). Dark bands represent CO-dominated areas, light bands are O-dominated areas. (Rose et al., 1996).
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8.4.5. R e l a t e d m e t h o d s
There are several direct and scanning imaging methods that may yet prove to be invaluable to surface scientists. In most cases, the beneficial aspect of the imaging mode may not be purely resolution. In some microscopies, the compatibility with reaction conditions such as high pressure and temperature may be deciding factors; possibly the detection of a single monolayer or less with reduced lateral resolution would be sufficient for some purposes. Figure 8.42 shows a PEEM image of a bulk-diffusion mediated surface reaction-diffusion front on Mo(310) at elevated temperature, 950 K (Kordesch, 1995b; Garcia and Kordesch, 1995). In principle, PEEM can achieve resolution in the lateral direction similar to or better than that of LEEM (because the optics involved are identical) (Kordesch, 1995a). Practically, however, a sufficiently intense source must be provided that allows high magnification PEEM and image acquisition in a reasonable time frame. Polycrystalline surfaces which do not fully exploit the LEEM contrast mechanisms may also be observed in other emission modes. Some very new optical methods, based on ellipsometry, have also been reported (Rotermund et al., 1995), in one case operating at atmospheric pressure. Figure 8.43 shows a Reflection Anisotropy Microscopy image of the CO oxidation reaction on Pt(110). There are new imaging methods based on synchrotron radiation that can detect magnetic layers (Stoehr et al., 1993), and of course spin polarized low energy electron microscopy has also been realized (Pinkvos et al., 1993). Spectromicroscopy is another area of growth, where an image is passed through the microscope with an energy analyzer, or microspectroscopy, where a small region of the image is analysed with a non-imaging method. Depth of field in rough specimens may require a scanning probe. Scanning LEEM (Kirschner et al., 1986) and UHV
Fig. 8.42. A PEEM image of carbon-sulfur deposition on oxygen covered Mo(310) at 950 K, from 10 -6 Torr 5% methane in hydrogen. The field of view is 260 lam. (Kordesch, 1995b).
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Fig. 8.43. A set of Reflection Anisotropy Microscopy images recorded during the catalytic oxidation of CO on Pt(ll 0), partial pressures: oxygen 4 • 10-n mbar, carbon monoxide 6.2 • 10-5 mbar, T = 494 K. These conditions are similar to those in Figs. 8.39-41. The image sizes are 3.1 • 3.9 ram2! (Courtesy of H.H. Rotermund.)
scanning low energy electron microscopy (Muellerova and Lenc, 1992) are also being investigated. There is also an effort under way to correct the aberrations in emission microscopes, pioneered by Rempfer (Rempfer and Mauck, 1992; Rempfer, 1990), and developed further by Tonner (Tonner, 1990), Rose (Rose and Priekzas, 1992) and Engel in Berlin. These efforts are aimed at increased resolution, but also for increased intensity at synchrotron sources by enlarging apertures while not reducing resolution. Acknowledgements:
M E K would like to thank Dr. Wilfried Engel of the Fritz Haber Institute in Berlin for many helpful discussions, and those who have generously provided figures for this work: A.M. Bradshaw, EA. Crozier, W. Engel, M. Gajdardziska-Josifovska, G. Lehmpfuhl, L.D. Marks, B. Rausenberger, H.H. Rotermund, R.M. Tromp and Y. Uschida.
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Robinson, M.T., 1981, in: Sputtering by Particle Bombardment I, ed. R. Behrisch. Topics Appl. Phys. 47. Springer Verlag, Berlin, p. 73. Rohrer, G., 1993, in: Scanning Tunneling Microscopy and Spectroscopy, ed. D.A. Bonnell. VCH, New York, p. 155. Rose, H. and D. Preikszas, D., 1992, Optik 92, 31. Rose, K.C., R. Imbihl, B. Rausenberger, C.S. Rastomjee, W. Engel and A.M. Bradshaw, 1996, Surf. Sci. 352-354, 258. Rotermund, H.H., G. Haas, R.U. Franz, R.M. Tromp and G. Ertl, 1995, Science 270, 608. Rotermund, H.H., 1993, Surf. Sci. 283, 87. Ruska, E., 1980, The Early Development of Electron Lenses and Electron Microscopy. Hirzel, Stuttgart. Sarid, D., 1991, Scanning Force Microscopy. Oxford University Press, New York. Sarikaya, M., ed., 1992, Ultramicroscopy 47, 1. Septier, A., ed., 1967, Focusing of Charged Particles, 2 vols. Academic Press, New York. Sigmund, P., 1981, in: Sputtering by Particle Bombardment I, ed. R. Behrisch. Topics Appl. Phys. 47. Springer, Berlin, p. 9. Skoczylas, W.P., G.F. Rempfer and O.H. Griffith, 199 I, Ultramicroscopy 36, 252-261. Smith, D.J., Z. G. Li, Ping Lu, M.R. McCartney and S.C.Y. Tsen, 1991, Ultramicroscopy 37, 169. Smith, S.T. and D.G. Chetwynd, 1992, Foundations of Ultraprecision Mechanism Design. Gordon and Breach, Amsterdam. Snyder, E.J., M.S. Anderson, W.M. Tong, R.S. Williams, S.J. Anz, M.M. Alverez, Y. Rubin, F.N. Diederich and R.L. Whetten, 1991, Science 253, 171. Souda, R., M. Aono, C. Oshima, S. Otani and Y, Ishizawa, 1985, Surf. Sci. 150, L59. Spatz, J.P., S. Sheiko, M. Moiler, R.G. Winkler, P. Reineker and O. Marti, 1995, Nanotechnology 6, 40. Spence, J.C.H., 1994, Ultramicroscopy 55, 293. Spence, J.H.C., 1988, Experimental High-Resolution Electron Microscopy, 2nd edn. Oxford University Press, New York. Stoehr, J., Y. Wu, B.D. Hermsmeier, M.G. Samant, G.R. Harp, S. Koranda, D. Dunham and B.D. Tonner, 1993, Science 259, 658. Teplov, S.V., V.V. Zastavnjuk, V. Bykov and J.W. Rablais, 1994, Surf. Sci. 310, 436. Tersoff, J. and D.R. Hamann, 1985, Phys. Rev. B 31,805. Tersoff, J., 1993, in: Scanning Tunneling Microscopy and Spectroscopy, ed. D.A. Bonnell. VCH, New York, p. 31. Tesmer, J.R. and M. Nastasi, eds., 1995, Handbook of Modern Ion Beam Materials Analysis. Materials Research Society, Pittsburgh, PA. Tonner, B.P., 1990, Nucl. Instr. Meth. A291, 60. Tortonese, M., R.C. Barrett and C.F. Quate, 1993, Appl. Phys. Lett. 62, 834. Tromp, R.M. and J.F. van der Veen, 1983, Surf. Sci. 133, 159. Tromp, R. and M.C. Reuter, 1992, Materials Research Society Proceedings 237, 349. Tromp, R., A.W. Denier van der Gon, F.K. LeGoues and M.C. Reuter, 1993, Phys. Rev. Lett. 71, 3299. Tsong, T.T., 1990, Atom-Probe Field Ion Microscopy. Cambridge University Press, Cambridge. Tsuno, K., 1993, Ultramicroscopy 50, 245. Turkenburg, W.C., W. Soszka, F.W. Saris, H.H. Kersten and B.G. Colenbrander, 1976, Nucl. Instr. Meth. 132, 587. Ueyama, H., M. Ohta, Y. Sugawara and S. Morita, 1995, Jpn. J. Appl. Phys. 34, L1086. Unertl, W.N., 1982, Appl. Surf. Sci. 11/12, 64. van der Veen, J.F., 1985, Surf. Sci. Rept. 5, 199. Veneklasen, L., 1992, Rev. Sci. Instrum. 63, 5513. Watt, I.M., 1985, The Principles and Practice of Electron Microscopy. Cambridge University Press, New York.
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Part III
Structure of Adsorbed Layers
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CHAPTER 9
Chemically Adsorbed Layers on Metal and Semiconductor Surfaces
H. O V E R Fritz-Haber-lnstitut der Max-Planck-Gesellschaft Faradayweg 4-6 D-14195 Berlin, Germany
S.Y. T O N G Department of Physics and Laboratory of Surface Studies University of Wisconsin-Milwaukee Milwaukee, WI 53201, USA
Dedicated to Prof. Dr. G. Ertl's 60th birthday
Handbook of Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B.V. All rights reserved
425
Contents
9.1.
Introduction
9.2.
Adsorption of carbon monoxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. Relationship between chemistry and surface science: Blyholder model . . . . . . . .
9.3.
9.4.
9.2.2.
Structural results
9.2.3.
Kinetic oscillations: CO-oxidation reaction
9.6.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................
427 428 429 434 440
O x y g e n adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
446
9.3.1.
Adsorption on close-packed surfaces: fcc(l 1 l ) , h c p ( 0 0 0 1 )
445
9.3.2.
Adsorption on open surfaces: f c c ( l l 0 ) , hcp(1010) . . . . . . . . . . . . . . . . . . .
449
9.3.3.
Developing of oxides
456
Alkali-metal/metal systems 9.4.1.
9.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..............
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L a n g m u i r - G u r n e y model and recent theoretical results
................
459 461
9.4.2.
Initial growth and adsorption geometry . . . . . . . . . . . . . . . . . . . . . . . . .
465
9.4.3.
Coadsorption R u ( 0 0 0 1 ) - C s - O and Ru(0001)-Cs--CO . . . . . . . . . . . . . . . . .
477
Metal/semiconductor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
486
9.5. I.
Adsorption of metals on Si(111) and Ge(111) substrates . . . . . . . . . . . . . . . .
486
9.5.2.
C o m m o n l y found models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
487
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
426
493
9.1. Introduction In this chapter we concentrate on chemically adsorbed layers on metal and semiconductor surfaces. Because of the volume of prior works and ongoing research, it is necessary to focus on a few topics. We have chosen to cover four topics. Each area has seen an influx of new ideas over the past several years. Even though none of these areas are completely u n d e r s t o o d - in fact, many of the microscopic details of chemisorption processes remain a mystery - nevertheless, a number of interesting trends have emerged. A further restriction is that we shall only discuss processes occurring on single-crystal substrates. Studies of chemisorption, whether on metal or semiconductor surfaces, are propelled partly by the interesting scientific questions they pose and partly by the technological importance of these areas. For example, the study of chemisorption on metal substrates is relevant to the understanding of catalytic reactions and the search for improved catalysts. Studies of chemisorption on semiconducting materials are of great interest to the electronic industry, in its ever increasing demand for miniaturization of devices. However, apart from noting some examples of technological applications, this chapter will draw attention to the basic physical and chemical properties of chemisorption. We shall concentrate on some illustrative models and discuss concepts which are relevant to a broad area of chemical bonding. The first section is concerned with the chemisorption of CO on transition metals. One may view CO as a paradigm of surface science, much like the hydrogen is in atomic physics. Many experimental and theoretical studies have been performed to elucidate the underlying adsorption process of CO. Carbon monoxide is one of the simplest molecules to a chemist and a molecule of great complexity to a physicist. The CO molecule exhibiting many interesting features is a stepping stone to the investigation of more complex molecular adsorbate systems. The study of CO on metal substrates also provides a link between carbonyl chemistry and solid-state physics because similar bonding mechanisms exist. In the second section we will concentrate on oxygen adsorption and the initial stage of oxidation. While on close-packed surfaces, the adsorption process seems to be quite simple, it is no longer true for the more open surfaces, e.g. fcc(110). With the latter systems the adsorption process is frequently accompanied by a reconstruction of the surface, indicating strong metal-oxygen bond formation. Oxidation, on the other hand, represents a final step in the chemisorption process and is associated with the incorporation of oxygen in the substrate layers to form, at least locally, oxide crystallites. The underlying process is often connected with a dramatic re-arrangement of substrate atoms during which the morphology is considerably changed.
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H. Over and S. Y. Tong
In the third section we will look at the adsorption of alkali-metal atoms on metal surfaces. This system has been regarded as a simple prototype to study the bonding of adsorbate atoms to the substrate. However, recent experimental and theoretical results belie this simplicity by showing that this system has curious properties which are inconsistent with the simple picture. Fortunately, theoretical calculations are tractable because the chemisorption process is predominantly determined by a single s-valence electron of the alkali-metal atoms. Therefore, the calculations are able to clarify the underlying mechanism as exemplified by the alkali-metal/AI(111) system. It is shown that the interrelation between the electronic and structural properties is the key to this mechanism. Another interesting adsorption system represents alkali-metal atoms on Ru(0001). Here, depending on the coverage, the adsorption site changes. A more complicated system consists of the coadsorption of oxygen and Cs on Ru(0001 ). This system exhibits several ordered structures reflecting the complex Cs-oxygen chemistry. We will compare this coadsorbate system with CO on Ru(0001)-Cs. The latter system is governed by an ionic interaction between the species. The last section of this chapter discusses the chemisorption of metal atoms on semiconductor substrates. A general principle is the desire of the interface to reduce the number of unpaired electrons in the adsorbate-substrate system. The balance between electronic and stress energies results in a number of models favored by different metals of the Periodic Table. We will present the electronic and structural properties of these models and explore the reasons why certain metals favor or reject particular models.
9.2. Adsorption of carbon monoxide CO oxidation and the Fischer-Tropsch reaction (catalytic CO hydrogenation) are of great practical importance, both in the chemical industry for making basic chemicals and for such environmental applications as automotive exhaust control. Manufacture of hydrocarbons and of other basic chemicals from coal is of particular interest since coal is an abundant mineral resource in many countries. Hot coal can be gassified by exposing it to water steam which forms a mixture consisting mainly of water and CO. This mixture can then be catalytically synthesized to different chemical compounds. Carbon monoxide, although a small molecule, provides a number of interesting features that qualify it as a stepping stone for the investigation of more complex molecular adsorbate systems. No other adsorbate has been as well studied as CO" hundreds of experiments and dozens of calculations have been performed on it, see for example the reviews of Campuzano (1990) and Hoffmann (1988). One important aim of surface science is to gain insight into the elementary steps of heterogeneous catalysis (Christmann, 1991 ). Although this objective is far from being achieved, the first steps in that direction have been taken, such as determining the geometric structure of small molecules at single-crystal surfaces. In addition, there are interesting analogies between surfaces and cluster complexes, like metal carbonyis, which append to our understanding of the bonding of molecules to metallic surfaces. Such comparisons will be used to understand the chemisorption process of CO on transition-metal surfaces (cf. w 9.2.1 and 9.2.2).
Chemically adsorbed layers on metal and semiconductor su~. "aces
429
A simple example of a chemical reaction on surfaces is presented in the last section (9.2.3) where we focus on the CO-oxidation reaction with special emphasis on the occurrence of kinetic oscillations. This reaction is readily catalyzed by transition metals, especially with platinum-group metals.
9.2.1. Relationship between chemistry and surface science: Blyholder model Let us first consider the molecular orbitals (MO) of CO for the simple case that the MOs are constructed by linear combination of atomic orbitals of C and O; the atomic orbitals are denoted in Fig. 9. l a on the right and left-hand side. Only those atomic orbitals with similar energies can be combined to give MOs. The full set of MOs are then filled with electrons of C and O, starting from the lowest energy level (1o) towards higher energies, according to Hund's rule. The most important MOs determining the reactivity of CO are the 5o and the 2n (frontier) orbitals. The 5o orbital, the highest occupied MO (HOMO, acting as the donor state) of CO, is a C-2pz-derived state which forms a lone-pair orbital concentrated at the carbon end of CO, while the 4o-bonding state results from a hybridization of O-2pz and C-2s orbitals; the z axis is directed along the molecule axis. The 2rt and l rt MOs are symmetric or anti-symmetric combinations of C-2px and O-2px or C-2py and O-2py states, giving rise to the bonding l rt and the anti-bonding 2rt orbitals. The antibonding 2rt MO of CO is empty and represents the lowest unoccupied MO (LUMO, acting as the acceptor state). The symbol o is used to denote an MO that has its charge density concentrated along the internuclear axis of the molecule, while the symbol rt is used for MOs that have zero-charge density on the internuclear axis. The bonding of CO on transition-metal surfaces is conventionally viewed in terms of a donor-acceptor mechanism (Blyholder model) analogous to that found in metal carbonyls (Blyholder, 1964). Upon chemisorption onto a metal surface the 5 ~ - C O molecular orbital (MO) interacts and hybridizes with the band states of the
Fig. 9.1. (a) Schematic molecular orbital diagram for a CO molecule; (b) conventional Blyholder scheme for the chemisorption of CO on transition metals.
430
H. Over and S. Y. Tong
metal substrate exhibiting proper symmetry, e.g. d,, states in Fig. 9. lb. Thus, the bonding 50. and the anti-bonding 50.* state of the combined metal-CO system are formed. This process is accompanied by electron donation from this MO to the metal which provides a part of the metal-carbon bond. In addition to this 0. donation, there occurs a back-donation of p and d electrons from the metal again with proper symmetry (e.g. d,~ in Fig. 9. l b) into the anti-bonding 2rt MO of the CO which strengthens the metal-CO bond and weakens the internal C - O bond. The two effects of 0. donation and 2rt back-donation are coupled because the more electrons are transferred into the 2rt MO of CO, the more electrons are donated from the 50. level of CO to the metal to maintain charge uniformity. It may be noted that the resulting dipole moment is rather small (around 0.5 Debye) suggesting nearly electroneutrality. In addition, both these CO MOs (50. and 2rt) have a larger amplitude on C, so the binding occurs with the C side down. The donation and back-donation of electrons have opposite effects on the strength of the C - O bond. The removal of charge from the 50. level slightly strengthens the C - O bond due to a more uniform distribution in the l rt level, while addition of charge to the anti-bonding 2rt level weakens the C - O bond (Hoffmann, 1988). As a net effect, the C - O bond is weakened. This can be seen from the vibrational spectra of metal-carbonyl complexes where the species with the highest metal-carbon stretching frequencies (strongest CO-metal bond) show the lowest carbon-oxygen stretching frequency (Emmett, 1965). Ultraviolet photoelectron spectroscopy (UPS) is a standard technique for studying electronic properties of (molecular) adsorption phenomena on solid surfaces. Figure 9.2 shows photoelectron-emission energy distribution curves of CO adsorbed on Ru(0001) (Menzel, 1975) in comparison with the spectra obtained for Ru:~(,CO)~2 and free CO molecules in the gas phase (Plummer et al., 1978). The UP spectrum of free CO molecules shows the 40., Ix, and 50. states expected from the molecular orbital diagram (Fig. 9.1); note that the 2rt MO of CO in the gas phase is empty and is therefore not seen in UPS. The spectrum obtained from Ru3(CO)12, however, exhibits only two broad peaks below the Ru-derived d levels. The easiest way to explain these data is that either the bonding interaction has lowered the energy of the 50. level to overlap the It t-derived state or Ix undergoes a bonding shift to overlap the 40. Theoretical studies of a Ni-CO cluster have demonstrated that 50. and l rt bands overlap and that bonding is achieved through hybridization of the 50. orbital (Ellis et al., 1977; Hermann and Bagus, 1977). Experimental verification of the orbital assignment is possible using angle-resolved UPS (ARUPS), as demonstrated with the system Ru(0001)-CO (Fuggle et al., 1975). The two spectra of Ru carbonyl and CO adsorbed on Ru(0001) (Fig. 9.2) are very similar. This means that the features characterizing the electronic properties of CO attached to only a small number of metal atoms (metal carbonyls) are essentially identical to the situation where CO is adsorbed on Ru(0001). The localized character of chemisorption, at least for this ligand, is strongly supported by this experimental result. For a systematic comparison of UP spectra of transition-metal carbonyl complexes to corresponding spectra of adsorbed CO the reader is referred to Plummer et al. (1978).
431
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I
Er
Fig. 9.2. Comparisonof UV (40.8 eV) photoemission spectra of free CO (a), Ru3(CO)12(b) (Plummer et ai., 1978), and CO adsorbed on Ru(0001) (c) (Menzel, 1975). The adsorption of CO on Ni(100) was studied with an all-electron local-densityfunctional method which confirms and refines the simple Blyholder model (Wimmer et al., 1985). In Figs. 9.3a and b the charge densities of the free molecule 5cy orbital and the free molecule 2~ orbital are presented. The changes in charge density that occur upon CO adsorption onto a Ni(100) surface are shown in Fig. 9.3c, from which a depletion of charge associated with the 5(~ orbital and a gain of charge density with 2~ character are evident to the eye. The B lyholder model predicts a strong correlation between the CO chemisorption energy and the degree of backbonding, i.e., with increasing back-donation the CO-chemisorption energy increases. The back-donation is favored by a large d-electron concentration close to the 2rt MO of CO (cf. Fig. 9.4). In Table 9.1 some selected initial heats of adsorption of CO on various metal single crystals are presented. The observed heats range from 58 to 160 kJ/mol, indeed revealing a strong variation of d-electron density close to the 2~ MO level. For Cu, e.g., the d-electron density near 2~ is small, hence back-bonding is weak associated with a small value of chemisorption energy. From the Blyholder model a further general feature can be deduced: the stronger the metal-carbon bond, the weaker the internal C - O bond, and therefore, the higher the tendency to dissociate. As depicted in Table 9.2 (Broden et al., 1976), at room temperature the adsorption of CO molecules onto transition metals in the left-hand region of the Periodic Table is characterized by dissociation. For metals on the right-hand side of this table, molecular adsorption takes place. It should be noted
432
H. Over and S. Y. Tong
Fig. 9.3. (a) Charge density of the free 5~ orbital of CO; (b) charge density of the free 2n orbital of CO; (c) change in charge density due to chemisorption of CO on Ni(001) (Wimmer et al., 1985). that at sufficiently low temperatures CO adsorbs molecularly on most transition metals on the left, while at higher temperatures CO dissociates on all of these metals. Furthermore, it is assumed that the back-bonding effect becomes more pronounced as the coordination number of the CO molecule on the metal surface
433
Chemically adsorbed layers on metal and semiconductor su~. "aces
Fig. 9.4. The energetic position of the center of the d band for the first transitions series. The positions of the CO 5~ and 2re levels are superimposed; after Hoffmann (1988). Table 9.1 Initial heats of adsorption (qst) of carbon monoxide on various metal surfaces Surface
qst (kJ/mol)
Reference
128 125 106 111 52 60 160 184 161 142 159
Bridge et al. (1979) Tracy (1972) Klier et al. (1970) Christmann et al. (1974) Tracy (1972) Horn et al. (1977) Pfntir et al. (1978) Brennan and Hayes (1965) Behm and Christmann (1980) Ertl and Koch (1970) Comrie and Weinberg (1976)
Co(0001 ) Ni(100) Ni(110) Ni(! 11) Cu(100) Cu(ll 0) Ru(0001 ) Rh( 111 ) Pd(100) Pd(l 11) It( 111 )
Table 9.2 Tcndency of CO to dissociate on transition metals at room temperature. Rapid dissociation takes place on the non-shadowed region. (Broden et al., 1976) IIIB
Sc Y La
IVB
VB
VIB
VIIB
VIII
VIII
Vlll
IB
Ti Zr Hf
V Nb Ta
Cr Mo W
Mn Tc Re
Fe Ru Os
Co Rh lr
Ni Pd Pt
Cu Ag Au
i n c r e a s e s , i.e., the 2rt o r b i t a l o f a C O m o l e c u l e in a h o l l o w or a b r i d g e site finds m o r e o c c u p i e d d states o f a p p r o p r i a t e s y m m e t r y (d,0 with w h i c h to i n t e r a c t . S i n c e the f r e q u e n c y o f the C O - s t r e t c h i n g v i b r a t i o n r e p r e s e n t s a g o o d i n d i c a t o r o f the d e g r e e to w h i c h this b o n d w e a k e n s (cf. T a b l e 9.3), H R E E L S a n d I R A S can f r e q u e n t l y be u s e d to a s s i g n the c o o r d i n a t i o n n u m b e r o f the a d s o r p t i o n sites.
434
H. Over and S. Y. Tong
Table 9.3 Frequencies of C-O stretching vibration in correlation with the coordination number of adsorption for various metal surfaces Surface
Adsite
Frequency (cm-l)
Reference
Ni(100) c(2• Ni(l 11), 0 < 0.2 0 = 0.3
on top 3-fold 2-fold
2070 1817 1910
Campuzano and Greenler (1979)
Andersson (1977)
Ru(0001 )(~f3-x'f3-)
on top
1980
Thomas and Weinberg (1979)
Rh( 111 )(~-•
on top
2070
Dubois and Somorjai (1980)
Rh(111) 0 > 1/3
on top bridge
2070 1870
Dubois and Somorjai (1980)
Pd( 111)(~-x~/3-)
fcc
1840
Bradshaw and Hoffmann (1978)
Pt(l 11) 0 > 1/3
on top bridge
2100 1875
Hayden and Bradshaw (1983)
Pt(ll 0) on ( I x I ), 0 < 0.3; on ( I x2), 0 < 0.3
on top on top
2094 2080
Bare (1982) Hofmann (1982)
9.2.2. S t r u c t u r a l results
On most of the transition metals detailed studies of the adsorption process of C O have been p e r f o r m e d . For an o v e r v i e w the reader is referred to C a m p u z a n o (1990) and H o f f m a n n (1983). In Table 9.4 the structural results of C O adsorbed on various metal surfaces are listed and partly c o m p a r e d to results k n o w n for metal c a r b o n y l s and C O on clusters. This c o m p a r i s o n reveals a l m o s t identical m e t a l - C and C - O bond lengths in adsorption systems and c o r r e s p o n d i n g c l u s t e r s / c a r b o n y l s . Striking is the variety of adsorption sites of C O on different metal surfaces" on-top, bridge and hollow sites; see Fig. 9.5 for an illustration of these adsorption sites. The adsorption sites vary not only from metal to metal, but also from one face to a n o t h e r face of the s a m e metal, and also from one c o v e r a g e to another c o v e r a g e on the s a m e face of the s a m e metal. T h e r e f o r e , it is clear that predicting the site o c c u p a t i o n of
Fig. 9.5. Illustration of CO adsorption in on-top, bridge, and hollow sites for the case of a densely packed surface (fcc(l I 1), hcp(0001)).
435
Chemically adsorbed layers on metal and semiconductor su~. aces
Table 9.4 CO adsorption structures on various metal (Me) surfaces in comparison with the results for respective metal carbonyls or clusters Structure
Site
Me-C bond
C-O bond
(A)
(A)
Reference
Ni(100)c(2x2)-CO
on top
1.7_-_-t-0.1
1.13+0.1
Kevan et al. (1981)
Ni(CO)4
on top
1.82
1.15
Sutton (1977)
Cu( 100)c(2x2)--CO
on top
1.9-~. 1
1.13+0.1
Andersson and Pendry (1980)
Cu cluster
on top
1.79-1.93
1.04-1.16
Chini et al. (1976)
Pd(100) (2~x ~-)R45-2CO
bridge
1.93_+0.07
1 . 1 5 + 0 . 1 Behm and Christmann (1980)
Pd cluster
bridge
1.85-1.94
1.09-1.17
Chini et al. (1978)
Pd( 110)(2xl )p2mg
bridge
1.94+0.02
1.1
Huang et al. (1993)
Pd(l 11) ('~-•
fcc
2.05_+0.05
1.15_+0.05
Ohtani et al. (1987)
Ru(0001) (ff-33xxf3-3)R30-CO
on top
1.90-&5).05 1.93_+0.04
1.16_+0.03 1.15_+0.05
Michalk et al. (1983) Over et al. (1993a)
Ru3(CO)i 2
on top
1.91
1.14
Mason and Rae (1968)
Rh( 1I 1) (~t3x'~.5,-)R30-2CO
on top
1.95+0.1
1.07+0.1
Koestner et al. ( 1981 )
Rh( 111 )(2x2)-3CO
on top
1.94+0.1
1.15+ 0.1
Van Hove et al. (1983)
and
bridge
2.03_+0.07
1.15+0.1
Rh(110)(2xl )p2mg
bridge
1.97_+0.09
1.13_+0.09
Batteas et al. (1994)
Rh6(CO)i 6
on top
1.96
1.10
Corey et al. (1963)
1.99
1.36
and
bridge Pt(l 11)c(4•
on top bridge
1.85_+0.10 2.08_+0.07
1.15_+0.05 1.15_+0.05
Ogletree et al. (1986)
C O on a p a r t i c u l a r m e t a l s u r f a c e is difficult, so that e a c h p a r t i c u l a r s y s t e m n e e d s its o w n c o m p l e t e s t r u c t u r e a n a l y s i s . T h e fine b a l a n c e b e t w e e n the a d s o r b a t e - s u b s t r a t e and a d s o r b a t e - a d s o r b a t e i n t e r a c t i o n s is d e m o n s t r a t e d in the d i v e r s i t y o f s t r u c t u r e s f o r m e d . E v e n on m e t a l s w h i c h e x h i b i t v e r y s i m i l a r c h e m i s t r y and are v e r y c l o s e in the P e r i o d i c T a b l e , s u c h as Pd and Rh, the C O a d s o r p t i o n site is d i f f e r e n t . T h e d e v i a t i o n of t e r m i n a l l y b o n d e d C O m o l e c u l e s f r o m 180 ~ for m e t a l c a r b o n y l s ( C - O - M e b o n d is not l i n e a r in this c a s e ) p o i n t s to b e n t b o n d i n g w h i c h m a y o c c u r a l s o on c o r r e s p o n d i n g C O - m e t a l a d s o r p t i o n s y s t e m s . F o r the c a s e o f Ru3(CO)i2 this a n g l e was f o u n d to be a b o u t 170 ~ ( M a s o n and R a e , 1968), see Fig. 9.6. T h i s m a y be the first r e f e r e n c e to b e n d i n g - v i b r a t i o n a l m o d e s , as d i s c u s s e d b e l o w .
436
H. Over and S. Y. Tong
Fig. 9.6. Molecular stereo-chemistryof Ru3(CO)I2with molecular symmetry not significantly different from D3h; after Mason and Rae (1968). In this section we will focus on only two of these CO-adsorption systems, namely Ru(0001)-CO and Pt(110)-CO, illustrating how different techniques in surface science provide vital contributions to determine the local adsorption geometry. Structural properties of the Pt( 110)-CO system are used in w9.2.3 for an understanding of kinetic oscillations during the CO-oxidation reaction on Pt(110) surfaces. Ru(0001)-CO has been studied with almost every surface-analytical technique, and, therefore, a wealth of information is available. Exposure of the Ru(0001) surface to CO at 110 K forms two ordered LEED structures, namely a (ffxff3-3)R30 at about 0 = 0.3 and a (2q-fx2q3-)R30 at 0 = 7/12 (Williams and Weinberg, 1979). A more detailed LEED study revealed that the (ff3-x{-f)R30 structure exists over a wide range of coverage due to the repulsive interaction between nearest-neighbor CO molecules and attractive interaction between next-nearest-neighbor molecules (Pfntir and Menzel, 1984). Using vibrational spectroscopy (HREELS), Thomas and Weinberg (1979) found only one vibrational band related to CO linearly adsorbed on top of Ru atoms (cf. Table 9.3: C - O stretch frequency of about 2000 cm -~ are assigned to CO in on-top position). The frequency of this C - O stretching mode shifts from 1984 cm -~ to 2061 cm -~ with increasing coverage (Pfntir et al., 1980). This shift was ascribed to dipole-dipole coupling of the CO molecules. Thorough UPS studies of Ru(0001)-CO (Steinkilberg et al., 1975; Hofmann et al., 1985; Heskett et al., 1985) showed that CO is adsorbed upright, even at high coverages. ESDIAD investigations by Madey (1979) arrived at similar conclusions. He found that the angular distributions for O + and CO + ions produced by electron impact are consistent with CO molecules centered about the surface normal with their widths being temperature-dependent. More specifically, the half angles at half
Chemically adsorbed layers on metal and semiconductor su~. aces
437
Fig. 9.7. Structure model (static) of Ru(0001)-(~x~/3-)R30-CO and the structural parameters for the best-fit arrangement. Small shaded circles: O atoms; small filled circles: C atoms. maximum of the ion cones (reflecting the average inclination of the molecular axis of the adsorbed CO molecules) are found to be about 16 ~ at 290 K and 12 ~ at 90 K. This finding was considered as evidence for the existence of a bending mode. The (4-3-3x4-3-3)R30 ~ structure of Ru(0001 ) - C O was also subjected to conventional LEED structure analyses (Michalk et al., 1983; Over et al., 1993a). These studies clearly favored that CO molecules are adsorbed through their carbon atoms over single Ru atoms (i.e. in 'on-top' position) with their molecular axes being parallel to the surface normal. The structural characteristics are summarized in Fig. 9.7: the Ru atoms coordinated with CO are displaced outwards by 0.07 + 0.03 ]k. The R u - C (1.93 + 0.04 A) and C - O (1.10 + 0.05 ]k) distances are within the expected range" cf. also Table 9.4. Under the influence of thermal excitation the oxygen atoms are expected to have more freedom to move parallel to the surface than perpendicularly, see Fig. 9.8. In fact, such anisotropic temperature effects have shown to be important in surface crystallography and were simulated in LEED by temperature-dependent distributions of O and C positions, according to the harmonic oscillator approximation (Gierer et al., 1996). This analysis indicated a bending-mode vibration of CO with an excitation energy of 5 + 1 meV (cf. Fig. 9.8); note that such low-frequency modes are related to large root mean-square displacements of oxygen by about 0.4 A at 150 K. Since these low-frequency modes are usually not so easily accessible with conventional techniques of vibrational spectroscopy, this represents a promising side application of LEED. The found value of 5 + 1 meV for the bending-mode vibration is also in nice agreement with the value found for a related system P t ( I l l ) - C O (CO adsorbed also in on-top position): the excitation energy of the CO-bending mode was determined to be 6 meV by means of He-atom scattering (Lahee et al., 1986).
438
H. Over and S. Y. Tong
Fig. 9.8. Illustration of the CO bending-mode vibration on Ru(0001) (Gierer et al., 1996).
Fig. 9.9. Phase transformation from the bulk-truncated Pt(l I0) surface into the missing-row (Ix2) reconstructed Pt(ll 0) surface. We now turn to the system Pt( 110)-CO. This substrate is qualitatively different from Ru(0001) since its thermodynamically stable surface consists of a (Ix2) missing-row structure (cf. Fig. 9.9). Exposure of the Pt(110)(Ix2) surface beyond 1 L CO at 300 K results, however, in a ( 1x I ) LEED pattern exhibiting a high degree of disorder. This indicates that CO lifts the reconstruction above a critical CO coverage of about 0.2 (Imbihl et al., 1988) and now renders the ( l x l ) phase energetically more favorable. With STM (Gritsch et al., 1989) the elementary step for this process has shown to be the creation of local ( I x l ) nuclei (holes) which required atomic motion of Pt only over a few lattice sites (activated process), as indicated in Fig. 9.10. This finding is also consistent with the observation of the adsorbate-induced transformation of the (2xl) into the ( l x l ) even at 250 K (Bonzel and Ferrer, 1982) which is too low to allow surface diffusion of Pt atoms over longer distances. Below 250 K this transformation is completely inhibited (Jackman et al., 1982). With increasing CO exposure the density of holes increases continuously up to about 3 L, creating a highly distorted P t ( 1 1 0 ) ( l x l ) surface. At elevated temperature (350 K), this transformation takes place via the movement of longer [ 110] strings in [001] direction, see Fig. 9.11. This leads to the formation of
Chemically adsorbed layers on metal and semiconductor surfaces
439
Fig. 9.10. At 300 K the elementary step for the CO-induced transformation of the (2x 1) into the ( 1x I ) phase consists of the creation of small holes (with lengths of the order of 10-15 ]~), whereby atoms are shifted only over short distances to missing-row sites. (a) Magnified STM image of one of the 'holes' and (b) corresponding ball model (Gritsch et al., 1989).
Fig. 9.11. At elevated temperatures Q50 K) the transformation of the (2xl) into the (lxl) phase proceeds through shifting of longer [ 110] rows in the [001 ] direction by one lattice constant. (a) STM image and (b) corresponding ball model (Gritsch et al., 1989). larger patches of the (1 x 1) d o m a i n s with typical d i m e n s i o n s of 2 0 - 5 0 ,~ in the [001 ] direction and 1 0 0 - 3 0 0 A in the [ITO] direction, thus giving a better ordered (1• structure as e v i d e n c e d by L E E D .
440
H. Over and S. Y. Tong
Pt(110)-(2xl)pg-CO .[OOl]
l-
giide plane --... [ l l O l
Fig. 9.12. The top view of the Pt(110)(2xl)p2mg--CO structure. The CO molecule is represented by closely spaced small circles. The glide plane is indicated. In order to form a well-ordered CO overlayer on this surface, one must heat the sample to about 500 K in 10 -7 mbar of CO resulting in a ( 2 x l ) p 2 m g structure with coverage 0 - 1 (Comrie and Lambert, 1976); a top view of real space model of this structure is shown in Fig. 9.12. Analogous to the system Pd(110)-CO and Ni( 110)CO (Pangher and Haase, 1993), the presence of the glide symmetry points towards the presence of zigzag chains along the [110] direction. Angular-resolved UPS studies (Hofmann et al., 1982) have indicated that the zigzag chains come from the CO molecules being tilted by about 26 ~. The on-top position for CO used in this model has been derived from RAIRS and HREELS measurements (Bare et al., 1982" Hofmann et al., 1982), cf. Table 9.3. Static tilting is frequently found for densely packed CO molecules, as for example for N i ( 1 1 0 ) ( 2 x l ) p m g - 2 C O (Hannaman and Passler, 1988; Wesner et al., 1988). The evolution of the Pt(110)(2xl)p2mg-2CO phase at T - 300 K has been studied by means of TDS, UPS, LEED, RAIRS, and EELS (Bare et al., 1982; Hofmann et al., 1982). At low coverages (0 = 0.1 ), isolated CO species are adsorbed in on-top sites on the reconstructed Pt(110)(Ix2) surface as found by infrared spectroscopy. UPS shows that these molecules are adsorbed with their axes perpendicular to the surface. For 0.1 < 0 < 0.3, the isolated CO molecules coexist with islands of CO adsorbed in on-top sites on the ( l x l ) surface. This finding has been confirmed by the inverse photoemission results of Ferrer et al. (1985) who found that the empty 2rt level of CO adsorbed on the ( Ix l) surface occurs at 5 eV above the Fermi energy, while this level for CO adsorbed on the (lx2) surface occurs at 3.4 eV above the Fermi level. With increasing coverages, islands grow in size (EELS), and the molecules in the islands are adsorbed with their axes tilted (UPS) due to the high density of molecules in the islands. At coverages greater than half a monolayer, the islands begin to coalesce, and eventually the ( l x 2 ) to ( l x l ) transition is completed. This transition was also the subject of a recent quantitative RHEED study (Schwegmann et al., 1995). 9.2.3. Kinetic oscillations: CO-oxidation reaction
The catalytic CO oxidation reaction on Pt-group metals proceeds through the Langmuir-Hinshelwood mechanism (Engel and Ertl, 1979) where both reactants are adsorbed on the surface before they react to form the product. Oxygen molecules
Chemically adsorbed layers on metal and semiconductor su~'aces
441
adsorb dissociatively requiring two neighboring, unoccupied sites. By contrast, CO molecules adsorb molecularly and tend to form densely packed layers which block further uptake of oxygen. As described in the last section, the clean Pt(110) surface is 'missing-row' reconstructed which transforms upon CO adsorption into a ( l x l ) structure when the CO coverage exceeds a critical value of about 0.2 (Imbihl et al., 1988). Oxygen, on the other hand, forms a relatively open network which allows CO to coadsorb into. The recombination of adsorbed CO with adsorbed O forms CO2 which is immediately released into the gas phase under typical reaction conditions. Usually, the rate of CO2 formation is stationary, and a large coverage of CO results in a low reaction rate. Under certain reaction conditions, however, the CO2-reaction rate is not stationary but exhibits an oscillatory behavior called kinetic oscillations. The crucial point for their occurrence is that the sticking probability of oxygen is about twice as high on a ( l x l ) phase as on a (lx2) phase. The origin of these oscillations has been explored in great detail with the Pt(110) surface and can be traced back to a close coupling between structure and reactivity of a surface (Eiswirth et al., 1986, 1989). A detailed and comprehensive report on kinetic oscillations can be found in Ertl (1990, 1991) and Imbihl (1989). The complete oscillation cycle will be described in the following. We start with a clean (lx2) surface which is exposed to a mixture of CO and 02 for conditions under which adsorption of oxygen is rate-limiting. Due to the larger sticking coefficient of CO compared to O2, CO is preferentially adsorbed and removes the (Ix2) reconstruction when it exceeds the critical CO coverage. Such an adlayer does not completely inhibit O2 adsorption, mainly due to the inevitable presence of surface defects. At these sites the sticking coefficient of 02 is considerably higher, so oxygen atoms adsorb and react off the adsorbed CO, resulting in two empty adsorption sites on the ( l x l ) phase. In an autocatalytic process, the oxygen uptake and hence the CO2 rate increase. This process continues until the CO concentration on the surface is depleted to such an extent that the reconstructed Pt(110)(Ix2) is restored and one reaction cycle is completed. Clearly, rate maxima occur after lifting the ( l x 2 ) reconstruction due to the larger amount of oxygen available on the surface for the CO2 reaction, and minima are related to the ( l x l ) structure. In Fig. 9.13 the CO2 reaction rate shows oscillatory behavior when the CO and oxygen partial pressures are kept fixed at a certain value. The work function A~ is proportional to the oxygen coverage and parallels the reaction rate. Temporal oscillations can be modeled by the numerical solution of (three) coupled non-linear differential equations, describing the temporal variations of the CO and O coverages (rate equations) as well as the fraction of the surface present as (lx2) or ( l x l ) phase (Krischer et al., 1991, 1992). All input parameters are taken from independent measurements. Corresponding results in the limit t ~ ,,o (socalled limit sets) are presented in Fig. 9.14 which agree qualitatively with data of Fig. 9.13. Another aspect consists in spatial self-organization which synchronizes the behavior of different local regions on a macroscopic scale and is therefore a necessary precondition for the occurrence of overall temporal oscillations. The
442
H. Over and S. Y. Tong
.
.
.
.
! .....
i
'!
_
el
{ 2o 100~01- . . . . . 0
, 200
- - -
~ . . . . 400
Time
~ -600
Is)
Fig. 9.13. Kinetic oscillations during the CO/O2 reaction at a Pt(110) surface. The CO2 partial pressure is proportional to the reaction rate, and the work function A ~ changes parallel to the oxygen coverage. Control parameters: substrate temperature T = 480 K and partial pressures: oxygen p(O2) = 2.1 x 104 mbar and p(CO) = 6.8• -~ mbar (Eiswirth et ai., 1989). "G' r
E
3 01
"~
0.7
\
:0i: _Q
4
-.
06[
~
0.4
.J
.,/" ""
/", ,,,.. "
--/,,, ""
/\~x~/,, -,,, ,.-
, /',,- , - :,',, "
"
"
""
...'.. .... ""
"
0
r
0.2
06
'
,o
20
3o Time
~o
50
6o
70
(s)
Fig. 9.14. Time series calculated by integrating the corresponding rate equations modeling the kinetics of CO oxidation on Pt(110). Particular control parameters: T= 540 K, p(O2) = 6.7• -5 mbar and p(CO) = 3.0• -5 mbar (Krischer et al., 1992).
Chemically adsorbed layers on metal and semiconductor surfaces
443
Fig. 9.15. (a) Spatial-temporal patterns of the standing wave type associated with harmonic temporal oscillations during the CO oxidation on Pt(110), cf. Fig. 9.9. The images are recorded in intervals of 0.5 s on a section of0.3x0.3 mm 2. Control parameters: T = 550 K, p(O2) = 4.1x10 -4 mbar and p(CO) = 1.75x10 -4 mbar. (b) Growth of a spiral wave. Width of the images are 0.2 mm recorded at 0, 10, 21,39, 56 and 74 s. Control parameters: T = 4 3 4 K, p(O2) = 3.0x10 ~ mbar and p(CO) = 2.8x10 -5 mbar (Jakubith et al., 1990).
444
H. Over and S. Y. Tong
through surface diffusion. In Fig. 9.15 two types of spatial patterns during the CO oxidation are depicted. To make local variations visible in the coverages of adsorbed species, a photoemission electron microscope (PEEM) is applied (Rotermund et al., 1990; Engel et al., 1991). This method, which yields a spatial resolution of about 1 l.tm, is based on the principle that the yield of photoelectrons depends sensitively on the local work function as one illuminates the sample with photons from a deuterium discharge lamp. The lateral intensity distribution of the photoemitted electrons is imaged through a system of electrostatic lenses onto a fluorescence screen. Since the work function of a clean Pt(110) surface increases by 0.3 eV and 0.5 eV when saturated with CO and oxygen, respectively, the areas covered with O appear dark in the images, while those covered by CO are brighter. The continuously growing spiral wave in Fig. 9.15b is diffusion-controlled (Jakubith et al., 1990). Its shape is elliptic, with the long axis along the [110] direction of the substrate single crystal for which the anisotropy of surface diffusion is responsible and in turn affects the propagation velocities of the fronts of the 'chemical waves'. The velocities in the two proper directions are 3.3 and 1.2 mm/s, respectively. Another type of pattern is related to CO2 rate oscillations (cf. Fig. 9.12) which consist of standing rather than propagating waves (Fig. 9.15a). The coupling mechanism proceeds through the gas phase, as can be seen from Fig. 9.12. The small modulation in the CO partial pressure (<1%) is a consequence of the varying reaction rate which takes place practically instantaneously (<0.0001 s). Apart from these two examples, several other spatial-temporal patterns have been observed, such as a 'solitary wave' (Rotermund et al., 1991) and 'chemical turbulence'. A quantitative description of propagating 'chemical' waves is based on a set of partial (instead of ordinary) differential equations of the reaction-diffusion type (Eiswirth et al., 1990). The properties of 'solitary waves' have been successfully modeled in this way (Bfir et al., 1992). For an overview of self-organization in reactions at surfaces the reader is referred to Ertl (1993).
9.3. Oxygen adsorption The chemisorption of oxygen on a metal surface is an important step in such fundamental processes like catalysis or corrosion. A knowledge of the atomic and electronic structure is necessary to understand these processes in detail and have therefore led many investigators to study elementary processes of the interaction of oxygen with transition metals. The crystallographically 'open' fcc(110) surface (w 9.3.2) should be particularly reactive, and this is why many previous studies have concentrated on the adsorption of oxygen on these surfaces. Much less work has been done on the 'equivalent' (1010) surfaces of hcp metals such as Ru, Re, Os, and Co, cf. Fig. 9.16. The chemisorption of CO and 02 is qualitatively different. While CO molecules adsorb usually in molecular form, 02 easily dissociates, and bond formation is achieved by single oxygen atoms. The main reason for this difference lies in the smaller dissociation energy of 02 and the much stronger chemical interaction of single O atoms with the transition-metal surface if compared to the case of CO. A w
445
Chemically adsorbed layers on metal and semiconducwr su~. aces
Fig. 9.16. A comparison of the hcp(10_10) (a) and the fcc(110) surface (b): top view (top) and side view (bottom). For the hcp(1010) surface both possible terminations are indicated.
2 0 + 2M
F~diss E[
~
02+ M
Distancez
Fig. 9.17. One-dimensional potential-energy diagram for the interaction of oxygen with a surface. The subscripts phys and chem refer to 'physisorption' and 'chemisorption', respectively. The interaction curve for an O atom formed by predissociation is shifted by Editsand leads to a pronounced minimum of depth Ed~ + Echem. Molecular interaction gives rise to a shallow minimum Ephys. The actually observed potential-energy curve is a superposition of the individual functions (shaded curve). simple one-dimensional diagram of the L e n n a r d - J o n e s type might help to elucidate this point (Fig. 9.17). As the oxygen molecule approaches the surface, the van der Waals interaction lowers the energy of this configuration and can give rise to a shallow potential minimum, Ephys, which is characteristic of the physisorption state
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far from the surface. By contrast, the potential-energy curve of two single oxygen atoms starts far away from the surface with a higher energy than an 02 molecule. This difference is determined by the dissociation energy of 02. Through strong chemical interaction with the surface, however, this potential-energy curve drops with decreasing separation from the surface far faster than the corresponding potential-energy curve of the oxygen molecule resulting in a deep potential energy well, Echem , with an equilibrium distance, Zchem, close to the surface. For impinging 02 molecules at the crossover point of these two potential curves, little more thermal/kinetic energy is required to overcome the small potential barrier, and dissociation takes place resulting in a much stronger chemisorption state. Because two O atoms are formed by a single dissociation event, the O - M (M = metal) bond energy is gained twice. This is one of the reasons why transition-metal surfaces are such efficient catalysts: they easily dissociate strongly bound molecules which then can further react to form desired species.
9.3.1. Adsorption on close-packed surfaces: fcc(111), hcp(O001) As can be seen from Table 9.5, the occupation by atomic oxygen occurs on the site with the highest coordination number with respect to the first substrate layer (so-called hollow sites) which is the common trend known from many different examples of structural investigations of atomic adsorbates on high-density surfaces of transition metals like fcc( 111 ) and fcc(100) of hcp(0001 ) surfaces (MacLaren et al., 1987; Van Hove et al., 1986); these high-coordinated adsorption sites are illustrated in Fig. 9.18. Furthermore, a comparison with structural studies of oxygen on close-packed transition-metal surfaces reveals a general property that strongly adsorbed chalcogens tend to occupy those sites that would be occupied by an additional metal layer. This was first mentioned by Marcus et al. (1975). For example, while on Ru(0001) the hcp site is occupied by oxygen, the fcc site is favored on Ir(111). Due to the strong interaction of oxygen with the metal surface, the minimization of the total energy is accomplished by displacements of atoms within the substrate lattice which are confined either to position changes within the unit cell, relaxations, or reconstructions characterized by breaking substrate bonds. For adsorption
Fig. 9.18. Illustration of high-coordinated adsorption sites on fcc(l 11), hcp(0001) and fcc(100).
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Table 9.5 Adsorption sites of oxygen on non-reconstructed metal surfaces System
Co( 100)c(2x2)-O Cu( 100)c(2• Cu( 100)c(2x2)-O Ni(100)c(2x2)-O Ni(100)c(2x2)-O Ni(100)(2x2)-O Ni( 111)('~r3-x#-3-)-O Ni(l 11)(2x2)-O Ni(ll 1)(2x2), (~-x#3-) lr(ll l)(2x2),(2xl )-O Ru(0001 )(2xl )-O Ru(0001 )(2x2)-O Rh(ll 1)(2x2)-O Rh( 100)(2x2)-O Pt(ll 1)(2x2)-O
Coordination and Site
Method
Reference
4-fold 4-fold 4-fold 4-fold 4-fold 4-fold 3-fold, fcc 3-fold, fcc 3-fold, fcc 3-fold, fcc 3-fold, hcp 3-fold, hcp 3-fold, fcc 4-fold 3-fold, fcc
LEED PED SEXAFS SEXAFS LEED LEED LEED LEED NEXAFS LEED LEED LEED LEED LEED TCT
Maglietta (1982) Tobin et al. (1982) DObler et al. (1985) St6hr et al. (1982) Tong et al. (1982) Van Hove and Tong (1975) Mendez et al. (1991) Grimsby et al. (1990) Pedio et al. (1990) Chan and Weinberg (1979) Pfntir et al. (1989) Lindroos et al. (1989) Wong et al. (1986) Oed et al. (1988) Mortensen et al. (1989)
TCT = Transmission Channeling Technique. PED = Photoelectron Diffraction. systems considered here, reconstructions of the close-packed surfaces have not been observed. For the system R u ( 0 0 0 1 ) - O , dissociatively adsorbed oxygen forms only an apparent ( 2 x 2 ) - L E E D pattern. During oxygen adsorption (at room temperature) at coverages of 0.25 and 0.5, the LEED intensities (at a fixed energy) of the oxygeninduced (1/2, 0) beam, however, exhibit m a x i m a (Madey et al., 1975) indicating two different ordered structures. H R E E L S measurements (Rahman et al., 1983) and investigations on a stepped Ru(0001) surface (Parrot et al. (1979) for the high-coverage phase reveal that the LEED pattern is explained by the superposition of three p ( 2 x l ) domains rather than a honeycomb structure. Thus, the two distinct m a x i m a in L E E D intensity can be ascribed to a p(2x2) and a p ( 2 x l ) structure. The local adsorption geometries of both structures were analyzed by L E E D (Pfntir et al., 1989a; Lindroos et al., 1989). It turned out that oxygen atoms occupy hcp sites in both phases, and even the bond lengths of oxygen to Ru are equal: 2.03 _ 0.04 A. This value is comparable to corresponding bond lengths in RuO2. The strong bonding of oxygen to the substrate is expressed in displacements (about 0.10 A) of the Ru atoms in the top layers from their bulk positions. Due to the different s y m m e t r y of both unit cells, the O-induced relaxations of the substrate are s o m e w h a t different. C o m p a r e d to the clean surface (Michalk et al., 1983), the 2% contraction of the first Ru layer spacing is removed by oxygen at 0 - 1/2 ( p ( 2 x l ) ) . This effect can be understood intuitively as follows: the adsorbate relieves, to some extent, the strong a s y m m e t r y that the bare surface creates for the outermost substrate atoms. The outermost substrate atoms now also have bonds on the external side, simulating a more bulk-like environment. At 0 - 1/4 (p(2• the oxygen coverage is not
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sufficient to remove this contraction. Such adsorbate-induced changes in the topmost substrate layers are frequently observed and represent a common reaction pattern in the adsorption of, e.g., O, S, and H on low-Miller-index faces of metals (Van Hove and Somorjai, 1994). On the Ru(0001) surface the use of NO2 allows one to produce a high-coverage (1• phase. A combined DFT and LEED investigation found that the first Ru interlayer spacing is expanded by about 3% (Stampfl et al., 1996). This expansion serves as a precursor state for oxygen penetration into the subsurface region which precedes the oxidation process (cf. w 9.3.3). In an investigation of the critical behavior of the order-disorder transition of the p(2• phase, the critical exponent lies in the universality class of the four-state Potts model (PfniJr and Piercy, 1989). This requires adsorption on the same highsymmetry sites in both the ordered and disordered phases which indicates a substantial energy difference between the two kinds of threefold sites, so that even in the disordered phase the hcp site is most favorably occupied. A similar result was obtained with the system Ni(100)-O - - but in a more direct way m in which oxygen atoms were shown to adsorb in fourfold-coordinated hollow sites independent of the coverage. For this investigation Heinz et al. (1991) utilize the diffuse LEED intensity spectra to be solely dependent on the local adsorption geometry (Maglietta, 1982; Yang et al., 1983). In a second example, we are concerned with the system AI(I 11)-O which has the benefit that several theoretical calculations are also available (Batra and Kleinman, 1984). Dissociative chemisorption of 02 on an AI(111) surface at 300 K takes place with a small sticking coefficient of about 0.005 (Brune et al., 1993). The observation of dispersion effects on the oxygen 2p-derived levels in angular-resolved photoemission was the first indication of the presence of ordered ( l x l ) islands on AI(I 11) (Eberhardt and Himpsel, 1979). This suggests the operation of net attractive forces between O atoms adsorbed on neighboring sites. The combination of XPS and cluster calculations also showed evidence for oxygen-island formation on AI(I 11). The two O ls binding energies observed in the chemisorption regime are consistent with an interpretation in terms of interior and perimeter adatoms (Bagus et al., 1991). The arrangement of oxygen adatoms in small islands and the basically random distribution of these islands across the surface were found to be characteristic of the O adlayer at low and medium coverages (Wintterlin et al., 1988). Island formation already starts at a coverage of 0.03, as pointed out by Brune et al. (1993). Brune et al. (1991) observed that part of the chemisorption energy can be transferred to kinetic energy of the dissociating oxygen molecule parallel to the surface: hot adatoms. For 0 = 0.0014, upon dissociation of an oxygen molecule at an AI(I 11) surface, the two O atoms move at least 80 A apart, while the energy is dissipated via phonon excitation and/or through electronic friction. The island growth observed for 0 > 0.03 at room temperature is intimately correlated with such hot oxygen atoms because at 300 K, oxygen atoms at equilibrium are practically immobile. This implies that if a ballistic O atom hits another adatom or ensemble which is already at rest, there is a high probability that it will become trapped.
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Oxygen atoms occupy threefold-fcc hollow sites as shown by SEXAFS (St6hr et al., 1982), LEED (Soria et al., 1978), XWS (Kerkar et al., 1992) and STM (Brune et al., 1993) at a height of about 0.7 .~ above the first A1 layer. Apart from these techniques, band-structure calculations of the ARUPS results are compatible with an O-AI layer spacing of 0.55-0.7/~ (Bylander et al., 1982). With regard to the bonding of oxygen to AI(111) one would expect that there will be a large charge transfer involved which should lead to a remarkable increase in work function as found with other systems. For AI(111), however, this has not been observed, the work-function change is quite small (Hofmann et al., 1979). Cluster calculations for All90 (Cox and Bauschlicher, 1982) showed, however, that the complicated charge-density distribution for covalently bound O in fcc sites of AI( 111 ) is related to a rather small dipole moment of only -0.09 Debye and is hence in line with the small work-function change observed in the experiment. m
9.3.2. Adsorption on open surfaces: fcc(llO), hcp(lOlO) The fcc(ll0) surfaces are well known for their strong tendency to reconstruct. While even the clean surfaces of 5d metals like Au(110) and Pt(110) form a (lx2) missing-row configuration, 3d metal surfaces like N i ( l l 0 ) and Cu(110) as well as the 4d metal surfaces like Ag(110) and Pd(110) exhibit this type of (Ix2) reconstruction due to alkali adsorption. Upon exposure to oxygen, on the other hand, N i ( l l 0 ) (Kleinle et al., 1990), C u ( l l 0 ) (Bader et al., 1986; Parkin et al., 1990; Feidenhans'l et al., 1990), Ag(110) (Becket et al., 1991" Taniguchi et al., 1992) and Fe(211) (Sokolov et al., 1986) undergo (2• missing-row type reconstructions. These two types of missing-row reconstruction are quite different: in the (1• phase every second close-packed row along [110] is missing, while in the (2• phase every other row perpendicular to the close-packed rows [001 ] is removed (cf. Fig. 9.19).
Fig. 9.19. Top view of the (2xl) missing-row structure in comparison with the (lx2) missing-row structure on the fcc(110) surface.
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Until the mid-eighties, there was an ongoing debate as to whether or not several of the fcc(110) surfaces reconstruct in (2x l) missing-row type and where the oxygen is located. Due to the problem of explaining the long-range mass transport involved in generating the missing-row arrangement at relatively low temperature (<500 K), several alternate models were proposed. For a historical survey the reader is referred to Brundle and Broughton (1990); for recent advancements on this field the reader is referred to Besenbacher and NCrskov (1993) and Koch et al. (1992). For the Ni(110)p(2xl)-O, He scattering (Engel and Rieder, 1984), low-energy ion scattering (Verheij et al., 1979), and medium-energy ion scattering (Van der Veen et al., 1979) suggested that reconstruction of at least part of the topmost Ni layer is involved in the formation of the p ( 2 x l ) structure; this list is not intended to be complete. The experimentally found dispersion of O-induced surface states in angle-resolved UPS (Pollak et al., 1991) has been found to agree with theoretical predictions for a missing-row type reconstruction (Noffke et al., 1990). The surface states were attributed to the interaction of the O 2p states with the next-neighbor Ni atoms along the [001 ]-directed Ni-O chains. Generally, the reconstruction of a surface during a chemisorption process occurs if the gain in chemisorption energy on a reconstructed surface (in comparison to the unreconstructed surface) is larger than the cost for breaking metal-metal bonds. Since the energy of reconstruction has to be raised first, this kind of process is activated and needs elevated sample temperatures. Several LEED structures have been observed with Ni(110)-O: p(3xl), p(2• and high-coverage p(3xl). The determination of the corresponding coverages was made by using medium-energy Rutherford back-scattering (RBS) (Smeenk et al., 1980) and turned out to be 0.25-0.30 for p(2• and 0.4 for the high-coverage p ( 3 x l ) indicating island formation with local coverages of 1/2 and 2/3, respectively. For the p ( 2 x l ) phase (first reported by Germer and McRae (1962)) a reconstruction of the substrate was proposed in order to explain the high intensities of the extra beams in the LEED pattern (Ferrer and Bonzel, 1982). Furthermore, the formation of the p ( 2 x l ) represents an activated process since for adsorption temperatures at about 150 K a c(2x4) structure is observed which transforms upon annealing to T = 450 K to (2x 1) or (3xl), depending on the actual oxygen coverage (Behm et al., 1986). The activation energy turned out to be 24 kJ/mole. A recent and very detailed LEED structural analysis clearly favored a missing-row structure as depicted in Fig. 9.20 (Kleinle et al., 1990). The adsorbed O atoms are located 0.2 above the long bridge sites in [001] direction, presumably with a slight displacement (0.1 A) in the [1T0] direction to an asymmetric adsorption site. The reduced bond strength of the topmost Ni layer (Ni-O chains) to the substrate induces subsurface lattice distortion: the first Ni layer spacing is expanded (in contrast to clean Ni(110)), and the third Ni layer is buckled. The authors found a strong bonding between Ni and O reflected by a very short distance of oxygen to the nearest-neighbor Ni atoms of 1.77 A compared to the sum of atomic radii of Ni ( 1.25 ,~) and O (0.74 ,~); a similar result was also found for the Cu(l 10)p(2x 1)-O structure (Bader et al., 1986; Parkin et al., 1990). The static structure of the oxygen-induced missing-row reconstruction on fcc(110) surface is only one part of the story. The main problem with the missing-
Chemically adsorbed layers on metal and semiconductor su~. "aces
Fig. 9.20. The (2•
451
missing-row structure of Ni(110)(2• as found by a LEED structure analysis (Kleinle et al., 1990).
row model concerns the massive transport of substrate atoms required to form the metal-oxygen chains. To that end, the transformation from the (1• to the p(2xl ) configuration was investigated by STM. This will be demonstrated first with the related system C u ( I I 0 ) - O . Upon adsorption of small amounts of oxygen on Cu(110), a streaky (2• LEED pattern becomes visible which indicates a lack of long-range order in the [110] surface direction. This can be explained in terms of a preferential growth of (2• islands ( O - C u - O strings) mainly along the troughs in [001] direction (Mundenar et ai., 1987) and with small dimensions in the [1101 direction. Recent STM studies confirmed this conclusion on the dynamics of the reconstruction of the Cu(110) surface induced by oxygen chemisorption (Coulman et al., 1990; Jensen et al., 1990). Cu atoms are removed from the step edges and diffuse out on the flat terraces where they react with oxygen atoms originating from the dissociative adsorption process. Both species are very mobile, so that short C u - O chains can condense to form (2• nuclei due to strong C u - O interactions, see Fig. 9.21. They grow rapidly along [001 ] and more slowly in the [ 110] direction in line with the afore-mentioned LEED experiments. The activated step in this process is the evaporation of Cu atoms from steps and less frequently from terraces. The stability of Cu-O strings increases if several chains group together providing a two-dimensional p(2• unit mesh. Weak interaction of C u - O rows to adjacent steps along [001 ] stabilizes the steps themselves, so that at higher oxygen coverages the supply of Cu atoms is blocked and Cu adatoms are generated by disruption of Cu atoms from flat terraces far from the steps. According to the underlying (dynamical) mechanism, this reconstruction should be denoted more precisely as an added-row (Coulman et al., 1990) rather than a missing-row structure. This assignment (added row) is also justified by recent theoretical studies (Jacobsen and NCrskov, 1990) which found that oxygen 2p states hybridize more
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Fig. 9.21. (a) STM image of (2• nuclei (dark streaks) on Cu(110): islands with two and three rows at the upper edge of steps along [001 ] and a single-row island on the flat terrace (Coulman et al., 1990); (b) corresponding hard-sphere model: top view.
453
Chemically adsorbed layers on metal and semiconductor surfaces
efficiently with low-coordinated metal atoms due to their energetically high-lying d states; the anti-bonding states of low-coordinated Cu atoms are less filled, so that d states are shifted upwards. Thus, oxygen chemisorbed in long bridge sites of an added row becomes energetically more favorable than adsorption in the first Cu layer. It is worth mentioning that the formation of C u - O - C u chains along the [001] direction has also been reported for the (2~-• phase on Cu(100) studied with STM during oxygen exposure (Jensen et al., 1991). An analogous reconstruction was also observed for N i ( 1 1 0 ) p ( 2 x l ) - O by STM (Eierdal et al., 1992). In addition, Eierdal et al. (1992) have shown that the low- and high-coverage (3x l) structures are stabilized by one and two O - N i - O chains per unit cell running along the [001] direction. In a very recent STM study of the O chemisorption on Ni(771), a (monatomic)-stepped (110) surface, the (2x l) reconstruction was found to proceed via a composite missing- and added-row mechanism. Driven by oxygen adsorption, Ni atoms move along the troughs to adjacent terraces underneath, leaving missing rows on the upper terraces and adding rows on the lower terraces (Haase et al., 1991 ; Koch et al., 1992). Accordingly, terraces are formed which are twice as large as for the clean surface, separated by double steps. Exposing the p(2xl)-reconstructed Cu(110) surface to much higher amounts of oxygen at T > 300 K initiates a second structural transformation during which a c(6x2) structure is formed. This structure, with a local oxygen coverage of 2/3, represents the counterpart to the p(3x I ) found with Ni( 110)-O and should therefore have a similar adsorption geometry. With a combination of STM, surface X-ray diffraction, and theoretical methods this complex structure was solved (Feidenhans'l et al., 1990). This solution has been confirmed also by a recent LEED study (Liu et al., 1996). A basic part of this structure consists again of Cu-O chains lying on top of a nearly undistorted substrate; these Cu-O chains are arranged with (3x l) periodicity similar to the high-coverage N i ( 1 1 0 ) p ( 3 x l ) - O phase. In addition to that, two 'super' Cu atoms are sitting above the chains, linking and stabilizing them, cf. Fig. 9.22. The c(6x2) structure nucleates preferentially at step edges and grows almost isotropically above 300 K, unlike the (2xl) configuration which nucleates at flat terraces and grows anisotropically in the [001] direction in the form of Cu-O chains. While the adsorption of oxygen on fcc(110) surfaces has been studied extensively, much less work has been done on the 'equivalent' (1010) surfaces of hcp metals, such as Co( 1010). Adsorption of half a monolayer of oxygen on this surface at 150 K and subsequent annealing at 230 K < T < 350 K result in the appearance of an ordered c(2x4) structure which transforms irreversibly into a p ( 2 x l ) structure upon warming to 350 K < T < 580 K (Schwarz et al., 1990b). The occurrence of an activated process and similar chemical properties of Ni and Co suggest that the p ( 2 x l ) structure may be related to a reconstruction of the substrate similar to that in N i ( 1 1 0 ) ( 2 x l ) - O . A structure analysis of the p ( 2 x l ) phase (Over, 1991b) based on LEED I - V curves clearly indicates that an added-row model can be excluded. Note that the Co(1010) surface has two different possible terminations due to the ABAB... stacking, whereby only the termination with a smaller corrugation represents the stable atomic configuration (Lindroos et al., 1990; Over et al. (1991 a); cf. I
!
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H. Over and S. Y. Tong
Fig. 9.22. Structure model of the reconstructed Cu(110)c(6x2)-O phase; (a) side view and (b) top view. Grey circles represent the 'super' Cu atoms, and the small black circles indicate the oxygen atoms. A c(6x2) unit cell is shown (Feidenhans'l et al., 1990). Fig. 9.16a. A missing-row model, on the other hand, is not very likely because the formation of this structure would need to remove Co atoms from a tightly bound double layer (layer spacing between the atomic plane in this double layer is only 0.7 A). It is worth mentioning that added row and missing row are no longer equivalent for this surface, as opposed to fcc(l 10) surfaces. From STM studies it has been indicated that this phase is formed by a double-layer missing-row reconstruction (Koch et al., 1993, 1994), which is illustrated in Fig. 9.23. The last system in this section which we focus on is oxygen adsorption on fcc-Rh(l 10). This system has gained some attention due to the ability of rhodium
Fig. 9._23. A hard-sphere model (top view) of the double-layer missing-row structure for the Co( 1010)(2x I )-O surface. Co atoms' large circles, Oatoms: small black circles. The arrows indicate the movements of Co atoms to transform the clean Co(1010) surface into the (2• 1)-reconstructed surface; after (Koch et al., 1993).
Chemically adsorbed layers on metal and semiconductor su.rlaces
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to reduce NOx and has thus made Rh an important constituent of the three-way automotive exhaust catalyst. Upon interaction with oxygen this surface shows a series of LEED patterns, namely (2xl)pmg, (2x2)pmg, c(2x8), np(2x3), c(3x6), c(3x8), and complex streaked c(2x2n) patterns, depending on parameters such as sample temperature, O coverage and reduction cycles (Tucker, 1966; Schwarz et al., 1990a; Comelli et al., 1992a,b; Dhanak et al., 1992). Evidence was found that the oxygen interaction is not only restricted to the outermost Rh atoms, but may also include deeper layers of the crystal (Schwarz et al., 1990a; Comelli et al., 1992a,b; Wohlgemuth, 1994). Further discussion will be confined to the structures (2x 1)p2mg and (2x2)p2mg. Exposure of Rh(110) to 1 L oxygen at low temperatures (125 K) results in a (2xl)pmg structure which transforms irreversibly upon annealing (470-970 K) into a (2x2)pmg structure (Comelli et al., 1992a). This transformation into the (2x2)pmg phase is thermally activated (the activation temperature is similar to the case of Ni(110)(2xl)-O), suggesting that its formation might be associated with a reconstruction process. Reacting the (2x2)pmg phase with CO or H2 at elevated temperatures (<450 K) removes all the oxygen on the surface leading to a metastable (lx2) structure; annealing this phase to 510 K ends up with a ( l x l ) LEED pattern (Comelli et al., 1992b). For the (2xl)p2mg structure Comelli et al. (1992a) proposed a model in which oxygen forms zigzag chains along the [ 110] direction (required to explain the glide plane) and occupies a threefold-coordinated site in the troughs. A HREELS study indicated a stretching frequency of 570 cm -~ for oxygen (Cautero et al., 1991 ). This value is close to the oxygen frequency on Rh(l 11), 540 cm -t, for oxygen in a threefold-coordinated site (Root et al., 1986) but significantly higher than on Rh(100), 403 cm -~, where oxygen resides in fourfold sites (Dubois, 1982). In a recent LEED m
Fig. 9.24. Hard-sphere model of the Rh(110)(2• structure (a)" in comparison with that of the Rh(110)(2x2)-p2g-20 surface (b) (Gierer et al., 1993).
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structure analysis this model has been confirmed (Gierer et al., 1993); cf. Fig. 9.24a. For the (2x2)p2mg structure Comelli et al. (1992a) suggested a model which involves (lx2) missing-row reconstruction of the surface with oxygen atoms arranged with a (2xl) periodicity. Evidence was found for the oxygen-induced ( l x 2 ) reconstruction by the higher thermal stability of the corresponding LEED spots in the [ ! 00] direction in conjunction with the formation of a metastable (! x2) structure after reduction of oxygen by CO or H 2. Structural studies of this system with He diffraction (Bellman et al., 1993) and STM (Leibsle et al., 1993; Murray et al., 1993) have confirmed that the (lx2) reconstruction of the Rh(110) substrate is indeed of the missing-row type. A complete LEED structure analysis found besides this missing-row type reconstruction that oxygen resides in threefold-coordinated fcc sites on the flanks of the I l l 0 ] troughs (Gierer et al., 1993; Comicioli et al., 1993)" cf. Fig. 9.24b. D
9.3.3. Development of oxides Bulk-oxide structures are in general thermodynamically more stable than overlayer structures. The observation of O overlayers therefore indicates the existence of an activation energy barrier which inhibits production of oxides, i.e., the formation of oxides is kinetically limited. The height of this barrier depends critically on the metal-metal bond energy, the metal-oxygen bond strength, the number of defects and the ability of O to diffuse into the bulk. Strongly reactive metals like Zr do not form stable surface layers of O. Even on the close-packed surface a subsurface-oxygen layer is created immediately, illustrating an early stage of compound formation (Hui et al., 1985). For Ru(0001)-O a high activation barrier for penetration and dissolution of atomic oxygen into the bulk was found. Surnev et al. (1985) observed oxygen dissolution into the bulk only at temperatures above 1200 K. Oxides are characterized by the formation of metal ions embedded in a lattice of oxygen ions, so that the process of oxidation necessitates the incorporation of O in the metal lattice. In a simple picture the corresponding penetration of oxygen atoms into the metal substrate (subsurface oxygen) along with the subsequent structural rearrangement of metal atoms constitute activated steps which usually have a lower barrier at steps and defect sites than on terraces. Thermal treatment can assist oxidation, but there are many examples, such as Ni(100)-O or A l ( l l l ) O, where oxidation of a stable O overlayer occurs even at room temperature simply by increasing oxygen exposure. There are several indications and surface-sensitive techniques which allow to monitor the onset of oxide formation: (a) LEED and RHEED: The crystallographic properties of metals and respective metal oxides are usually different, so that the LEED patterns are different, once ordered oxide islands are sufficiently large. (b) XPS, UPS, AES: An onset of electronic features of the corresponding metal ion involved in the metal-oxide arrangement can become visible. (c) SIMS: At the nucleation point of oxides the yield of positive and negative ions varies.
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(d) MEIS: The onset of large displacements of surface metal atoms from their original lattice sites can be observed. (e) IPES: During oxidation the density of states above the Fermi level (empty states) changes. (f) STM: Usually, the density of metal ions in oxides is different from that of the atoms in the substrate, and accordingly the surface topography is expected to change during oxidation. In the following we concentrate on the oxidation of the Ni(100) surface which has been studied extensively for over 20 years (Brundle and Broughton, 1990). The Ni(100) surface exposed to oxygen gives rise to two distinct ordered overlayer structures observed in LEED, namely a p(2• and a c(2• at coverages of 1/4 and 1/2, respectively. For both overlayers the fourfold-coordinated site is occupied by oxygen, and the O-Ni layer spacing is around 0.8 A (Chubb et al., 1990; Oed et al., 1988), cf. Fig. 9.25. The substantial expansion of the first substrate spacing found with Ni(001)c(2x2)-O is consistent with a softening of the surface-force constants which are required to explain the shift of surface-phonon frequencies upon oxygen adsorption (Rahman et al., 1984). The rather small Ni-O layer spacing was explained by a bonding of O to the Ni atom in the second layer directly underneath Ni(2) (Chubb et al., 1990) which weakens the metal-metal interlayer bond. The bond between O and Ni(2) is characterized by a broadening of Ni d bands and a charge transfer to the region between O and the subsurface Ni. Cluster calculations (Bauschlicher and Bagus, 1984) reveal that the primary bonding interaction is between Ni 4s, 4p and O 2p levels with about unit charge transfer from Ni to O in conjunction with a strong covalent element to the bonding. These calculations reproduce the decrease in the O-Ni stretching frequency from 54 to 38.5 meV as the coverage increases from 1/4 (p(2x2)) to 1/2 (c(2x2)) (Andersson, 1979), thus showing that chemisorption properties arise principally from the local environment. Compared to NiO (stable oxide), the major differences are a slightly higher charge transfer and a much stronger Ni 3d involvement for the oxide. The rise in sticking probability before the c(2x2)-O structure covers the surface entirely was associated by Holloway and Hudson (1974) with the onset of NiO nucleation. NiO(I 11) and NiO(100) LEED patterns could be observed beyond the completion of the c(2x2)-O structure. This indicates that the ordering of NiO is poor in the initial stage of oxidation. The work function increases upon oxygen exposure up to the completion of the p(2• phase indicating overlayer oxygen. Beyond this point, the work function decreases monotonically, which is consistent with oxygen penetrating into the bulk. Several other techniques confirmed the onset of oxidation at this stage. For example, in LEED (Holloway and Hudson, 1974) the NiO pattern was observed; XPS and UPS (Hopster and Brundle, 1979a,b) detected the onset of oxidation via the Ni2+-derived electronic feature; MEIS found NiO by the onset of large displacements of surface Ni atoms (Smeenk et al., 1981 ). In IPES, the onset of oxidation showed up in the density of states above the Fermi level (Scheid et al., 1982). The oxidation rate can be increased by damaging the surface as observed by Miranda et al. (1980). This observation is in line with the view that the penetration
458
H. Over and S. Y. Tong
of oxygen atoms is activated in the initial stage of oxidation and that the corresponding energy barrier is reduced at defects. In an STM investigation the conversion from stable overlayer to oxide was ascribed to the presence of defects at the surface which act as nucleation centers for oxide growth (Wilhelmi et al., 1991). In this study the oxidation starts even when the LEED pattern shows a sharp c(2• structure in accordance with the observations made by Holloway and Hudson (1974). For oxygen exposure of 5000 L at a temperature of 540 K, STM images show that the Ni(100) surface is completely covered by NiO islands with typical lateral dimensions of 50 ]k. These islands are oriented in the [ 1T0] directions. A certain degree of order within the NiO crystallites is evident from LEED experiments. We can summarize the initial stage of adsorption of oxygen on Ni(001) in the following one-dimensional potential energy diagram (sketched in Fig. 9.25), which was originally developed (Andersson, 1979) to explain the drastic change in the O-Ni stretching frequency for the p(2x2) and the c(2x2) structure. For the particular anharmonic potential Andersson (1979) used to model this system, the value of U, (barrier for diffusion into the bulk) is the adjustable parameter which turns out 0 0
|
2:
o ('~"
U(z)
Ue Ua i.....,..,........-.~
Z
t !
--0.8 A Fig. 9.25. One-dimensional potential energy diagram of O on Ni(001). dL is the Ni-O layer spacing, and U~,is the potential barrier to O~,jpenetration of the lattice (Andersson, 1979; Brundle and Broughton, 1990).
Chemically adsorbed layers on metal and semiconductor su~. aces
459
to be U a - - 1.5 eV (p(2x2)) and U a - " 0.8 eV (c(2• As pointed out by Andersson, this interpretation is supported by the fact that O dissolves readily into the bulk, indicating a low potential barrier. In addition, it is known that the nucleation of oxides starts from the c(2x2) structure, suggesting that the barrier is lower for that structure. LEED structural analyses of both ordered structures have supported this conclusion. Oxygen atoms in the c(2x2) structure penetrate more deeply into the surface by 0.05 A (Chubb et al., 1990) compared to the p(2• phase (Oed et al., 1990). Recalling that the first layer spacing of the clean surface is contracted by 1%, the total expansion induced by adsorption is 3.5% for the p(2x2) and 7% for the c(2• structure which may be attributed to 'buried' oxygen atoms associated with a first step towards oxidation.
9.4. Alkali-metal/metal systems The adsorption of alkali-metal atoms on metal substrates has attracted significant interest in experimental as well as theoretical research since the pioneering work of Langmuir (1923, 1932) and Gurney (1935). This interest has been prompted, on the one hand, by the technological importance of alkali adsorbates for efficient (i.e. low work function) photocathodes with negative electron affinity and efficient sources of highly spin-polarized electrons (GaAs treated with Cs and O) (Spicer, 1977; Pierce and Meier, 1976). On the other hand, alkali metals find widespread application in the manufacture of heterogeneous catalysts in order to achieve a superior selectivity and/or reactivity, see for example (Mross, 1983). For the ammonia synthesis reaction, the dissociation of N 2 is the rate-limiting step, so that an increase of the dissociative Nz adsorption will increase the overall synthesis rate. Ertl et al. (1983) found that preadsorbed potassium on Fe(100) increases the rate of dissociative nitrogen chemisorption by two orders of magnitude. The dominant effect on transition-metal surfaces is viewed to be a local increase in the electron density around the adsorbed alkali-metal atom induced by the large dipole moment (NCrskov et al., 1984). This effect becomes stronger with increasing atomic number of the alkali-metal atoms, as demonstrated in Fig. 9.26. The rate of ammonia production over a carbon-supported Ru catalyst for various alkali metals is a function of the amount of alkali metal in the catalyst (Ozaki and Aika, 1981). Catalysts are usually employed at relatively high temperatures where evaporation of pure alkali metals would occur. Therefore, adsorbed alkali-metal atoms in catalysts are generally stabilized by formation of a complex with oxygen (cf. w 9.4.3). Alkali-metal adsorption on metal surfaces serves also as a prototypical system to study the basic mechanisms of chemisorption because it is generally assumed that alkali-metal adsorption is not complicated by adsorbate-substrate mixing, and the alkali-metal/metal interaction is dominated by partial charge transfer (or possibly polarization) of the single s-valence electron. Reviews of many aspects of physics and chemistry of alkali-metal adsorption and excellent sources of references can be found in the monograph by Bonzel et al. (1989). Aruga and Murata (1989) published a review on alkali-metal adsorption on metals with an emphasis on correlating electronic properties with ordering properties.
460
H. Over and S. E Tong
30
-
!
20-
~10 2:
-
Na 00 2 4 6 8 10 alkali metal/mmol(g-cat) 1 Fig. 9.26. The rate of ammonia production over a carbon-supported Ru catalyst for various alkali metals as a function of the amount of alkali metal in the catalyst. With increasing atomic number of the alkali-metal atoms, the production rate increases; after Ozaki and Aika (1981).
Structural properties of alkali metals on metal surfaces are compiled and discussed in the review by Diehl and McGrath (1996). In the following we begin with some (theoretical) remarks on alkali-metal adsorption on metal surfaces (w 9.4.1), which go back to the fundamental work of Langmuir ( 1923, 1932) and Gurney (1935). The adsorbed alkali atom is considered to be ionic at low coverages and becomes neutral with increasing coverage. Recent theoretical investigations have questioned this traditional view and suggested that the net charge of the alkali adatom is rather insensitive to coverage and that there is little or no net charge transfer from the alkali metal to the metal substrate (Ishida, 1990). This initiated a heavy controversy about the question whether the bonding of alkali metals to the metal substrate has to be considered as being ionic or covalent in nature which eventually ended up with a 'revival' of the ionic picture for the limit of small alkali-metal coverages (Scheffler et al., 1991). In w 9.4.2 we proceed to the structural properties of adsorbed alkali-metal atoms mainly on Ru(0001 ) and AI(111) which show that for these systems the adsorption geometries found are quite varied. A knowledge of the adsorption geometry provides essential information towards the understanding of the electronic and chemical properties of these surfaces because theoretical models are based on the geometry of adsorption, particularly the adsorption site and the adsorbate-substrate bond length. Many theoretical studies of alkali-metal adsorption until recently have assumed that adsorption takes place in hollow sites and that the substrate is unperturbed. This is frequently not the case, as will be demonstrated in w 9.4.2.
Chemically adsorbed layers on metal and semiconductor su~. "aces
461
The last section is devoted to the properties of alkali-metal and oxygen overlayers coadsorbed on Ru(O001), in which the complex C s - O chemistry is reflected by a large variety of mixed phases with long-range order. By contrast, the direct interaction between Cs and CO is only weak, however, when coadsorbed, e.g., on a Ru(O001 ) surface, they interact heavily. The structural properties of this coadsorption system in relation to its electronic properties will finish this section. 9.4.1. Langmuir-Gurney model and recent theoretical results
One of the prominent features observed in alkali-metal/metal systems is the substantial decrease of the electronic work function. The work function of a metal is defined as the minimum energy required to extract one electron from a metal. It can be divided into contributions from the bulk and from the surface. For the case of simple metals (described in the Jellium model) (Lang and Kohn, 1970), the surface contribution to the work function is caused by the 'spilling out' of electrons into the vacuum region which creates an electrostatic dipole layer at the surface (cf. Fig. 9.27). Only this contribution is sensitive to the charge redistribution caused by the adsorption of atoms and molecules, so that monitoring of the work-function change A ~ yields information on electronic properties of adsorbates. After the discovery of the work-function lowering and the desorption of positive Cs ions Langmuir (1923, 1932, 1933) proposed the following model. Since the ionization potential of an unperturbed alkali-metal atom like Cs (3.9 eV) is less than the work function of a bare transition metal like W (4.63 eV), Cs is adsorbed on W as a positive ion, donating one electron to the substrate: bonding to the surface is thus purely ionic in nature. Assuming that the metal substrate is a perfect conductor with an atomically flat surface, the Cs + ion and its negative image charge form a surface edge ~ vacuum e l
metal 1.0
.
.
~0.5 -
.
.
.
.
.
.
.
.
.
~ ~
I ideal charge 1/distributi6n
dipole moment
umU,~
[~ i\ 0 -10
I
I
"~-
/
realistic char~:e distributioff
~
-5 0 5 Distance (atomic units)
.,
10
Fig. 9.27. Electron-density distribution at the edge of a metal surface. The 'spilling out' of charge density into the vacuum region creates a dipole moment perpendicular to the surface.
462
a) attractive interaction between the positive Cs ion and its image charge (vertical interaction)
H. Over and S.Y. Tong
b) repulsive dipole-dipole interaction between the Cs-atoms (lateral interaction)
Fig. 9.28. The main energy contributions determining the behavior of alkali-metal adsorption on metal surfaces. (a) The partial charge transfer from the alkali-metal atom to the metal substrate along with its image charge (screening charge) induce an electric dipole moment It. The bonding of the alkali-metal atom to the substrate can be described in a very simplistic view by the attractive electrostatic force between the alkali-metal ion and its image charge. (b) The lateral interaction is dominated by the repulsive dipole-dipole interaction. dipole g which opposes the clean substrate charge spill-out dipole, hence reducing the work function of the surface. In addition, the attraction between the alkali-metal ion and its image charge was considered to be responsible for the bonding force (cf. Fig. 9.28a). For low alkali-metal coverages, each adatom contributes individually to the work-function change, so its decrease is linear in coverage. At higher coverages (decreasing average separation of dipoles) the mutual depolarization of dipoles reduces the effective dipole moment per adatom, resulting in a non-linear variation of the work-function change. A~ goes through a minimum and increases from there on to the high coverage value (see Fig. 9.29). This electrostatic model (depolarization proceeds via charge back-donation) also explains the decrease of binding energy with increasing coverage. This ionic picture is similar to the 'harpooning' mechanism which is well known in the reaction dynamics of gaseous particles with strongly different electronegativity (Polanyi, 1932). However, for a solid surface, this view is certainly oversimplified. Gurney (1935) criticized this ionic picture of alkali-metal adsorption by pointing out that barium also lowers the work function of tungsten, even though its first ionization potential is greater than the work function of W. In a recent investigation of the Ru(0001)-Mg system (Over et ai., 1993b) the variation of the work function was shown to be similar to that of Ru(0001)-Na, yet the ionization energy of Mg (7.65 eV) is significantly higher than the work function of Ru(0001) (5.4 eV). Figure 9.30 illustrates this point. Gurney showed in a simple quantum-mechanical model that the valence level of an atom approaching a metal is shifted upon interaction with the surface and is lifetime-broadened into a band, which means that electrons can tunnel between the adatom and the solid. The combined influence of the level mixing (lowers the energy), image shift (raises the energy), and surface barrier shift (lowers the energy) determines the ultimate energy position of the
Chemically adsorbed layers on metal and semiconductor surfaces
of'
I
i
I
i
I
i
Cs/Ru(0001) =
463
I
T=80 K
= ring
Ru O _
>
(U v
t
~
~ p(2 x 2) ,
~ 013- 1
.
-
*-" ( V i x ~ ) - a 3 0
_
_
o-2 t3
~-3 o -4
_
,
I
,
0.1
1
,
,
I
,// multilayer
I
0.2 0.3 0.4 Cs coverage
Fig. 9.29. The change in work function depending on the coverage for Cs adsorbed on Ru(0001); after Hrbek (1985).
>
Na-Ru(0001)
0.0-
Mg-Ru(0001)
0.0-
1.0-
-0.5 -
(D
2.0-
@ -1.0 9
qlp 9
-1.5 -
O0
O 0
9
O @ osm
3.0 -
9
0
~ Oip
9
.,p 9 o op
-2.0 -
! ,.
0.0
. . .
I
0.5
. . . .
I
. . . .
1.0
I
. . . .
1.5
coverage / ML
I
2.0
4.0--
""
0.0
"
'
I
'
"
'
"
I
"
'
"
"
I
"""
0.5 1.0 1.5 coverage / ML
'
'
2.0
Fig. 9.30. Comparison of the work-function change observed with Ru(0001)-Na and Ru(0001 )Mg. The coverages are given in physical monolayers ML. a d s o r b a t e level at the equilibrium a d s o r b a t e - s u b s t r a t e separation. Typical values for the shifting (0.5 ... 1.0 eV) and b r o a d e n i n g of this valence level (0.3 ... 1.5 eV) e s t i m a t e d by using a simplified H a m i l t o n o p e r a t o r ( A n d e r s o n , 1961; G a d z u k et al., 1971; M u s c a t and N e w n s , 1978) are shown in Fig. 9.31. The m o s t i m p o r t a n t
464
H. Over and S. Y. Tong
Fig. 9.31. The electronic energy diagram as a single alkali-metal atom approaches a metal surface according to the model of Gurney. Upon interaction of the alkali metal with the substrate the atomic s level empties and broadens.
consequence of this model is that even a single alkali-metal atom is not completely ionized since the tail of its energy distribution reaches below the Fermi level, and hence this level is partly occupied. A recent investigation of K adsorbed on Cu(110) by metastable He* atom deexcitation spectroscopy (Woratschek et al., 1985) has shown that the K 4s state is partially occupied even at very low coverages (1.2%), i.e. for isolated adsorbed K species. A similar effect but for higher coverages was observed by Horn et al., 1988) using UPS with the system A I ( I I I ) - K . With increasing alkali-metal coverage each dipole depolarizes its neighbors. The energy level (of a single adatom) drops in energy and consequently, the occupation of the s resonance increases. This results in a diminished effective dipole moment, and a more metallic bond replaces the original strongly ionic bond. This scenario manifests itself in the nonlinear variation of the work-function change. It should be emphasized that this so-called Langmuir-Gurney model does not predict the value of the polarizabiltity and hence gives no insight into the microscopic electronic structure of the adatoms, such as the change in alkali metal-induced electron density and the density of states of the valence resonances. The electronic structure is better described by using the density-functional formalism (Hohenberg and Kohn, 1964; Kohn and Sham, 1965; Lang, 1973) where self-consistency is applied to study trends in chemisorption phenomena (Lang and Williams, 1978; NOrskov et al., 1981 ; Hjelmberg, 1978; Bormet et al., 1994). Recently, theoretical studies for the system AI(001)-Na (Ishida and Terakura, 1988; Ishida, 1990) concluded that the effective electron transfer is small. Even at low coverages a strong covalent character of the alkali metal-substrate bond was inferred from a hybridization of Na s and Pz states. The main effect is an internal polarization of the alkali adatom or, in other words, the alkali atom is bonded to the substrate with practically zero charge transfer. This conclusion seemed to be supported by experiments of Rifle et al. (1990) and Astaldi et al. (1990) in which the energetic position of UPS and HREELS features, respectively, showed no significant dependence on the alkali-metal coverage. In this 'covalent picture', the decrease of the dipole moment with coverage is ascribed to the direct N a - N a interaction due to orbital overlapping and is accompanied by a weakening of the Na-substrate bond followed by a strengthening of the N a - N a bond.
465
Chemically adsorbed layers on metal and semiconductor surfaces
Very recently, Scheffler et al. (1991) pointed out that the electron-density change for the system AI(001)-Na at low coverages can be explained by a response of the metal surface (screening charge) to an external point charge. This indicates that a hybridization of Na 3s and 3pz orbitals plays no significant role. The adsorption of a single alkali-metal atom on AI(001) is instead described by a charge transfer from the alkali-metal atom to the substrate (ionic bonding), wherein the slightly positively charged alkali atom induces an image-charge density (screening charge) in the metal surface. Using both UPS and inverse photoemission, Frank et al. (1989) have studied the downward shift of both s and unoccupied Pz resonances for Na, K, and Cs on AI(111) as a function of coverage. They found for potassium in the zero-coverage limit that the center of.gravity of the s resonance is 0.7 eV above the Fermi level which is consistent with an ionic bond to a 'single' K atom. The corresponding Pz resonance turned out to be 2.7 eV above the Fermi level (2 eV apart from the s resonance), so that a considerable sp hybridization has to be excluded at low alkali-metal coverages.
9.4.2. lnitial growth and adsorption geometry From a structural point of view, the adsorption of alkali-metal atoms on metal surfaces can be roughly divided into three categories. In a first class of systems, alkali-metal atoms occupy high-symmetry sites (so-called hollow sites) on the unreconstructed substrate as frequently observed for fcc(100) and hcp(1010) surfaces, see Table 9.6. These systems reflect the generally 'accepted' view, wherein for atomic adsorbates adsorption sites with high coordination are preferred (MacLaren et al., 1987). In this context, however, it is worthwhile to note that these results are currently under debate for AI(100), where an intermixing of alkali and substrate-atom layers has recently been reported (Aminpirooz et al., 1992). Secondly, several fcc(l 10) surfaces (like Ni, Cu, Pd, and Ag) undergo a reconstruction into (lx2) (missing-row) structures upon adsorption of even a small number of alkali-metal atoms; Gerlach and Rhodin (1968, 1969) performed first experimental investigations with Na, K, and Cs adsorbed on Ni(110). Missing row-type reconstructions have been experimentally verified by ion-scattering studies (Cu(110)-Li (Copel et al., 1985) and Ag(110)-K (Frenken et al., 1987)) and LEED intensity analyses (K and Cs on P d ( l l 0 ) (Barnes et al., 1985, 1988). In a detailed STM investigation with Cu(110) Schuster et al. (1991, 1992) have demonstrated the strictly local character of this charge-induced reconstruction process. They found that the critical nucleus is a single adsorbed K atom. According to theoretical studies, the corresponding driving force results from an increase in adsorption energy, due to the higher coordination number which overcomes the small energy barrier for the transition from the (Ix 1) unreconstructed structure to the (lx2) reconstructed structure (Jacobsen and NCrskov (1988). The remaining section focuses on the adsorption of alkali metals on closepacked surfaces (third category), namely hcp(0001) and f c c ( l l l ) , particularly AI(111) and Ru(0001), because on these two metal surfaces detailed structural D
466
H. Over and S. Y. Tong
Table 9.6 Structural determination for alkali-metal adsorption on metallic surfaces. Rad = hard-sphere radius. 0 = coverage. System
Rad
0
Method
Adsite
Reference
Ru(0001)('~3 x,13-)-Li
1.39
0.33
LEED
hcp
Gierer et al. (1995)
Al(100)c(2x2)-Na AI( 111 )(,]3-3x,~-3)-Na AI(I 1l)-Na
1.43 1.88 1,67
LEED SEXAFS NISXW
4-fold 6-fold 6-fold
Van Hove et al. (1976) Schmalz et al. ( 1991 ) Kerkar et al. (1992a)
Ni( 100)c(2x2)-Na Ru(0001 )(2x2)-Na Ru(0001 )(~3x,]3-3)-Na Ru(0001)(3x3)-Na
1.41 1.58 1.58 1.67
0.50 0.33 0.120.33 0.50 0.25 0.33 0.44
LEED LEED LEED LEED
4-fold fcc hcp on top & bridge
Demuth et al. (1974) Hertel (1994a) Hertel (1994a) Hertel (1994a)
AI(I 11)(',]3-3x,,/-j)-Na AI(I 11)(~3x,J3-3)-Na Co( 1010)c(2x2)-Na Cu( 100)-Na, incomm. Ni( 100)c(2• Ni( 100)-Na Ni(l 1 l)(2x2)-Na Ru(0001 )(2x2)-Na Ru(0001)(4'3-x~/3 )-Na
1.80 2.15 1.87 1.53 1.96 2.0 !.45 1.94 1.98
0.33 0.33 0.50 0.18 0.50 <0.25 0.25 0.25 0.33
LEED LEED LEED X-ray LEED DLEED LEED LEED LEED
on top 6-fold 4-fold 4-fold 4-fold 4-fold on top fcc hop
Stampfl et al. (1992) Stampfl et al. (1992) Barnes et al. ( 1991 ) Meyerheim (1993) Muschiol et al. (1992) Wedler et ai. (1993) Fisher et al. (1992) Gierer et al. (1992) Gierer et al. (1992)
AI(I i l)(x/3 x,,/3 )-Rb Ai( 111 )(~t3• )-Rb Ru(O001 )(2x2)-Rb Ru( 0001 )( ~.~ • )-Rb
1.93 2.31 2.03 2.03
0.33 0.33 0.25 0.33
LEED LEED LEED LEED
on top 6-fold fcc hcp
Nielsen et al. (1994) Nielsen et al. (1994) Hertei (1994b) Hertel (1994b)
Ag( ! 1 l)-Cs,incomm Ag( 111 )--Cs,incomm Cu( ! 11)(2x2)-Cs Rh(100)c(4x2)-Cs Ru(0001 )(2x2)-Cs Ru(0001 )(J3-• ~/.~)-Cs
1.79 2.06 1.73 2.10 1.90 2.17
0.15 0.30 0.25 0.25 0.25 0.33
SEXAFS SEXAFS LEED LEED LEED LEED
unknown unknown on top 4-fold on top hop
Lamble et al. (1988) Lamble et ai. (1988) Lindgren et al. (1983) Eggeling et ai. (1989) Over (1992a) Over (1992a)
i n f o r m a t i o n is a v a i l a b l e for a l m o s t all a l k a l i - m e t a l s p e c i e s . W e will start w i t h a close inspection of the system Ru(0001)-Cs which shows several general features c o m m o n to alkali m e t a l s a d s o r b e d on m e t a l s u r f a c e s . F i g u r e 9 . 2 9 g r a p h s t h e w o r k - f u n c t i o n c h a n g e as a f u n c t i o n o f t h e C s c o v e r a g e ( H r b e k , 1985). T h e c o v e r a g e 0 is d e f i n e d as the ratio o f the n u m b e r o f a d s o r b a t e a t o m s to t o p - l a y e r s u b s t r a t e a t o m s . U p to a c o v e r a g e o f 0 = 0 . 0 4 , the w o r k f u n c t i o n d e c r e a s e s l i n e a r l y w i t h c o v e r a g e , r e f l e c t i n g a c o n s t a n t d i p o l e m o m e n t o f a b o u t 10.5 _+ 0 . 2 D e b y e . In t h e Langmuir-Gurney p i c t u r e this d i p o l e m o m e n t is a s s o c i a t e d w i t h a p a r t i a l c h a r g e
Chemically adsorbed layers on metal and semiconductor sudaces
467
transfer to the substrate. With increasing coverage the mutual depolarization of the dipoles formed by the adsorbate complex gives rise to the non-linear variation of the work function with coverage, indicating a transition of the adsorbed particles from an 'ionic' to a more 'metallic' state. Around 0 = 0.25 (corresponding to the formation of the (2• structure observed in LEED), AO reaches a minimum, followed by a continuous increase until saturation near the value for the bulk Cs is reached. At the minimum the effect of the increase in dipole density is almost canceled by the decrease of the dipole moment per adatom. The enhanced singlettriplet conversion of metastable He* atoms found in MDS measurements for the (2x2)-Cs structure compared to lower coverages was interpreted in terms of a metallization of this layer (B6ttcher et al., 1994) consistent with the interpretation of corresponding HREELS measurements (Jacobi et al., 1994). The bonding works mainly through charge transfer of the loosely bound Cs-6s electron to the substrate, so that in the depolarization regime 0 > 0.04 a back-flow of charge from the substrate results in a decrease of the adsorption energy (together with a decrease of the dipole moment) with coverage, as demonstrated by a series of thermal desorption spectra (Fig. 9.32) taken for various initial Cs coverages (Bludau, 1992; Over et al., 1992a). The spectra show a broad desorption peak, labeled o~, (submonolayer regime) which is, for 0 < 0.04, solely attributed to the operation of strong dipole-dipole repulsion as a consequence of the large (constant) dipole moments (cf. Fig. 9.28b). For coverages higher than 0.04, the A - A interaction (A = adsorbate) and the A-S interaction (S = substrate) become coupled |
0.85
t
!
'l
r
IL
I
"
'
'
t
'
" ' ' CS -TDS
]
c o
-o-..;
13_ L
o 0 o (3 o o
200
400
I~00
800
1000
1200
--O.56
Q~
200
400
600
800
Temperature/K
Fig. 9.32. Series of thermal desorption spectra for different initial coverages of Cs on Ru(0001). The inset shows the c~peak in a magnified representation. The numbers are presenting the initial Cs coverages (Over et al., 1992a).
H. Over and S. Y. Tong
468
because the operation of depolarization is accompanied by a weakening of A-S bonding via charge back-donation (note that the A-A interaction is dominated by the dipole-dipole repulsion and therefore the energetic position of the s-resonance is shifted towards the Fermi level). At a coverage of 1/3, this desorption state saturates, corresponding to the optimum ('~-• structure. Higher Cs coverages desorb in the second layer peak and the multilayer desorption peak, exhibiting activation energies for desorption close to the value for Cs-bulk sublimation (79 kJ/mol). Due to the dominant dipole-dipole repulsion which acts laterally to the surface and the sufficient mobility of adsorbed alkali-metal atoms (Naumovets and Vedula, 1985), (T, 0) phase diagrams usually reveal a rich variety of different phases, as shown in Fig. 9.33. They are also found in other alkali-metal adsorption systems, such as Ru(0001)-Na (Doering and Semancik, 1983), Pt(111)-K (Pirug and Bonzel, 1988), N i ( l l l ) - K (Chandavarkar and Diehl, 1988). For low coverages, the typical alkali-induced ring structure is observed in the LEED pattern, with the ring diameter being a function of coverage. The ring indicates a liquid-like structure with a constant mean spacing (forced by the dipole-dipole repulsion) between the adsorbate atoms but no long-range azimuthal ordering with respect to the substrate. The occurrence of this phenomenon suggests that the corrugation of the substrate potential is fairly weak, otherwise lattice gas structures with unit cells larger than (2• would be expected for lower coverages, as reported in other cases (Pirug and Bonzel, 1988). With increasing coverage, the (2• phase develops with maximum LEED intensity of its half-order beams at 0 = 0.25 which is stable up to 325 K. A LEED structure analysis (cf. Fig. 9.34a) of this structure (Over et al., 1992a)
z.O0
-
C s / R u ( 0 0 1 ) ... :--:_ \ Disorder
300
t
-
I
I I
c_
ca. 200 E
-
'
,
Ring
I
100 _. ,
0
~R
I I ,I I
II II
U II ~-, s... I ,
I' II
l ' .... I I '; YWI I
I
II
m
II
o 0{','}
I I "-"
i,Lnl!
i;~12x2), E p-
I
i ,
~|
/
!,
,
,I
I
.
t
I
E I
O
--t
:
9
3
I--
:
t
D e s o r p t ion
~_Rlng :- f ~
I
~61 n"l
~
I
I
0.2
>,
L~
:- I
I
__j:_LL 0.1
I/I L.
h,
~
O -
__
=
-5
-I
0.3
I
0.4
0.5
Fig. 9.33. Experimental phase diagram for the system Ru(0001)--Cs.The cross-hatched area represents a qualitative result (Bludau, 1992; Over et al., 1992a).
Chemically adsorbed layers on metal and semiconductor su~. aces
469
revealed that Cs resides in on-top position with a Ru-Cs bond length of 3.25 + 0.08 ~, corresponding to a hard-sphere radius of 1.9 /~ for the adatom, closer to its Pauling ionic radius (1.69/~) than to its covalent radius (2.35 ,~). The adsorption process in the p(2x2) phase induces a shift of 0.10/~ in the Ru atom coordinated with Cs towards the bulk. Also for the p(2x2) structures formed by Cu(11 l)-Cs (Lindgren et al., 1983), Ni(111)-K (Fisher et al., 1992), AI(111)-K (Stampfl et al., 1992), and AI(111)-Rb (Nielsen et al., 1994) adsorption in on-top sites was determined by several methods, see Table 9.6. With increasing temperature, a transition from the (2x2) phase to disorder proceeds through a range characterized by a ring-like diffraction pattern (cf. Fig. 9.33) intersecting the (1/2, 0) position (melting transition). This indicates that the dipole-dipole interaction is still sufficient to force the Cs atoms into a constant interatomic spacing, while the azimuthal ordering by the corrugation of the substrate is weaker and disappears first. The transition to the (f3-x4-3-)R30 structure (saturation coverage) takes place via a phase regime denoted as 'rotated structures' with a continuous spot splitting as a function of coverage. The decrease in length of the unit-cell vectors in real space and the corresponding increase in coverage indicate a fully relaxed Cs overlayer. The repulsive lateral interactions between the adatoms are strong enough to prevent the Cs atoms from locking in at highly symmetric adsorption sites. In contrast to liquid-like structures, the A-S interaction is still able to align the overlayer unit cell with a particular (coverage-dependent) angle with respect to the unit cell of the substrate. A similar phenomenon of 'rotational epitaxy' (predicted by the theory of Novaco & McTague (1977)) was also found in other systems, such as Li and Na on Ru(0001) (Doering and Semancik, 1984, 1986), P t ( I I I ) - K (Pirug and Bonzel, 1988) and Rh(100)-Cs (Besold et al., 1987). Near saturation coverage of the monolayer, the polarization of the alkali atoms has almost disappeared, and the formation of an adlayer with a metallic binding c h a r a c t e r can be o b s e r v e d (e.g. as s u r f a c e - p l a s m o n e x c i t a t i o n ) . The Ru(0001 ) ( ' ~ - x f f ) R 3 0 - C s structure provides the completion of the first monolayer. It can be observed up to desorption temperature. During the multilayer adsorption the long-range order is destroyed. In the (~-x'~-)R30 overlayer (cf. Fig. 9.34b), Cs occupies the hcp site, inducing no relaxations of the substrate as derived from a LEED structure analysis (Over, 1992a). The derived Cs-Ru layer spacing of 3.15 + 0.03 ]k corresponds to a hard-sphere radius of the adsorbed Cs atom of 2.17 + 0.02 ]k, probably reflecting a more metallic bonding. Alkali ions have effective radii typically 1 /~, smaller than the effective radius of the respective alkali-metal bulk value. This observation might offer the prospect of finding an increase in bond length with increasing coverage which could point to a transition from more 'ionic' to more 'metallic'. In comparison with the Cs radius found with the (2x2)-Cs system (1.90 ~), the increase of the effective radius of the adparticle with coverage might therefore be superficially interpreted as such a transition; this issue will be addressed in more detail below. These data are in qualitative agreement with the structural parameters for the Ag(111)-Cs system as determined by SEXAFS (Lamble et al., 1988); note that the corresponding adsorption sites were not determined in this study.
470
H. Over and S. Y. Tong
Rul0001)/Cs-I~]• ~
~
Ru{0001)/Cs
,
~
~
(2,2) '
0.0z,~
1 2.1z,• 0.05/~
0.07~
2.13~.07A
Cs - rodius" 2.2~
- rodius: 1.95 A
I
.
f Fig. 9.34. Structural models lot Ru(O(X)l)(x/3-3xx/-f)R30--Cs (a) and p(2x2) (b) with optimal structural parameters as found by LEED analyses (Over et ai., 1992a). The results of the analyses of both ordered Cs structures are summarized in Fig. 9.34. Essentially, two important structural features have emerged: apart from the variation of the Cs radius with coverage, the Ru(0001)-Cs system exhibits a switching of adsorption sites from on top to hcp (hollow) as the coverage increases from 0.25 to 0.33. For the (~3-x,f3-)R30 structure, the corresponding dipole moment of Cs (-- 2.8 Debye) is significantly lower than for the p(2x2) phase (= 2.0 Debye), due to mutual depolarization of Cs atoms as indicated by an increase in the work function. Consequently, a change in coverage is paralleled by a modification of the electronic properties of the adatoms and therefore the interaction between adsorbate and substrate. In general, when the coordination number increases, the bond length increases, and the strength per bond decreases. For ionic bonding (Kittel, 1976), a switching from coordination number 3 (hcp) to 1 (on top) should result in a decrease in bond length by about 0.3 /k, so that the observed change in bond length for the system Ru(0001)-Cs (and we expect the same for Ag(l 1 l)-Cs) is more related to the local adsorption site than to the degree of ionicity. This conclusion is supported by structural analyses of the systems Ru(0001)-K (Gierer et al., 1992), Ru(0001)-Na (Hertel et al., 1994a) and Ni(100)-K (Wedler et al., 1993) for which the coordination numbers of the adsorbed particles are independent of the coverage and no variation of the alkali-metal radii has been found. Furthermore, the structural analysis of the coadsorbate system Ru(0001)(,f3-xff)R30-Cs-O (Over et al., 1992b) exhibits a
Chemically adsorbed layers on metal and semiconductor su~. "aces
471
salt-like structure in which the effective Cs radius reduces only by 0.1 /~ with reference to the clean Ru(0001)-('~-x~4~-)R30-Cs phase. It should be noted that the geometry of the Cs-adsorption site is not altered and that in this ionic Cs-O structure the Cs adatom donates easily its 6 s-valence electron density to the substrate and to the electronegative O atom. This result might thus be regarded as an upper limit for the variation of effective Cs radii expected as a function of coverage. However, even a constant bond length does not preclude the possibility of an ionic to a metallic transition, as this conclusion would necessitate identifying bond length with ionicity. It appears that a simple comparison of hard-sphere radii with ionic, covalent or metallic Pauling radii is not sufficient to fully characterize the nature of bonding. For example, the covalent radius of Cs is given by the bond length of Cs2, while the bonding of Cs on Ru(0001) results from a complicated interplay of A - A and A-S interaction. The behavior of occupying different adsorption sites in the (2• and (4-f3xg3-)R30 ~ structures is more peculiar. Both phases are commensurate with the substrate lattice and would permit identical adsorption site geometries without affecting the unit cell if the A - A and A-S potentials are uncoupled. As mentioned before, the A - A interaction is dominated by dipole-dipole repulsion. Generally, the interaction energy between two dipoles decreases with an increase of electron densities between the Cs ions. This electron density could, for instance, be supplied by the substrate. Figure 9.34 shows that, in the (2x2) phase, better screening between the dipoles can be achieved if neighboring Cs atoms have a substrate atom directly between them, as occur at the on-top sites but not at hollow sites. The fact that these substrate atoms between adatoms are raised by 0.1/~ (and thus enhancing the screening ability) is consistent with this model. As can be seen from Fig. 9.34, with the (f3-3x'~r3-)R30 phase, occupation of the on-top sites would no longer improve the screening, and instead, the hollow sites are preferred. The energy difference between high-symmetry sites is small for a Cs atom due to its large size, which implies that it experiences a rather small substrate electrondensity corrugation (Neugebauer, (1992a). To study this latter effect, Over et al., 1995f) analyzed the temperature dependence of LEED I-V curves by applying the concept of 'split positions' (Over et al., 1993a). This technique has been shown to be sensitive to lateral (harmonic) discursions of adparticles around their equilibrium positions associated with small excitation energies. On the one hand, the resulting 'split positions' as a function of the temperature can be compared with the mean-square deviation derived from the harmonic-oscillator approximation (twodimensional and isotropic), where only the excitation energy is used as a fitting parameter. The corresponding value turns out to be 1.2 + 0.3 meV. On the other hand, when the energy potential relief is dominated by the dipole-dipole (D-D) interaction (for the (2x2) phase the dipole moment is about 2.8 Debye), the curvature of the D - D energy-potential surface at the equilibrium position (evenly dispersed (2x2) overlayer) is directly correlated to the excitation energy of this vibrational mode. The corresponding excitation energy turns out to be 1.0 meV. The result of 1.2 + 0.3 meV (split position) in comparison with 1.0 meV (D-D i~ateraction) may yield an estimate for the contribution of the A-S interaction to the
472
H. Over and S. Y. Tong
excitation energy of roughly 0.1-0.5 meV, which indeed is consistent with a very small A - S corrugation. A similar analysis can also be carried out for Cs motions perpendicular to the surface. The resulting excitation energy of about 9 meV is in very good agreement with an energy loss found in HREEL spectra (7.8 meV) for the Ru(0001)(2x2)-Cs system (Jacobi et al., 1994). The actual adsorption site occupied in alkali-metal systems will be a result of a sensitive balance between corrugation of the substrate potential, magnitude of the dipole moment, interatomic spacings, and electrostatic screening. Therefore, no prediction of the adsorption site can be made, as demonstrated, for example, by the system Ru(0001)-K. For this system the same two ordered overlayers as those for Ru(0001)-Cs have been observed, namely a p(2x2) and a ('~-x,~-)R30 structure, for which LEED analyses were performed. In the ( ' ~ x , ~ - ) R 3 0 phase the K atoms reside in threefold hcp sites, while in the p(2x2) phase the fcc site is favored. In both phases the K hard-sphere radii are nearly equal and close to the covalent Pauling radius, cf. Table 9.7. In contrast to the Ru(0001)p(2x2)-Cs system, potassium d o e s n o t occupy an on-top site. The effective corrugation of the substrate potential is expected to be larger due to the smaller K radius, so that the influence of the repulsive dipole-dipole interaction might be not as dominant as with the Ru(0001 )-Cs system, and screening via the substrate should play only a minor role. This explanation is supported by the fact that the initial dipole moment for Cs with about 10.5 Debye (Hrbek, 1985) is larger than for K with 7.5 + 0.3 Debye (Uram et al., 1986). Concerning the occupation of the different threefold-hollow sites, it should be noted that the difference in the binding energy per adatom for adsorption on the hcp and the fcc sites is presumably very small. For example, total energy calculations Table 9.7 Structural results found for alkali-metal adsorption on Ru(0001). The hard-sphere radii are compared with corresponding Pauli radii (Over et al., 1995d). Pauling radii (]k)
Alkali metal/Ru(0001): LEED analysis Alkali
Structure
Site
Radii (A)
Ionic
Covalent
Metallic
on top hcp
1.90 2.17
1.69
2.35
2.67
Cs
(2x2) ('f3-•
Rb
(2• (q-3-xq-3-3)R30
fcc hcp
2.03 2.03
1.48
2.16
2.48
K
(2x2) (,fJ-3x,/3-)R30
fcc hcp
1.94 1.98
1.33
2.03
2.35
Na
(2• (q-J-x,,/3-)R30
fcc hcp
1.58 1.58
0.95
1.54
1.90
Li
(,]3-xq-J-)R30
hcp
1.39
0.60
1.23
1.55
Chemically adsorbed layers on metal and semiconductor surfaces
473
performed for ordered structures of the AI(111)-K system (Neugebauer and Scheftier, 1992b) led to the conclusion that the binding energies per adatom for the two types of threefold-hollow sites are practically degenerated. Nevertheless, a switching of large domains from one type of adsorption site to another one represents a phase transition. In this case, energy differences between two islands occupying exclusively fcc and hcp sites of the order of kT would suffice to stabilize one phase over the other. It is worth noting that the occupation of different adsorption sites during the formation of a crystal layer has been observed even for a homoepitaxial system (Wang and Ehrlich, 1989): investigations with field ion microscopy of the It( 111 )-Ir system demonstrated that single Ir adatoms favor the hcp sites, while for clusters of Ir atoms the probability of occupying the bulk-like fcc sites rises rapidly with increasing cluster size, reaching nearly 100% for clusters with seven Ir atoms. The self-adsorption of Ir is governed by strong lateral attraction (as reflected by the nucleation in islands at low coverages) which accounts for a coupling of the adsites of neighboring adparticles. For ordered alkali-metal/metal systems, such a correlation is likewise given by dipole-dipole repulsion, so that the site switching determined for Ru(0001)-K is not surprising, although it is not completely understood. An alternative view could be that a cluster of seven Ir atoms builds up a metallic state. The transition from p(2x2) to ,f3-x,f3 is also accompanied by a transition from more 'ionic' to more 'metallic', so that one can speculate that the 'expected' site (fcc site for fcc(l l l) and hcp site for hcp(0001)) will always be occupied if metallization occurs. In this line of reasoning, one would expect that alkali metals smaller than potassium, namely sodium and lithium, should occupy hollow sites. Indeed, structural analyses by LEED (Gierer et al., 1992; Hertel et al., 1994a) confirm this speculation. As with K adsorbed on Ru(0001 ), Na atoms occupy threefold-fcc sites in the p(2x2) phase, while for the ('~-x4-3-)R30 phase the hcp site is favored. The hard-sphere radii found in this study are nearly constant (1.58 ~) and close to the covalent Pauling radius. For the system Ru(0001 )-Li Gierer et al. (1995) could only observe one ordered structure, namely the ('43-x'43-)R30 phase; Doering & Semancik (1986) have reported a p(2x2) structure in addition to the (,13-x,~-)R30 phase. A quantitative LEED analysis of the (,]3-x,f3-)R30 structure indicated that Li atoms reside also in hcp sites with a hard-sphere radius of 1.39/~. Besides the two phases of Ru(0001)-Na mentioned above, the high-coverage (3x3)-4Na overlayer was analyzed by Hertel et al. (1994a). They found that the nearest-neighbor distances are almost uniform and Na is not adsorbed in high-symmetry sites. This demonstrates a strong adsorbate-adsorbate interaction, presumably attractive, which dominates over the small 'substrate' corrugation potential. Last, we briefly describe the results of the system Ru(0001)-Rb which provides a connecting link of Cs and K in various respects (Hertel et al., 1994b). The radius of Rb as well as the initial dipole moment are between the values given for Cs and K. Especially for the p(2x2) structure, the question where the Rb atom will reside is of particular interest. A LEED study revealed, similar to Ru(0001)-K, a twisting of adsorption sites from fcc for p(2x2) to hcp for (~r3-x,f3-)R30, so that the on-top adsorption on Ru(0001) is restricted to Cs. For the Rb(2x2) phase, also an analysis
474
H. Over and S. Y. Tong
of the lateral vibrations was performed, applying the concept of split positions. It turned out that the excitation energy of this lateral vibration is 2.0 + 0.5 meV which is in agreement with values found for the Rb-graphite system employing the technique of inelastic He scattering (2.8 meV) (Cui et al., 1993). A comparison of different alkali metals on Ru(0001) shows that the hcp site is always favored for the (~f3-x~/-3-)R30 phase; cf. Table 9.7. This site represents the 'expected' site if growing of Ru on metallic Ru(0001) is considered. One might speculate that the favored adsorption site in metallized alkali films should be the 'expected' one. A more complicated situation arises for the (2• phase where the fcc site is preferred for all alkali metals but Cs (on top). This means that each of these alkali-metal/metal systems undergoes a switching in adsorption site with varying coverage. In order to resolve the underlying mechanism, especially for the transformation from hcp to fcc sites, extended a b - i n i t i o calculations as a function of the overlayer density are required. Another property shared by all the investigated systems is that the alkali-metal radii found by experiment, although not subject to a deeper understanding of the type of bonding, nicely reflect the tendency of corresponding Pauling radii. In all these cases, the alkali-metal radii found by LEED are independent of the coverage. Several conclusions can be drawn by comparing the local geometry of alkali metals adsorbed on different metal substrates. Top sites have been observed only on hexagonal surfaces, and the effective radius of alkali-metal adatoms in on-top positions is less than when adsorbed in higher coordination. This effect can be traced back to the general experience that with increasing coordination number the bond length increases. All alkali radii are close to the respective covalent Pauling radii if corrected for the effect of different coordination numbers, and, last, the adsorption of alkali metals on 'open' surfaces generally takes place in highly coordinated sites, presumably due to a stronger corrugation potential of the substrate. We will now briefly review the structural properties of alkali metals adsorbed on AI(111), for which results from different experimental techniques as well as theoretical investigations are available. Theoretical calculations of the adsorption geometry and electronic structure from first principles for A! (s-p band metal) are in general less demanding, compared to alkali adsorption on transition-metal surfaces where d electrons of the substrate play an important role. For alkali-metal adsorption on metal surfaces it is commonly assumed that alkali atoms reside on the surface (see Table 9.6) and that no intermixing occurs with the substrate. This seems plausible since alkali metals have a low solubility in most metals (Miedema and Niessen, 1988). Recently, however, the formation of a Na-AI surface alloy has been reported. Using polarization-dependent SEXAFS measurements, Schmalz et al., 1991) showed that the Na atoms for coverages of 0 = 0.16-0.33 occupy an unusual sixfold-coordinated substitutional site on AI( 111 ) at room temperature (cf. Fig. 9.35a). Subsequently, a b - i n i t i o density-functional calculations have elucidated the underlying mechanisms (Schmalz et al., 1991; Neugebauer and Scheffler, 1992b), proving that this adsorption geometry is indeed energetically favorable due to the small energy for the formation of surface vacancies (0.41 eV) and a better substrate-mediated screening of the direct A - A repulsion. The explanation for the
Chemically adsorbed layers on metal and semiconductor su~. aces
475
Fig. 9.35. (a) The atomic structure of AI(111)(~3-x~-)R30-Na room temperature phase. The Na atom substitutes the AI atom; (b) the underlying vacancy structure of AI(111). low vacancy formation energy for the (43-x43-)R30 structure is that for this geometry the group III A1 substrate can create a favorable spZ-bonded surface layer (cf. Fig. 9.35b). The Na adatoms at the substitutional site behave similarly to isolated adatoms, and the ionic type of bonding can therefore develop more strongly than on the unreconstructed surface. High-resolution core-level spectroscopy clearly supports the occurrence of intermixing between Na and AI (Andersen et al., 1992). This technique utilizes the fact that the core-level binding energy of an atom is depending on its coordination sphere, and thus it is possible to determine the number of different sites involved in this system. Once a special feature of a core-level binding spectrum can be assigned to an adsorption site (by an alternate technique like LEED or SEXAFS), this method can be used for a fingerprinting. Another intriguing system, AI(I 1 I)-K, was analyzed by LEED (Stampfl et al., 1992) which shows that the adsorption of K on AI(I 1 1) at 90 K and 300 K forms a (~/-f3• structure. However, the adsorption sites are significantly different; both structures are displayed in Fig. 9.36. At 90 K the adatoms occupy on-top sites
Fig. 9.36. The atomic geometry of low temperature (a) and the room temperature adsorption structure of K on AI( 111 ). For low temperatures, K occupies the on-top site, while for room temperature the substitutional site is occupied.
476
H. Over and S. Y. Tong
on a buckled surface. The A1 atom coordinated with K is displaced downward by 0.25 A! These sites convert irreversibly to a configuration with K residing in a substitutional site on warming to 300 K (activated process). The on-top site, metastable phase, thus serves as a precursor for the equilibrium adsorption site. Neugebauer and Scheffler (1992b) showed that in contrast to Na, the on-top position might become favorable. This result was explained as a consequence of the energy gain due to substrate relaxation for the on-top geometry and the bigger size of K compared to Na. Andersen et al. (1993) performed a comprehensive study of Na, K, Rb, and Cs adsorbed on AI(111) at 100 K and at room temperature using high-resolution core-level photoemission spectroscopy in combination with qualitative LEED. They found that island formation is more the rule than an exception for alkali adsorption on AI(111), a conclusion which is at odds with the commonly accepted picture of alkali-metal adsorption (compare, e.g., the growth properties of alkali metal/Ru(0001)). More specifically, at 100 K, for small coverages, a dispersed phase for Na, K, and Rb was observed which transformed beyond a coverage of about 0.1 in island growth. This indicates that the net alkali-alkali interaction changes from repulsive to attractive. For Cs adsorption at 100 K no island formation was found at any coverage. This is attributed to a strong repulsive interaction between Cs atoms which is reasonable due to their large dipole moments and bigger sizes. Theoretical studies by Neugebauer and Scheffler (1993) confirmed that for AI(11 l)-Na at 0 > 0.1 the island formation (with a metallic, attractive adsorbate interaction) is energetically advantageous. For the dispersed phases they predicted the threefold-hollow position. Brune et al. (1995) obtained images of the island formation and the substitutional site for Na by STM. Similar results were obtained with the system AI(111)-K. Again, at very low coverages, calculations Neugebauer and Scheffler (1993) predicted that K atoms adsorb in hollow sites, so that the island formation has to be accompanied by a switching of the adsites from threefold to ontop position. In the on-top geometry the AI atom coordinated with K moves down by about 0.25/~, (Stampfl et al., 1992) and thus makes screening work better (this is presumably the driving force of this process). Andersen et al. (1993) pointed out that all the ordered structures formed by alkali adsorption at room temperature involve removal of the AI atoms where the vacancies are occupied by alkali-metal atoms. It is also interesting to note that for all the systems there exists a threshold alkali coverage which has to be exceeded before alloy formation can begin. In this section we have demonstrated that the adsorption of alkali-metal atoms on metal substrates reveals a variety of interesting physical and chemical properties which are beyond a simple adsorption system suggested by the Langmuir-Gurney picture. The drastic increase of recent publications clearly demonstrates that this issue is of current interest. The main difference between Ru and A1 is that AI represents a 'soft' material which allows removal of Al-surface atoms during the adsorption process and hence the formation of surface alloys. Several observations especially on AI(111) ~ could be explained either by simple physical pictures or by involved theoretical investigations. Nevertheless, many questions remain to
477
Chemically adsorbed layers on metal and semiconductor surfaces
be answered. For AI(111), e.g., no method has yet been found which predicts the details of the reaction path through which the surface A1 atoms are kicked out in the process of substitutional adsorption. While in the last few years much of the theoretical work has been done for the A1 substrate, theoretical studies on transition metals are almost non-existent. It therefore appears that a generalized theoretical view on the entire variety of alkali-metal/metal adsorption systems is required. 9.4.3. Coadsorption Ru(OOO1)-Cs-O and Ru(OOOI ) - C s - C O
The properties of alkali-metal and oxygen overlayers coadsorbed on transition-metal surfaces have been the subject of numerous investigations (see, e.g. Surnev (1989), Bonzel (1988)), not only because of fundamental interest but also for the role these systems play as actual promoters in catalytic reactions such as, e.g., ammonia synthesis (Ertl, 1991 a). Surface analysis of a catalyst used in industrial ammonia synthesis (Ertl et al., 1983) revealed that its active surface is uniformly covered by a submonolayer of a composite K+O phase with nearly 1:1 stoichiometry. Oxygen is not reduced under reaction conditions, but strongly interacts with K atoms. This interaction thermally stabilizes potassium while preserving its promoter efficiency. In the 'electrostatic' model by NOrskov et al. (1984) the electronic field formed by the dipole of the 'ionic' alkali-adsorbate complex predominantly stabilizes an electronegative molecule (such as CO) adsorbed in its vicinity. In this section we will mainly be concerned with thin films formed by coadsorption of Cs and O atoms on a Ru(0001) surface, starting with multilayer-Cs films. In addition, a few comments will be made on the interaction of Cs with CO. This specific coadsorbate system is particularly useful since it allows to use most of what we have learned in the preceding three sections. But let us start with the discussion about the (Cs+O)-Ru(0001) system. The oxidized films nicely reflect the complex chemistry of corresponding Cs oxide-bulk materials whose structures had been studied extensively (Simon, 1971; Simon and Wasterbeck, 1977; Vannerberg, 1962). If the alkali overlayer is in the 'metallic' state, i.e. at higher coverages, interaction with other molecules lead frequently to compound formation where
3 =2 r
1 0
IA 0
IB 1
t~'~l 2 3
I 4
1 5
t 6
I C,, t 7 8
O2-Exposure[k]
9
10
2O
Fig. 9.37. Variation of the work function of a 3-4 ML thick Cs film as a function of 02 doses (Woratschek et al., 1987).The regions markedby A, B, C denote the rangesover which the oxides Csl iO3, Cs20, CsO2, respectively, are formed.
478
H. Over and S. Y. Tong
Fig. 9.38. Structural unit of cesium suboxide Cs1103 (Simon, 1971). solely the valence orbitals of the alkali atoms are involved. One of the pronounced attributes found in these systems is the lowering of the work function upon interaction with oxygen. This has several practical applications in manufacturing high-efficiency photocathodes. Figure 9.37 shows the variation of the work function of thick Cs films with 02 exposure. Different regions denote the ranges over which various oxides are formed as identified by UPS and MDS (Woratschek et al., 1987). De-excitation of metastable noble-gas atoms (e.g. He*) (MDS) occurs at the surface with low work function (such as Cs-covered surfaces) via Auger de-excitation. The excitation energy of He* (21.6 eV) serves to emit an electron from the target. In contrast to UPS which probes a layer of about 5-10/~ thickness, MDS is extremely surface-sensitive and probes only the valence-electronic levels at the outermost atomic layer (Ertl and K~ippers, 1987). The clean Cs surface exhibits an intense band in MDS from the 6s states just below the Fermi level whose intensity even increases with small 02 doses. This initial stage of oxidation corresponds to nucleation and growth of the cesium suboxide Cs~ ~O3 (the structural unit is depicted in Fig. 9.38). The combination of UPS and MDS studies give clear evidence for the penetration of doubly charged O 2- ions below the surface, while the topmost layer still consists of metallic Cs atoms. The enhanced Cs 6s emission observed in MDS (surprisingly since part of the formation of O2-consumes part of the 6s electrons) was interpreted as a confirmation of the quantum-size effect (Woratschek et al., 1986) predicted by Burt and Heine (1978), to explain the lowering in work function. The conduction electrons tend to avoid 02- ions and are thus confined in space. As a consequence of the uncertainty principle, their kinetic energy rises, i.e., the work function is reduced. In this case, the 6s wave function should 'leak' further into the vacuum, which should increase in turn the 6s-derived MDS signal. Continuing 02 deposition (about 0.4 L) causes a decrease in intensity of the Cs 6s emission (MDS) due to the onset of the formation of cesium peroxide (Cs202) at the surface; this compound consists of Cs § ions (with empty 6s levels) and peroxide ions O~-. This process is then followed by transformation into cesium superoxide CsO2 which contains Cs § and hyperoxide ions 02. Switching to submonolayer-Cs films, one cannot expect to find Cs-oxide compounds (but at least it might be possible to identify similar 'planar' structural elements). The coadsorption of Cs and O on Ru(0001) results in a formation of a
Chemically adsorbed layers on metal and semiconductor surfaces
479
large wealth of mixed phases with long-range order (Bludau et al., 1995), see Fig. 9.39. The general property of stabilizing alkali-metal adlayers by the addition of oxygen is reflected by the substantial increase of the desorption temperature for Cs in the presence of coadsorbed oxygen which suggests strong interactions between both adsorbates. Furthermore, the sticking coefficient S (defined as the ratio of the number of actually adsorbed particles over the number of impinging particles) for oxygen adsorption is substantially enhanced by Cs pre-adsorption: S changes from 0.4 (0c, = 0) to 1 (0c., = 0.3) (Kiskinova et al., 1986; Bludau et al., 1995). It is generally assumed that the increase of the oxygen-sticking coefficient is due to the enhanced electron flow into the anti-bonding molecular orbital accompanied by a reduction of the activation barrier of dissociation. Starting with an ordered R u ( 0 0 0 1 ) ( ' ~ - • structure, i.e. one monolayer of Cs, already the addition of small doses of oxygen (0.05 L) leads to the appearance of a new incommensurate superstructure and a gradual disappearance
Fig. 9.39. Schematic phase diagram for the Ru(000l)-Cs-O coadsorption system; after Bludau et al. (1995).
H. Over and S. Y. Tong
480
of the ('~-x'~-)R30-LEED pattern. This clearly indicates a strong interaction between oxygen and the Cs-adsorbate layer. Oxygen exposures between 0.3 and 0.9 L and subsequent annealing give rise to an appearance of a new (~-3-x'~-)R30 structure. The corresponding stoichiometry is Cs:O of 1:1. A respective LEED structure analysis (Over et al., 1992b) has provided a model in which Cs and O atoms are located in hcp sites with respect to the Ru(0001)-substrate lattice; structural parameters are summarized in Fig. 9.40. Thus, a 'salt-like' structure is developed which represents the optimum arrangement allowing the greatest number of oppositely charged 'ions' to touch without requiring any squeezing together of 'ions' with the same charge. With respect to the positions of the Cs atoms, atomic oxygen resides in threefold sites below the plane formed by the alkali-metal adlayer, which is consistent with work function and MDS data (B6ttcher et al., 1991 ). The appearance of atomic oxygen instead of molecular oxygen (as contained in the stoichiometric equivalent Cs202 compound) is confirmed by HREELS measurements (Shi et al., 1992). Compared to the structures of the respective pure adsorbate phases, the Ru-O and Cs-Ru bond lengths are modified in a way consistent with a net transfer of electronic charge from Cs to O: the oxygen hard-sphere radius increases by 0.12 A, and the effective Cs radius decreases from 2.2 A in the clean Ru(0001)-Cs phase to 2.1 A with the coadsorbate phase (Over et al., 1992b). Moreover, oxygen atoms in the Cs-O layer at the hcp sites experience Ru (O001)/Cs/O -(~3 x~-3} R30 ~
b)
O- radius: 0.8/~
ygen
Fig. 9.40. Structural model for Ru(0001)('43-x'~-)R30-Cs-O with optimal structural parameters as found by a LEED analysis (Over et al., 1992b).
481
Chemically adsorbed layers on metal and semiconductor surfaces
~
7.28
(3x2~/3)rect Cs-O / Ru(0001 )
CsO2-bulk
Fig. 9.41. Planar structure model for the system Ru(0001)(3x2"~-)rect-Cs-O. (a). The main structural element Cs-O2-Cs zigzag chain (b) is compared with the CsO2 bulk structure (c) (Bludau et al., 1995). an enhanced charge density induced by the Cs layer, as expected by comparison with calculated charge-density distribution of the related AI(I 11)(,f3-x,~-)R30-Na system with Na residing in a threefold-hollow site (Neugebauer, 1992a). This nicely reflects their electron affinity. As summarized in Fig. 9.39, the coadsorption of Cs and O leads to an overwhelming number of different ordered structures; for a comprehensive representation of the geometric structures, see Bludau et al. (1995). We will restrict ourselves here to two illustrative examples which point to similarities between Cs oxide-bulk structures and Cs-O surface species. Exposure of 02 to a Cs monolayer (0c~ = 0.33) lead to a (3x24-3-)-rect. structure at 0o = 0.68, corresponding to a stoichiometry Cs:O = 1:2. Both the appearance of a glide-plane symmetry and the dominance of certain LEED beams in the LEED pattern give evidence for a model exhibiting distinctive Cs-O2-Cs zigzag chains. The interatomic Cs spacing in these chains (7.2 ~ ) agrees well with that found in crystalline CsO2 (7.28 ]k) (Fig. 9.41); note that the stoichiometry in the (010) surface of CsO2 is also Cs:O = 1:2. A similar result was found with the ('(7-• structure which was prepared by oxidizing a submonolayer Cs film (0c., = 0.28). Again, the stoichiometry turned out to be Cs:O = 1:2. A comparison of this surface structure with the (010) surface of CsO2 is presented in Fig. 9.42 from which the structural element of now linear C s - O - O Cs chains becomes evident. The formation appears to be a compromise of attaining a Cs-O bond length comparable to CsO2 and the interaction of the Cs-O complex with the substrate. The higher density of Cs atoms in the (3x24-3-)-rect. structure (0.052 atoms//~, 2) compared to that in the (010) surface of CsO2 bulk (0.044 atoms/]k 2) causes a large strain in the Cs-O overlayer resulting in the formation of zigzag instead of linear C s - O - O - C s chains. The Cs density in the ~ structure (0.045 atoms/~ 2) is almost the same as in the corresponding oxide. Here, the strain induced by the substrate's corrugation only is uniaxially relieved: C s - O chains 2 and 3 (Fig. 9.42) are parallel-shifted by 3.6/~ (half length of the @-- periodicity) and compressed more tightly in the direction perpendicular to the chains. The most striking feature of the phase diagram (Fig. 9.39) represents the existence of the (x/-7-x@--)R19.1 structure over a wide range of the Cs coverage,
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Fig. 9.42. Structure model for Ru(0001)(~--• 19. l-Cs-O in comparison with the (010) surface of CsO2 bulk material (Bludau et al., 1995).
while the respective optimum oxygen coverage is determined by the stoichiometry of two oxygen atoms per Cs atom. The presence of the (~-• 19.1 LEED pattern down to coverages as low as 60% of the nominal coverage indicates island growth and underlines the strong driving force to build up the CsO2 surface species. Upon raising the sample temperature, these (~r7--x'~--)R19.1 islands dissolve as observed in LEED by the transformation of a (~--x4-7-)R 19 ~ pattern into a ring-like pattern intersecting the nominal ~ spots (Trost et al., 1995). The ring-like LEED pattern indicates that still the ~ distance is abundant, and since Cs is a much stronger scatterer than oxygen, this might be attributed to the persistence of C s - O - O - C s clusters, while losing the azimuthal order. Even for coverages higher than the density of the CsO2 bulk (010) plane, the Cs-O layer forms a (3x2~f3-)rect. lattice, now exhibiting C s - O - O - C s zigzag chains instead of linear chains. Altogether these findings give strong evidence for the formation of two-dimensional Cs oxides with
483
Chemically adsorbed layers on metal and semiconductor surfaces
ITDS
Cs/Ru(0001), CO/Ru(0001)
Oc~-0.2s
Oco~sat.=0.54
,~
a)
CO/Cs/Ru(0001) Oc~=0.25 |176
=~"
~.
[ I
b)
ffl co
400
600 800 1000 1200 temperature (K)
Fig. 9.43. Thermal desorption spectra of the systems: (a) Ru(0001 )-CO: saturation of CO at T = 300 K (0co = 0.54) and Ru(0001 )(2• (0c.,= 0.25); (b) Ru(0001 )(2x2)-Cs-CO: Cs precoverage = 0.25; CO coverage = 0.50 (Over et al., 1995c). a stoichiometry of two oxygen atoms per Cs atom. The commensurability of the qff and the (3x2q-3)rect phase, on the other hand, reflects the influence of the corrugation potential of the substrate on the formation of these structures. While the interaction of Cs with oxygen is dictated by the chemical reactivity between these species, as seen by the formation of structures which are consistent with a surface species of Cs oxide, the situation changes drastically when we proceed to the coadsorption of Cs and carbon monoxide on Ru(0001). The direct interaction between Cs and CO is very weak. However, when brought together onto a Ru(0001) surface, they interact strongly with each other, as demonstrated by the effect of thermal stabilization of each adsorbate. As indicated in the thermodesorption spectra (see Fig. 9.43) of singly adsorbed Cs and CO in comparison with the compound system, not only both species are thermally stabilized, but also a coincident desorption of both species takes place at about 600 K. This thermal stabilization can readily be explained by a substrate-mediated interaction between these species, i.e., the alkali metal /'lushes the surface with electron-charge density that the CO molecule in turn uses in order to form a stronger back-bonding (cf. w 9.2.1 ). This would explain why CO is more strongly bound to the substrate. Cs is also more strongly bound to the Ru(0001 ) surface since this mediated charge transfer from Cs to CO re-ionizes the Cs atom which strengthens the bond to the substrate according to the L a n g m u i r - G u r n e y model (cf. w 9.4.1). The coincident desorption, on the other hand, is rationalized by an autocatalytic reaction process: with the release of CO from the surface the cause of the stabilization of Cs also ceases (and vice versa).
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This interpretation is supported by HREELS measurements which showed that indeed the C - O stretch frequency is reduced substantially from 252 meV (clean Ru(0001 ) surface) (Thomas and Weinberg, 1979) to 203 meV for the (2x2)-Cs-precovered Ru(0001) surface (Jacobi et al., 1994) consistent with an increased backdonation into the anti-bonding CO 2n-orbital. The re-ionization has also been demonstrated by HREELS by monitoring the Cs against substrate vibration. For very small Cs coverages this vibration showed up in HREEL spectra at about 8 meV. This vibration disappears, however, for higher Cs coverages beyond 0.25, thus indicating that the Cs layer becomes metallic and the dipole excitation is screened. If onto such a metallic (2x2)-Cs surface CO is coadsorbed, the Cs-Ru vibration reappears (Jacobi et al., 1994) consistent with the interpretation of an re-ionized Cs overlayer. Since the Cs-Ru vibration appears at nearly the same energy as for low Cs coverages, comparable bond strengths between Cs and Ru are expected, so that the addition of CO increases the effective strength of the Ru-Cs bond. What remains to be clarified is the actual adsorption geometry of this coadsorbate system. As we have already shown in the last sections, both CO and Cs sit in on-top positions on Ru(0001) when adsorbed separately. If Cs and CO form
Fig. 9.44. CO (a) and Cs (b) both reside in on-top position in their pure phases formed on Ru(0001 ). In the mixed (2x2) phase of these species it is not possible that both can retain on-top site. Either CO or Cs has to leave its accustomed adsorption site (c, d). The actual adsorption geometry found is model (c).
Chemically adsorbed layers on metal and semiconductor surfaces
485
a mixed (2x2) phase, it is not possible that both species can retain their 'natural' adsorption site for steric reasons (cf. Fig. 9.44). A recent LEED analysis (Over et al., 1995b,c) revealed that Cs remains in on-top position, while CO switches from on-top (clean surface) to a threefold-coordinated hcp site (Cs-precovered surface). The corresponding structure is depicted in Fig. 9.45. The site change of CO can be explained if one recalls that high coordination sites are characterized by an improved back-donation, while on-top occupation favors the mechanism of ~ donation (cf. the discussion in w 9.2.1). Hence, the actual adsorption site of CO will be a competition between both effects determining the energetically lowest adsorption geometry. This mechanism is sufficient for a proper description of the energetics of this coadsorption system since it is known from theoretical studies of related alkali-metal/metal systems that the adsorption energy difference for Cs at different adsorption sites is only very small (about 20 meV) (Neugebauer and Scheffler, 1992b). Therefore, the total energy of the mixed (CO+Cs) system is determined by the adsorption site of CO. The presence of coadsorbed alkali-metal atoms improves the capability of back-donation, due to the enhanced electron charge density at the surface, and hence forces CO to change its adsite from on-top to high-coordination sites. Theoretically, this site switching has been proposed by MiJller (1993) for a related system CO+K on Pt(1 11 ). Another aspect of the bonding geometry consists in the enhanced Cs-induced buckling (0.18/~) found in the topmost Ru layer when compared to the clean (2x2)-Cs (0.10 A) surface. This effect can be traced back to the additional bonding of the remaining three Ru atoms in the (2• unit cell to the
Fig. 9.45. The atomic geometryof the coadsorbate phase Ru(0001)(2x2)-Cs--CO (Over et al., 1995b).
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H. Over and S. Y. Tong
CO molecules, what in turn weakens the R u - R u bonding to the single Ru atom beneath the Cs atom and hence allows for an enhanced rumpling of the topmost Ru layer. Probably the most apparent feature of the ( 2 • structure is the 'saltlike' arrangement of Cs and CO: the interaction in the mixed overlayer of Cs and CO adsorbed on Ru(0001) seems to be described by a two-dimensional lattice of adsorbates interacting electrostatically. The charge transfer from Cs, likely to be mediated by the Ru(0001) substrate, to the 2n: CO anti-bonding orbital is responsible for the observation of a check pattern of anions and cations. The electrostatic interaction between CO and Cs dominating the lateral interaction is supported by the observed island growth of this mixed phase. If one starts with a liquid-like Cs overlayer (coverages <0.17 where no ordered LEED pattern can be observed), the addition of CO leads to the formation of a well-ordered ( 2 • structure (Over et al., 1995c). This points towards the electrostatic attraction between oppositely charged species which stabilizes this ordered ( 2 • structure instead of an arrangement of single C s - C O complexes.
9.5. Metal/semiconductor systems
9.5.1. Adsorption of metals on Si(l 11) and Ge(l 11) substrates Bulk-terminated semiconductor surfaces are inevitably unstable because of the large number of unpaired sp3-type dangling bonds on them. The dangling bonds are fl~rmed with the creation of a sharp vacuum-solid interface (i.e. an ideal surface). To reduce the number of unpaired bonds, surface atoms rearrange in position to form new bonds. The new bonds often introduce elastic stress in the surface region. The balance between electronic and stress energies gives rise to an array of superlattices found on reconstructed semiconductor surfaces of different materials, e.g., (2xl), (2• (~13-• (7x7), c(2• etc. The reader can find a discussion of a number of reconstructed structures and the reasons why they are formed on clean Si(111), Ge(l 11 ), GaAs(110) and GaAs(111) surfaces in (Tong et al., 1990). In this last section of the chapter we explore some of the structures formed at a metal-semiconductor interface. It is particularly useful to understand the nature of bonding at a metal-semiconductor interface because the performance and lifetime of many technologically advanced devices depend on the perfection of atomic order at an interface. A metal-semiconductor interface is at once interesting because it involves the bonding of different materials: (i) semiconductor atoms whose electrons are relatively localized and they hybridize in directional configurations, and (ii) metal atoms whose itinerant electrons form bands with long-range influence. The end mixture is often a hybrid of the two pictures. While the general situation is far from being well understood, in the case of low coverages (i.e. at or below one monolayer (1 ML)) of metal atoms, a number of important trends have emerged. This section discusses the better understood models and examines the bonding between different metal and semiconductor atoms.
Chemically adsorbed layers on metal and semiconductor surfaces
487
In the following we restrict the discussion to metal-semiconductor interfaces involving the (111) face of Si and Ge. There are reviews, by Kono (1994), Nagayoshi (1994), Nogami (1994), and Mtinch (1995), on this subject which offer excellent reading and useful insights.
9.5.2. Commonly found models The most commonly formed structures on Si(111) or Ge(111) with metals are:
Model 1: The 3-fold site adatom model; periodicity (~f3•
coverage 1/3 ML
This is an adatom model. On a bulk-terminated (111) surface of Si or Ge, there is one unpaired sp 3 bond per surface atom. By placing a metal adatom at a 3-fold site, the metal electrons pair with the lone electrons in the dangling sp 3 orbitals. At 1/3 ML coverage, the ordered structure has a (~/-3-• periodicity. Group HI metals: This model is favorable for the group III metal atoms because each metal atom contributes three electrons to the interface. At 1/3 ML coverage, all the dangling sp 3 orbitals become paired. However, the metal-semiconductor bonds are at angles very different from the 109028 ' angle of a typical tetrahedral bond. Thus, considerable stress energy is introduced to the interface. Nevertheless, we expect this model to be favored by most group III metals because all the unpaired orbitals at the interface are eliminated. Studies by many surface characterization methods and ab-initio theory have confirmed this expectation. It is now established that this model is formed for A1, Ga and In on Si(111) (Northrup, 1994; Nicholls et al., 1985; Kawazu and Sakawa, 1988; Huang et al., 1990b; Woicik et al., 1993; Hamers and Demuth, 1988; Nogami et al., 1988; Wu et al., 1993). There are two types of 3-fold adatom sites, named respectively T4 and H 3, where the former (latter) is directly above a Si atom (vacancy) in the third layer of the interface (see Fig. 9.46). Characterization techniques and theory have established that the group III metal atoms prefer the T4 site, due primarily to resonant bonding between the metal atom and the Si atom directly below. Electronic calculations of
Fig. 9.46. Schematic structure of Si(l 11)(~/3-•
with Ga at a "I"4site.
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the interface show a band gap at the Fermi level, consistent with the fact that all interfacial bonds are paired. Although less work is done on Ge(l 11), we expect a rather similar general picture. The situation with the smallest group III atom, B, is more complicated. Because of its size - - B has an atomic radius of 0.98 ]k - - the B-Si bond length of 2.15 is shorter than the distance from the center of the 3-fold hollow to one of the corner atoms (this distance is 2.22 A). Without sublattice distortion, B at the 3-fold site would fall through the opening. Studies have found that the B atom interchanges its position with the third layer Si atom (see Fig. 9.47) (Bedrossian et al., 1989; Headrick et al., 1989; Lyo et al., 1989; Grehk et al., 1992; Huang et al., 1990a,b; McLean et al., 1990; Thibaudau et al., 1989; Nogami et al., 1995). This site is known as B 5 because of the five near-neighbor Si atoms surrounding each B atom. More than anything else, this novel structure illustrates the fact that while the trends discussed in this section provide very useful guidelines, the actual structure formed for a particular metal atom is the result of a delicate balance between electronic and stress energies. Therefore, each interface structure must be independently determined by one or more characterization methods. The B on S i ( l l l ) system fully reflects this delicate balance: besides the B5 site, other variations of this model have also been reported (Lyo et al., 1989; Nogami et al., 1995; Kumagai et al., 1994; Cao et al., 1993). The remaining dangling bond of the outermost Si atom empties through charge transfer to the B atom (Lyo et al., 1989). Group IV metal atoms: With 1/3 ML of group IV metal atoms adsorbed at a 3-fold site, while the semiconductor's sp 3 bonds at the interface are paired by metal electrons, a new unpaired metal electron is left on each metal atom. Since the coverage is 1/3 ML, this configuration still reduces the dangling bond density by 2/3. In addition, if the adsorption is at a T4 site, the unpaired metal electron can form a resonant bond with the semiconductor atom directly below. This structure has been reported to form for both Sn and Pb on Si(l 11 ) and for Pb on Ge( 111 ) (Nogami et al., 1989; Kinoshita et al., 1987; Conway et al., 1989; Ganz et al., 1991; Hwang and Golovchenko, 1992, 1993).
Fig. 9.47. Schematic structure of Si(111)(,~x'ff)R30-B, with B at a B5 site. Bond lengths are taken from Huang et al. (1990a).
Chemically adsorbed layers on metal and semiconductor su~. "aces
489
Group V metal atoms: The adsorption of a group V metal atom at a 3-fold hollow site, either T4 or H 3, pairs up three substrate sp 3 bonds, leaving a pair of metal electrons. For the metal electrons, the bond energies are lowest if the three metalsemiconductor bonds resemble Px, Py or pz-type bonds which are separated by 90 ~ This configuration favors the larger group V metal atoms, where the adatom sits high at a 3-fold site. Studies have shown that this structure is formed at the T4 site for the biggest two group V metal atoms: Sb and Bi on Si(111) and Bi on Ge(111) (Elswijk et al., 1991 ; Wan et al., 1991 ; Takahashi et al., 1987; Park et al., 1993; Shioda et al., 1993). Interestingly, this structure is not found for the smaller group V element As, which prefers model 3 (see below). Model 2: The metal trimer overlayer; periodicity "~13x~]-3R30~ coverage 1 M L In this model we start with a full monolayer of metal atoms on unreconstructed Si(111) or Ge(111 ). The metal atoms then trimerize in the plane (see Fig. 9.48). The metal trimers are centered above either a T 4 or n 3 site, depending on which configuration has a lower total energy. Within a trimer each metal atom forms two bonds with its nearest neighbors in the plane. A third bond is formed between the metal atom and a semiconductor atom in the layer below. While the coverage is one full monolayer, the periodicity of the ordered structure is "~-x'~-R30, same as in model I. Group 111 metal atoms: At 1 ML coverage, model 2 for the group III metal atoms has no unpaired bonds at the interface. Interestingly, no group III metal atoms have so far been found to form this structure on either Si( 111 ) or Ge(111 ). One explanation is that in model 1 the three metal-semiconductor bonds are all equivalent. Therefore, the 6 electrons can resonate among the three bonds, further reducing the total energy. On the other hand, the three bonds involving the metal electrons in model 2 are d i f f e r e n t - two are metal-metal bonds, and one is a metal-semiconductor bond. The bond angles are also different. This may explain why no group III metal atoms have been found to form model 2 at the interface. However, we know of only a few cases where this model has been tested for the group III metal atoms. Whether this model is truly unstable for all group III metal atoms awaits to be established. Group i V metal atoms: For the group IV metal atoms, model 2 produces one unpaired metal electron per surface atom. With such a high density of unpaired electrons at the interface, this model is not a good candidate for the group IV metals on Si(111) or Ge(111). So far, only Pb is reported to form trimers centered on the H 3 site on G e ( l l l ) (Hwang and Golovchenko 1992, 1993). On S i ( l l l ) , Pb is
Fig. 9.48. The metal trimer model: the metal trimers (cross-hatched circles) are shown centered above T4 sites.
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H. Over and S.Y. Tong
reported to form an incommensurate layer with coexisting domains of trimers centered on T4 and H 3 sites (Seehofer et al., 1994). However, these results are inferred from STM images. No atomic coordinates have been determined. These results have not been corroborated by quantitative structural techniques. Group V metal atoms: At a 1 ML coverage, model 2 is favorable for the group V metal atoms. The five electrons on each metal atom can hybridize to form three p-type bonds and a doubly occupied orbital. In this sense, the ordered interface with a "~r3-x4-3-R30 periodicity has no unpaired electrons. Studies have shown that both Bi and Sb form this structure at a 1 ML coverage (Elswijk et al., 1991; Wan et al., 1991 ; Takahashi et al., 1987; Park et al., 1993; Shioda et al., 1993; Abukawa et al., 1988" Nagayoshi, 1991" Kinoshita et al., 1988; Woicik et al., 1991" Mfirtensson et al., 1990; Watanabe et al., 1993). Interestingly, the As atom, whose radius of 1.2 ]k is close to that of Si (1.175 ]k), is found to prefer a different ordered structure at the 1 ML coverage (Bringans et al., 1985; Olmstead et al., 1986; Uhrberg et al., 1987; Becker et al., 1988). The structure preferred by As is described below.
Model 3: The substitutional model; periodicity (lxl), coverage 1 ML In the substitutional model the outermost layer semiconductor atoms on unreconstructed S i ( l l l ) or G e ( l l l ) are replaced by metal atoms, one metal atom per semiconductor atom (see Fig. 9.49). The number of unpaired electrons remaining at the interface depends on the valence of the metal atoms. Group !11 metals: For the group III metal atoms, the metal electrons rehybridize into three sp 2 bonds, leaving no unpaired electrons at the interface. Unfortunately, the sp 2 bonds prefer a 120 ~ planar geometry. For the larger group III atoms, this model requires substantial bond distortions in the substrate. However, because of the (1 x I ) periodicity, lateral distortions of the substrate are not allowed. Therefore, to release elastic strain, an incommensurate phase is often formed. For example, Ga, with a radius of 1.32/~, is reported to form a 6.3• incommensurate phase on S i ( l l l ) (Zegenhagen et al., 1988; Meade and Vanderbilt, 1989). Within each domain, some studies suggest that the local structure is the substitutional model (Zegenhagen et al., 1988; Meade and Vanderbilt 1989).
Fig. 9.49. Side view in the (110) plane of the Si(l 1l)(l• the top Si layer.
surface. The As atoms substitute for
Chemically adsorbed layers on metal and semiconductorsurfaces
491
Group IV metals: For the group IV metal atoms, the substitutional model produces one unpaired metal electron per metal atom. Therefore, electronically, there is little reason for this structure to form. There have been reports that Pb forms a ( l x l ) structure on Si(111); however, it has not been established whether its local arrangement is the substitutional model (Estrup and Morrison, 1964; Heslinga et al., 1990). Group V metals: For group V metal atoms, the substitutional model allows three p-like bonds and a doubly occupied orbital to form at each metal atom. This is a favorable model if the stress energy is small. The substitutional model is found for 1 ML of As atoms on Si(111) and Ge(l 11) (Bringans et al., 1985; Olmstead et al., 1986; Uhrberg et al., 1987; Becker et al., 1988). The As radius, 1.20/~, is close to the Si(111) radius of 1. ! 75 ~; therefore, the overall stress energy is small. On the other hand, for the larger group V metal atoms, Sb with a radius of 1.40 ]k and B i with a radius of 1.45/~, the substitutional model produces considerable stress. The larger group V metal atoms instead prefer model 2. Observation: Before we describe a fourth model, one which involves mainly group IA and IB metal atoms, we would like to present a few observations on the three models described so far. Group III metal atoms, at the 1/3 ML coverage, generally prefer the bonding structure of model 1, except if the metal radius is too small, so that the metal atoms would fall through the 3-fold hollow space, as in the case of B. At the 1 ML coverage, group III metal atoms do not seem to prefer either model 2 or 3, although more structural work is needed to confirm this observation. Models 1 and 2 are viable for the group IV metal atoms at 1/3 ML and 1 ML coverages, respectively, while model 3 seems to be unfavorable for these metal atoms. For group V metal atoms at the 1/3 ML coverage, model 1 is favored for the larger atoms. At the I ML coverage, the larger group V metal atoms prefer model 2 while the smallest metal atom, As, prefers model 3. These general trends should provide helpful first steps in a search for an unknown interface structure. The dependence of structure on the size of the metal atoms, besides their valence, is an indication of the importance of stress energy at the interface. Model 4: The substitutional model; periodicity ( ~ x ~ ) R 3 0 , coverage I ML In this model the semiconductor atoms in the top layer are replaced by one monolayer of metal atoms, just as in model 3. Then, either the semiconductor atoms in the second layer of the interface trimerize, in a model known as HCT, or the metal atoms in the top layer trimerize, in a model known as CHCT. The two models are favored by group IA and IB metal atoms on Si( 111 ) and Ge(111 ). We describe these two models separately below: (a) The H C T model: The HCT model is formed if, at an interface, the semiconductor-semiconductor trimer bond is stronger than either the metal-semiconductor or metal-metal bond. This, for example, is the case for Ag on Si(111), where the Si-Si bond is the strongest, followed in order by the S i - A g bond and the A g - A g bond. As a result, the Si atoms in the second layer trimerize. The trimerization pairs up two dangling bonds on each Si atom, a third Si bond is connected to a Si atom in the third layer. The remaining Si electron is paired with the s-valence electron on
492
Fig. 9.50. Si(111 )(~ff•
H. Over and S. Y. Tong
)R30-Ag: The Si and Ag bonds are doubly occupied on this interface.
the Ag atom (see Fig. 9.50). Empty state STM images show a honeycomb array of bright spots, hence the name of the model: honeycomb-chained-trimer HCT. Because each Si atom in the trimer has one unpaired electron to bond to a metal atom, this model favors the IA or IB metal atoms. Ab initio theory (Ding et al., 1991, 1992) and a host of experimental techniques (Wan et al., 1993; Takahashi et al., 1988; Katayama et al., 1991; Wan et al., 1992; Huang et al., 1994; Over et al., 1993c, 1995d) have established that 1 ML of Ag atoms forms this structure on either S i ( l l I) or Ge(l I1). LEED has further shown that 1 ML of Li atoms forms this structure on Si(l 11) (Over et al., 1993c). We expect some other group I metals to also form this structure, although at present no quantitative analysis exists for the other elements. (b) The CHCT model: This is a companion model to the HCT model. In this case, the metal-metal trimer bond is the strongest at the interface. For example, for Au on Si( 1 I 1), the A u - A u bond is stronger than either the Au-Si or Si-Si bond. As a result, the Au atoms trimerize in the plane. The bonding picture is more complex and the surface is metallic. While the Si atoms are no longer nearest neighbors in the plane, they nevertheless form bonds through the Au trimers, resulting in six atom clusters of three Si atoms and three Au atoms each (see Fig. 9.51). The clusters are held in fixed orientations on the surface via Si-Au bonds. STM images of this surface show one bright spot per ~ unit cell for both filled and empty states. The bright spots of STM correspond to the centers of the six atom clusters. This model has been found for Au on Si(111 ) and Ge( 111 ) (Ding et al., 1992; Quinn et al., 1992;
Chemically adsorbed layers on metal and semiconductor surfaces
493
Fig. 9.51. Si(l 1l)('~--xfff)R30-Au: The Au trimer and the three nearest Si atoms form a triangular cluster of 6 atoms. The clusters are bonded together by Au-Si bonds. Chester and Gustafsson, 1991 ; Dornisch et al., 1991 ; Salvan et al., 1985; H a s e g a w a et al., 1990; Nogami et al., 1990; Takami et al., 1994; Higashiyama et al., 1986; Over et al., 1995a). The fixed orientation of the Au trimers on Si(l 11) has been observed directly by a surface-imaging technique: Kikuchi electron holography (Hong et al., 1995). Acknowledgements
We are grateful to E. Wimmer, R. Feidenhans'l, J. Wintterlin, and H.H. Rotermund, who kindly provided illustrations/figures of their work to use them in this chapter. H.O. acknowledges critical reading of the manuscript by D. Fetter and H. B ludau and fruitful discussion with G. Ertl, H. Bludau, H. Wohlgemuth and M. Gierer. Thanks go to M. Richard who carefully prepared most of the line drawings, and special thanks go to I. Reinhardt who processed and proofread the manuscript. SYT thanks for partial support by NSF Grant no. D M R 8805938, DOE Grant no. D E - F G 0 2 - 8 4 E R 4 5 0 7 6 and ONR Grant no. N00014-90-J-1749.
References Abukawa, T., C.Y. Park and S. Kono, 1988, Surf. Sci. 201, L513. Aminpirooz, S., A. Schmalz, L. Becker, N. Pangher, J. Haase, M.M. Nielsen, D.R. Batchelor, E. BOghand D.L. Adams, 1992, Phys. Rev. B 46, 15594. Andersen, J.N., M. Qvarford, R. Nyholm, J.F. van Acker and E. Lundgren, 1992, Phys. Rev. Lett. 68, 94.
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CHAPTER 10
The Structure of Physically Adsorbed Phases J. S U Z A N N E and J.M. G A Y Facultd des Sciences de Luminy Ddpartement de Physique Marseille, France
Handbook o]Su#, ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
503
Contents 10.1. Principles of physisorption interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. The nature and the magnitude of the forces involved in physisorption . . . . . . . . .
10.1.1.1. I n t e r a c t i o n p o t e n t i a l s
10.1.2.
505 505
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
505
10.1.1.2. M e t a l substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
507
10.1.1.3. Ionic substrates
507
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physisorbed layers as model systems to check basic concepts . . . . . . . . . . . . .
508
10.1.2.1. T w o - d i m e n s i o n a l (2-d) phases; 2-d phase transitions and critical
10.2.
phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
508
10.1.2.2. D y n a m i c s of 2-d phases . . . . . . . . . . . . . . . . . . . . . . . . . . .
511
C o m m e n s u r a t e and incommensurate structures of submonolayers
...............
515
10.2.1. C o m m e n s u r a t e and higher order c o m m e n s u r a t e structures . . . . . . . . . . . . . . .
517
10.2.2. Incommensurate structures - - the N o v a c o - M c T a g u e Rotation . . . . . . . . . . . . .
518
10.2.3. The c o m m e n s u r a t e - i n c o m m e n s u r a t e transition - - misfit dislocations and domain walls 10.3.
10.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
519
Other phase transitions in adsorbed monolayers . . . . . . . . . . . . . . . . . . . . . . . . .
521
10.3.1.
521
2-d solid/solid transitions; 2-d polymorphism
.....................
10.3.2. Order--disorder transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
521
10.3.3. The 2-d melting transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
523
Experimental techniques and substrates in physisorption studies . . . . . . . . . . . . . . . .
531
10.4.1.
Low Temperature set-up
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
531
10.4.2.
Powder experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
532
10.4.2.1. T h e r m o d y n a m i c m e a s u r e m e n t s : v o l u m e t r y , c a l o r i m e t r y . . . . . . . . .
533
10.4.2.2. N e u t r o n scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
535
10.4.2.3. X - R a y scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
543
10.4.2.4. M t i s s b a u e r s p e c t r o s c o p y
545
10.4.3. Single crystal experiments
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.3.1. E l e c t r o n scattering: L E E D , R H E E D , T H E E D , A E S 10.4.3.2. A t o m scattering
...........
548 548
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
555
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
557
10.4.3.4. E l l i p s o m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
560
10.4.3.3. X - R a y diffraction
10.5.
Review of experimental results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
561
10.6.
The structure of multilayer films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
561
10.6.1.
From 2-d to 3-d; wetting
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
562
10.6.2.
Layering and surface roughening . . . . . . . . . . . . . . . . . . . . . . . . . . . .
563
10.6.3.
Surface melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
565
10.6.4.
Experimental
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
565
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
567
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
567
10.7.
504
I0.I. Principles of physisorption interactions 10.1.1. The nature and magnitude of the forces involved in physisorption When atoms or molecules are said to be physisorbed onto the surface of a solid, it means that there is no apparent chemical bond with the atoms of the surface (the substrate) but rather, the adsorbate is bound to the substrate by the London dispersion forces or van der Waals interactions. These interactions arise from attraction between the electric dipoles mutually induced by the fluctuating charge distribution of the atoms. The attractive part of these interactions for each pair of atoms is, to a first approximation, proportional to the product of the atomic polarizabilities and varies as the minus sixth power of the separation r of the atoms. Besides the attractive term, a repulsive interaction occurs at close separation which comes from the exchange-overlap forces. For two closed shell atoms, in the Hartree Fock approximation, the repulsion varies approximately exponentially with separation. However, for practical reasons, a minus twelfth power of the separation is often used to describe this repulsive interaction. The van der Waals interaction can be expressed in terms of the electrodynamic response properties of the separated atoms. Calculations of the van der Waals interaction between two hydrogen atoms have been made a long time ago (Pauling and Wilson, 1935; Hirschfelder et al., 1965). The repulsive overlap energy comes mainly from the Pauli exclusion principle. Its mathematical calculation is always complicated, even when the charge distribution around the atom is known (Pauling and Wilson, 1935). At larger distances, retardation effects due to the finite transmission time of the fluctuation signal cause the interaction to decrease more rapidly than the minus sixth power. The magnitude of the interaction energies involved in physisorption ranges from a few tens of a Kelvin for the lighter quantum gases (He, H2) to about 2500 K (Vidali et al., 1991) for the heaviest rare gases or small molecules (CH4, CzH6...). These values are at least ten times smaller than the energies involved in chemisorption (see Chapters 2 and 9).
10. I. 1.1. Interaction potentials The considerations discussed above lead to the classical 12-6 Lennard-Jones (LJ) analytical form for the pair interaction between two atoms or molecules separated by the distance r (Steele, 1973, 1974; Vidali et al., 1991):
505
J. Suzanne and J.M. Gay
506
V lls
-1 I
0
..
P
1
I
r/(j
~
2
t
i
J
3
Fig. 10.1. 12-6 Lennard-Jones potential energy for the van der Waals interaction of a pair of atoms or molecules as a function of the reduced distance rio between them (see Eq. (10.1)). 12
V(r)=4eI(~/
6
-(~/
]
(10.1)
where r~ and ~ are adjustable parameters fitting thermodynamic data, transport properties or spectroscopic data of the bulk material (Bruch, 1983). Such a potential is represented in Fig. 10.1. The dispersion forces are assumed to be pairwise and additive. Hence, the interaction between a neutral, nonpolar gas atom or molecule at a position defined by r and the surface of a van der Waals solid is a simple sum over all pair interactions
V(r) = ~_~ Vi(Ir - r;I)
(I 0.2)
i
where ri defines the position of a substrate atom i. For a plane semi-infinite solid and a gas atom at large distance z above the surface the interaction potential is"
V(z) =
C3 z3
(10.3)
If the distance z becomes smaller than a few lattice spacings, the atom starts to feel the atomic structure of the surface. Then the continuous approximation, Eq. (10.3), fails. When the atom "touches" the surface, the interaction energy may be written as the sum of a laterally averaged term vo(z) and terms having the 2-d periodicity of the substrate surface:
V(r) = v,,(z) + E
eic'p vc~(z)
(10.4)
G~O
where G represents the reciprocal lattice vectors of the surface, z the adatom height above the surface, and p the adatom position parallel to the surface.
The structure of physically adsorbed phases
507
The coefficient C3 in Eq. (10.3) involves the dynamic polarizability of the adatom o~and the dynamic linear dielectric function e of the substrate (Bruch, 1983; Vidali et al., 1991). Theoretical values of the C 3 coefficients for various adsorbates and substrates are given by Bruch (1983) and Vidali et al. (1991) and are compared, when available, to experimental data (Vidali et al., 1991) such as the isosteric heat of adsorption, virial coefficient measurements from scattering experiments, LEED, AES studies, etc. The total interaction energy of an assembly of van der Waals atoms in a condensed phase includes the sum of the potential energies of atomic pairs which represents about 90% of the total cohesive energy. However there is a term coming from the multibody interaction like the triple dipole energy of Axilrod and Teller (1943) and Muto (1943), or the substrate mediated dispersion energies of Sinanoglu and Pitzer (1960) and MacLachlan (1964). In the case of xenon adsorbed on graphite, for instance, the two contributions are weak and represent only 5% of the heat of condensation (cohesion energy) of a xenon monolayer. 10.1.1.2. Metal substrates As long as the separation between a molecule and the surface is sufficiently great, the continuum model is appropriate for metals. The result is the universal behaviour that the interaction between a neutral atom and the surface varies as l/d 3 at moderate separation changing to 1/d 4 due to retarding effects at long range. The C3 coefficients for the interaction energy are again determined by the electromagnetic responses of the solid substrate and the molecule. There are apparently differences for the adsorption of inert gases on transition metals such as tungsten (Engel et al., 1979; Wang and Gomer, 1980) and palladium (Palmberg, 1971; Horn et al., 1978a; Hermann et al., 1980) where mechanisms involving some degree of chemical bonding have been proposed (Flynn and Chen, 1981; Lang et al. 1982). For the interaction of rare gases or molecules with metals, the jellium continuum models are used for the long range part of the interaction energy (Bruch, 1983). The long-range dipole moment induced by the metal in the adatom has been analysed by several authors using the continuum model (Galatry and Girard, 1983; Girard, 1986). However, the short range counterpart of this dipole remains difficult to evaluate although semi empirical potentials have been used which introduce the discrete nature of the substrate at short range (Girard and Girardet, 1987). 10.1.1.3. lonic substrates Interesting studies have been performed on the (100) face of ionic substrates such as NaCI or MgO. The interaction energy of the adatoms with the surface can be written (Meichel et al., 1988; Girardet and Girard, 1989):
v=vM+ vs + VMs
(10.5)
where VM defines the adatom-adatom interaction potential within the monolayer, Vs is the ion-ion interaction in the dielectric substrate that has to be considered in dynamical calculations, and VMS is the holding potential between adatoms and substrate atoms.
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J. Suzanne and J.M. Gay
In the case of rare gas atoms, VMSis the sum of two major contributions (Girardet and Girard, 1989), a potential Vh of the classical Lennard-Jones type describing the pairwise interactions between the monolayer and each of the crystal atoms, and a substrate-mediated energy term. If one considers the adsorption of a molecule having a permanent electric multipole (dipole, quadrupole or octopole), there is a supplementary term to the interaction energy, that is the electrostatic contribution VE due to the ionic substrate charge distribution and the permanent multipole (Lakhlifi and Girardet, 1991).
10.1.2. Physisorbed layers as model systems to check basic concepts Among the attractive features of physisorbed layers is the important fact that they can be observed under true thermodynamical equilibrium conditions on the one hand and that the simple interactions involved between adatoms and between adatoms and the substrate make calculations, modeling and molecular dynamics an accessible challenge to physicists. The equilibrium conditions are reached because of the weak interactions. A supplementary attraction of these systems is that they can be considered as models of two-dimensional matter. The loss of one dimension not only leads to a simplification of the theoretical models but gives rise to a new state of the matter which has new properties. Also critical phenomena or phase transitions of two dimensional systems belong to new universality classes characterized by specific sets of critical exponents. It is an exciting challenge to check experimentally the theoretical predictions and vice versa.
10.1.2.1. Two-dimensional (2-d) phases; 2-d phase transitions and critical phenomena In the early 1930s, Peierls (1937) and Landau (1937) demonstrated that there can be no long-range crystalline order in a 2-d phase at any finite temperature. It is noteworthy that the divergence of the displacement autocorrelation function is so weak for reasonable temperatures that crystal order still persists in systems less than astronomic size; 2-d solid systems display only quasi long-range crystalline (or translational) order. However, as Landau pointed out, even if the long-range translational order does not exist, "bond orientational" long-range order is transmitted infinitely far, that is the crystalline axes could still be well defined (see w 10.3.3). As in three dimensions, two-dimensional matter may appear in solid, liquid or gas phases. 2-d solids and 2-d liquids are dense phases with positional order or disorder, respectively. A 2-d gas is a dilute disordered phase localised in a plane. The similarity between 2-d and 3-d is preserved by the existence of phase transitions and coexistence domains. In particular, a two-dimensional triple point, at temperature T~D, is defined by the coexistence of 2-d gas, 2-d liquid and 2-d solid phases. The variance v, that is the number of independent parameters, is given by the phase rule (Lupis 1983): v = c + 2-~
(10.6)
where c is the number of constituents (c = 1 if there is one type of adsorbate only)
The structure of physically adsorbed phases
509
and ~ is the number of coexisting 2-d phases. Physisorbed systems are in fact in equilibrium with the adsorbate 3-d gas phase. Considering the 3-d gas phase results in replacing the number 2 by 3 in Eq.(l 0.6). Good examples of 2-d systems are physisorbed monolayers of rare gases or small molecules on energetically smooth surfaces such as graphite (Dash, 1975; Bienfait, 1980, 1982) or some metal surfaces (Unguris et al., 1979, 1981 ; Kern and Comsa, 1988). They are the first studied systems showing 2-d gas ---> 2-d liquid ---> 2-d solid transitions in the monolayer and 2-d phase diagrams analogous to their 3-d counterpart have been drawn. The existence phase domains are shown in pressure-coverage (Fig. 10.2), temperature-coverage (Fig. 10.3), or temperaturepressure (see further Fig. 10.15) diagrams. The pressure-coverage phase diagram can be directly deduced from sets of volumetric adsorption isotherms (see w 10.4.2.1) as shown in Fig. 10.2. When two 2-d phases coexist, the system is monovariant according to Eq. (10.6); this results in a vertical isotherm at a pressure P function of temperature. A triple point (coexistence of 2-d gas, 2-d liquid, and 2-d solid phases) is represented by a single vertical line in the pressure-coverage diagram at temperature T~D. For T < T~D, one can see the 2-d dilute phase (gas) --) 2-d dense phase (solid I) transition with increasing coverage. For T~t D < T < TED, two sharp transitions can be seen in the series of isotherms: the first step characterizes the 2-d dilute phase (gas) --) 2-d dense phase (liquid) transition, whereas the second small substep, is the signature of the 2-d dense phase (liquid) ---) 2-d dense phase (solid I) transition. At T > T~D, the first transition rounds up whereas the second remains vertical. ~D is the 2-d critical temperature. The phase diagram is classical
2d S
A ..I W O <
/ 2d G + 2d S !
ul O O
T1
I
T2
2d H
.T5
Tt
PRESSURE
Fig. 10.2. Schematic coverage vs. pressure phase diagram of 2-d adsorbed phases from adsorption isotherms at seven increasing temperatures T~, T2, ~ D , 7"3,~ C , T4 and 7"5.~ D and ~D are the 2-d triple point and critical temperature respectively. 2d G, 2d S~, 2d S~, 2d L and 2d H, represent the two-dimensional gas, solid I, solid II, liquid and hypercritical phases, respectively. This phase diagram is typical of the case of a monolayer of rare gas atoms or molecules such as CH4 on the graphite (0001 ) surface. Coverage of unity corresponds to the completion of the solid I phase, just before the solid l-solid II coexistence (from Bienfait, 1980).
J. Suzzmne and J.M. Gay
510
in the sense that there are regions of phase coexistence and regions with single phases. (See Chapters 2 and 13 for additional discussion of phase diagrams.) Increasing further the coverage within the monolayer range may result in another kind of transition: a solid-solid transition between two different solid phases. Such a transition (solid I ---> solid II) was first observed between commensurate and incommensurate solids in the krypton/graphite system and is evidenced by a small substep in the isotherm at higher coverage. Since then, many transitions of this kind have been reported and studied in numerous other systems (see w 10.2 and 10.3). Volumetry measures the number of adsorbed molecules and the definition of the coverage is then conventional. For chemisorbed layers, coverage one is usually defined as that corresponding to the areal density that is equal to the density of substrate adsorption sites. In physisorbed systems, molecules do not necessary sit on lattice sites of the substrate surface. As a consequence, coverage one is specified with respect to a well defined dense monolayer phase. It results that, for a given substrate, the areal density of physisorbed monolayers at coverage one will depend upon the adsorbate under consideration. In the example of Fig. 10.2, coverage one corresponds to a layer of solid I just before the solid I-solid II transition. In most of the cases, when no substep is observed, the monolayer coverage is defined by the inflection point of the plateau after the step. When the 2-d solid phase structures are known, coverage one can be determined as that corresponding to a given phase. For ethane on graphite shown in Fig. 10.3, coverage one is assigned to the commensurate ("~3-x",f3) $3 phase.
1.2
S 3 . 2 nd
1.0
S2" S3 ~5
2 $1 S 2
0.8 .< c~
layer
I 12+
~J3
12
t.
S I T ' 21
_
L
-I"1,
.... S 1 0.6
o u
F 0.4
I 1 + 2D Gas
S 1+ 2D Gas
0.2
I
I
I
I
I
I
20
4Q
60
80
1 O0
120
Tic
I 140
TEMPERATURE (K)
Fig. 10.3. Phase diagram of ethane monolayer adsorbed on graphite. Coverage of unity corresponds to that of a completed $3 (commensurate ~-x~/-3-) solid phase. (From Gay et al. 1986b).
The structure of physically adsorbed phases
511
Perhaps, the clearest and most straightforward signature of 2-d behavior in an adsorbed monolayer comes from heat capacity measurements. It is expected from statistical thermodynamics that in 2-d, the lattice heat capacity C of a solid follows a C ~ T2 law whereas in 3-d solids a C ~ T 3 law prevails; e.g., the exponent of the temperature is equal to the dimensionality of the system. Indeed, measurements have shown that the heat capacity of a monolayer of 3He or 4He on graphite obeys the quadratic law (Goodstein et al., 1965; Dash, 1975; Hering et al., 1976; Van Sciver and Vilches, 1978). After experimental evidence of the existence of 2-d liquids from adsorption isotherm measurements (indirect) and mobility measurements using M0ssbauer spectroscopy (see w 10.4.2.4.), another convincing indication came from the shape of the liquid-vapor phase boundary near the critical point. In bulk neon and nitrogen (Pestak and Chan, 1984), the density difference between liquid and vapor along the boundary near Tc is described by (P~iq- P~,p) " (To - T) 1~
(10.7)
where 13 = 0.32 close to the theoretical value 0.315 (Stanley 1971). The density p of a 2-d system at the liquid 2-d-vapor 2-d boundary also varies as a power law, but with [3 = 1/8 in theory (Stanley, 1971). Measurements have been performed in the case of a methane monolayer adsorbed on graphite (Kim and Chan, 1984) and the value 13 = 0.13 has been found, again very close to the theoretical prediction. Another experimental measurement on the second layer of argon on cadmium chloride (Larher, 1979a) has given 13 = 0.16, also in pretty good agreement with the theoretical value.
10.1.2.2. Dynamics of 2-d phases Two kinds of motions animate an adsorbed atom: vibration and translation above the substrate surface. When molecules are concerned, rotations may also occur. Measurement of vibrational modes or translational and rotational motions is very useful to understand the microscopic properties of the system, particularly the adsorbate-adsorbate and adsorbate-substrate interactions. Average vibrational properties appear in the Debye-Waller factor which can be deduced from measurements of the temperature dependence of the elastic scattering of electrons (LEED), atoms (He), neutrons, or X-rays. The eigen frequencies of individual molecules or the density of states of collective excitations (phonons) can be determined through surface spectroscopies.
Individual vibrations of adsorbed molecules.
The D e b y e - W a l l e r factor gives the mean square displacement of adsorbed molecules. A typical value is around 0.015 A,2 for a xenon atom above the graphite surface at T = 60 K (Coulomb et al., 1974). This value is about five times smaller than the component of vibrations of surface atoms perpendicular to the (111) face of a bulk xenon crystal. This reduction indicates a partial hindering of the xenon vibration when adsorbed onto the graphite surface. Modeling of this effect gives interesting information about the potential and the force constant of the xenon-graphite interaction.
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J. Suzanne and J.M. Gay
More important is the measurement of the vibrational frequencies which are directly related to the adatom (or molecule)-surface force constant through the second derivative of the potential. Here again, it is worth comparing the various modes of vibration of an adsorbed molecule to those occurring in the bulk. As an example, we consider the molecular vibrations observed in butane molecules adsorbed on graphite (Taub et al., 1977a, 1978). The incoherent inelastic neutron scattering spectrum (see w 10.4.2.2) obtained within the monolayer coverage range is shown in Fig. 10.4. It shows a rich structure with four well-defined peaks which are compared with model calculations. The first three peaks are due to intramolecular CH 3 and CH2 torsions of the bulk solid. There is no appreciable change in the frequencies due to physisorption of the butane molecules. The fourth peak at 112 cm -~ (i.e. 14 meV) is interpreted as a rocking surface mode about an axis parallel to both the surface and the hydrocarbon chain direction. Model calculations also predict two other modes consistent with the broad peak centered at 50 cm -~ (i.e. 6.2 meV). The first one is a rocking mode with an axis perpendicular to the hydrocarbon chain and the second is a bouncing mode of the entire molecule perpendicular to the surface. The models suggest that the butane molecule is adsorbed with its carbon skeleton parallel to the graphite surface. The important conclusion of these studies is that physisorption produces only a weak perturbation of the internal molecular modes. Besides, a bouncing mode specific to the adsorbed state appears. III
I
I
I
I
!
j
/
CH2-CH2 CH 3 TORSION
I
P,
~\tt~/'k~
t
C4HIo MONOLAYER ON CARBOPACK B T = 77K
,11 111 I i 400300 200
I
I I00
I 50 AE
I 25 (cm "1)
Fig. 10.4. Incoherent inelastic neutron scattering spectrum of a butane monolayer adsorbed on graphite. The background has been subtracted from the spectrum. The arrows indicate the energy of the three lowest-lying modes of the bulk solid. The vertical lines at the bottom of the figure are calculated modes. The inset shows the proposed orientation of butane with respect to the graphite basal plane. Only the four coplanar hydrogen atoms (O) on one side of the carbon skeleton have been included for clarity. (From Taub et al., 1977a).
The structure of physically adsorbed phases
513
Collective motion. An ideal 2-d monoatomic crystal exhibits two vibrational modes involving lateral displacements of the atoms within the plane of the crystal. The two modes are orthogonally polarized either in the direction of propagation or perpendicular to it; these are the longitudinal mode and the transverse mode respectively. One can get the dispersion relations c0(k) from standard models treated in solid state physics textbooks. The calculations are similar to those used in the model of the linear chain of atoms (Kittel, 1976). Furthermore, upon adsorption, a third mode appears due to vibrations of the adatoms perpendicular to the substrate surface. If the substrate is supposed to be rigid and smooth, the mode normal to the surface is simply given by
coI -
(10.8)
where k0 is the force constant of the adatom-substrate interaction. This mode is independent of the wavevector k parallel to the layer. It is referred to as an Einstein mode. In reality, a dynamical coupling occurs between the substrate (surface wave and bulk phonons) and the adsorbate. Atom scattering experiments on rare gas monolayers adsorbed onto graphite or on fcc metals (Mason and Williams, 1983; Gibson and Sibener, 1985; David et al., 1986), or on a methane monolayer on MgO(100) single crystal surface (Jung et al., 1991) have shown that the normal modes are nearly dispersionless. Experimental phonon dispersion curves are shown further in w 10.4.3.2 (see Fig. 10.31). The results of the theoretical models taking into account the adlayer-substrate dynamical coupling (see for instance, Hall et al., 1985) have shown an overall quantitative agreement with helium atom scattering experiments in the case of rare gases adsorbed on graphite (Toennies and Vollmer, 1989) and on Pt (111) surfaces (Kern et al., 1986b, 1987c; Zeppenfeld et al., 1990b). Translational and rotational motions. The existence of 2-d plastic and fluid phases is deduced from the observation of translational and rotational diffusion in adsorbed layers. Rotational diffusion is of course particular to molecular adsorbates. We will see in w 10.2 that adsorbed layers may form solids that are either commensurate or incommensurate with the underlying substrate lattice. Similarly, 2-d adsorbed fluids can behave as lattice fluid or as normal isotropic fluids. In the former case, the atoms or molecules are strongly correlated and keep a partial positional and orientational order. They perform a jump diffusion process between surface sites with a residence time in each site that is long compared to the time required for the jump between sites. In the latter case, the molecules display a translational brownian motion leading to a 2-d isotropic diffusion; that is, the fluid does not feel the substrate corrugation any longer. The system may pass from one type to the other by increasing the temperature. Diffusion has been investigated in various types of hydrocarbon molecules physisorbed on graphite: CH4 (Coulomb et al., 1981), C2H 6 (Coulomb and Bienfait, 1986), CzH 4 (Grier et al., 1984), and MgO substrates: CH4 (Bienfait et al., 1987b).
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J. Suzanne and J.M. Gay
Table 10.1 Translational diffusion coefficient Dt (in 10--6cm 2 S-1) of CH4 adsorbed on MgO(100) and graphite(0001), and bulk methane. (From Bienfait et al., 1987b) Temperature
Coverage
CH4 bulk
(K) CH4/graphite 0.63 72 88 91.5 97
0.72
CH4/MgO 0.90
83 115
0.8 =0 5-6
=0 --0
12
30
40
Quasi elastic neutron scattering is a very powerful tool for these studies, as explained in w 10.4.2.2. Methane adsorbed on graphite and MgO, and ethane adsorbed on graphite provide two interesting examples described below. Methane on graphite and methane on MgO(lO0). Table 10.1 gives the translational diffusion coefficient D t o f C H 4 molecules on graphite and MgO (100) surfaces (2-d melting temperatures are 56 K and 82 K, respectively). It clearly appears that the 2-d solid is stabilized on the MgO substrate to temperatures higher than on graphite. Both methane solid layers are commensurate, ('~3-x'~-) on graphite and c(2x2) on MgO. This means that the substrate corrugation is larger for the MgO surface than for graphite. On MgO, the methane molecules in the 2-d fluid phase continue to experience the strong modulation of the square substrate and the motion is slowed down. Between 87 K and =100 K, jump diffusion takes place between equivalent sites 4.21 ]k apart. The mean residence time t on site has been experimentally evaluated to be t = l x l 0 -~~ s at 88 K and 4x10 -~ s at 97 K. The corresponding translational diffusion coefficient Dt (given in Table 10.1 by Bienfait et al., 1987b) can be deduced from t with Eq. (10.23) given in w 10.4.2.2. At 72 K, the methane molecules perform an isotropic rotation around their center of mass which remains located at surface sites. Ethane on graphite. Ethane on graphite also presents lattice liquid phases that will be described in more detail in w 10.3.3. Because of the rod-like shape of the molecule, various rotational motions can occur as found experimentally in quasi elastic neutron scattering studies (Coulomb and Bienfait, 1986). In the low density plastic phase I~ (see phase diagram in Fig. 10.3), ethane molecules are animated by rotational and translational motions. Analysis of the neutron scattering data shows that the rotational motion is isotropic around the center of mass of the molecule. Up to T = 87 K, the molecules perform a jump translational motion on the graphite sites in agreement with LEED experiments which show a strongly correlated fluid with a (2x2) commensurate structure (Gay et al., 1985). Table 10.2 gives the rotational and translational diffusion coefficients in the I~ phase versus temperature (Coulomb and Bienfait, 1986).
The structure of physically adsorbed phases
515
Table 10.2 Translational diffusion coefficient Dt and rotational diffusion coefficient Dr for the isotropic motion of the C2H6 molecule in the I1 phase for different coverages and temperatures.(From Coulomb and Bienfait, 1986) T (K)
66.4
71.4
76.2
84.1
87
122
Coverage
0.54
0.54
0.54
0.54
0.4 and 0.63
0.4 and 0.63
D r in 101~ -l D t in 10-6 cm2s -l
5-t-1 <0.1
5+1 0.4_+0.3
5+1 1.7+0.3
5+1 3.4~0.6
5-t-1 5+1
50-&10
Table 10.3 Rotational residence time "t and translational diffusion coefficient Dt for the ethane $3 phase adsorbed on graphite and its corresponding liquid at three different temperatures. (From Coulomb and Bienfait, 1986) T (K)
67
87
122
z in 10-12s D t in 10-6 cm 2 s -I
50 =0
7.5 2:k-0.2
<3 14-1-2
At higher density, the ethane monolayer presents a triangular commensurate (4-3-3x'~-3) solid phase $3 where molecules perform an uniaxial rotation about the C - C axis. Table 10.3 presents the values of the rotational and translational diffusion coefficients for $3 and its corresponding liquid (Coulomb and Bienfait, 1986).
10.2. Commensurate and incommensurate structures of adsorbed monolayers One of the basic ideas in studying physisorbed layers is to model two-dimensional matter. However, it is worthwhile to clarify the limits of this approach. The structure of the so-called 2-d, physisorbed films are indeed determined not only by a d a t o m adatom interactions in the adsorbed layer, but also by adsorbate-substrate interactions competing with them. Depending on the relative strengths of both types of interactions, the structure of the adsorbed layer may be either in or out of registry with the structure of the substrate surface. Quantitatively, the concept is described by a matrix S relating the basis vectors of the adsorbate to those of the surface (Park and Madden, 1968) (see Chapter 1, w 1.4). One distinguishes three cases: commensurate adlayer (simple lattice), high-order commensurate adlayer (coincident lattice) and incommensurate adlayer. It is also of interest to compare the natural symmetry of the densest planes of the bulk structure of the adsorbate to the symmetry of the adsorbed layer. In some particular cases of commensurate or high-order commensurate films, the substrate field forces the adatoms to pack in an unnatural way, considering their bulk structure. This is termed a symmetry frustration phenomenon.
516
J. Suzanne and J.M. Gay
~
!!iii!ii~
Fig. 10.5. Unit cell vectors a, b and a', b' for the graphite (0001) surface and the (x/-f• commensurate adlayer, respectively, a = 2.456/~, a = 4.26 A. The Ar, Kr and Xe atoms are represented with their respective van der Waals radii, sitting in the hexagonal hollow sites occupied by the (x/3-• structure. The model systems that have been most widely studied, both theoretically and experimentally, have been rare gases physisorbed on graphite (0001). Other substrates with non-hexagonal symmetry have been also investigated. For theoreticians, monoatomic gases are undoubtedly easier to simulate. On the other hand, graphite is a very convenient substrate for experimental investigations. In Fig. 10.5 is shown the basal (0001) plane of graphite with Ar, Kr and Xe atoms represented with their van der Waals diameters. The distance between adsorption sites is 4.26 A for the commensurate (x/-3xx/-f) R30 structure. This commensurate hexagonal structure has to be compared to the dense 3-d ( 111 ) plane of the fcc structure of rare gases. The lattice mismatch is defined as m=
ORG
--
~ a'
a'
(10.9)
where a' and aRC are the unit mesh vectors of the (x/3-• R30 and the 2-d hexagonal adlayer structures, respectively. If the 2d-adlayer structure was that of the dense 3 d ( l l l ) plane of the fcc structure of rare gases, the lattice mismatch would be m = +2% for Xe a n d - 6 % for Kr, at 50 K , - 1 2 % for Ar at 30 K and - 2 6 % for Ne at 10 K. With the weak lattice mismatches of Xe and Kr, one might expect commensurate (q-3-• R30, or incommensurate phases, depending on temperature and pressure, and a commensurate-incommensurate phase transition. Molecular adsorbates are more complicated owing to the superimposition of orientational and positional order effects. Among the various phase transitions in physisorbed layers, the commensurateincommensurate transition (CIT) has motivated many studies. Frank and Van der Merwe (1949) solved the one-dimensional model of a chain of atoms interacting with elastic forces and exposed to the periodic field of a substrate. They showed that incommensurability appears only for lattice mismatches beyond a critical
The structure of physically adsorbed phases
517
mismatch. This critical mismatch is a function of the relative weights of adsorbateadsorbate and adsorbate-substrate interactions. Moreover, the incommensurate chain is, in fact, made of commensurate domains separated by localised incommensurate walls. For 2-d systems, although no exact calculation was done, one usually agrees that the CIT occurs with walls separating commensurate domains. For increasing lattice mismatches, the system can tend to a uniformly modulated incommensurate structure, or to a higher order commensurate phase. The latter case may be a stage of a "devil's staircase" (Bak, 1982). Note that distinction between uniform incommensurate phase and a very high order commensurate phase is sometimes tiny. 10.2.1. C o m m e n s u r a t e and higher order c o m m e n s u r a t e structures
Commensurability or incommensurability of an adsorbed monolayer depends on the balance of the competing adsorbate-adsorbate and adsorbate-substrate interactions. Simple commensurate monolayers have been observed: (,/3-x'~-) R30, or non-rotated (2x2) on graphite, (,~-x~4-3) R30 on Pt(ll 1), (,~-x~2--) R45 on MgO (100), for instance. Higher order commensurabilities can occur considering large monolayer unit cell with several molecules. These structures are stabilized by a lock-in contribution to the monolayer-substrate energy, due to some molecules sitting in coincident sites on the substrate. An example of such high-order commensurability is given in Fig. 10.6 by the E-phase of CF3CI monolayer adsorbed on graphite (Weimer et al., 1988). Fuselier et al. (1980) have developed a coincident site lattice (CSL) model for a triangular monolayer adsorbed on hexagonal lattice. They show that a discrete set of large CSL unit cells, that are multiple of the primitive unit cell, can be accommodated with an orientational rotation 0 of the adsorbed layer with respect to the substrate. This is called high-order commensurability. In the case of a very large CSL unit, with a large number of molecules
Fig. 10.6. Suggested arrangement of the CF3CI molecules, represented by their center of mass (e) in the E-phase, after Weimer et al.~1988~. The smallest cell is rectangular containing four molecules. In the coincidence hexagonal ('q73x~/73)R5.8 cell, four out of the sixteen molecules are located in the centers of the graphite hexagons.
5 18
J. Suzanne and J.M. Gay
per unit, the lock-in part of the adsorbate-substrate interaction energy becomes insensitive to specific positions on the substrate. It may then be said that the layer is incommensurate. 10.2.2. I n c o m m e n s u r a t e s t r u c t u r e s ~ The N o v a c o - M c T a g u e rotation
In the approximation of a rigid substrate, the elastic deformation energy of the substrate upon adsorption is neglected. Consequently, only adsorbate-substrate and adsorbate-adsorbate interaction energies are considered. The former energy has been described by Steele (1973) as a Fourier series over the reciprocal lattice vectors (see Eqn. (10.4) in w 10.1.1). Most often, the first few terms of the Fourier development are sufficient for a correct description of the adsorbate-substrate interaction. The zero-order term, v0, is the averaged interaction, whereas the firstorder term, v~, measures the corrugation of the substrate potential. When v~ is small compared to the adsorbate-adsorbate interactions, calculations in the harmonic approximation show that static distortion waves are present in the incommensurate phase and that the energy minimum of the layer can be reached by a definite rotation relatively to the substrate, which is, in general, not a symmetry angle. This is called the Novaco-McTague rotation (Novaco and McTague, 1977; McTague and Novaco, 1979). Figure 10.7 shows the energy of the mass density wave phase relative to the undistorted lattice with the same parameter versus the angle of rotation for various adsorbate layers adsorbed on graphite. Ne and Ar/graphite have large angles of rotation, as confirmed experimentally by the LEED pattern of Ne (Calisti et al.,
z; O
ca
~-z
13
:50
ANGLE OF ROTATION
8 (deg)
Fig. 10.7. Novaco-McTague rotation of Ne, Ar, Kr and Xe incommensurate layers on graphite. The minimum energy of the mass-density wave, calculated for the lattice parameter that is equal to the distance of adatom-adatom pair potential-energy minimum, indicates the angle of rotation. (From McTague and Novaco, 1979).
The structure of physically adsorbed phases
519
[] 9
9
[]
9
9
[]
[]
9 9
[]
Fig. 10.8. LEED pattern of Ne/graphite, from Calisti et al. ( 1981 ). The sketch shows the graphite first order spots (e) and the twelve Ne superstructure spots (m and I"1) from two domains rotated 0 = +I 6.6 ~ with respect to the commensurate ('~r3-x~)R30 structure. The lull and dashed lines represent the Ne and the ('~3-• mesh vectors, respectively. 1982) in Fig. 10.8. The N o v a c o - M c T a g u e model is valid as long as the modulation of the substrate potential can be seen as a perturbation applied to the uniform adsorbate layer structure. Close to the CIT, the influence of the modulation of the substrate field is important, since it is responsible of the commensurability and anharmonic effects should be considered. The N o v a c o - M c T a g u e model is therefore valid only for rather large incommensurabilities.
10.2.3. The commensurate-incommensurate transition (CIT) The CIT is characterized by the formation of various incommensurate regions far from each other, separating commensurate domains. Prokrovskii and Talapov (1979) extended the Frank and Van der Merwe theory to 2-d systems, but kept the domain walls aligned in a fixed direction. In reality, the walls may cross. Figure 10.9 shows heavy and light domain walls corresponding to incommensurate compressed and expanded phases, respectively. Depending on the wall crossing energy, a striped
J. Suzzmne and .I.M. Gay
520
Fig. 10.9. Schematic diagrams showing (a) hexagonal and (b) striped arrangements of domain walls (supcrlight walls). Heavy (h) and superheavy (sh) walls can be also found in incommensurate layers (c). (From Kern and Comsa, 1989). incommensurate phase or an incommensurate phase with a hexagonal network of walls can be favoured (Bak, 1982). In addition, Bak predicted a continuous CIT in the former case whereas a first order CIT is expected in the latter one. A numerical study developed by Shiba (1979, 1980) shows that the incommensurate phase can be stabilized by rotating it with respect to the substrate surface axes. Beyond the CIT, the sequence: striped phase ---) non-rotated hexagonal phase --~ rotated hexagonal phase, is possible with increasing lattice mismatch. This full sequence is indeed experimentally observed for the Xe monolayer adsorbed on P t ( l l l ) (Kern et al., 1988). The domain wall model is well suited to incommensurate structures as long as the commensurate domain dimensions remain much larger than the domain wall width. With increasing incommensurability, the incommensurate regions grow at the expense of the commensurate ones. Figure 10.10 illustrates this continuous evolution in a series of computer simulations of Krypton on graphite (Koch et al., 1984).
0 =1.028
0=1.042
O = 1.073
0 = 1.088
Fig. 10. I0. Snapshot pictures of the incommensurate krypton monolayer on graphite for different coverages 0. The domain walls are shown as solid lines. Individual atoms are not shown. The commensurate domains gradually shrink with increasing incommensurability. After the simulations of Koch et al. (1984).
The structure of physically adsorbed phases
521
10.3. Other phase transitions in adsorbed monolayers 10.3.1.2-d Solid/solid transitions. 2-d Polymorphism. As in three dimensions, different solid phases may be present in an adsorbed layer (polymorphism). These phases are characterized by their densities. The positional order of the center of mass of the molecules and the orientational order are additional characteristics of these phases. Transitions have been observed between phases in which the molecules pack in different structures and have different orientations. An example is given by the system ethane (C2H6) adsorbed on graphite (Suzanne et al., 1983; Gay et al., 1986b). The phase diagram of this system in the monolayer range is shown in Fig. 10.3. There are three low temperature solid phases S~, $2 and $3 of molecular area 21, 17 and 10.7/~2, respectively. The molecules in the less dense solid phase $1 have their C-C axes approximately parallel to the surface and aligned in a herringbone pattern with a commensurate c(4• rectangular mesh. The C - C axes are perpendicular to the surface in the denser phase S 3 which has a commensurate ('~-• triangular packing. Between these two solid phases, there is an intermediate incommensurate phase $2 featuring a rectangular cell for which two different unit cells have been proposed: a (10• rectangular cell with 12 molecules (Suzanne et al., 1983) or an incommensurate slightly rotated rectangular mesh with a herringbone packing and two molecules per mesh (Osen and Fain, 1987). The ethane/graphite system also is a good illustration of a transition from an orientationally ordered (S~) to an orientationally disordered (I~) solid state and a continuous 2-d melting transition which will be described in the next section. 10.3.2. Order-disorder transitions Various types of disordering may occur in physisorbed systems and can be classified depending on what order is lost. Three kinds of ordering are indeed usually distinguished in uniform solid phases: (i) positional order of the center of mass of the adsorbed molecules; (ii) bond orientational order which is characterized by the orientation of a bond in two nearest-neighbor molecules; and (iii) molecular orientational order related to the orientation of each individual molecule. Molecular orientational order-disorder transitions. This kind of phase transition is well known in bulk molecular solids, like methane for instance (Press, 1972). It has been observed in physisorbed monolayers as well. The first experimental observation was for nitrogen adsorbed on graphite. At low temperature, nitrogen molecules feature a herringbone packing in a 2('~-• commensurate orientationally ordered solid phase. Beyond 25-30 K, the molecules become orientationally disordered resulting in a smaller ( ~ - x ~ - ) unit cell (Diehl and Fain, 1983). Similar behavior occurs in ethane monolayers on graphite (see Fig. 10.3). The transition from the S~ (4• herringbone solid to the (2x2) 11 phase at 65 K or from the $2 incommensurate herringbone to the 12 phase at 57 K are both examples of
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J. Suzanne and J.M. Gay
orientational o r d e r - d i s o r d e r transitions that are accompanied by a structural change. The ethane/graphite system is particularly interesting since rotational diffusion of the molecules has been measured by quasi elastic neutron scattering (Coulomb and Bienfait, 1986). The mobility change between the uniaxial rotation of the ethane molecules about their C - C axis in the S~ phase and their isotropic rotation in the I1 phase is clearly associated with the phase transition. Positional disordering in c o m m e n s u r a t e 2-d solids. Positional disordering from an ordered phase which is commensurate with the substrate to a disordered or fluid-like phase can occur over a finite temperature range. Such a transition has been observed with quantum gases on graphite. The transition from the commensurate (~3x'~-) phase to the fluid state belongs to the universality class of the three-state Potts model I because there are three equivalent sublattices of sites on the graphite surface for the molecules. This kind of transition is clearly shown by heat capacity measurements of 4He monolayers; a critical exponent cx = 0.36 has been found in good agreement with the theoretical value 1/3 (Bretz, 1977). If the nature of the surface is changed, for instance, by first adsorbing a monolayer of krypton onto the graphite surface, the number of equivalent adsorbing site sublattices decreases to two and an Ising behavior with c~ = 0 is then expected and is found experimentally (Tejwani et al., 1980). The krypton/graphite (Specht et al., 1987) and nitrogen/graphite (Kjems et al., 1976) systems also feature a transition from an ordered commensurate solid to a disordered state which is a fluid. The former system has received much attention since the krypton monolayer is the only one to form a commensurate solid, stable
~:'~.~a . Fig. 10. I 1. Computer simulation of the wandering domain walls observed for a krypton monolayer on graphite. The walls separate commensurate domains in which the Kr atoms occupy one of the three adsorption sublattice A, B, or C of the graphite surface (Abraham et al., 1982).
1 The three-state model is introduced in Chapter 13.
The structure of physically adsorbedphases
523
to high temperatures. However, krypton on graphite provides a complex phase diagram and various types of order-disorder transitions occur depending on the range of temperature and krypton pressure (Specht, et al., 1987). In particular, besides the ordered commensurate solid ---) fluid transition, which is first order, there is a transition from the commensurate solid to the incommensurate solid which passes through a disordered fluid like phase, the so-called "reentrant fluid". It is believed that this disordered phase is due to domain walls fluctuations which occurs for small incommensurabilities. This effect has been explained theoretically (Coppersmith et al., 1981, 1982; Caflisch et al., 1985; Halpin-Healy and Kardar, 1985; Huse and Fisher, 1984a) and it has been shown in computer simulations (Abraham et al., 1982) (see Fig. 10.11). In brief, the energetics are such that the domain walls are unstable to the formation of free dislocations at all temperatures for small incommensurabilities. For a 2-d system, free dislocations are a fluid-like state since shear cannot be sustained.
10.3.3. The 2-d melting transition
Physisorbed monolayers are good model systems to study the melting of 2-d matter and to check the theoretical approaches. In the ideal case, the theory considers 2-d melting in the absence of a periodic substrate potential; i.e., melting on a smooth substrate. Exploring the ideas of Kosterlitz and Thouless (1973), Halperin and Nelson (1978), and Nelson and Halperin (1979) developed a theory that a 2-d solid melts in a two stage process, involving a hexatic liquid crystal that intervenes between the solid and the isotropic fluid phases within the [Tin- T/] temperature range. - For T < Tm, the solid has long range bond orientational order and quasi-long range positional order (see w 10.1.2.1); that is, the positional pair-correlation function decays algebraically as a power law of distance. Pairs of dislocations do not alter the quasi-long range order of the low temperature solid, as shown in Fig. 10.12. Tm < T < T/, the pairs of dislocations dissociate that transforms the solid into a hexatic liquid crystal. The hexatic phase has algebraic decay of sixfold bond orientational order and exponentially decaying translational order, i.e., quasi-long range bond orientational order and no longer quasi-long range positional order. The free dislocations can be seen as pairs of disclinations. Disclinations are defects which modify the bond orientational order, as shown in Fig. 10.12. 7',. < T, the dislocations dissociate into free disclinations. Now bond orientational order is lost. An isotropic fluid phase then appears, where both bond orientational and translational correlations fall off exponentially. However, in experiments, the monolayers are physisorbed onto a periodic substrate. Long-range translational order occurs for 2-d commensurate solids. Incommensurate solids, however, are found to have translational order which decays algebraically just as predicted for a smooth substrate below Tm. Phases of this kind can exist over a range of temperatures. Their periodicity may "accidentally" match that of the substrate resulting in a Novaco-McTague orientational -
-
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J. Suzanne and J.M. Gay
Disclination
Dislocation
P a i r of d i s l o c a t i o n s
Fig. 10.12. Schematic view of point defects in a sixfold symmetry 2d-lattice. Disclinations modify the local symmetry (fivefold or sevenfold, for instance). A dislocation can be seen as a pair of bounded disclinations. The Burger's vector is shown for each dislocation.
epitaxy (see w 10.2.2). These effects are responsible for particular behavior above the melting temperature. According to theoretical models (Nelson and Halperin, 1979), there are two interesting cases depending on the symmetry of the substrate periodic field. (i) For triangular 2-d lattices on hexagonal substrates, long range orientational order should be present even above Tm because of the substrate potential. An Ising transition may possibly take place, if the incommensurate solid presents a N o v a c o McTague rotational epitaxy with two symmetric degenerate orientational minima, a few degrees off perfect alignment. (ii) For triangular lattices on square substrates, there should be an Ising transition above T,n even without the N o v a c o - M c T a g u e effect. The substrate presents a 12-fold symmetric potential, which acts as an Ising perturbation on the 6-fold symmetry of the 2-d incommensurate solid. Hence there should be an Ising-like phase transition above which the orientational order vanishes. The mechanism described above leads to a continuous melting process. Whether it is the mechanism involved in the melting of physisorbed monolayers is not clear up to now. Evidence of bond orientational order in a 2-d fluid has been found in xenon adsorbed on both graphite (Nagler et al., 1985) and silver (Greiser et al., 1987), and in ethane on graphite. The ethane/graphite system is unique in the sense that not only the "structure of the liquid" (Gay et al., 1985, 1986b), but also the
The structure of physically adsorbedphases
525
mobility of the molecules within the adsorbed layer have been probed (Coulomb and Bienfait, 1986). However, among the above-mentioned systems, only xenon undergoes a melting transition from an incommensurate solid monolayer, on both substrates. Furthermore, the transition and the correlations in the fluid phase have been explored in great detail. A first order melting transition has been observed in xenon on graphite at submonolayer coverages (Thomy and Duval, 1970b; Thomy et al., 1981, Litzinger and Stewart, 1980; Dimon et al., 1985; Colella and Sutter, 1986) with a 2-d triple point at T = 99 K (see Table 10.4). The melting evolves to a continuous transition at higher density and higher temperatures (Dimon et al., 1985; Nagler et al., 1985; Colella and Sutter, 1986). Although melting is clearly first order at submonolayer coverage (Litzinger and Stewart, 1980), the nature of the continuous melting at higher coverages is less clear and various explanations have been given. Computer simulations show that the dislocation core energy must play an important role (Saito, 1982; Swendsen. 1982). They also emphasize the effect of the substrate corrugation on the order of the transition (Abraham, 1983), particularly when the lattice mismatch between the 2-d solid and the substrate varies with temperature and coverage. Finally, substrate imperfections have been invoked to explain continuous transitions which otherwise would be first order (Abraham, 1984; Hulburt and Dash, 1985). One point seems to be clear. The xenon liquid phase is a well correlated fluid with an appreciable degree of translational and orientational order. The solid melts into an orientationally ordered fluid around 138 K with positional correlations over distances of the order of 100 ]k. The behavior of the orientational and positional order is consistent with the hexatic picture of the melting transition. However, since the graphite substrate has the 6-fold symmetry, one may question whether the hexatic order observed in the xenon fluid is provoked by the substrate field. Nagler et al. (1985) claimed that their data indicated much stronger orientational order than would be given by the graphite substrate field. In order to make a more definitive statement, new experiments are needed on a smoother substrate. For this purpose, high resolution X-ray scattering experiments were performed on silver (111) single crystal surfaces (Greiser et al., 1987). They showed that, slightly above the melting temperature, the xenon fluid has large positional correlation lengths and hexatic orientational order. Besides, under similar experimental conditions, the melting temperature and the lattice constant of the xenon monolayer on silver are very close to those on graphite. These results support the idea that the hexatic order is intrinsic to the xenon monolayer and that the Halperin-Nelson model of 2-d melting applies to these systems. Xenon monolayers on graphite and silver.
The transition from the S~ (4• herringbone solid to the I~ phase at 65 K or from the $2 incommensurate herringbone to the I2 phase at 57 K is an orientational order-disorder transition, (see w 10.3.2 and Fig. 10.3). LEED measurements (Gay et al., 1985) have shown a triangular (2• commensurate structure for Ij with a short positional order correlation length (=50 at 70 K). The bond orientational order gradually vanishes with increasing Ethane s u b m o n o l a y e r on graphite.
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J. Suzanne and J.M. Gay
Table 10.4 Experimental results on physisorbed monolayers System
~D (K)
T2c~ (K)
Structure
Melting
C,UIC,HIC, HICR Continuous C,IC Continuous ? C,IC
H2,HD,D2/ graphite He/graphite He4/graphite
1.3
Ne/graphite
13.5
15.8
HICR
1st order
Ar/graphite
47.2
59
HICR
Weakly 1st order or continuous
Kr/graphite
84.8
85.3
C,HIC, HICR
I st order
Xe/graphite
99
C,HIC, HICR
1st order and continuous
N2/graphite
49
C,HIC, UIC
I st order
O2/graphite
25.5
IC,HICR
I st order
117
65
C,IC
CO/graphite
COz/graphite
122
127.5
NO/graphite N20/graphite
55 102
87 !18.5
C2N2/graphite
189
190
IC C
CH4/graphite
56.5
68.8
C,HIC
I st order
CF4/graphite
74
99
CF3H/graphite CF3Cl/graphite
90 55
CH3Cl/graphite
118
C,UIC, I st order HIC UIC,HIC,IC C,UIC,HIC 1st order continuous UIC,IC I st order
Technique (Ref.)
Remarks
Ca(l,2), ND(3-5), LEED(3,6) ND(5,7) Ca(8-10), ND(ll,12) Ca(13,14), ND(15), LEED(16) Vo(17), Ca(18), Rotated ND(19,20), correlated fluid LEED(21), XR(22,23) Vo(17,24,25), Incipient triple Ca(26), LEED point ? (27,28), AES(27), THEED(29), XR(30,31), He(32) Vo(17,33,34), Hexatic fluid Ca(35,36), LEED, AES(37), THEED (38), XR(23,3942), H(43) Vo(44), Ca(45-47), Herringbone ND(48-50), LEED structure (51,52), XR(53) Ca(54), ND(55), Magnetic LEED(56), XR ordering. (57,58) Correlated fluid LEED(59), XR(53) Pinwheel and herring-bone structures Vo(60) Herringbone structure Vo(61), ND(62) Vo(60) Herringbone structure Herringbone Vo(60) structure Vo(25), Ca(63), ND(64-66), QENS (67), LEED(68), NMR(69) Vo(70), Ca(71 ), ND(72), XR(73) Electric ordering XR(74) Electric ordering XR(75) XR(76), He(77)
Electric ordering
527
The structure of physically adsorbed phases
System
T~D (K)
Tc2D (K)
Structure
C2H2/graphite
118
IC
C2H4/graphite
120 155.5 110
C,IC
Continuous
C2H6/graphite
114.2 130
C,HIC,IC
Continuous
IC C
I st order
C,UIC HIC C,HIC C,IC
1st order
C4H l~/graphite C6H6/graphite
116
C6H 14/graphite 151 NH3/graphite SF6/graphite 131.4 Sn(CH3)n/graphite 95 Fe(CO)5/graphite C4H6Fe(CO)3/ graphite
168 254
C,IC
170 146
I st order
UIC,HIC
H2,D2/MgO(100) He4/MgO(100) Ar/MgO(100)
38
1 63
C,UIC
Kr/MgO(100)
66.6
86.8
HIC
Xe/MgO(100)
100.8
119
HIC
N2/MgO(100)
C,UIC
CO/MgO(100)
C
CH4/MgO(100)
80
C2H6/MgO(100)
133
C C,IC
HIC C C
Xe/NaCI(100-610) CO/NaCI(100) CO2/NaCI(100) H20/N aCl(100) Kr/BN Xe/BN
102
Melting
87 119
Technique (Ref.)
Remarks
Vo(78), ND(79) Vo(80) Ca(81), ND(82,83), Correlated fluid QENS(84), LEED (85) Vo(86) Vo(87), Ca(88), Herringbone ND(89), Q E N S structure. (90), LEED(91-93) Correlated fluid ND(94) Vo(95), ND (96, 97), LEED(98), XR(99), NMR(100) ND, LEED(101 ) He(102) Vo(103), XR(104) Vo(105), MOs, XR(106) M6s, ND(107) M6s(108)
Vo, Ca(109), ND(110) Ca(109) Vo(ll 1,112), ND (112), LEED(113) Vo(ll 1,112), LEED(114) Vo(111,112), LEED(115) Vo,ND(112), LEED(114) Vo(112), LEED(116) ND(II 2), LEED (113), He(117) Vo,ND(118), LEED(119) LEED, AES(120) He(121), IR(122) LEED(123), IR(124) IR(125) Vo(126) Vo(126)
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J. Suzanne and J.M. Gay
Table 10.4 (continued) System
tD
cD
(K)
(K)
Technique (Ref.)
87 89 77 113 235.5
Vo(61) Vo(126) Vo(126) Vo(126) Vo(126)
NO/BN
50
CH4/BN C2Hn/BN C6Hi2/BN
64
Structure
Xe/Ni( 111 ) Kr/Cu(110) Xe/Cu(100) Xe/Cu(110)
C,HIC C,UIC HIC C,UIC
Xe/Cu(211 ) Kr/Pd(100) Xe/Pd(100) Ar/Ag( 111 ) Kr/Ag( 11 I) Xc/Ag( 111 )
HIC HIC HIC HIC HIC HIC
Xc/W(110) Xc/Ir(100) Xe/Pt( ! 11 )
C,UIC,IC C,HIC C,UIC,HIC, HICR IC
Kr/Pt(! 1 i) Lameilar halides Xc/CoCI 2 CH4/Col 2 Xc/CoBr 2 Xe/FeCi 2 CH4/FeC! 2 CH4/Fel 2 Kr/NiC! 2 Xe/NiCI 2 CH4/NiC! 2 Ar/CdC! 2 Kr/CdCI 2 Xc/CdC! 2 CH4/CdCI 2 NO/CdC! 2 CHn/Cdl 2 NO/Cdl 2 CH4/CdBr 2 NO/CdBr 2 CH4/Pbl 2
Melting
LEED(127) LEED, AES(128) LEED(129) LEED, AES(120,128,130) LEED( 131 ) LEED(132) LEED(132) LEED(133) LEED(131,133) LEED(131,134), XR(135), He(136) LEED(137) LEED(138) He(139) He( 136,140)
99 77.5 99 73 82.5 81 99 96.5 56 78 90
95.7 78.6 96.8
75.5 96 91 88 79 100 92
HIC HIC
Continuous I st order
Vo(141) Vo(142) Vo(141) Vo(141) Vo(142) Vo(142) Vo(142), XR(143) Vo(141) XR(143) Vo(141) Vo(142) Vo(142) Vo(141) Vo(142) Vo(61) Vo(142) Vo(61) Vo(142) Vo(61) Vo(142)
Remarks
The structure of physically adsorbed phases
System
T'~D (K)
T2c~ (K)
Structure
529
Melting
Technique (Ref.)
Mixed films Ar+Xe/graphite Kr+Xe/graphite
c,ic C,IC
Kr+SF6/graphite Xe+SF6/graphite CO+Ar/graphite
C,IC
N2+Ar/graphite
C,IC
LEED(151 )
Kr+C6H !2/graphite
C
Vo( 152), N D, XR(153)
XR(144) Vo(145), XR(146,147) Vo(148) Vo(149) ND(150), LEED( 151 )
Remarks
Pinwheel and herringbone structures Glassy pinwheel and herringbone structures
Structure: C = commensurate; IC = incommensurate; UIC = uniaxially incommensurate; HIC = hexagonal incommensurate; HICR = rotated hexagonal incommensurate. Techniques: Vo = adsorption volumetry; Ca = calorimetry" ND = neutron diffraction; QENS = quasielastic neutron scattering; LEED = low energy electron diffraction" AES = Auger electron spectroscopy; THEED = Transmission high energy electron diffraction; XR = X-ray scattering; H = hydrogen atom scattering; He = helium scattering; NMR = nuclear magnetic resonance; M6s = M6ssbauer spectroscopy; IR = infrared spectroscopy. References: 1: Motteler and Dash 1985; 2: Freimuth and Weichert 1985, 1986; 3: Cui et al. 1988; 4: Freimuth et al.. 1990; 5: Nielsen et al.. 1977; 6: Cui and Fain 1989; 7: Feile et al. 1982; 8: Ecke et al. 1985; 9: Campbell and Bretz 1985; 10: Matecki et ai. 1974b, 1978; 11: Weichert et al. 1982; 12: Carneiro et al. 1981 ; 13: Huff and Dash 1976; 14: Rapp et al. 1981 ; 15: Tiby et al. 1982b; 16: Calisti et al. 1982; 17: Larher and Gilquin 1979b; 18: Migone et ai. 1984; 19: Taub et al. 1977b; 20: Tiby and Lauter 1982a; 21: Shaw et al. 1978, 1980; 22: McTague et al. 1982; 23: D'Amico et al. 1990; 24: Larher 1974; 25: Thomy and Duval 1970b; 26: Butler et al. 1979; 27: Kramer and Suzanne 1976b; 28: Fain et al. 1980; 29: Schabes-Retchkiman and Venables 1981; 30: Specht et al. 1987; 31: Guryan et al. 1988; 32: Chung et al. 1987; 33: Thomy et al. 1981; 34: Colella and Sutter 1986; 35: Jin et al. 1989; 36: Litzinger and Stewart 1980; 37: Suzanne and Bienfait 1974; 38: Hamichi et al. 1989; 39: Hong et al. 1986, 1989; 40: Nagler et al. 1985; 41: Mowforth et al. 1985; 42: Dimon et al. 1985; 43: Ellis et al. 1981; 44: Larher 1978; 45: Chan et ai. 1984; 46: Zhang et al. 1988; 47: Chung and Dash 1977; 48: Kjems et al. 1976; 49: Eckert et al. 1979; 50: Wang et al. 1987; 51: Diehl and Fain 1983; 52: You and Fain 1985a; 53: Morishige et al. 1985; 54: Stoltenberg and Viiches 1980; 55: Nielsen and McTague 1979; 56: Toney and Fain 1984, 1987; 57: Morishige et ai. 1987; 58: Mochrie et al. 1984; 59: You and Fain 1985b; 60: Terlain and Larher 1983; 61: Matecki et al. 1974b, 1978; 62: Coulomb et ai. 1980; 63: Kim et al. 1986a; 64: Giachant et al. 1979; 65: Dutta et al. 1980; 66: Beaume et al. 1984; 67: Coulomb et al. 1979b, 1981; 68: Gay et al. 1986a; 69: Quateman and Bretz 1984; 70: Dolle et al. 1980; 71: Zhang et al. 1986; 72: Croset et al. 1982; 73: Kjaer et al. 1982, 1983; 74: Knorr and Weichert 1988; 75: Weimer et al. 1988; 76: Shirazi and Knorr 1991; 77: Ruiz-Suarez et al. 1988; 78: Peters et al. 1986; 79: Thorel et al. 1980; 80: Menaucourt et al. 1980; 81: Kim et al. 1988; 82: Satija et al. 1983; 83: Larese et al. 1988; 84: Grier et al. 1984; 85: Eden and Fain 1991; 86: Bockel et ai. 1984; 87: R6gnier et al. 1981b; 88: Zhang and Migone 1989; 89: Coulomb et al. 1979a; 90: Coulomb and Bienfait 1986; 91: Osen and Fain 1987; 92: Gay et al. 1985, 1986b; 93: Suzanne et al. 1983; 94: Trott et al. 1981; 95: Khatir et al. 1978; 96: Monkenbusch and Stockmeyer 1980; 97: Meehan et al. 1980; 98: Bardi et al. 1986; 99: Gameson and Rayment 1986; 100: Tabony et al. 1980; 101: Krim et al. 1985; 102: Rowntree et al. 1990; 103: Bouchdoug
530
J. Suzanne and J.M. Gay
Table 10.4 (continued) et al. 1984; 104: Marti et al. 1986; 105: Brener 1985; 106: Shechter 1982; 107: Wang 1983; 108: Shechter 1976; 109: Ma 1987; 110: Degenhardt et al. 1987; 111: Coulomb et al. 1984b; 112: Coulomb 1991; 113: Meichel etal. 1986; 114: Angot and Suzanne 1991; 115: Meichel et al. 1988; 116: Audibert et al. 1992; 117: Jung et al. 1989; 118: Trabelsi and Coulomb 1992; 119: Sidoumou et al. 1992; 120: Bardi et al. 1980; 121: Schmicker et al. 1991; 122: Heidberg et al. 1991b; 123: Schimmelpfenning et al. 1990; 124: Heidberg et al. 1991a; 125: Heidberg and H~iser 1990; 126: Dupont-Pavlovsky et al. 1985; 127: Fargues et al. 1989; 128: Glachant et al. 1981; 129: Chesters and Pritchard 1971; 130: Glachant et al. 1984; 131: Roberts and Pritchard 1976; 132: Moog and Webb 1984; 133: Unguris et al. 1981; 134: Cohen etal. 1976; 135: Greiser et al. 1987; 136: Gibson and Sibener 1985;137 Engel et al. 1979; 138: Ignatiev et al. 1972; 139: Kern et al. 1986a, Kern 1987a; 140: Kern et al. 1987b; 141: Teissier and Larher 1980; 142: Nardon and Larher 1974; 143: Morishige et al. 1988; 144: Bohr et al. 1983; 145: R6gnier et al. 1981a; 146: Ceva et al. 1986; 147: Stephens et al. 1986; 148: Bouchdoug 1986; 149: Menaucourt and Bockel 1990; 150: You et al. 1986b; 151: You and Fain 1986a; 152: Razafitianamaharavo et al. 1990a; 153: Razafitianamaharavo et al. 1990b.
t e m p e r a t u r e f r o m 70 to 95 K. B e y o n d 95 K, an i s o t r o p i c fluid is o b s e r v e d ( s e e F i g . 10.13). T h e i n t e r m e d i a t e I~ p h a s e (65 K < T < 95 K ) is d e s c r i b e d as a l a t t i c e fluid, s i n c e q u a s i e l a s t i c n e u t r o n s c a t t e r i n g s h o w s t r a n s l a t i o n a l d i f f u s i o n in I~ w i t h e t h a n e m o l e c u l e s j u m p i n g f r o m site to site o f a ( 2 x 2 ) lattice ( C o u l o m b and B i e n f a i t , 1986). T h e t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t D t is s m a l l e r t h a n 10 -7 c m 2 s -a c l o s e to the S~ to I~ t r a n s i t i o n but i n c r e a s e s to O t = 5 x 1 0 -6 c m 2 s -! at 87 K, as r e p o r t e d in T a b l e 10.2. T h e high t e m p e r a t u r e v a l u e s o f D t are o f the s a m e o r d e r o f m a g n i t u d e as in the bulk. W e m a y c o n c l u d e that the r e d u c e d d i m e n s i o n a l i t y and the g r a p h i t e s u r f a c e c o r r u g a t i o n do not a p p r e c i a b l y c h a n g e the t r a n s l a t i o n a l m o b i l i t y o f the m o l e c u l e in the i s o t r o p i c liquid.
Fig. 10.13. LEED patterns of the ethane submonolayer adsorbed on graphite showing the continuous change in positional and bond orientational orders of the lattice fluid I! between T = 70 K (a) and T = 102.7 K (h). (From Gay et al., 1985).
The structure of physically adsorbed phases
531
10.4. Experimental techniques and substrates in physisorption studies Most of the techniques used to characterize surfaces and chemisorbed phases can be used in physisorption studies. However special precautions have to be taken because of the weak bonding of the adsorbate to the surface. The main consequence of the weak adsorbate-surface interactions is the necessity to work with the substrate at low temperature, usually well below room temperature for the light atoms or molecules. Typical values of the investigated temperature ranges can be estimated from the 2-d triple point temperatures ~D and critical point temperatures T~D given in Table 10.4 for various gases and substrates. A large number of studies on the thermodynamics, structure, and dynamics of physisorbed phases have been performed on well characterized, uniform, defect free surfaces of powder substrates. The interesting feature of these substrates is that, due to their large surface to volume ratio, they allow the use of techniques which are otherwise devoted to the analysis of bulk matter. An important part of our knowledge on the behavior of 2-d physisorbed phases comes from these experiments. However, a few shortcomings are encountered; e.g., the distribution of orientations of crystallites in the powders prevent any determination of the orientational epitaxy of the monolayer. Also, size effects or capillary condensation can occur and perturb the analysis of monolayer or multilayer behavior. For these reasons, the use of single crystals is also very important. lO.4.1. Low temperature set-up
The low temperature set-up will be different for powder and single crystal studies. Usually, the powder experiments do not require ultra high vacuum (uhv) since the very large adsorbing area makes contamination almost negligible even with a base pressure around 10 -6 Torr. On the other hand, single crystals need a uhv environment, i.e., a base pressure around 10-~~ Torr. Also, in the latter case, rotation and translation of the sample is required for structural studies and to move the sample in front of the various analysis probes used in the uhv chamber. Furthermore, cleaving of the crystal under uhv may be necessary which makes the realisation of the low temperature set-up a complicated challenge. Various low temperature systems for studies using powders (Thorny and Duval, 1969; Bretz et al., 1973) and single crystals (Chinn and Fain, 1977; Cohen et al., 1976; Calisti et al., 1982; Drir and Hess, 1986a) have been described. They usually use liquid nitrogen or liquid helium as the cooling fluid. In the case of quantum gases (He, H2) very low temperatures below 1 K may be required (Van Sciver and Vilches, 1978). For these cases, more sophisticated cryostats are necessary; e.g., 3He, 3He/4He dilution and adiabatic demagnetization (White, 1979) refrigerators. With such designs, temperatures as low as 2 mK may be reached. More recently closed-cycle refrigerators have been used. These can be adapted to uhv systems and allow temperatures down to 30 K to be reached (Angot and Suzanne, 1991).
532
J. Suzanne and J.M. Gay
10.4.2. P o w d e r experiments
The powders used for physisorption studies are constituted of small crystallites presenting a single type of face. Thus all metal powders are excluded. The quality of the surface must usually be very good; i.e., well-ordered with a small number of defects or impurities. The behavior of the adsorbates on these surfaces is the same as on single crystal surfaces except size effects which may occur if the crystallites are smaller than 100 A, or capillary condensation when multilayer films are condensed as we shall discuss later (see w 10.6). Graphite has been the most studied surface for physisorption experiments. Since the pioneering work of Thomy and Duval (1970b), very uniform, defect-free exfoliated graphite (0001) surfaces have been prepared and used in numerous studies. On these surfaces, major contributions have been achieved for the understanding of the properties of 2-d adsorbed matter. These outcomes are due to the excellent quality of the substrate and to the large number of techniques which can be used. Indeed, due to the large surface to volume ratio, typically 2 0 - 4 0 m2/g, many techniques used for analysing the properties of bulk matter can be adapted to adsorbed monolayers as said above. There are commercial forms of exfoliated graphite: grafoil, ZYX and foam from Union Carbide and papyex from Le Carbone Lorraine. They are constituted of crystallites either randomly oriented or presenting some orientational distribution defined by its half width at half maximum (HWHM). Table 10.5 describes the various types of graphite substrates which have been used in physisorption experiments. Grafoil, papyex and ZYX consist of sheets of exfoliated graphite crystals, with primarily basal planes exposed and preferential orientation of the planes parallel to the plane of the sheet. ZYX is the most uniform powder substrate. Its orientational distribution is the smallest with a large coherence length. However its specific adsorption area is smaller than grafoil or papyex. Other uniform powder substrates include lamellar halides (Larher, 1971a; Nardon and Larher, 1974; Matecki et al., 1974a) like PbI 2, CdI2, FeI2, CoI 2, CdCI2, FeCI2, CdBr2, NiCI2, but also BN (Matecki et al., 1978; Delachaume et al., 1983; Table 10.5 Characteristics of the most commonly used graphite powder substrates Type of powder
Vermicular* (Passell et al., 1986) Foam* (Birgeneau et al., 1982) Grafoil* (Taub et al., 1977b) ZYX* (Dimon et al., 1985) Papyex 1.1"* (Marti et al., 1986) *Union Carbide; **CarboneLorraine.
isotropic isotropic partially oriented oriented oriented
Orientational Specificarea Crystallitesize (m2/g) (/~) distribution HWHM (o) 10 27 28
500 1000 100
15
1-3 20
2000 300
9 19
533
The structure of physically adsorbedphases
Bockel et al., 1984), and MgO (Coulomb and Vilches, 1984a; Coulomb et al., 1984b). The various adsorbates which have been studied on these substrates are summarized in Table 10.4. 10.4.2.1. Thermodynamic measurements: volumetry, calorimetry Volumetry. Adsorption isotherm measurements are made using an experimental set-up like that shown in Fig. 10.14. The number of adsorbed molecules is determined from standard volumetry (Thomy et al., 1981). It is often expressed by the volume of these molecules in standard temperature and pressure conditions; i.e., 273 K and 760 Torr. Basically, this number is plotted versus the equilibrium pressure P at constant temperature T to obtain a set of isotherms. The main thermodynamic quantities that can be derived from the isotherms are: the global 2-d phase diagram with the phase boundaries and coexistence regions, the triple point and critical temperature of the monolayer, the heat and entropy of adsorption from the Log P versus 1/T curve (Clausius Clapeyron line). These determinations are very useful prior to any structural studies. They indicate the domains of existence in temperature and pressure of the various phases as previously shown in Fig. 10.2. Along the vertical step of the isotherm, two 2-d phases coexist, except at a triple point temperature where three 2-d phases are present. From the slope of the Clausius-Clapeyron line, one gets the isosteric heat of adsorption -
-
-
q~t=- R
(~ log P ; ~ I/T
(10.10)
where the subscript F stands for constant surface concentration.
VALVE
1 l
CRYOSTAT
CALIBRATED VOLUME
GAUG~
SAMPLE PUMPS
Fig. 10.14. Sketch of the experimental set-up used in volumetric measurements and neutron scattering experiments. The gas (CH4,here) is introduced from a calibrated volume. (From Coulomb et al., 1977).
J. Suzanne and J.M. Gay
534
Within the first adsorbed layer, when one phase is dilute (2-d gas) and the other one is dense (2-d liquid or 2-d solid), the isosteric heat of adsorption is simply related to the latent heat of the transition 2-d gas - 2-d liquid or 2-d gas - 2-d solid. We have"
(I0.I I)
qst=Hg-H~D
where Hg is the enthalpy of the 3-d adsorbate gas and H~D the enthalpy of the 2-d dense phase (Larher, 1971b). The intercept of the log P vs 1/T lines gives the entropy of 2-d vaporization or 2-d sublimation which is the difference between the entropy of the 3-d gas and that of the 2-d liquid or solid (Larher, 1971b, 1974). When a first order transition occurs between two dense phases, the meaning of qst is less simple since the relation between the enthalpies of the phases and the isosteric heat involves the surface concentration of each coexisting phase (Larher, 1971b, 1974). When condensation occurs in a layer-by-layer fashion (see w 10.6), the equilibrium pressure p(n) of the nth layer adsorbed onto the ( n - l ) underlying layers at temperature T is given by (Larher, 1971 b)
Log
P
Bulk S O L I D
~v
ID' \ 2d (;AS
X I/T
Fig. 10.15. Schematic log P vs. 1/T phase diagram for physisorbed rare gas atoms or molecules on uniform solid surfaces such as graphite (0001 ). The domains of existence of the various 2-d monolayer phases and bilayer are shown together with that of the bulk solid phase. The solid lines are regions of phase coexistence.
The structure of physically adsorbedphases
A (.) log pC,) = _ ~ + B~,) T
535
(10.12)
A ~") = A H / R where AH is the enthalpy difference between the gas phase and the condensed phase. B~") = (~g) - ~n))/R where ~(g) is the entropy of the 3-d gas at
pressure unity and cr~n)is the entropy of the nth layer. When n ~ oo, Eq. (10.12) is then the bulk vapor pressure equation and A ~**)= AHo/R where AH0 is the enthalpy of vaporisation of the condensed 3-d phase. The transitions (or steps) are characterized by straight lines in a log P versus 1/T phase diagram as shown in Fig. 10.15. Besides the domains of stability for the various 2-d surface phases, one can see the monolayer-double layer transition and the 3-d vapor pressure (sublimation) curves. There are two kinds of experimental set-up: Isothermal calorimetry where the adsorbate is introduced continuously inside the adsorption cell located in a calorimeter kept at constant temperature; it measures directly the isosteric heat of adsorption. The introduction is slow enough to be always in quasi-equilibrium conditions (Rouquerol, 1972; R6gnier et al., 1975). Adiabatic calorimetry at constant coverage or specific heat measurements, where a small dose of energy is transferred to the sample cell at fixed amount of gas adsorbed onto the surface of the substrate; the increase of temperature is measured and related to the heat capacity of the adsorbed film. Different techniques can be used on powder substrates (Bretz et al., 1973; Elgin and Goodstein, 1974; Chan et al., 1984). A calorimetry technique for measuring small samples has been developed which allows the use of near single crystal surfaces (Campbell and Bretz, 1985). Adiabatic calorimetry is a valuable technique by which the thermodynamics of the adsorbate can be fully characterized. It is well suited to determine the temperatures of the phase transitions occurring in the adsorbed layers and the phase boundaries. Particularly, as said before (w 10.1.2.1), it has shown clearly the 2-d nature of adsorbed monolayers and has also been used to measure the critical exponents at order-disorder transitions. Calorimetry. -
-
10.4.2.2. Neutron scattering
Neutrons have proven to be a powerful microscopic probe for the study of structural and dynamical properties of bulk materials. This was due to the large intensity and the wide spectral range of high flux reactors. The application of neutron scattering to surface studies began only in the seventies. The principles and specific characteristics of this technique applied to physisorbed layers were described widely by the experimentalists who developed them (Kjems et al., 1976; Taub et al., 1977b). The main difficulty in using neutrons is in their weak interaction with matter, unlike electrons or He atoms, for instance. In investigations of surface layers, the problem is to discriminate the signal from the layers against that from the substrate. This obliges one to choose systems with strongly scattering adsorbates and rather transparent substrates. Most often, the experimental surface spectra are determined
J. Suzanne and J.M. Gay
536
Table 10.6 Neutron scattering and capture cross sections of some gases and substrates (from White, 1977) Gas
Cross-section in 10-28m 2 Coherent
4He
1.13
Substrate
Incoherent -
Ne 36Ar
2.66 74.20
0.24 -
Kr Xe H2 D2 N2 02 C! 2 CD 4 CH4 ND 3 NH 3 D20 H20
6.88 2.90 7.03 22.36 44.41 16.90 46.13 139.6 8.68 108.7 0.42 46.0 0.36
159.4 4.02 0.6 0.2 6.84 8.0 318 6.33 239.4 4.03 159.5
Cross-section in 10-28 m 2 Coherent
Graphite MgBr 2 MgO FeI 2 Pbl 2 SiO 2 AI203
5.56 44.4 16.0 50.76 50.3 31.4 74.8
Incoherent 0.9 0.04 1.0 0.84 -
from the difference between the spectra recorded with and without the adsorbed layers. In Table 10.6 are reported the coherent cross-sections for a selection of gases commonly used for physisorption studies. The large coherent cross-sections of the deuterated molecules is noteworthy, although nitrogen and the isotope 36Ar are also attractive candidates. On the other hand, normal hydrogen is undeniably the strongest incoherent scatterer. Even with carefully selected systems, the experiments are not possible without large surface area samples of a few m 2. This is the reason why neutron scattering is limited to powder experiments. Different types of powdered substrates with various specific areas and textures have been used. A summary of the most commonly used powders is given in Table 10.5. In this kind of experiment, neutrons are scattered from the whole sample. Highly absorbing materials cannot therefore be used. This is one of the reasons why there are no neutron studies of physisorbed systems on powdered metals, for instance. The standard set-up is a thin aluminium or stainless steel container, a few cm high, loaded with the adsorbent powder. This cell is mounted in the tail of a cryostat with access for the gas provided by a thin capillary. The experimental set-up is similar to that of volumetric measurements shown in Fig. 10.14. Sample temperature is measured with a thermometer (Pt, Ge, carbon resistors, etc.) embedded in the container. An absolute temperature calibration can be precisely performed measuring the bulk vapor pressure of the adsorbing gas. The effective surface area of the sample is calibrated with an adsorption isotherm of Kr, or any other gas with a well defined monolayer structure. The area is deduced from the known density of the adsorbed layer and the measured quantity of adsorbed gas. The classical
The structure of physically adsorbedphases
537
cleaning procedure consists of baking the powder for several hours at high temperature (--1000~ though a thermal treatment at 200~ could be sufficient for recycling a graphite powder which has been kept in a clean atmosphere.
Neutron diffraction from powdered samples.
Coherent elastic neutron scattering (diffraction) is used to investigate the structure of the adsorbed layers. The problem of diffraction from a surface is developed in Chapter 7. It is easy to draw an analogy between a film adsorbed on a substrate and the reconstructed layer on top of a crystal. Both give rise to adsorbate (superlattice) reciprocal lattice rods normal to the surface with an interference term when the adsorbate (superlattice) and substrate (bulk crystal) reflections coincide or are close. The position of the rods is determined by the size and symmetry of the adsorbed layer unit mesh, whereas the modulation along the rods is given by the distribution of the adatoms perpendicular to the surface. This modulation is thus very sensitive to the film thickness. The intensity is uniform along the rod for a monolayer and becomes more and more peaked around the Bragg reflections of the adsorbate bulk with increasing thickness. These results are equally applicable to neutron or X-ray diffraction. It is worthwhile pointing out that the kinematical theory of scattering can be applied to X-rays and neutrons. This makes the quantitative analyses for these techniques rather easy compared to the case of electrons. The Laue conditions for diffraction from a 2-d crystallite require that the scattering vector Q ends on a reciprocal lattice rod. Each rod is defined by (i) its projection onto the crystallite surface around the reciprocal lattice v e c t o r Ghk, (ii) its cross section Shk (e.g. Gaussian, Lorentzian, etc.), and (iii) its modulation (e.g. along the rod) Fh,. The scattered intensity may be written as a function of Qll and Q• components of Q parallel and normal to the crystallite plane, respectively
I(Q) = Shk (Qn- Ghk) Fhk(Q9
(10.13)
In powders, the orientation of a given crystallite is described in terms of its Eulerian angles 03, X and ~ relative to a frame (e~,e2,e3) chosen such as Q = Qe2"x and co are polar and azimuthal angles defining the crystallite c axis (normal to the layer) direction, and ~) characterizes the orientation of the crystallite about its c axis. The powder-averaged intensity Ip(Q) is obtained by integrating over 03, X and ~. Considering the case of a powder-orientation distribution probability P(X), which is function of the polar angle only, we have: 2n
~
2~
Ip(Q) = I d03 1 d~ I d~ P(~)l(Q2cos2~ + G~k- 2QGh, cosgt cos~) F(Qsinv) ()
0
0
(10.14) where sin~ = sinco sinx. The most visible consequence of the orientation distribution of 2-d scatterers is in the asymmetric "sawtooth" shape of the diffraction lines. The origin of this
538
J. Suzanne and J.M. Gay
ca ~
zI/
Qol/Q, 7 ~ j/~2
IDetecti~planel~
r
o,IM
QoQ~ Q2 Scattering vector Q Fig. 10.16. Diffraction lineshape from a disoriented powder of 2d crystailites. Scattering is measured in a fixed plane of the diffractometer. In the reciprocal space (origin f~), lattice rods of three different crystallites are shown with respective tilt (polar) angles 0, Xi and X2 with respect to diffractometer plane. In this plane, the Laue condition for diffraction implies that the scattering vector is Q0, QI and Q2 respectively for the three represented crystallites. For a full distribution of 2d crystallites, scattering from a given rod is observed over a continuous Q-range starting at Q0. The intensity damping with increasing Q is due to a reduction of the intersection of the rod with the Ewald sphere for tilted rods. The detailed shape of the trailing edge depends on the orientation distribution. particular shape can be schematically understood as shown in Fig. 10.16. This effect was first demonstrated by Warren (1941) for randomly oriented powders of 2-d crystallites. The profile is, in reality, influenced by preferential orientation of the 2-d crystallites in some powders. Detailed calculations of the diffraction profiles can be found in the literature (Kjems et al., 1976; Ruland, 1967; Ruland and Tompa, 1968; Stephens et al., 1984; Larese et al., 1989). As an example, Fig. 10.17 shows typical diffraction spectra for CD4 films of different thicknesses adsorbed on M g O powder. The line extending in the 1.4-2.05 /~-~ range comes from the (10) 2-d Bragg reflection. Its shape changes dramatically with the film thickness. The (11) peak is observed around 2.1 ~-~. The calculation of the diffraction lines is based on Eqs. (10.15) and (10.16). More explicitly, Fhk(Q_O can be written as Fhk(Q• = I s e i2rt~hx+ky) eiQlz 12
(lO.15)
539
The structure of physically adsorbed phases
3. 103
C D 4 / M g O ,.
,
T=77K
d) 0=4 l a y e r s
O.
2. 103
c) 3 layers
t..._
o t... t.....
o.
r
'~.
1. 103:
~Ab)
~
0=2
layers
0 0
1.103
1~
a)
tt)
0=1
layer
I
s.
0
,5 ~
'Q(A-1)
Fig. 10.17. Neutron diffraction spectra of CD4 adsorbed on MgO powder at 77 K. (a) 1 layer, (b) 2 layers, (c) 3 layers and (d) 4 layers. The shape of the diffraction line is representative of the film thickness. (From Madih et al., 1989). w h e r e the sum is calculated o v e r all the m o l e c u l e s within a v o l u m e d e f i n e d by the surface unit mesh and including the total film thickness, z is the m o l e c u l e h e i g h t a b o v e the surface, x and y are its c o o r d i n a t e s parallel to the surface. In the case o f CD4 films on M g O , the m o l e c u l e s are alternatively located at x = 0, y = 0 and x = 1/2, y = 1/2 in the s u c c e s s i v e layers. A s s u m i n g a c o n s t a n t interlayer spacing d, it results: - for the m o n o l a y e r Fhk(Q• = 1 , for both (10) and (11) reflections. A f t e r integration a c c o r d i n g to Eq. (10.14), the two peaks have the s a m e " s a w t o o t h " shape with a m a x i m u m for Q• = 0 (i.e. Q = Gi0 and Q = Gli
J. Suzanne and J.M. Gay
540
- for the bilayer, Flo(Q.O = 4 sin 2 Q"2 d and F ll(Qa.) = 4 c o s 2 Q•2 d The rod modulations are clearly distinct. F~(Q• is still m a x i m u m for Q . = 0 and the (11) peak m a x i m u m has not moved. In contrast, F~o(QO = 0 for Q• 0 so that there is no more intensity for Q = G~0. The diffraction line reaches a m a x i m u m at higher Q. - for the trilayer, Fio(Ql) = (2 cos Q• d - 1)2 and Fll(Q.O = (2 cos Q• d + 1 )z. The rod modulations are again different, that results in the different lineshapes of the (10) and (11) reflections. Neutron diffraction experiments are usually performed on a triple-axis spectrometer or a two-axis spectrometer with a multidetector, using thermal neutrons (Z, -- 1.3 - 2. 6/ ~ ) . Typical Q resolution is about 2 - 3 . 1 0 -2 ,/k-~
Inelastic neutron scattering. Inelastic neutron scattering is used to probe the dynamics of adsorbed layers; see w 10.1.2.2 (Marshall and Lovesey, 1971). It is worth listing its advantages in comparison to other surface spectroscopy techniques. They include (i) the absence of selection rules unlike infrared or Raman spectroscopies, (ii) the possibility of easy investigation of any hydrogenated compound (incoherent scattering), and (iii) the relative ease of calculating vibrational mode cross sections. Coherent inelastic neutron scattering allows investigations of the dynamics of physisorbed layers, and particularly of the collective modes. As for diffraction, the models should take into account the appropriate orientational distribution of the
z 0fj ~:: o iz o ~z
_
o3 pz
[
i
l
1
~
Q=5.5~
i -I
l
1
l
500 io~
ee
, {..
i".
I00 o -I
rr "' a_
[ ~ i Q = 2 . 7 5 ~,-I
500
o 300 ~D LtJ (J Z I00 rr LIJ I.,L G
--
e'e
Q = 2.75 A 2 / 3 isotropic, --113 gaussian
2
4
le
~ t',/ /
---.-gaussian
"
\t
9 ~o~
o-j Q=3,5A 2 / 3 isotropic, I / 3 gQussian -.-goussian
2
6
AE
4
6
(meV)
Fig. 10.18. Inelastic neutron scattering spectra from Ar adsorbed on partially oriented graphite. The scattering vector Q is parallel to the graphite foils. Computer simulations (solid and dashed lines) are compared to the experimental spectra. (From Taub et al., 1977b).
541
The structure r$physically adsorbed phases
powder. The spectra measured experimentally are more nearly representations of the density of phonon states than the actual phonon dispersion curve, that would be obtained with a 2-d single crystal. Advantage is frequently taken from the use of highly oriented powders. Contributions of the in-plane and out-of-plane modes can be distinguished in this way (Taub et al., 1977b). Figure 10.18 shows inelastic spectra from Ar 36 monolayers at 5 K, adsorbed on grafoil. The experimental data are compared to a simulation of the different modes taking into account the orientational powder distribution of grafoil. Incoherent inelastic neutron scattering is basically related to single particle correlations. Due to the strong scattering from H atoms, incoherent scattering is also used for investigating the collective excitations. This neutron scattering technique is well suited for studies of low-frequency modes of adsorbed layers (Hansen and Taub, 1987). The analysis of such measurements relies on the comparison of the experimental data with calculated inelastic spectra based on the phonon dispersion relations. Quasielastic incoherent neutron scattering (QENS) is particularly useful for studying orientational order in physisorbed phases. Characterisation of lattice fluid layers can be done as well. At very low energy transfers, quasielastic neutron scattering evidences molecular rotational or translational diffusion (Bienfait, 1987a; B6e, 1988). The scattering law, which is directly proportional to the differential scattering cross section, has two components SR(Q,o3) and ST(Q,c0) describing these two motions. The resulting law S(Q,co) is the convolution of these components
S(Q,co)
= SR(Q,r
(10.16)
) (~ ST(Q,0~)
where Q is the scattering vector, h o the neutron energy transfer; | denotes the convolution operation. The rotational scattering law may be written as the sum of an elastic term S~(Q) and a quasielastic term ~ i ( Q ) : SR(Q,(.o ) - ~R~(Q) +
(10.17)
SqRe'(Q,o)
The Elastic Incoherent Structure Factor (EISF) defined as:
Ss~(Q)
(10.18)
EISF(Q) = ,.~(Q) + ~Re,(Q)
is often used in the analyses of rotational diffusion experimental data (B6e, 1988). Tractable analytical forms of SR(Q,o~) are available for simple models (isotropic rotation, rotation about one molecular axis, etc.). For uniaxial rotational diffusion of the nuclei on a circle of radius r, oo
1
SR(Q,co) = J~(Q r sinx) 8(c0) + 2 ~_~ J~ (Qr sinx) 9- 9
D~m 2
r~ (D~m2) 2 + 032
m=l
(10.19)
542
J. Suzanne and J.M. Gay
where Dr is the rotational diffusion constant and Jm is a m-order Bessel function of the first kind (B6e, 1988). The QENS study of the rotational diffusion of the rod-like ethane molecule adsorbed on graphite (Coulomb and Bienfait, 1986) has been very helpful in the determination of the orientation of the molecules in the various phases observed in the monolayer range (see w 10.1.2.2 and Fig. 10.3). The rotational diffusion coefficients of the ethane monolayer on graphite at different temperatures are reported in Table 10.2. For all known models, the translational component is a Lorentzian function 1
ST(Q,o) ) = --
f(Q
f2(Q) +
0)2
(10.20)
whose width f(Q) depends on the model. For a 2-d isotropic liquid with brownian mobility, f(Q) is a function of Dr, the translational diffusion coefficient, "c the angle between the 2-d system and the scattering vector Q.
f(Q) = Dt Q2 sin 21:
( 10.21 )
Correlated liquid phases have been also widely discussed in the literature (Bienfait, 1987a; B6e, 1988). Among them, lattice fluid phases with translational jump diffusion have been observed in physisorbed layers. Their "structure" images the 2-d crystalline structure of the underlying solid, and various models are available. For example, the square symmetry of MgO(100), which is responsible of the c(2x2) structure of the methane solid monolayer, also induces square symmetry order in the fluid phase (Bienfait et al., 1987b). Assuming jumps into a square lattice with unit mesh vectors (a,b),f(Q) is written as
f(Q) = [2 - cos Q.a - cos Q.b]/2t
(10.22)
where t is the mean residence time on a site. The parameters t and a, length of the jumps, are related to the diffusion coefficient Dr: a2
D , - 4t
(10.23)
In Fig. 10.19, are reported some QENS spectra from a C2H 6 monolayer adsorbed on graphite (Coulomb et al., 1985). The measured spectra describe jump translations of molecules (lattice fluid) performing isotropic rotation. Some typical values of both translational and rotational diffusion constants deduced from QENS measurements are reported in Tables 10.1, 10.2 and 10.3. Neutron time-of-flight spectrometers are commonly used for inelastic scattering. The energy resolution depends on the wavelength and can be as narrow as 27 l.teV, for instance for ~, = 8 ]k with IN5 spectrometer of the Institut Laue-Langevin. Triple axis spectrometers are also employed for the inelastic studies. Inelastic
The structure t~'physically adsorbed phases
543
S(Q,w) 0 = 0 . 4 monolayer
T=87 K
6 103 410
Q(A")
1
9
210-
.........
,r
0
0.2
',",~ .'f..._'~-~,_
Jr"
,c"
0.6
~
de"
15w(meV)
Fig. 10.19. Incoherent quasielastic neutron scattering of 0.4 monolayer of ethane adsorbed on graphite
at 87 K. The experimental data are fitted against a model including a jump translational motion plus an isotropic rotational diffusion. (From Coulomb et al., 1985). scattering is typically performed with incident neutron energies of some tens of meV and an instrumental resolution of about 1-2 meV. To summarize, the elastic and inelastic neutron scattering techniques are useful tools to investigate the details of the microscopic structure and dynamics of physisorbed layers, though they are not basically surface-sensitive techniques. Nevertheless, they can be employed with strongly scattering species adsorbed on high surface-area substrates. Among the different adsorbates, it is worth mentioning that hydrogenated molecules can be studied both by elastic coherent scattering using deuterated molecules, and by incoherent scattering with normal hydrogen. One of the essential experimental advantages of neutron techniques is that they may be used under relatively high gas pressure. The large penetration depth of neutrons allows investigations of the whole thickness of multilayer films, that cannot be done with He atoms or electrons, for instance. In addition, no desorption effects perturb the physisorbed layer under the neutron beam. Additionally, data analysis is usually relatively easy, but the properties experimentally probed are powder-averaged.
10.4.2.3. X-ray scattering Like neutrons, X-rays are not strongly interacting probes and are therefore not normally surface sensitive; see Chapter 7. The use of X-rays or neutrons has had almost the same development since the late seventies. The experimental techniques are similar; they require high surface area substrates, and they can be operated
544
J. Suzanne and J.M. Gay
with a relatively high vapour pressure. The diffraction profiles can be calculated using the same models. Nevertheless, the scattering cross-sections of the different adsorbates determine the specificity of X-rays or neutrons. Whereas X-ray scattering strength is directly proportional to the atomic number, neutron scattering cross-sections do not depend so smoothly on it. Most of the X-ray scattering experiments are diffraction studies (McTague et al., 1982; Stephens et al., 1984; Morishige et al., 1985; Marti et al., 1986; Knorr and Weichert, 1988), but some EXAFS studies have also been performed (Bouldin and Stern, 1982; Guryan et al., 1988). Besides the powder X-ray diffraction technique, a new field of experiment is now opened with experiments on single crystal surfaces (see w 10.4.3.3). X-ray diffraction experiments are performed on the same types of powders as neutron diffraction. Sample thickness is however reduced to 1-2 mm due to the smaller penetration depth of X-rays. The experimental set-up is slightly different depending of the X-ray source. With rotating anode sources, the beam is usually focused by the monochromator. With synchrotron sources, the beam is unfocused. As for neutron experiments, the powder is contained in a cell connected via a capillary to a standard filling volume. The cryostat as well as the cell has X-ray transparent windows of beryllium or mylar. The diffractometer is a classical two-axis or triple-axis spectrometer. Figure 10.20 shows a schematic view of a typical X-ray scattering set up for use with synchrotron. Resolution is typically I x l 0 -2 ,/k-~ with a rotating anode source, but it can be as high as 3x10 -4/~-~ with synchrotron light. The latter high resolution allows very precise structure determination. Extended X-ray absorption fine-structure studies of physisorbed layers probe the local structure (more information about this technique can be found in w 7.5 of Chapter 7). More precisely, EXAFS has been applied to characterize the adsorption sites and the atom's local environment in incommensurate phases. The EXAFS signal can be measured by the transmission of X-rays as a function of the energy or by fluorescence.
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The structure of physically adsorbed phases
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10.4.2.4. MOssbauer Spectroscopy Mrssbauer spectroscopy is well known for its example the exploration of vibrational spectra Waller factors and thermal shifts (Frauenfelder, successfully been used to study the dynamical
application in bulk systems, as for through measurements of D e b y e 1960; Nussbaum, 1966). It has also behaviors of physisorbed systems.
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Fig. 10.22. M6ssbauer resonant absorption signal of butadiene-iron-tricarbonyl (BIT) adsorbed on graphite (0001) for various coverages in the monolayer range: (a) Measured in the direction perpendicular to the layer: a, 0.3 ML; b, 0.6 ML; c, 1.0 ML; d, bulk crystal. (b) Measured in the direction parallel to the mean direction of the graphite basal panes: a, 0.3 ML; b, 0.6 ML; c, 1.0 ML. The sharp drop in the signal indicates an abrupt transition to a mobile state of the adsorbed BIT molecules parallel to the graphite surface signature of a 2-d melting transition (From Shechter et al., 1976). However the technique is limited to adsorbed molecules containing an atom which presents a M6ssbauer (resonant) transition. The most c o m m o n M6ssbauer elements are Fe sv and Sn"9 having a resonant transition at 14 keV and 23.8 keV, respectively. Indeed, molecules with these elements have been used in physisorption experiments. They are butadiene iron tricarbonyl (BIT) C4H6Fe(CO3) (Shechter et al., 1976, 1979), tetramethyl tin (TMT) Sn(CH3)4 (Shechter et al., 1980, 1982, 1990) and iron pentacarbonyl (IPC) Fe(CO)s (Wang et al., 1983; Shechter et al., 1988) monolayers adsorbed on graphite powders. This technique is very sensitive for probing the mobile states of the adsorbed films and observing changes in mobility. It showed for the first time, evidence of 2-d melting (Shechter et al., 1976). Figure 10.22 shows the MOssbauer intensity of BIT adsorbed on graphite (grafoil) at different coverages versus temperature. The abrupt drop of the signal at a fixed temperature (150-165 K) when the g a m m a ray propagation vector is parallel to the grafoil sheets (Fig. 10.22b) was interpreted as the signature of the 2-d melting of the monolayer. The drop is considerably reduced when the propagation direction is perpendicular to the grafoil planes (Fig. 12.22a). Indeed, this result means that the adsorbed molecules are still on the graphite surface but in a mobile state with a motion confined in a direction parallel to the surface.
The structure of physically adsorbed phases
547
Other dynamical behavior has been investigated. Dynamical changes have been observed in submonolayers of TMT on graphite below the 2-d melting temperature (T2m~ --- 95 K). They were attributed to an incommensurate-commensurate (~-• R19 transition occurring with increasing coverage around 0.65 monolayer. This hypothesis was supported by X-ray diffraction experiments (Shechter et al., 1982). An abnormal gradual decrease of the spectral intensity was also observed as T approaches T2m ~ which was interpreted as the occurrence of 2-d edge melting (Shechter et al., 1990), the analog of the 3-d phenomenon known as surface premelting. The same observation has been made for the IPC/graphite system which was also attributed to edge premelting effects. Also, for this latter molecule, abnormal behavior in the Mtissbauer intensity versus temperature between 120 and 150 K was considered as the precursor of a transition to an orientationally disordered phase which is completed by T = 167 K as shown by neutron diffraction experiments (Wang et al., 1983). Attempts have been made to measure the 2-d diffusion coefficients of the adsorbed molecules in the fluid phase. Difficulties arise because the line broadening, when it occurs, sets in abruptly and is too large to be measurable in the parallel direction. However, a quantitative estimate can be based on the fractional change in the perpendicular cross section (see Fig. 10.22a). These changes can be understood as arising from a sampling of the surface parallel signals, due to the distribution of crystallite orientations in grafoil or papyex (see w 10.4.2) and finite detector geometry (angular collimation). In the case of BIT molecules (Shechter et al., 1979), the diffusion coefficient D has been estimated to be around 10 -9 cm2/s. This value is more characteristic of a highly viscous media than a typical fluid. Normal liquids have diffusion constants in the range of 10 -5 cm2/s. The same order of magnitude has been deduced from quasi elastic neutron scattering measurements in a methane liquid monolayer adsorbed on papyex (Coulomb, 1979b, 1981). Another estimate can be made by measuring the reduction of the M6ssbauer fraction while changing the direction of propagation of the gamma ray with respect to the plane of the graphite sheets by rotating the sample cell. In this way, a diffusion constant of the order of 10 -8 cmE/s was found for TMT molecules above the melting temperature (Shechter et al., 1980). This value of D is still three orders of magnitude below that of methane on graphite. It has been questioned if these unusual low values of D are intrinsic to BIT or TMT on graphite systems or if there are unexpected difficulties involved in the M6ssbauer technique or its analysis. In order to answer this question, other techniques like quasi elastic neutron scattering should be used on the same systems. Finally, an estimate of the binding energy of a molecule above the surface can be obtained by M~3ssbauer spectroscopy (Shechter et al., 1979). Since the adsorption cross section is in general proportional to the mean square displacement of the resonant atom in the gamma ray propagation direction, one can measure this mean square displacement in the direction perpendicular to the surface. For small vibrations about the minimum, the potential can be expanded in a power series and truncated after the quadratic term. In this approximation, one can obtain a simple expression for the binding energy perpendicular to the substrate in terms of the
548
J. Suzanne and J.M. Gay
frequency and equilibrium position of the atom. For a 9-3 Lennard-Jones potential, a molecule of mass m, at equilibrium distance zo from the surface and frequency co has a binding energy to ~:,, = m co2z~/27
(10.24)
The Debye-Waller factor for the same simple model is related to the M6ssbauer intensity f b y the relation
4~2kBT log f = - m 6o2~,2
(10.25)
where kB is the Boltzman constant, T the temperature and L the wavelength of the gamma ray. The value of Zo is estimated from the sum of the van der Waals radius of the substrate surface atoms and that of the adsorbed molecule. For TMT (Shechter et al., 1982) a value to = 7.3 kcal/mol has been found at monolayer coverage whereas the isosteric heat of adsorption measured from adsorption isotherms gives q~t = 13.2 kcal/mol. The difference may be attributed to the lateral interactions with the neighboring admolecules since the M6ssbauer signal mainly probes the motion perpendicular to the surface. It is clear, however, that given the rough approximations, one gets only a crude estimation of the binding energy.
10.4.3. Single crystal experiments 10.4.3.1. Electron scattering." LEED, RHEED, THEED, AES LEED (Low Energy Electron Diffraction).
LEED is a classical technique for surface structure studies, as shown in Chapter 7 of this handbook. In physisorption, it particularly allows determinations of the epitaxial relations between the substrate surface and the adsorbate layer. We recall here a few important features of this technique. The range of electron energies involved in LEED is typically 20-500 eV. A major limitation of LEED comes from the dynamical behavior of the electronsurface interactions leading to multiple scattering which complicates the analysis of the structure. LEED usually gives easily the unit mesh of the monolayer (2-d) lattice and its orientation relatively to the underlying surface. The number of molecules in the unit mesh can be then deduced from the van der Waals diameter of the adsorbed species, if the coverage is known. The molecular orientation and position inside the cell cannot be easily obtained from the LEED data. A LEED multiple scattering analysis of the data could be performed, but it has never been done in physisorption studies. Another severe limitation of LEED comes from its strong interaction with the molecules of the 3-d gas which surrounds the adsorbate for experiments performed under equilibrium conditions between this gas and the sample surface. LEED experiments are limited to pressures below 10-5 Torr. Many interesting phase transitions in monolayer or multilayer physisorbed films unfortunately occur in a temperature range where the equilibrium pressure is well above this value. This is
The structure of physically adsorbed phases
549
the case, for instance, for the melting transition in rare gas monolayers adsorbed on graphite or MgO(100). However, other diffraction techniques using neutrons or X-rays as probes can work even at pressures larger than one Torr. Finally, another problem with LEED comes from the perturbation of the adsorbed layers by the electrons due to stimulated desorption by thermal effects or to dissociation of the adsorbed molecules. These effects are negligible with the heaviest atoms (Xe, Kr) and remain weak for the other rare gases on graphite. On metal surfaces stronger perturbations are observed even at currents as low as 10 -7 A (Unguris et al., 1981). The situation may be more dramatic with molecules. For instance, CH 4 adsorbed on graphite suffers dissociation under the electron beam of a standard LEED gun whose intensity is around 1 ktA. The consequence of this dissociation is an irreversible contamination of the graphite surface by amorphous carbon (Suzanne and Bienfait, 1977). This perturbation of the adsorbate by LEED has been greatly reduced by using electron guns operated at low currents, typically 10-9A, and using a channel plate intensifier to get enough intensity at the fluorescent screen (Chinn and Fain, 1977). In addition, better resolution is obtained in the LEED systems using low current guns because of the reduction of space charge effects. The beam size at the sample surface is around 0.1-0.3 mm diameter and the coherence length is of the order of 100 to 200/~ (Fain, 1982). Still, even with such low electron beam currents, perturbation of the adsorbate may occur. It has been reported for ethylene C2H4 adsorbed on graphite (Eden and Fain, 1991 ) or for ethane C2H6 adsorbed on MgO(100) (Sidoumou et al., 1992). Irreversible contamination of the surface may be reduced by keeping exposure times as short as possible. This can be achieved by exposing the sample to the electron beam only during the time necessary to acquire the data with a video camera linked to a computer. This time is usually of the order of a few seconds (Eden and Fain, 1991; Sidoumou et al., 1992). Nevertheless, LEED remains the easiest surface diffraction technique, in spite of the above mentioned drawbacks. In physisorption studies, it has been used to characterize various phases and phase transitions, as described in w 10.2, 10.3 and 10.6. Among the most spectacular results, there are the Novaco-McTague epitaxial rotation of incommensurate monolayers of rare gases or molecules on graphite, the observation of 2-d polymorphism, and the study of a continuous melting transition in a monolayer of ethane molecules on graphite. An important side application of LEED in physisorption studies is the measurement of adsorption isotherms on single crystal surfaces. These measurements, using a method described below, complement the results obtained by classical volumetry as described above (w 10.4.2.1) by extending them to lower temperatures and pressures. They are very useful to characterize the domains of stability of the various 2-d phases, to show the existence and the order of phase transitions occurring in the monolayer and to give the heat of adsorption of the molecules on the surface. Attenuation of the intensity of a substrate LEED spot can be measured versus equilibrium pressure of the adsorbate at a fixed substrate temperature (isotherm) or versus temperature at fixed pressure (isobar) (Unguris et al., 1979, 1981; Diehl and Fain, 1982; Gay et al., 1986b). The fractional coverage is proportional to the
J. Suzanne and J.M. Gay
550
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PRESSURE (Tort) Fig. 10.23. Monolayer LEED equilibrium isotherms for ethane adsorbed on graphite (0001) at the 2-d gas-I ~transition (see text and phase diagram in Fig. 10.3)" electron energy is 137 eV. The intensity of the graphite (01 ) spot has been plotted against ethane vapor pressure at different temperatures: (1) 79.5 K, (2) 84.5 K, (3) 88.2 K, (4) 91.9 K, (5) 96.4 K, (6) 99.7 K. The vertical step is a signature of a first-order transition. (From Gay et al., 1986b).
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I'RESSURE (Torr) Fig. 10.24. Multilayer LEED equilibrium isotherms for ethane adsorbed on graphite (0001 ). (1) 63.9 K, (2) 67 K, (3) 70 K, (4) 75.4 K. The two vertical steps are due to second layer and bulk condensation. Intensities 0 and 1 correspond to background (thick ethane film) and bare graphite spot intensities respectively. (From Gay et al., 1986b). l o g a r i t h m of the attenuation. In a stepwise isotherm, calibration is d o n e by a s s u m ing that c o v e r a g e is one m o n o l a y e r at the knee which p r e c e d e s the plateau ( U n g u r i s et al., 1979). As a matter of fact, accuracy in the absolute quantitative m e a s u r e m e n t is not required in most o f the studies. W h e n a sharp first o r d e r transition occurs b e t w e e n a dilute 2-d gas phase and a dense solid (or fluid) phase a sharp vertical
The structure of physically adsorbed phases
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12 14 16 IO00/T Fig. 10.25. Log P versus l/Tphase diagram for ethane adsorbed on graphite as determined from LEED equilibrium isotherms and classical volumetric measurements. As in Fig. I0.15, the lines represent phase transitions and coexistence regions: I1 (2-d gas-l~ transition), $3 (12-$3 transition), 2 (second layer condensation) and 3-d (bulk condensation) are deduced from LEED isotherm measurements; (a) (2-d gas-2-d liquid transition), (b) (12-$3 transition), (c) and the dotted line (second layer condensation), (d) and (e) (bulk condensation around the bulk triple point T3D) come from volumetry studies. (From Gay et al., 1986b).
step is observed in the isotherm as shown in Fig. 10.23 for ethane on graphite (Gay et al., 1986b). Condensation of successive layers yields more steps as shown in Fig. 10.24. Usually it is not possible to observe more than two or three steps. Above these coverages, the LEED intensity attenuation is such that the substrate spots are no longer visible. As for volumetric measurements, a set of isotherms at different temperatures on the same system allows to plot a Clausius-Clapeyron line whose equation is given by formula (10.12) and shown in Fig. 10.25. The heat of adsorption of the first and second layers can be calculated from the slope of the line (see w 10.4.2.1). Some precautions have to be taken in these kinds of measurements. The pressure measurement with an ionization gauge (Bayard-Alpert) is indeed dependent on the nature of the gas, as a consequence of its own ionization cross section for a given ionization potential. Usually, the gauge calibration is performed by the manufacturer with nitrogen. The gauge should be calibrated for other studied gases. This can be simply done using the relative ionization cross sections given in tables, but also calibrations against non specific gauges such as capacitance manometers can be performed. In the latter case, the pressure ranges overlap around 10-2-10 -5 Torr. Expanding gas from a small volume where pressure is measured with a capacitance gauge to a larger volume and using the law of perfect gases, allows a calibration of the ionization gauge down to the 1 0 - 6 1 0 -7 Torr pressure range. In any event, the
552
J. Suzanne and J.M. Gay
accuracy is typically 20% in pressure measurements. If the gauge is linear, this uncertainty does not affect appreciably the value of the slope of the ClausiusClapeyron line and consequently the determination of the heat of adsorption. Another precaution concerns the fact that the uhv chamber is usually not in thermal equilibrium with the substrate. The chamber is at room temperature Tr whereas the substrate is at low temperature Ts. In order to take this situation into account, a thermomolecular correction should be applied (Edmonds and Hobson, 1965; Diehl and Fain, 1983) to the reading of the pressure gauge P(measured) that is: (10.26)
P(corrected)= P(measured)" " XT-2"s ~ l r
Equation (10.26) assumes perfect accommodation of the gas on the surface, e.g. a sticking coefficient equal to one whatever is the gas temperature. If one takes into account the sticking coefficients Sr and S, for the molecules at room and substrate temperature respectively, it has been shown that the pressure should be further multiplied by the ratio Sr/S, (Golze et al., 1986). If the substrate temperature is low enough, the equilibrium pressure of the adsorbed layer may be too low to be measured, that is below 10-~~ Torr. Still, it is very useful to foilow the decrease of intensity of a substrate spot versus time in order to control the coverage change upon adsorption. The completion of a monolayer is usually clearly indicated by a knee in the curve followed by a plateau where the decrease of intensity is much slower. Also, phase transitions taking place within monolayer coverage give a substep in the plateau. An illustration is presented in Fig. 10.26 for ethane adsorbed on graphite (Gay et al., 1986b).
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2.6x 10 -9 torr
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curve represents the variation of the graphite (01 ) spot intensity as a function of time at constant ethane pressure (p = 2.6x10-9 Torr) and temperature (T = 50 K). Electron energy is 138 eV. The arrows indicate the onset of visual LEED observation of S ~, $2 and $3 2-d solid phases (see phase diagram in Fig. 10.3). (From Gay et al., 1986b).
The structure of physically adsorbedphases
553
Fig. 10.27. LEED patterns of ethane monolayer adsorbed on graphite at T = 50 K. The S~ (a), $2 (b) and $3 (c) solid phases observed along the kinetic isotherm of Fig. 10.26. Electron energy is (a) 142 eV, (b) 90 eV and (c) 142 eV. (From Suzanne et al., 1983).
The LEED superstructure due to the adsorbate can be monitored all along the isotherms and related to equilibrium pressures or exposure time. Figure I0.27 shows the LEED patterns of ethane monolayer on graphite observed at different exposure times along the kinetic isotherm of Fig. 10.26. Each LEED pattern corresponds to a given solid phase called S~, $2 and $3 (see Fig. 10.3). Their structures are discussed in w 10.3.1. R H E E D (Reflexion High Energy Electron Diffraction). The capabilities of this technique have been described by Lagally (1985, 1988), particularly for diffraction from disordered surfaces. The electron energy range is 10-50 keV and the beam hits the sample surface at grazing incidence; e.g., 0.5-3 ~ from the surface. Although the energy is large, the penetration is only a few layers in the direction perpendicular to the surface which makes RHEED a surface sensitive technique. Since the wavelength of the electrons is about 0.1/~,, the Ewald sphere has a large radius. This causes the diffraction pattern of a very flat surface or of a uniform monolayer to consist of streaks rather than sharp spots. These streaks are the intersection of the 2-d reciprocal lattice rods with the almost flat Ewald sphere (Passell et al., 1986) (see Chapter 7 for a more complete discussion of the Ewald sphere). It should be mentioned that the streaks are observed because the coherence length of the crystalline surface (long range order) is finite and also because there is an instrumental broadening due to energy spread, angular divergence of the electron beam, etc. (Lagally, 1985, 1988). In the ideal case of a perfect crystal and perfect instrument, the RHEED pattern consists of spots. In any case, transmission and reflexion patterns can be distinguished by the respective arrangements of diffracted beams (Lagally 1988). For thicker layers, the reciprocal lattice rods become modulated in the direction perpendicular to the surface. This gives rise to modulated streaks, or cigar-shaped spots on the observation screen. In most of the cases with real surfaces, streaks are observed when the surface is flat and when a uniform flat film is adsorbed on the surface. If 3-d islands are
554
J. Suzanneand J.M. Gay
formed, spots appear on the screen due to transmission diffraction. These features of RHEED have been used in investigations of the mode of growth (see w 10.6.1) of physisorbed films on graphite surfaces: rare gases and nitrogen (Seguin et al., 1983; Bienfait et al., 1984; Venables et al., 1984), CF4 (Venables et al., 1984; Gay et al., 1988), CH4 (Krim et al., 1986) and ethane (Gay et al., 1988). The RHEED pattern from a thick xenon film adsorbed on graphite is shown in Fig. 10.36 of w 10.6.
THEED (Transmission High Energy Electron Diffraction). It is only recently that transmission high energy electron microscopes with uhv environments are available. Nevertheless, the physisorption of rare gases on graphite has been investigated by this technique since 1976 (Venables et al., 1976) and the studies have continued up to the present with great improvements. First, the experiments were possible because graphite is an inert surface. Second, many precautions were taken in the experimental set up and sample preparation: - The graphite sample was heated to 1550~ prior to the adsorption experiment - uhv was locally obtained by cryopumping at the sample stage -impurities were prevented from being deposited onto the graphite surface by the electron beam by letting the beam pass through a formvar film 500/~ thick, then through a series of cooled apertures. With the precautions mentioned above, THEED has been successfully applied to determinations of the structure of the xenon (Faisal et al., 1986; Venables et al., 1988; Hamichi et al., 1989, 1991) and krypton (Venables and Schabes-Retchkiman, 1977; Schabes-Retchkiman and Venables, 1981) monolayers and the growth mode of thin films of these adsorbates (Price and Venables, 1975; Kramer, 1976a). Accurate measurements of the commensurate-incommensurate transition of xenon monolayer and misorientation of the incommensurate phase versus misfit have been achieved. Analysis of the spot shapes could be performed using kinematic diffraction calculations. It was concluded that threading dislocations and associated disorder in the domain-wall network are both present in the incommensurate xenon monolayer (Hamichi et al., 1991). AES (Auger Electron Spectroscopy). Auger electron spectroscopy allows qualitative and quantitative chemical analysis of surface layers. AES is a three electron process. The sample is bombarded with high energy (3-30 keV) electrons (primary electrons). This results in the ionization of the material due to the emission to secondary electrons from core levels (energy E~). Electrons from more external levels (energy Ez) then fall on the level Ej giving the energy EI-E 2. This excess energy is responsible for the emission of a third electron, called Auger's electron, from the energy level E3. The kinetic energy of the Auger's electron is E = E~ - E 2 - E3; it is characteristic of the electron energy levels and therefore of the chemical composition of the material. The intensity of the Auger transition characteristic of the adsorbate can be used to measure the amount of adsorbate on a substrate surface. AES adsorption isotherms have been reported for xenon (Suzanne et al., 1974) and krypton (Kramer and Suzanne, 1976b) monolayers adsorbed on graphite single crystal surfaces.
The structure of physically adsorbedphases
555
However, it should be mentioned that the electron beam intensity used in AES measurements which is > 100 nA is large enough to perturb appreciably most of the physisorbed species. The heaviest rare gases are probably the only adsorbates which allow the use of AES for measuring equilibrium adsorption isotherms. Molecular adsorbates such as methane or other hydrocarbons would suffer electron stimulated desorption or dissociation. 10.4.3.2. A t o m scattering
Light atom (H, He) beam scattering has become a powerful tool for investigations of the structure and dynamics of adsorbed layers. Major characteristics of He scattering are given in Chapter 7. Therefore, we mention only some particular properties of this technique that make it well suited to investigations of physisorbed layers. First, the low energy of the He atoms and their inert nature ensures that He scattering is a completely nondestructive probe, especially with delicate phases, like physisorbed layers. A thermal He nozzle beam has a wavelength (1.09-0.46 ,A,) comparable with the interatomic distances in the adsorbed layers and is therefore well suited for diffraction studies. In addition, the energy of the He atoms (17-100 meV) is in the same range as those of collective excitations in overlayers. The advantage of He scattering over inelastic neutron scattering (w 10.4.2.2) is in the use of single crystals and thus the capability to measure phonon dispersion curves in different crystallographic directions. An example of time-of-Ilight He spectrometer is shown in Fig. 10.28 (David et al., 1986). In such a set-up, the momentum resolution is about 0.01 A,-~ for ~ = 1 A,; this resolution is sufficient for most usual qualitative discussions. Energy resolution can be about 0.4 meV FWHM, for 20 meV He beam energy, but it can be brought to ~0.1 meV with lower beam energies. Due to the large atomic scattering cross-section (e.g., >110 /~2 for HeXe/Pt(l 11) at 18 meV after Poelsema et al. (1983) atom scattering is sensitive to very low coverages. This property is used for detecting impurities, like hydrogen or for studying the gas ---> (2-d solid+gas) transition in adsorbed layers. In a way similar to LEED isotherms, the attenuation of the specular beam upon adsorption changes at the onset of 2-d islanding. The coverage dependence of the attenuation is used to determine the lateral interaction energy between the adatoms (Poelsema et al., 1983). Figure 10.29 shows the specular intensity reflected from a Pt(111) surface upon condensation of Kr. Like any diffraction technique, H or He coherent elastic scattering allows determination of the real space lattice and, in particular, the symmetry and orientational epitaxy of adsorbed layers (Ellis et al., 1981; Chung et al., 1987; Kern et al., 1988). The power of the He diffraction technique is illustrated by studies by Kern et al. (1988). They investigated in detail the Xe monolayer physisorbed on Pt( 111 ). Using a well ordered defect-free single crystal surface, they identified unambiguously commensurate ---->striped incommensurate ---->hexagonal aligned incommensurate ----> hexagonal rotated incommensurate transitions with increasing incommensurability. Figure 10.30 shows the (1,2) diffraction lineshape for incommensurate Xe layers. The intensity analysis involves calculations of the elastic diffraction probability based on the He-adsorbed Xe layer potential (Ellis et al., 1981; Schwartz, 1987).
J. Suzanne and J.M. Gay
556
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k
I
;i L,-,3
/
I\ C
S
Scattering chamber
Fig. 10.28. Schematic view of a high resolution He time-of-flight spectrometer used tar physisorbed layer studies. N is the nozzle beam source, S I and $2, and A I ..... A5 are skimmers and apertures, respectively. Gas is introduced by the gas doser (G) in front of the sample (T). (CMA) represents a cylindrical mirror analyzer for Auger spectrometry. The chamber is also equipped with an ion gun (IG) and a LEED (L). The scattered beam is detected by a quadrupole mass analyzer with channeltron (QMA). (C) is a chopper. (From David et al, 1986).
Ts- 54 K
y
0.96 o
!1
0.92
0.88 1
0
I 1 L, L I t &O 80 120 160 exposure time (secl
Fig. 10.29. Specularly reflected He beam intensity from a Pt(l 11) surface upon exposure of Kr (PK,- = 2. I x 10-9 Torr) at 54 K. (From Kern et al., 1988).
The structure of physically adsorbed phases
557
Fig. 10.30.3-d plot of the (1,2) He diffraction peak of incommensurate Xe layers on Pt(l 11) at various incommensurabilities corresponding to (a) a striped phase, (b) coexistence of a striped phase and a hexagonal aligned_phase and (c) a hexagonal rotated phase. Q denotes the reciprocal lattice vector in the F' K direction, while q~denotes the azimuthal angle. (From Kern, 1987a). Surface lattice dynamics (see w 10.1.2.2) has been studied by inelastic He scattering. The change of dynamical behavior from the monolayer to thicker layers is informative of the growth mode of the adsorbed films. Figure 10.31 shows the phonon dispersion curves of Kr overlayers (1, 2, 3 monolayer thick) on Ag(l 11). Very detailed studies can also be performed to reveal the adlayer-substrate dynamical coupling (Zeppenfeld et al., 1990a). The attractive part of the He-surface interaction potential is usually neglected (hard wall approximation), since it changes only weakly the corrugation shape of a surface. It is, however, related to the resonant scattering effect or selective adsorption. These resonances appear in azimuthal scans of the He specular reflection intensity. An example has been recently given by Jung et al. (1989) for the orientational configuration of the methane molecule adsorbed on MgO (100), see Figure 10.32. The resonance lineshapes allow the determination of the binding energies.
10.4.3.3. X-ray diffraction X-ray diffraction from layers physisorbed on single crystal substrates benefits from the advantages of X-ray scattering (see Chapter 7). The use of simple kinematic scattering theory for quantitative interpretation of the diffraction lines makes the technique particularly attractive. The high resolution of the experimental set-up enables accurate investigations, specially by the use of synchrotron radiation X-ray sources, although conventional sources can also provide sufficient flux to allow
558
J. Suzanne and J.M. Gay
I
-
v
I
v , "
I
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_
"
.
.
.
d~V
.
r
A
z
1 8UL
a
F
O. 25
13. 5
O/Omax
O. 75
I
M
Fig. 10.3 !. Phonon dispersion curves of mono-, bi-, and trilayers of Kr overlayers on Ag(l 11) from inelastic He scattering measurements. The dispersion curve of bulk krypton is also reported. (From Gibson and Sibener, 1985). valuable experiments on single crystals. The experimental studies using powdered substrates (see w 10.4.2.3) are more easily tractable, but they suffer from the lack of direct orientational structural information. Only single crystal studies can provide unambiguous results about the orientational structure and epitaxy of physisorbed layers. Single crystal studies require a surface substrate that is easy to clean and that can be made with relatively large uniform regions. Natural graphite (Specht et al., 1987; Hong et al., 1989) or silver single crystal (Greiser et al., 1987) have been used to date. The surface area is much smaller than in powder experiments, so that the surface coverage can dramatically change with a weak variation of pressure in the cell. This difficulty is overcome by putting a mass of powdered material in the cell that acts as a stabilizing ballast (Hong et al., 1989). Great care should be taken with the cleanliness of the surface. A common experimental set-up consists of a uhv chamber with a device allowing periodic cleaning; e.g., thermal flash, sputtering, etc. The sample is oriented in a way that the scattering vector lies nearly in the surface plane. This grazing incidence geometry yields a lower background level and higher peaks than a transmission geometry, as used with powders. The experimental studies of layers physisorbed on single crystals, reported to date in the literature, essentially deal with Kr (Specht et al., 1987) and Xe (Hong and Birgeneau, 1989; Hong et al., 1989) on graphite and Xe on silver (Greiser et
559
The structure of physically adsorbed phases
I
I
I
I
I
c
El GEl A Fi/;?~ A~'.b B
M/ V ~'" 'q/~a8 =71.5~
A
/
c=.
^
..
\
70.5
~
03
c--.
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>.. F--cO
Z W
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z_ C15.
6.5 ~
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1
-30 -15
!
I
0 15 ~ (deg)
1
30
Fig. 10.32. Azimuthal scans of the He specular intensity from monolayer CH4 on MgO, at different fixed incident angles 0. The capital letters label resonances. (From Jung et al., 1989).
al., 1987). The specificity of the single crystal experiments is used for orientational structural studies; e.g., rotated incommensurate phases and hexatic phases. Figure 10.33 is a comparison of data from a xenon monolayer on powdered and single crystal graphite substrate. The single crystal study clearly shows the sequence of Xe adsorbed phases: rotated incommensurate solid ~ reentrant aligned incommensurate solid ~ commensurate solid (Hong et al. 1989). The angular o-scans around the graphite <110> direction show the rotation of the Xe monolayer as function of temperature. The ('~-x#-3-)R30 commensurate phase is observed below 60 K, as deduced from the peak positions c o - 0 ~ and Q - 1.70 ~-~. At high
560
J. Suzanne and J.M. Gay
b
i i9
:OoOOF 3~176176 IK
o
:
119.12 K
20C
1
I
I
o
J
c- 1000f 0
119.12K
loc J
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79.78K
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_
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I
I
~
0
'
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,
i
,
l
,
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I
I
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,
!
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I
i
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'59.
'
50-
~ 1.6
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,
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~~176
I
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0
---
1__1__
1.62
~_l~_J___a_L_s
1.66 Q(A')
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-0.4
0 0.4 ( degl ees )
0.8
Fig. 10.33. Comparison ot'diffraction lines from 1.16 layers of xenon adsorbed on vermicular (a)and single-crystal (b) graphite substrates. Azimuthal to-scans are significant only on a single-crystal. The commensurate structure ('~-3-x~/3-)R30is characterized by Q = 1.70 ~-~ and the azimuthal angle to = 0 ~ (From Hong et al., 1989). temperatures, the xenon monolayer is incommensurate (Q = 1.62 ,/k-~) rotated (co = + 0.6~ The rotation angle and the lattice parameter decrease continuously as the temperature is lowered. At 79.78 K, Q = 1.67/~,-~, the rotation angle goes to zero; then the realigned incommensurate phase appears. At comparable temperatures, the diffraction lines from the powder experiment are consistent with the single crystal study. Their typical sawtooth shape has been already explained in w 10.4.2.2 in the similar case of neutron scattering. The power of the single crystal technique is nicely demonstrated by this example. Besides the surface X-ray diffraction experiments that investigate the projection of the reciprocal lattice onto the surface, other studies were dedicated to measuring the Bragg rod profiles (Hong and Birgeneau, 1989). These latter experiments provide information about the growth of multilayer films.
10.4.3.4. Ellipsometry Ellipsometry is a non-perturbative tool for studying physisorbed layers. It measures changes in polarization upon reflection of a light beam from a surface. Its applicability has been considerably improved by the availability of uhv techniques and by
The structure o.]physically adsorbed phases
561
the computerized automation of the measurements. The principle of the technique has been described in numerous papers (Quentel et al., 1975; Bootsma et al., 1982; Nham and Hess, 1989). Experimental details have been given by Jasperson and Schnatterly (1969) and by Volkmann and Knorr (1989). Adsorption isotherms can be measured through the change of the ellipsometric parameters versus temperature at constant pressure: - A, the phase change between the incident and the reflected beam, - p, the ratio of the amplitude reflection coefficients for incident light linearly polarized parallel and perpendicular to the plane of incidence, i.e., s and p polarization. The Fresnel theory for reflection by plane parallel continuum layers can be used to relate these ellipsometric quantities to substrate and overlayer properties. Particularly, the surface coverage can be measured with a sensitivity of the order of 0.01 monolayer (Quentel et al., 1975; Bootsma et al., 1982). A great advantage of the technique lies in the weak interaction of the light with atoms or molecules which produces a negligible influence on the adsorbed monolayer. In situ measurements at high gas pressures are possible which gives the capability to explore the whole 2-d phase diagram on single crystal surfaces. Experiments have been performed on various adsorbed layers on graphite: xenon (Quentel et al., 1975; Faul et al., 1990), krypton (Faul et al., 1990; Nham and Hess, 1989), argon (Faul et al., 1990; Nham and Hess, 1989; Youn and Hess, 1990a), oxygen (Drir and Hess, 1986a; Youn and Hess, 1990b), nitrogen (Faul et al., 1990; Volkmann and Knorr, 1991), methane (Nham and Hess, 1989), tetrafluoromethane CF 4 (Nham et al., 1987) and ethylene (Drir and Hess, 1986a). Most of the recent experiments have studied multilayer adsorption of these adsorbates (see w 10.6.1) and have shown that ellipsometry is a powerful tool for studying growth modes and wetting behavior on single crystal surfaces. 10.5. R e v i e w o f e x p e r i m e n t a l results
Table 10.4 reports most of the physisorbed systems experimentally investigated in the monolayer range. It is certainly not exhaustive and the reader is invited to refer to the references given in the papers that are here mentioned. Some of the systems have been extensively characterized; only a few properties are known for others. The authors' intention is to provide the essential references on the experimental work available in the literature. For each system, the structural information indicates the presence of commensurate, incommensurate, etc. phases without any specification about their domain of existence. Similarly, the melting behavior lists only whether the transition is first order or continuous and does not include possible changes with coverage within the monolayer range. 10.6. T h e s t r u c t u r e o f m u l t i l a y e r f i l m s
While studies of physisorbed monolayers continue, recent years have seen a shift to multilayer films. The role of the third dimension of these systems, that is not equivalent to the two in-plane dimensions, has attracted the interest of many
562
J. Suzanne and J.M. Gay
theorists and experimentalists. The physics of molecular films evolves from two to three dimensions as the films thicken. The systematics of the 2-d ~ 3-d transition relies on how the bulk gas-solid (T < T~d) or gas-liquid (T~d < T) coexistence is approached. New phenomena emerge and they evolve, with increasing thickness, to become the surface transitions of bulk matter. Adsorbed multilayers are indeed useful as test systems for studies of some bulk surface properties such as surface melting and surface roughening. The multilayer behaviour is related to wetting. Excellent reviews of multilayer physisorption have been recently done by Dash (1985, 1988), Dietrich (1988), Schick (1990) and Hess (1991). 10.6.1. From 2-d to 3-d; wetting
The various growth modes can be classified as: (i) "complete wetting" or type 1; at low temperature, the adsorbed film deposits as a succession of distinct layers, detected in a stepwise adsorption isotherm. The layer-by-layer deposition continues ad infinitum as the equilibrium pressure rises to the bulk vapor pressure, i.e., the film grows asymptotically to bulk. This is shown in Fig. 10.34. (ii) "Incomplete wetting" or type 2; the thickness of the adsorbed film is limited until vapor pressure of bulk saturation is reached, at which point any additional adsorbate condenses into bulk crystallites or droplets. (iii) "Nonwetting" or type 3 is the particular case with a zero layer limited thickness. The wetting behavior was studied systematically by Pandit, Schick and Wortis (1982). This work presents a valuable classification scheme for a comprehensive picture of wetting. It relies on the relative strengths of the adsorbate molecule-substrate potential u and the adsorbate-adsorbate molecular potential v. It was theoretically predicted that complete wetting might be expected for sufficiently large u/v, whereas smaller u/v ratio would induce incomplete wetting. Experimental studies of various films adsorbed on graphite (Bienfait et al., 1984) showed reentrant incomplete wetting on the high u/v side, at low temperatures. Complete wetting would be therefore limited to a range of intermediate u/v ratios. Other theoretical works (Muirhead et al., 1984; Gittes and Schick, 1984; Huse, 1984b) have introduced substantial changes in the former theory. In particular, they consider the effect of structural mismatches between the first monolayer of solid adsorbate and any plane of the bulk crystal. The continuity of the growth depends strongly on how the bottom layers can restructure. Even if the lattice structure and molecular orientations in the monolayer and bulk solid are compatible, a strongly attractive substrate (large u/v) will compress the few first layers inducing stress energy that can prevent complete wetting. Complete wetting by a liquid film is much more common than by a solid film. Indeed, numerous systems exhibit a triple point wetting transition characterized by incomplete wetting of a solid film for T < T~d, but complete wetting by liquid for T > 7',3d. The thickness dc of the film versus temperature at coexistence follows a power law (Krim et al., 1984) i
d~
,
(10.27)
The structure of physically adsorbed phases
563
77.3 K
,.,,! ILl r,r'
ILl 0 0
!
1
0.2
!
1
0.4
I
I
0.6
I
0.8
1
P/Po .Fig. 10.34. Volumetric adsorption isotherm of krypton adsorbed on graphite at 77.3 K. P,, is the bulk vapour pressure ( 1.75 Torr) at 77.3 K. The layer-by-layer condensation causes the stepwise isotherm, up to 5 layers. (From Thorny and Duval, 1970a). The exponent (-1/3) has been predicted for long range interactions which is the case for van der Waals systems. Critical wetting when the temperature is increased at three phase coexistence has also been considered theoretically (Dietrich and Schick, 1985). In this case the thickness of the film at coexistence varies as
dc= (Tw- T)-'
(10.28)
which shows that complete wetting should occur at Tw the critical wetting temperature. Hess (1991) has summarized the wetting behavior of various adsorbates on graphite (see Table 10.7). Most of the experimental work has investigated wetting on graphite (see reviews by Suzanne (1986) and Hess ( 1991 )). Studies on gold ( 111 ) (Krim et al., 1984) should also be mentioned as well as on Ag (111) (Gibson and Sibener, 1985; Suzanne, 1986), Pt (111) (Kern et al., 1986b), Pd (Miranda et al., 1984) and MgO (100) (Gay et al., 1990).
10.6.2. Layering and surface roughening Adsorption of a thick film may proceed layer-by-layer (first order condensation) or continuously with increasing vapor pressure. The ellipsometric isotherms of argon/graphite reported in Fig. 10.35 (Youn and Hess, 1990a) illustrate first order and continuous condensation with the particular situation of reentrant first order layering transitions in this system. For each layer, a critical temperature is estimated
J. Suzanne and J.M. Gay
564
Table 10.7 Wetting properties of various adsorbates on graphite.The relative strength of the adsorbate-substrate and adsorbate-adsorbate interactions is measured by u/v. The wetting modes 1 or 2 are defined in the text. Parentheses indicate uncertain interpretations or extrapolations. (From Hess, 1991) Adsorbate
u/v
H20 CO 2 C2H 4 CF 4 O2 CzH 6 Xe Kr Ar N2 Ne CH 4 CO H2
Wetting mode
0.31 0.46 0.42-0.79 0.73 0.93 0.99 1.04 1.17 1.23 1.32 1.39 1.51 1.55 4.6
Low-T solid
Higher-T solid
Liquid
2 2 2 2 2 1 1 1 2 2 1 2 I
(1) 1 1 1 1 -
2 1 1 1 1 1 1 1 1 (1) 1 I (1)
. m
c'-
5
7'
t 0.8
t
>.., 'L
.,Q ,< v
79.6
K
72.7
K
68.8
K
66.9
K I
4 0.4
I
[ ~- [IJ,,o[ -1/3
I 1.2
(K) -1/3
Fig. 10.35. Ellipsometric isotherms of argon condensed on graphite. The ellipsometric signal is reported versus Ila-l.to1-1/3, where l.t and ~ are the multilayer chemical potential and the bulk chemical potential, respectively. Sharp layering transitions are observed up to 67 K and around 73 K (reentrant first order layering transitions). Smooth isotherms are indicative of continuous condensation. ( F r o m Youn and Hess, 1990a).
The structure of physically adsorbed phases
565
above which there is a continuous adsorption of the layer. The layer critical points may fall either below or above the bulk melting point. This is a major determinant of the character of multilayer film growth. This results, in the former case, in the existence of a range of temperatures below the melting point where the growth of solid films is continuous whereas layered liquid films may exist in a range above the melting point, in the latter case. For wetting solid films, it has been theoretically demonstrated that the sequence of the layer critical temperatures converges to the roughening temperature of the bulk adsorbate material (Nightingale et al., 1984). At the roughening transition of a semi-infinite crystal, thermally excited steps and kinks disorder the surface (see Chapters 2 and 13). The same can also occur in adsorbed multilayers when the surface of the film is delocalised, i.e. for temperatures above the critical points. It is noteworthy that the substrate field limits the surface disorder, except for very thick films. To date, the experimental studies of layering and surface roughening of thick films are based on measurements of the layer critical points, essentially for various adsorbates on graphite (Hamilton and Goodstein, 1983; Kim et al., 1986b; Larher and Angeraud, 1988; Nham and Hess, 1988; Zhu and Dash, 1988; Youn and Hess, 1990a).
10.6.3. Surface melting The surface melting phenomenon consists in the appearance of a liquid-like disordered surface layer on a solid in equilibrium at temperatures below the bulk melting point (Dash, 1989; Nenow and Trayanov, 1989). Like surface roughening, surface melting is basically a surface transition of a semi-infinite crystal (see Chapter 13). This phenomenon may be viewed as wetting of the crystal by its melt. Mean field theory has been developed predicting the occurrence of surface melting (Trayanov and Tosatti, 1988; Dash, 1988). Theoretical models also show a power law temperature dependence of the quasiliquid layer thickness for long range forces (van der Waals systems, for instance). Thick physisorbed films have been extremely valuable for surface melting investigations. As for surface roughening, the substrate field may hinder the surface disorder. In particular, substrate-induced freezing has been observed above the bulk melting point. This is an effect of the substrate field which stabilizes a solid layer at the substrate interface. Surface melting in thick solid films has been extensively studied in krypton, argon and neon films on graphite (Zhu and Dash, 1988; Gay et al., 1991; Pengra et al., 1991), in oxygen on graphite (Krim et al., 1987), in methane films on MgO (Bienfait et al., 1988; Gay et al., 1990) and on graphite (Bienfait et al., 1990; Gay et al., 1992), and in deuterium hydride HD on MgO (Zeppenfeld et al., 1990c).
10.6.4. Experimental Most of the experimental techniques described in w 10.4 and used for studies of physisorbed monolayers, have been employed in investigations of thick films as well. The quartz microbalance has been more particularly used in the latter cases. Its characteristics are briefly reported at the end of this section.
566
J. Suzanne and J.M. Gay
Fig. 10.36. RHEED patterns of xenon films adsorbed on graphite around 50 K. (From Venables et al., 1984). (a) Bilayer film giving rise to uniform streaks. (b) Uniform wetting 10 layer thick film responsible for the streak modulations.
The first evidences of film growth have been provided by volumetric isotherm measurements (Thomy and Duval, 1970a; Menaucourt et al., 1977; Ser et al., 1989). The resolution is limited to 4-5 layer thick films, above which the actual equilibrium pressure of the thick film becomes undistinguishable of that of bulk. In addition, as for all classical tools requiring powdered materials (calorimetry, neutron scattering, etc.), the volumetric technique is severely limited by capillary condensation; it has been observed in graphite and MgO powders beyond 4-5 layers (Set et al., 1989; Larese et al., 1989, Gay et al., 1990). This effect may completely hide or modify intrinsic multilayer phenomena (wetting, surface melting, etc.). Experimental techniques using single crystal substrate surfaces overcome this problem. Among them, LEED and RHEED (Venables et al., 1984; Krim et al., 1986; Gay et al., 1984, 1988), elastic and inelastic helium scattering (Gibson and Sibener, 1985; Kern et al., 1986b), ellipsometry (Nham and Hess, 1989; Faul et al., 1990) and quartz microbalance (Krim et al., 1984) are the most widely used techniques in multilayer studies. Figure 10.36 shows typical RHEED patterns for a xenon film deposited on graphite around 50 K; the uniform streaks for the bilayer evolve to strongly modulated streaks for a 10 layer thick film, indicating a uniformly wetting thick film.
Quartz micro-balance. High frequency microbalances have been used as thickness monitors for deposited thin films or to measure the amount of gas adsorbed on metal surfaces (Lu and Czanderna, 1984). The resonant frequency of a crystal oscillating in its thickness shear mode is lowered by mass loading. The frequency shift is proportional to the mass deposited. The sensitivity is adequate to detect coverage changes of less than one monolayer (Krim et al., 1984). The metal substrate also serves as the oscillator electrode. It is first evaporated onto the quartz. The result is not a single crystal surface. In the case of gold, it is polycrystalline with (111) faces parallel to the quartz substrate. Adsorption isotherms have been measured for various gases (rare gases and light molecules) onto this surface (Krim et al., 1984) to determine the growth habits of multilayer films of these molecules.
The structure of physically adsorbed phases
567
It has been shown that triple point wetting takes place. The thickness of the film at coexistence obeys Eq. (10.27). 4He films on Ag and Au(l 1 1) surfaces have also been studied by this technique (Migone et al., 1985). The results show incomplete wetting with a thickness at coexistence dc which varies according to Eq. (10.28). 10.7. Conclusion Physically adsorbed phases have provided the physico-chemist with a good understanding of the matter in reduced dimensionality. Unlike chemisorbed systems, physisorbed films can be studied under true thermal equilibrium because of the weak interactions. In the monolayer range, the adsorbed phase has a quasi 2-d behavior. Phase diagrams featuring 2-d gas, 2-d liquid and 2-d solid phases with coexistence regions, triple point and critical points have been constructed as in bulk. Commensurate or incommensurate solids are observed within the monolayer depending on the relative lattice mismatch between the adsorbate and the substrate. Furthermore, 2-d polymorphism can be associated with different orientational orders of molecules above the surface. Various first order and continuous phase transitions have been identified: commensurate-incommensurate transitions, solid l-solid 2 transitions, order-disorder transitions and 2-d melting transitions. The mechanisms of these transitions have been analysed and the role of domain walls, dislocations and disclinations has been emphasized. Model calculations are an accessible challenge due to the simple adsorbate-adsorbate and adsorbate-substrate interactions involved. This special feature of physisorbed phases has allowed the modelling of the statics and the dynamics of the monolayers leading to a better understanding of the mechanisms governing the monolayer behavior. Experiments have been performed either with single crystal surfaces under uhv conditions or with well characterized, uniform powders under less restrictive conditions. Techniques and probes normally devoted to the study of bulk matter, like neutron or X-ray scattering have been used with these latter systems. Finally, physisorbed films with increasing thickness can be used to study the change in the properties of the matter when going from 2-d to 3-d. The analysis and modelisation of the growth mode of films or wetting behaviors is an important challenge in many aspects of thin film processing. A new phase transition, namely surface melting, has been inferred from thin physisorbed film studies, and here again, model calculations could be performed and compared to experiments to test the role of the various parameters. References Abraham, F.F., 1983, Phys. Rev. B 28, 7338. Abraham, F.F., 1984, Phys. Rev. B 29, 2606. Abraham, F.F., S.W. Koch and W.E. Rodge, 1982, Phys. Rev. Lett. 49, 1830. Angot, T. and J. Suzanne, 1991, in: The Structure of Surfaces III, S.Y. Tong and M.A. van Hove, eds. Springer Series Surf. Science 24, Springer, Berlin, p. 671.
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Audibert, P., M. Sidoumou and J. Suzanne, 1992, Surf. Sci. Lett. 273, L467. Axilrod, B.M. and E. Teller, 1943, J. Chem. Phys. 11,299. Bak, P., 1982, Rep. Prog. Phys. 45, 587. Bardi, U., A. Glachant and M. Bienfait, 1980, Surf. Sci. 97, 137. Bardi, U., S. Magnanelli and G. Rovida, 1986, Surf. Sci. 165, L7. Beaume, R., J. Suzanne, J.P. Coulomb, A. Glachant and G. Bomchil, 1984, Surf. Sci. 137, L117. B6e, M., 1988, Quasielastic Neutron Scattering. Hilger, Bristol. Bienfait, M., 1980, in: Phase Transitions in Surface Films. NATO ASI B51, J.G. Dash and J. Ruvalds, eds. Plenum, New York, p. 29. Bienfait, M., 1982, in: Dynamics of Gas-Surface Interactions. G. Benedek and U. Valbusa, eds. SpringerVerlag, Berlin, p. 94. Bienfait, M., 1987a, in: Dynamics of Molecular Crystals, J. Lascombe, ed. Elsevier, Amsterdam, p. 353. Bienfait, M., J.L. Seguin, J. Suzanne, E. Lerner, J. Krim and J.G. Dash, 1984, Phys. Rev. B 29, 983. Bienfait, M., J.P. Coulomb and J.P. Palmari, 1987b, Surf. Sci. 182, 557. Bienfait, M., J.M. Gay and H. Blank, 1988, Surf. Sci. 204, 331. Bienfait, M., J.M. Gay and P. Zeppenfeld, 1990, Vacuum 401,404. Birgeneau, R.J., P.A. Heiney and J.P. Pelz, 1982, Physica B 109/110, 1785. Bockel, C., J. Menaucourt and A. Thomy, 1984, J. Physique 45, 1391. Bohr, J., M. Nielsen, J. Als-Nielsen and K. Kjaer, 1983, Surf. Sci. 125, 181. Bootsma, G.A., L.J. Hanekamp and O.L.J. Gijzeman, 1982, in: Chemistry and Physics of Solid Surfaces IV, R. Vanselow and R. Howe, eds. Springer-Verlag, Berlin, p.77. Bouchdoug, M., J. Menaucourt and A. Thomy, 1984, J. Chim. Phys. 81, 381. Bouchdoug, M., J. Menaucourt and A. Thomy, 1986, J. Physique 47, 1797. Bouldin, C. and E.A. Stern, 1982, Phys. Rev. B 25, 3462. Brener, R., H. Shechter and J. Suzanne, 1985, J. Chem. Soc., Faraday Trans. I, 81, 2339. Bretz, M., 1977, Phys. Rev. Lett. 38, 501. Bretz, M., J.G. Dash, D.C. Hickernell, E.O. McLean and O.E. Vilches, 1973, Phys. Rev. A 8, 1589 Bruch, L.W., 1983, Surf. Sci. 125, 194. Butler, D.M., J.A. Litzinger, G.A. Stewart and R.B. Griffiths, 1979, Phys. Rev. Lett. 42, 1289. Caflisch, R.G., A.N. Berker and M. Kardar, 1985, Phys. Rev. B 31, 4527. Calisti, S., J. Suzanne and J.A. Venables, 1982, Surf. Sci. 115, 455. Campbell, J.H. and M. Bretz, 1985, Phys. Rev. B 32, 2861. Carneiro, K., L. Passell, W. Thomlinson and H. Taub, 1981, Phys. Rev. B 24, 1170. Ceva, T., M. Goldmann and C. Marti, 1986, J. Physique 47, 1527. Chan, M.H.W., A.D. Migone, K.D. Miner and Z.R. Li, 1984, Phys. Rev. B 30, 2681. Chesters, M.A. and J. Pritchard, 1971, Surf. Sci. 28, 460. Chinn, M.D. and S.C. Fain, 1977, J. Vac. Sci. Technol. 14, 314. Chung, S., A. Kara, J.Z. Larese, W.Y. Leung and D.R. Frankl, 1987, Phys. Rev. B 35, 4870. Chung, T.T. and J.G. Dash, 1977, Surf. Sci. 66, 559. Cohen, P.I., J. Unguris and M.B. Webb, 1976, Surf. Sci. 58, 429. Colella, N.J. and R.M. Sutter, 1986, Phys. Rev. B 34, 2052. Coppersmith, S.N., D.S. Fisher, B.I. Halperin, P.A. Lee, F. Brinkman, 1981, Phys. Rev. Lett. 46, 549; ibid. p. 869; 1982, Phys. Rev. B 25, 349. Coulomb, J.P., 1991, in: Phase Transitions in Surface Films 2, H.Taub, G. Torzo, H.J. Lauter and S.C. Fain, eds. Plenum, New York, p. 113. Coulomb, J.P. and M. Bienfait, 1986, J. Physique 47, 89. Coulomb, J.P. and O.E. Vilches, 1984a, J. Physique 45, 1381. Coulomb, J.P., J. Suzanne, M. Bienfait and P. Masri, 1974, Sol. State Commun. 15, 1623. Coulomb, J.P., M. Bienfait and P. Thorel, 1977, J. Physique C 4, 31.
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This Page Intentionally Left Blank
CHAPTER
11
Interactions Between Adsorbate Particles
T.L. E I N S T E I N Department of Physics University of Maryland College Park, MD 20742-4111, USA
Handbook of Su.rface Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
577
Contents
11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1.2. General features of lateral interaction energies . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1. Fundamental ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2. Electronic indirect interactions in simple tight-binding model . . . . . . . . . . . . . 11.2.2.1. M o d e l H a m i l t o n i a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
589 581 581 586 586
11.2.2.2. C a l c u l a t i o n of c h a n g e in o n e - e l e c t r o n e n e r g i e s using G r e e n ' s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2.3. S i m p l e r illustration: pairs on a ring 11.2.3.
....................
Multisite interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3.1. T h r e e - a d a t o m (trio) interactions . . . . . . . . . . . . . . . . . . . . . .
597 597
11.2.3.2. C o m p l e t e o v e r l a y e r s
598
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.4. Coulombic effects: self-consistency and correlation, and other improvements
11.3.
11.2.5.
Lattice indirect interactions: phonons and elastic effects
11.2.6.
Asymptotic form of the indirect interaction between atoms and between steps
Attempts to model real systerns
...............
Embedded cluster model
! 1.3.3.
Effective medium theory and embedded atom method - - semiempiricism
11.3.4.
Empirical schemes
599 601
....
.....................
11.3.2.
607 611 611
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
613 615 619
11.3.5.
Field-ion microscopy, modern tight-binding, and more on semiempiricism . . . . . .
620
11.3.6.
Scattered-wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
628
Further implications of lateral interactions
. . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1. Ordered overlayers and their phase Boundaries
11.5.
....
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1. Tight-binding, jellium, and asymptotic-ansatz
11.4.
588 592
....................
631 631
11.4.2.
Local correlations and effects on chemical potential . . . . . . . . . . . . . . . . . .
632
11.4.3.
Surface states on vicinal and reconstructed fcc(110) surfaces
635
Discussion and conclusions
.............
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
578
638 642
11.1. Introduction
Progress in computing the interactions between small numbers, even pairs, of chemisorbed atoms has been remarkably slow because of the very low symmetry of the problem. In contrast, the energetics of monolayers of adatoms, which have the full two-dimensional symmetry of the substrate, can now be characterized with impressive precision. However, even treatment of partially completed adlayers with (2xl) or c(2x2) symmetry doubles the size of the surface primitive cell, but quadruples the size of secular matrices, raising computer time requirements by a factor of order 43. At the other end of the scale, a single adatom (in a symmetric site) will at least have the point-group symmetry of the substrate. Associated with these symmetries are conserved quantities ("good quantum numbers") which make calculations simpler. As a result, a variety of elaborate many-body techniques have been applied successfully to these systems. For two adatoms on a surface, there is little or no symmetry, typically just a two-fold rotation or mirror plane (leading often to splittings of levels). Few systems have been treated in a satisfactory way. Sophisticated computations attempting to assess these interactions tend to resort to studies of ordered overlayers (Tom~inek et al., 1986). Desjonqu6res and Spanjaard (1993) signaled the difficulty of the problem by placing it as the final topic in their recent text. Reviews stressing various aspects of the problem have been presented by Einstein (1979a, 1991), Muscat (1987), March (1987, 1990), Braun and Medvedev (1989), Feibelman (1989a), and NCrskov (1993). This chapter will explore the many mechanisms by which chemisorbed atoms interact with each other. To set the stage early, it is useful conceptually to distinguish between direct and indirect interactions. Direct interactions would occur even if there were no substrate; they are, thus, sometimes called "through-space". Examples include van der Waals, dipolar, and electronic hopping (between the adatoms). The substrate, however, will generally provide at least some degree of perturbation. The alternative is indirect interactions, in which to lowest order there would be no interaction without the substrate. The coupling can be by electronic states (usually predominant), elastic effects, or vibrational coupling (usually insignificant). Since the coupling to the substrate is crucial, these are sometimes called "through bond". Special emphasis will be given to the indirect electronic ("pair") interaction between two light gas or transition series atoms on a (transition) metal. Moreover, we thoroughly explore a simple model of these interactions. The motivation is not so much to explain specific data but rather to give a theoretical framework in which to understand the relative magnitudes and qualitative behavior of the interactions. Without this sort of picture, it is difficult to make sense of the results that emerge from more realistic attempts to describe the adsorption systems. 579
580
T.L. Einstein
We also present a thorough summary of the methods that have been applied to make progress in understanding this general problem. As we fix ideas, it may be helpful to describe our problem in oversimplified terms by speaking of three characteristic energies: (1) Eat is the binding energy of an isolated atom to the most attractive site on the surface. Typically this is a high-symmetry site; e.g., on a square lattice, such sites (cf. Fig. 1.12) can be called A (atop, or linear, or on-top, the latter two inconsistent with the abbreviation), B (bridge, between two substrate atoms), and C (centered, or hollow, above the middle of a square). Identical terminology applies to substrates with triangular symmetry. Ed denotes the diffusion barrier, due to variations in the adatom-substrate potential, between adjacent most-favored sites. Usually this is a saddle point in a potential energy surface; e.g., if the C site is most attractive, one might expect Ed = E c - Ea, although substrate relaxation can sometimes lower this barrier significantly. The corrugation of the substrate potential provides an upper bound for Ed. Finally, E.,,, is the magnitude of the characteristic energy of interaction between nearby atoms. In physisorption, Ea, is comparable to E~, both being much greater than E0. (For dense overlayers, the actual diffusion barrier increases significantly due to adatom-adatom effects.) In contrast, in chemisorption E,~ > Ed >> E,a. Thus, in this ideal scenario, adatoms all sit in the most favorable site; their lateral interactions are relatively small. In this case it is fruitful to couch the discussion in terms of a classical lattice gas picture. (Cf. Chapter 13 by Roelofs.) In setting the stage for what we will find regarding electronic lateral interactions, it may be helpful to divide (somewhat artificially) the physics into a few regimes, depending on the separation between adsorbates. (1) In the near regime, the adatoms may be close enough to have non-negligible direct interactions. If not, they typically still "share" one or more substrate nearest neighbors, so that the bonding of one adatom to this substrate atom is strongly influenced by the presence of the second adatom. This regime is the most important for applications: in chemistry it determines the details of dissociative adsorption; in surface transport it enters problems of surface diffusion mechanisms. The strong impact of one adatom on the other may alter their binding sites, weaken bonding to the substrate, etc., as emphasized in a review by Feibelman (1989a). In short, in this regime Eaa may be comparable to E,.,. (2) In the intermediate regime, these effects fade and the lattice-gas approximation improves. The individual-adatom adsorption process is largely immune to the interactions. Most of the interesting physics can be isolated in the question of how the disturbance produced by a point defect at some position propagates to another. All the occupied band states in the substrate are involved in a complicated way. This regime is important in describing the formation of ordered fractional monolayers of adsorbates and in characterizing the chemical potential and the correlation functions of these adsorbates, even at higher temperatures at which there is little order. Thus, these interactions play a role in understanding thermal desorption spectra, vibrational line shifts, etc. (3) The asymptotic regime is reached when the adatoms are several spacings apart. The interaction is dominated by the substrate Fermi surface. Analytic expressions, albeit complicated, can be derived. Until recently, there was little evidence of experimental impact of
Interactions between adsorbate particles
5 81
this regime, but there may be implications for the interactions between steps on metal surfaces. We caution that as with most simple pictures of complicated phenomena, it is easy to point out ways in which the broad-brush rendition is oversimplified. For example, in the case of weak, non-directional bonding, the adatoms may slip out of high-symmetry binding sites even when a couple spacings apart, as suggested by Persson (1991) for some cases of CO adsorption. Moreover, for weak bonding, the Fermi-surface electrons may dominate at close spacings, inviting simple descriptions in terms of frontier orbitals (Hoffmann 1963, 1988). A major motivation for studies of pair interactions is to understand the origin of the wide variety of ordered overlayers at fractional coverages on metal surfaces (see Chapter 13). These have been tabulated by Ohtani et al. (1987), Van Hove et al. (1989), Watson (1987, 1990, 1992), and Watson et al. (1993). Consider a c(2z2), i.e. a checkerboard pattern on a square lattice (Chapter 1, Fig. 1.1). This pattern could arise simply because of a strong nearest-neighbor repulsion: E~ > 0 for an overlayer with about half the sites occupied. If additionally there is a next-nearestneighbor attraction, one finds islands of adatoms with c(2x2) symmetry at low temperatures and coverages. Sometimes these lie at temperatures so low that the equilibrium local configuration is not attained during the time of an experiment, at most several hours. But when islands are present, they provide strong evidence of an attraction. There are many more complicated phases. The explanation of most of their ground-state energies in terms of pair interactions is fairly obvious (cf. Suzanne and Gay, Chapter 10, and Roelofs, Chapter 13), but for troublesome cases, exhaustive tabulations have been published by Kaburagi and Kanamori (1974, 1978) and Kaburagi (1978). By attraction or repulsion here, we mean that the lattice site is favorable or unfavorable. There is no implication about the direction of a force acting on an adatom sitting in a lattice site; in the lattice gas picture, this force is assumed to vanish. In the near region, this assumption may often be questionable, but in the intermediate region it should be reasonable. Some experimental data on these systems appear in Chapters 10, 12 and 13 to which we shall refer. We will dwell mostly on theory. Progress in the field has come in the form of study of self-consistency and correlation effects (which seem to be less important than might be expected) and of multi-parameter-model attempts to describe real systems. Next we shall show in a single simple model how pair, three-adatom, etc., interactions combine to produce ordered overlayers. We will briefly consider changes in density of states (DOS) caused by two-adatom interactions from a similar viewpoint, and also show the more dramatic effects that arise when these combine to produce an ordered overlayer. In closing, we shall speculate on areas ripe for development.
11.2. General features of lateral interaction energies 11.2.1. Fundamental ideas
If chemisorbed atoms are sufficiently close to overlap each other, there will be a strong direct interaction. This interaction is essentially a chemical bond, compara-
582
T.L. Einstein
ble in strength to the chemisorption bond. (There is interesting physics in the degree to which these bonds are not simply the equivalent of bulk bonds. We shall explore these effects further in later sections.) For larger lateral interadatom distance R, the interaction falls off exponentially along with the overlap so that for R more than a few A, it is negligible. Most of the physics of this problem comes from the two adatoms and their substrate nearest neighbors; hence a cluster calculation can be appropriate. These interactions are important for supersaturated or even monolayercovered surfaces. They also arise in the problem of dissociation and reassociation of adsorbing molecules (e.g., the question of whether there is an activation barrier) which has been studied both schematically and in great detail. If the chemisorption b o n d involves charge transfer, electric dipole moments l.t will develop. Kohn and Lau (1976) showed that the non-oscillatory part of the dipole-dipole interaction energy on metals behaves as Edip_dip '~
2~all-ttCt4~EoR3
(11.1)
for large R. The novel aspect of this expression is the factor of 2, for which they give the following qualitative explanation: For either adatom, say a, ~ta is the product of the charge transfer qa to the adatom and the distance Za between the adatom and the surface/image plane, at which the induced charge of--qa lies. However, the potential experienced by a second adatom is determined by the first adatom and its image at-z~,, and so is 2~aZb(4~Eo)-lR-3. Hence, the work in bringing the second charge from z = +oo to z = Zb, and so Ed~p.d~~ contains the novel factor of 2. Inserting numbers, we find this interaction energy to be 1.25 eV times the two dipole moments in units of debyes divided by R 3 in A 3. N0rskov (1993) reviews the direct electrostatic interaction in some detail. The effect is generally larger for electropositive than electronegative adsorbates because the latter tend to bond closer to the substrate; consequently, they are better screened and so have a smaller dipole moment. For alkali adatoms, dipolar effects dominate the interactions which determine the 2D phase diagram (Bauer, 1983; Mtiller et al., 1989). Pre-adsorbed alkali-metal atoms increase both the binding energy and the dissociation rate of light gas dimers like CO, NO, N2, or 02 on metals, while preadsorbed electronegative atoms do the opposite: Typically adsorption of these dimers involves some charge transfer to them. (Back donation to the anti-bonding molecular orbital exceeds donation from the bonding orbital.) In the simplest approximation, the resulting energy is the product of the admolecule-induced dipole moment normal to the surface and the gradient of the electrostatic potential due to the preadsorbed atoms (or the extra charge times the potential itself). To support this picture, NCrskov et al. (1984/5) explored the form of this potential, for several different preadsorbed atoms on jellium, as a function of the height of the dimer above the surface. For the particular case of N2 on Fe( 111 ) with pre-adsorbed K, NCrskov (1993) finds an interaction of 0.08 eV, which can be used to account for most of the measured shift in adsorption energy due to predosing. The second-order correction, proportional to the square of the potential, is always attractive. Thus, for cases in which the charge transfer is from the dimer, he notes
Interactions between adsorbate particles
583
that the long-range interaction with a pre-adsorbed alkali can be repulsive while the short-range interaction is attractive. In addition to producing static repulsions, dipole-dipole interactions can raise the vibrational frequency of adsorbed molecules. For example, Scheffler (1979), using just dipole-dipole coupling, accounts for the coverage dependence of the shift of the C - O stretch frequency of CO on Pd(100) and Pt(l 11) measured by IR absorption reflection spectroscopy. In the process, he derives a coverage-dependent (as well as frequency dependent) effective polarizability which depends significantly on the distance of the dipole from the reference (image) plane of the metal (assumed to be jellium). In focusing on the cases of CO on Cu(100) and Ru(0001), Persson and Ryberg (1981) advanced the treatment of these questions by treating the adsorbate polarization as a single entity rather than trying to split it into admolecule and an image; they, furthermore, used the coherent potential approximation (CPA) (Soven, 1966) to consider interactions for a dense but not ordered overlayer. They find that the dipole-dipole interaction is enough to account for the coverage-dependent frequency shift for the Ru substrate, but that on Cu there is a counteracting chemical shift of nearly the same magnitude. NCrskov (1993) discusses the shifts of dimer vibrational frequencies due to interactions between pre-adsorbed (non-neutral) atoms and the dimer. The van der Waals interaction always produces a weak attraction between two adatoms and is the dominant contribution in the case of physisorption. The leading term is the dipole-dipole contribution, which goes as -C/R6; C in turn is proportional to the square of the polarizability. According to Hirschfelder et al. (1954), C is roughly 30 eV-~ 6 for Ar, N 2, and 02, and five times as great for Xe. For physisorbed gases, this mechanism dominates the interaction, and hence the details of the interatomic potential have been studied extensively. To fit gas-phase data, one must go beyond a simple R -12 Lennard-Jones repulsion (to some exponential description) to avoid overestimating C by nearly a factor of two. Two higher-order gas-phase effects are non-negligible: (1) The R -8 dipole-quadrupole force which increases the depth of the well-minimum by roughly 10% and (2) the repulsive (in all important cases) R -9 triple-dipole (Axilrod-Teller (1943)-Muto (1943)) interaction, the magnitude of which is at most 3% (for Ar) to 5% (for Xe) of a pair interaction if all distances are set at their equilibrium values. While this effect is of little concern here, there have been interesting applications (Klein et al., 1986). A variety of calculations of rare gas adsorption onto jellium (Sinanoglu and Pitzer, 1960), continuous dielectrics (McLachlan, 1964), Xe crystals (MacRury and Linder, 1971), and graphite (Freeman, 1975) all show that physisorption reduces the gas-phase pair attraction by roughly 20%. As an example of the state of the art in this refined subject, Barker and Rettner (1992) produce an accurate "empirical" (actually more semiempirical, in the language we will use later) potential for Pt(111)-Xe as a "benchmark." For the lateral interactions, they include, in addition to the van der Waals potential, the "nonadditive" McLachlan modification, the interaction of adsorption-induced and image dipoles, and the triple-dipole term, citing as reference Bruch's (1983) clear and comprehensive discussion of the significant contributions. (This classic review of lateral interactions in physisorption,
584
T.L. Einstein
as well as of the single-atom holding potential, provides an account of the general features of this problem that is evidently still timely a decade later. March (1987, 1990) presents more recent reviews. Vidali et al. (1991) have produced a useful compilation of potentials for physisorption.) The substrate-mediated dispersion energy is the largest contribution to the lateral interaction at the intermediate separations of ordered overlayers, accounting for slightly over half the (repulsive) corrections to the gas-phase interaction for two sample Xe overlayers (Bruch, 1983). The effect of the substrate on the interadatom interaction was first tackled using perturbation theory by Sinanoglu and Pitzer (1960). McLachlan (1964) calculates in second-order perturbation theory the interactions of adatom dipoles and their images in the substrate, including a frequency-dependent response for the substrate. Explicit expressions for the substrate-mediated dispersion energy and tables of the attendant coefficients are given by Bruch's (1983) review; a key issue is determining the distance of the adatoms from the image plane. Freeman (1975) approaches the problem using the Gordon-Kim (1972) version of density functional theory (for Ar adatoms) and obtains fair agreement with the preceding formalism; similarly, Vidali and Cole (1980) apply both methods to He on graphite. The next largest contribution to the substrate-related interaction, perhaps half the size of the preceding, is the interaction of adsorption-induced dipoles. The role of the surface was noted earlier in Eq. (ll.1). Bruch (1983) reviews the many contributors to this subject. He and Phillips (1980) showed how to compute these effects for an overlayer lattice. Other effects include triple-dipole (Axilrod-Teller (1943)-Muto (1943))interactions within the overlayer and changes in zero-point energy. In this framework, lateral interactions can be computed to an accuracy that makes those working on chemisorption truly envious. Nonetheless, there are some differences between calculations on particular systems, e.g. the above-mentioned benchmark (Mtiller, 1990; Gottlieb and Bruch, 1991 ; Barker and Rettner, 1992). To apply the van der Waals perspective to chemisorption, we can invoke the surface molecule picture to posit that the interaction between, say, two chemisorbed O atoms (coupled to their substrate neighbors) is similar to that between two O2 molecules (although now the molecules are oriented), i.e. roughly -25 eV/~6/(R[/~])6. At second and third neighbor separations on Ni(100), for instance, this yields an interaction o f - 1 3 meV a n d - 2 meV, respectively, which is usually negligible compared to the electronic indirect interaction. For heavy adsorbates (e.g., W or Re) these numbers could possibly be several times greater; no firm data exists. A curious application, to Ni(100)-O, of van der Waals ideas by Gallagher and Haydock (1979) suggested that by virtue of large overlap with the attractive Ni potential of the substrate, the O 2p orbitals become larger and far more polarizable, dramatically increasing the associated interaction. There has been little follow-up work on this viewpoint. The first proposal that adatoms might interact indirectly was made by Kouteck)5 (1958). The essence of this interaction is seen in Fig. 11.1, taken from the pioneering work by Grimley (1967) on this problem, which even now begins most discussions. Consider two atoms, each with an atomic potential producing some (relatively high-lying) bound state. In free space (and at moderate separation), each
Interactions between adsorbate particles
585
I I
Fig. 11.1. Classic schematic of the indirect interaction between pairs of adatoms. (a) Potential and wave thnctions Ibr two atoms in vacuum separated so far that there is no overlap and so no direct interaction. (b) The same atoms, now chemisorbed on a simple metal surface. From Grimley (1967a), with permission. of the bound-state wavefunctions will remain confined near its atomic site; the vacuum barrier is insurmountable. If, alternatively, they are adsorbed onto (or absorbed into) a metal, both atomic wavefunctions can tunnel through the narrow potential barrier to the metal and couple with propagating metal wavefunctions. Figure l l . l b shows how both atomic wavefunctions might couple to one such background eigenstate. If the coupling places the two atomic wavefunctions in (out of) phase, the interaction is attractive (repulsive), lowering (raising) the energy of the participants. From the oscillatory nature of the intermediate wavefunction, the electronic indirect interaction should be oscillatory in sign as a function of interadatom distance. It should be (two-dimensionally) isotropic if and only if the (surface of the) metal background is. Such isotropy is expected only for substrates which can be well approximated by free-electron or jellium models. Furthermore, the two adatomic orbitals can couple through not just one, but any of the occupied states (including surface states). As adatom separation increases, fewer substrate wavefunctions will match well with the atomic orbitals, causing a rapid decay in magnitude of the interaction energy. As discussed at the outset, our discussion assumes that Ed >> E~,, which should be a good approximation for strong chemisorption at low to moderate coverage. Under these circumstances, the most favorable adsorption sites will be filled or vacant, and when nearby sites are filled, the associated interaction energy will modify the total energy of the system. In this lattice gas picture, the Hamiltonian of the adatoms takes the form:
H=E, Zninj+E2 Z ninj+... (ij),
(6)2
(11.2) "~" Z ET Z llinjllk'Jr" Z EQ Z trlitlJ llkrlldr ''" T ((jk).r Q (ijkl)tI
Each site of the net of most-favored substrate sites (labeled i) can be occupied (ni = 1) or vacant (ni = 0). Here the pair interaction energies are denoted E,,, for mth
586
T.L. Einstein
neighbors; Ev is the "trio" (three-adatom, non-pairwise) interaction energy, with the index running over the possible trimer configurations; E o is the "quarto" energy; and so forth. For this formulation to be useful, the pair energies should fall off relatively rapidly in magnitude with increasing m, so that only a few need be considered. Furthermore, the multisite terms should be small; at worst, only a small number of the most closely spaced multiadatom terms should contribute. We shall see that the pair interactions do decay rapidly. The multisite terms are smaller but not always negligible. Moreover, there may be several different configurations with comparable magnitude. Nonetheless, cancellations typically occur such that the energies of ordered submonolayer overlayers are often adequately described by the pair energy of the closest pair(s) found in the overlayer. Before delving into specific simple models, it is worth stating the underlying philosophy motivating them. 11.2.2. Electronic indirect interactions in simple tight-binding model
To gauge roughly the relative magnitudes and general behavior of these interactions, it is convenient and customary to study a simple model, in this case a tight-binding model in which the substrate is a single-band, simple-cubic solid. (See LaFemina, Chapter 4, for a discussion of tight-binding models.) This model was adopted two decades ago (Einstein and Schrieffer, 1973) (hereafter ES) to embody the idea that the d-bands of the substrate were primarily responsible for the interactions and, unlike jellium, allowed one to consider the dependence of the interaction on the type of adsorption site in a simple way. 11.2.2.1. Model Hamiltonian The model, as well as many subsequent discussions of interactions between adatoms, is couched in terms of an Anderson (1961) (magnetic) model in which the adatoms are represented as dilute impurities at sites r (= a,b for pairs) in an unperturbed host: m
me,~,+ Z (H'~ + H'r)
__ n ~
(I 1.3)
r
The first term in the parentheses represents the atomic factors of adatom r, while the second is this atom's coupling to the metal. To include a direct interaction, one would add terms of the form H',~, coupling atoms a and b. Until recently, most work on the problem has amounted to taking progressively more realistic expressions for various of these terms and solving the resulting system to varying levels of approximation. To simplify notation, we assume that the adatoms are identical. Over the last decade the coadsorption problem has attracted some interest; it is straightforward to extend the formalism. Some of the simplicity of the above ansatz comes from the use of an atomic orbital picture. While this formulation makes it easy to do initial calculations, it neglects such effects as orbital deformation and local distortion, which may often be important. The adatom part of the Hamiltonian is
Interactions between adsorbate particles
H'~ = G'~~ n~,~+ Unit n~,
587
(1 1.4)
(y
and similarly for H~. This expression can be generalized to include degenerate orbitals, multiple levels, etc. As a first approximation, one might set e] at the ionization level - I of the adatom and take as U the difference between -1 and the affinity level. For greater accuracy, e ~ should be raised and U reduced by correlation effects (screening and image charges). In a (restricted) Hartree-Fock approach, one neglects U entirely and replaces G~ by G,, = G~ + U(na`,) where (n~`, is the mean occupation of the adatom for either spin direction. For neutral chemisorption, (n,,`,) is 1/2, suggesting G,, be the (negative) average of the ionization and affinity energies, as in many chemical molecular orbital calculations (where this is called the Mulliken (1934) electronegativity) (cf., e.g., Pople and Beveridge, 1970) Using the idea of chemical transferability, Pandey (1976) adjusted the adatom and coupling parameters so that cluster computations of small molecules fit the levels found in photoemission experiments; presumably the same parameters carry over to the chemisorption system. Brenig and SchOnhammer (1974), Hertz and Handler (1977), and Bell and Madhukar (1976) went beyond Hartree-Fock in the case of single atom adsorption. The first group also showed in the pair problem that correlation effects are relatively unimportant, compared to the single atom case, as we shall discuss below. On the other hand, using self-consistent Hartree-Fock and resorting to mean-field theory, Gavrilenko et al. (1989) explored the parametric conditions for magnetic ordering of the adatoms. Davydov (1978) considers a similar question, including direct interactions for a chain of adatoms. The simplest approximation for the substrate assumes a single band of one-electron states with energy ek. (A band index would also be needed if more than a single band were considered.) Many-body effects could also be included by putting a diagonal Coulomb term like U on each substrate site. Since only the component of crystal momentum parallel to the surface is conserved for a slab or semi-infinite crystal, k merely labels the states in some suggestive fashion. It is usually convenient to work in a mixed representation of kll and a layer index. In general there can be a different coupling between each k state and the adatom. For most purposes, it is adequate to consider, in the case of bonding at an atop site, a single coupling constant V between adatom a (or b) and its nearest neighbor on the substrate: H,,s - - Vc+~,co,, + h.c
(11.5)
where c § and c are creation and annihilation operators, respectively, for electrons in the state indicated by the substrate. For bonding in a bridge or centered site, Co,, is replaced by a symmetric normalized combination of c-operators for the number of substrate neighbors of the adatom. In principle this coupling should also consider an overlap term between atoms and metal. This question has been discussed at length by, among others, Sch6nhammer et al. (1975), Grimley (1974), and Einstein (1973). The usual approach is to "renormalize" previously stipulated natural orbitals (and resulting energies) with L6wdin ( 1 9 5 0 ) o r Gram-Schmidt (Birkhoff and MacLane, 1965) schemes.
T.L. Einstein
588
11.2.2.2. Calculation of change in one-electron energies using Green's functions Our goal is to find interaction energies between chemisorbed atoms, which in a one-electron framework can be expressed in terms of the associated change in density of states Ap: El:
AW=2f
( E - EF) Ap(~) dE
(11.6)
The factor of 2 comes from spin degeneracy, and the use of ~ - eF rather than just s indicates that the number of electrons rather than the chemical potential is being fixed (Grimley, 1967; Newns, 1969; ES). (The contribution due to the integral over svAp(s) is the result of an infinitesimal shift in the Fermi energy.) In the calculation presented below, the essential idea is that the interaction between adsorbates can be obtained by finding the underlying shifts in the one-electron energies of the system. It is convenient to do so in terms of one-electron Green's functions. We present enough detail below to show that this procedure is not so daunting as novices might suspect. Nonetheless, we follow this subsection with an explicit simple illustration using a ring as the "substrate" and analyzing the results in terms of level shifts to make contact with those readers more comfortable thinking in terms of quantum chemical models. To obtain the change in density of states A 9 needed so that the integral can be evaluated, we adopt a method used earlier in the theory of dilute alloys by Lifshitz (1964). Suppose the unperturbed (H '= H,,m= 0) and perturbed Hamiltonians, H ~ and H - H ~ + H', have eigenvalues sj and Ej, respectively. Then ~ (1 1.7) J But this can be rewritten as Ap(s) = -
'Imp/.
rc
_- llm Z ~
l
E-Ej.+i8
' /
e-13j+i8
ln(E-H+iS)-ln(E-H,,+iS)
J
_- _ l l m rc
O--~--Lndet( 1 )(e-H+iS) ~ ~- H,, + i8
1x Im ~oq In det (1 - G oAV)
I
Rememberthat the units of a delta function are the inverse of its argument.
(11.8)
Interactions between adsorbate particles
589 ^
where G ~ is the unperturbed retarded Green's function (e - Ho + i8) -~ and V =- H~s was given in Eq. (11.5). (Since we choose here to follow convention by using retarded functions, the signs of all imaginary quantities will be the opposite of those appearing in ES and subsequent papers in that series, which used advanced Green, s functions, with infinitesimals of the opposite sign.) In scattering theory det (1 - G~ is familiar as the Fredholm determinant (of the 0th partial wave) (Gottfried, 1966). Furthermore, ^
- I m In det (1 - G~
= rl(e)
(11.9)
where rl(S) is the s-wave phase shift. (This identification is perhaps clarified by the observation that "Im In" is an arctangent, yielding an angle that amounts to a scattering phase shift.) This approach makes optimal use of the higher symmetry of the unperturbed system and the locality of the perturbation associated with adsorption. It is generalized in the scattering theory approach (Feibelman, 1989a). In terms of q, one writes the interaction energy simply as El:
A w- - (2/,) j" n(s) ds
(J 1.~0)
The Fredholm determinant contains a dense set of alternating poles and zeros, which turns into a branch cut in the continuum limit. Dreyss6 and Riedinger (1983) pointed out that one can circumvent numerical difficulties with this sort of integral by adopting the contour-integration approach (in the T = 0 limit) developed for temperature-dependent fermion Green's function problems. The result is basicalJ~y an integration of the real part of the analytic continuation of in det (1 - G~ from s~ + i0 § to s~ + ioo. This integration can be cast into a finite interval by making a substitution for the imaginary part of the energy integration variable (Liu and Davison, 1988). To evaluate the phase shift for the two-adatom problem, we arrange the matrix so that the adatom sites (a and b) and the substrate nearest neighbors to which they couple (o and^n) come first (o, a, n, b) and then all other substrate sites. The matrix (1 - G ~ V) then differs from a unit matrix only in the upper left hand 4• block:
- G x V,,,
'~
-o,,~,, v,,o
o
1
0
o
-G,\v,,.
-G~x, v,,o
1
-o,,\ v.~ 0
-c~x v~
(11,11)
1
The superscript X indicates that the substrate Green's functions can be easily generalized, for adsorption in B or C rather than A sites, to represent a (normalized) hybrid (cf. remarks after Eq. (11.5)) of substrate orbitals (Einstein and Schrieffer,
590
T.L. Einstein
1973). ~ T h e m a j o r result c o m i n g from the possibility o f c o u p l i n g to c o m b i n a t i o n s o f orbitals is that G B or G c is g e n e r a l l y very different from G A, so that the pair interaction will d e p e n d very strongly on the adsorption site. This feature arises naturally in L C A O models, in contrast to the other simple starting point, j e l l i u m m o d e l s (see below). (Braun (1981), h o w e v e r , argues that the effective c, b e c o m e s a d s o r p t i o n - s i t e d e p e n d e n t , possibly mitigating the variation with site s y m m e t r y . ) T h e d e t e r m i n a n t o f this matrix can be written as ^
det(l _ G " V ) = (1 - G,,,,G,,x, IVo,,I 2) (I - Ghh G,XlV~hl2) _ Go~G,,,,G~,~,G,,,,IV,,,,12V,,hl x x 2 (11.12) It is n o t e w o r t h y in this e x p r e s s i o n that the p a r e n t h e s e s e n c l o s e the c o n t r i b u t i o n to det(l - G ~ f r o m the adsorption of an isolated a d a t o m a at o (or b at n). F a c t o r i n g out these terms, and a s s u m i n g a d a t o m s a and b are identical, as are sites o and n, we find A
det(l-G
~
^
det (1 - G ~ V) -
[det ( 1 -
G~ ~')sing,e] 2
--x 2 x 2 = l - ( G o , , ) (G,,,) V ~
(11 . 13)
where G,,,x is a Green's function for a single adatom renormalized to account for its adsorption: G,,,x - ( I
- G .... G,,X,,IV .... 12)-' G,,,, -
[ c - c,,- V 2 G,X,,,(c)l-'
(11.14)
B e c a u s e of the logarithm, the phase shift (and hence the c h a n g e s in D O S and e n e r g y ) c h a r a c t e r i z i n g the "pair" interaction o_.fthe a d a t o m s can be o b t a i n e d directly from the phase shift associated with (1 V4(G,,X,, ) 2 ( G , ,x, ) ) 2 rather than from explicitly subtracting twice the s i n g l e - a d a t o m - p h a s e shift from the t w o - a d a t o m shift. For any n u m b e r of a d a t o m s , the single a d a t o m adsorption part factors out of the matrix (ES" G r i m l e y and W a l k e r , 1969). On the other hand, as s h o w n b e l o w in Eq. (11.20), for more than two a d a t o m s there is no way to factor out the pair effects from the h i g h e r order ones. (The feature that the s i n g l e - a d a t o m part factors out is a p l e a s a n t c o n v e n i e n c e , but with m o d e r n c o m p u t a t i o n a l p o w e r it does not p r o d u c e a significant i m p r o v e m e n t in numerical results, except perhaps in the a s y m p t o t i c r e g i m e . ) -
cos(~. R,) Explicitly, G,,An = ~ - - , where g,, is the vector in the surface plane from site o to site n, ej J e - ej+ i8 denotes the eigenvalues of H,,, and the notation on the wavevector reflects the fact that only crystal momentum in the surface plane is a good quantum number. If a single adatom sits in a bridge site between surface atoms 0 and 1, then G B.... = G A.... + GAl. If a second adatom sits between n and n+l (assuming all ,B A A A four sites colinear for simplicity, then G,,,, = G,,,, + (l/2)(G,,.~+t + G .... -l). To complete the description, one must make some statement about how the adatom-substrate hopping depends on the adsorption site, which will involve some at-least-implicit assumption about dependence on bond angles, bond lengths, local relaxations, etc. The parameter Vthat appears in the formalism corresponds to ~ times the hopping parameter between the adatom and one of the z members of the hybrid to which it couples.
Interactions between adsorbate particles
591
In the LCAO framework, the formula for the pair interaction energy E,, between the adatoms adsorbed on sites o and n (which we identify as nth nearest neighbor sites on the surface) is, from Eqs. (11.9), (11.10), and (11.13), F-,F
2 f I m In [1 -(G,,X(~;)) 2 (GX,,,(e)) z V a] de E,, = rc
(11.15)
To gain some understanding of this interaction, we first expand the logarithm and consider the lowest order term (Kim and Nagaoka, 1963), which becomes a good approximation for weak coupling (small V) or large separation (small G,,.)" El:
E,, - - - - 2 Im f V4 (GX(~;)) 2 (Go,,(c)) x 2 de
(11.16)
/1; -oo
If G is neglected (which is generally a poor approximation at small separations), expression (11.16) is just the RKKY interaction energy (Ruderman and Kittel, 1954" Yosida, 1957), in which two localized spins (here localized defects) interact via coupling to a bulk conduction electron sea. If this sea is viewed as a free-electron gas, the propagator G,,, reduces to a continuum G(IRI;~), where R goes from one bulk spin/defect to the other. This bulk interaction is proportional to (x cos x sin x) x -a, where x = 2kFIRI. It is thus oscillatory in R and decays asymptotically as R -3, characteristic of Fermi surface domination. We shall discuss the decay on surfaces in the section on asymptotics. A physical interpretation of Eq. ( l l . 15) is that an electron in an occupied state starts at one adatom, hops back and forth to the substrate many times, then propagates to the second adatom, hops back and forth again for a while, then propagates back to the starting site. Alternatively, one can describe the process as a particle and a hole propagating from one adsorption site to the other (Zangwill, 1988). While Eq. (11.16) suggests that the interaction is proportional to V4, such behavior only obtains in the limit of weak coupling. For stronger coupling, the V-dependence in G eventually cancels the leading V4. This strong-adsorption case is the limit of the "surface molecule", in which the adatom and its substrate partner form a dimer which rebonds perturbatively (with the bulk coupling strength) to the substrate. The interaction between the adatoms then comes from the interference between the two dimers in the rebonding process, which does not depend on V. Grimley (Grimley, 1967; Grimley and Walker, 1969) was the first to apply the Anderson model to chemisorption, using as a substrate a semi-infinite single-band crystal with a phenomenological surface reactivity. This adjustment highlights a problem with tree-electron gas substrates, namely how to allow coupling with adatoms. If the adatom sits beyond an infinite barrier, e.g., there will be no coupling whatsoever. To avoid this problem, to put in site specificity in a natural way, and to reflect the belief that the d-bands were primarily responsible for the lateral interactions, ES modified Grimley's model by using as the substrate the (100) face
592
T.L. Einstein
of a single-band simple cubic crystal ("simple cubium") in the nearest-neighbor tight-binding approximation. Eq. (11.15) was then evaluated numerically. Table 1 1.1 capsulizes the results of ES. The energies are measured in units of one-sixth of the bandwidth (i.e., twice the hopping parameter). For the typical transition metal d-band being modeled, this unit is of order 1 to 2 eV. The Fermi energy and adatom level are measured relative to the center of the band. As the table illustrates, the pair interaction is highly anisotropic, oscillatory in sign, and rapidly decaying. At close separations the decay is precipitous, more exponential than inverse power like, dropping roughly by 1/5 with each lattice spacing, while asymptotically it decays as R-5. While asymptotic behavior is discussed in more detail in w 11.2.6, we note here that it is characteristic of dominance by a single k-state on the Fermi surface. The more complicated behavior at shorter range shows that many electronic states participate in the pair interaction. The pair interaction is comparatively insensitive to changes in to and V, somewhat more sensitive to shifts in the Fermi energy (especially for larger interadatom separation), and very sensitive to the adatom binding site. Typical values of the magnitude of the nearest, next nearest, and third nearest pair energies are 1xl0 -~, 2x10 -2, and 8• -3 units, although each of these can vary over a range of an order of magnitude. As presented here, this formalism implies that the substrate is essentially rigid during the adsorption process. In fact local distortions certainly do occur. Feibelman (1987, 1989, 1990) has emphasized that these distortions can play a crucial role, particularly at near-neighbor spacings. At farther separations, it does not seem unreasonable to believe that the distortions essentially renormalize G~,, while leaving G,,, relatively unaffected. Thus, over this range, the distortions might be taken into account by tuning the atomic and coupling parameters. n
X
11.2.2.3. Simpler illustration: pairs on a ring Many of the ideas in the preceding section may be couched in a (Green's functions) language unfamiliar to some readers. In an attempt to make the key ideas clearer to people more comfortable with the language of quantum chemistry, we present in this section results of an explicit calculation, done with Mathematica TM, in which the substrate is taken to be a ring of 50 atoms. For a system of such limited size, we can keep explicit track of what happens to all of the molecular orbitals. While I D models are a typical starting point in similar studies (Hoffmann, 1988; Whitten, 1993), we caution that consequently they contain some anomalous features which are not characteristic of most 3D substrates. In attempting to keep the following discussion uncluttered, we do not dwell on such unpleasantries as the inevitability of split-off states (due essentially to the divergence of the density of states at the band edge) and the anomalously slow decay of the interaction with separation. After exploring interaction energies from the perspective of shifting molecular orbitals, we show how the problem can be recast in the Green's function formalism presented above. The Hamiltonian of the ring itself (Hmc,~~of Eq. (11.3)) can be represented by a 50x50 matrix with non-zero entries (taken as -1/2) only along the two diagonals next to the main diagonal (i.e. entries {n, n+l }) and at the corners ({ 1,50 }, {50,1 }) to close the chain into a ring. By analogy, e.g., to benzene rings, it is well known
Table 11.1 Display of the pair interaction energy En suggesting the sensitivity of adatom arrays to changes in the Fermi level, the hopping potential V, the adatom energy level &a, and the binding-site symmetry A, C, B, and BP. (For bridge-site adsorption, there are two nearest-neighbor configurations: in B, the vector R between adatoms is in the plane formed by the adatom and its two substrate neighbors; in BP, R is perpendicular to it. Note that, e.g., for E2 there is no difference between B and BP.) One adatom sits at the origin "0"; the pair energy is for a second adatom at the nth nearest-neighbor site. The magnitude of the number given is 10 plus the common logarithm of the magnitude of the interaction. A plus (minus) sign indicates that the interaction is repulsive (attractive). Thus, table entries of . The energy unit is one-sixth the substrate band width, roughly 1-2 eV. +8.9, -7.7, and 6 . 6 represent interactions of +8x10-~,-5x10-~,and 4 x 1 0 ~respectively. Each chart is labeled by the symmetric adlayer pattern predicted. Adapted from Einstein and Schrieffer (1973) and Einstein (1979).
5
a
2. b
t
'a-
$ rn
C
% 2g 2 L
A
Binding site n =
3 1 0
C 4 2 1
5 4 3
3 1 0
B 4 2 1
5 4 3
6
9
13
3 1 0
BP 4 2 1
5 4 3
3 1 0
4
5
2
4
1
3
3.
F
594
T.L. Einstein
that the 50 eigenstates are traveling w a v e s with w a v e v e c t o r s k = nrt/25a, w h e r e a is the n e a r e s t - n e i g h b o r spacing. T h e r e are just 26 distinct e i g e n e n e r g i e s - c o s ( k a ) ; all but two (viz. 1 and - 1 ) are d o u b l y d e g e n e r a t e (due to the s y m m e t r y o f c l o c k w i s e and c o u n t e r c l o c k w i s e travel). T h e b a n d w i d t h in these units is, thus, 2, and the b a n d is c e n t e r e d about 0. T h e prescription in this Htickel m o d e l for adding extra atoms essentially f o l l o w s that in the t i g h t - b i n d i n g model. An a d a t o m h a v i n g e n e r g y ~,, ( i . e . - c ~ ' in Htickel l a n g u a g e ; ~ = 0) couples, via the h o p p i n g p a r a m e t e r - V (i.e. -13' in Htickel l a n g u a g e ; recall 13= 1/2) to an orbital o f the ring (as in atop bonding). C o n s e q u e n t l y , o n e adds a row and c o l u m n to the 5 0 • matrix. T h e d i a g o n a l entry { 51,51 } is ~,,; the only o t h e r n o n - z e r o entries are pairs o f - V ' s at { 1,51 } and { 51,1 }. F o r simplicity, to retain s y m m e t r y in ~ and to focus on c o v a l e n t effects, we take e,, = 0. As m i g h t be e x p e c t e d , the c o m p u t e d e i g e n v a l u e s m o v e away from E,, = 0, by an e n e r g y ,,,: V2 in the p e r t u r b a t i v e regime. (Actually, this interaction splits the d e g e n e r a c i e s o f the ring. T h e c o m b i n a t i o n o f the two eigenstates with an a n t i n o d e at the a d s o r p t i o n site gets shifted, while the other c o m b i n a t i o n with a node keeps its original value o f - c o s ( k a ) . ) This d o w n w a r d shifting o f orbitals increases the a d s o r p t i o n e n e r g y (absolute value o f the c h a n g e in total e n e r g y due to adsorption) as the Fermi e n e r g y a p p r o a c h e s ~,, = 0; thereafter, the a d s o r p t i o n e n e r g y decreases, e v e n t u a l l y r e a c h i n g zero (as particle-hole s y m m e t r y d e m a n d s ) . To assess pair interactions, we add a second a d a t o m n sites away from the first.' T h e matrix b e c o m e s 52• with ~,, = 0 at {52,52} and - V at {n+1,52} and {52,n+1 }. To c o m p u t e the pair interaction at close spacings, we c o m p a r e the e i g e n v a l u e s when the a d a t o m s are at n e i g h b o r i n g or n e x t - n e a r e s t n e i g h b o r sites with those w h e n they are at o p p o s i t e sides of the ring. To m a k e sense of these results, we first c o n s i d e r the situation of a d a t o m s at o p p o s i t e sides o f the ring. For an infinitely large ring one would e x p e c t results to be similar to the s i n g l e - a d a t o m case, but with shifts twice as large. For the finite case here, we note that this will occur only for states with an even n u m b e r of nodes, so that the ring eigenstates to which the a d a t o m s c o u p l e will have the s a m e a m p l i t u d e s on the two a d s o r p t i o n sites. T h e pair interaction arises from the shifts in the e n e r g y levels w h e n the a b o v e w i d e l y - s e p a r a t e d pair o f a d a t o m s are m o v e d to nearby sites. 2 W e e x p e c t that the
Note that on the ring, there is really a second pair interaction over separation (50-n)a. Because of the periodic boundary conditions used in the previous section, this effect exists implicitly in the formalism developed there. For a large ring, this second interaction is negligible, but this effect prevents us from using a small ring. The alternative of using a chain rather than a ring is undesirable because the "substrate" sites are inequivalent. 2
There are alternative definitions. Burdett and Ffissler (1990) start with ligands (viz. CO) attached to 1, 2, or 3 metal atoms, using the extended Htickel model, and seek to explain the structure of the ligand "pair potential" for a monolayer. Since it is impossible to move ligands far apart, they define the pair energy as the sum of the energy of the system with both ligands present and the energy with both absent, minus twice the energy with just one ligand attached. Some thought shows that this definition is equivalent to the one we use, assuming that our adatoms are far enough apart that they do not interact. While this perspective may be appealing, it is a chore to keep track of the electrons as they are added, and tricky to trace the evolution of the levels.
Interactions between adsorbate particles
595
eigenenergy will decrease (become more energetically favorable) if the coupling is in-phase and increase if it is out-of-phase. More explicitly, we consider the eigenvectors of the antecedent of each energy level, from the original ring (without adatoms). If the eigenvector has the same sign on the two nearby sites, we expect the shift to be attractive (i.e. the level lowers in energy when the adatoms are brought to the nearby sites). The pair interaction comes from the shift of the occupied levels. How many levels are occupied is, of course, determined by the Fermi level. We generally expect that near the bottom of the band, the shifts will be attractive (negative) because the adatoms couple in phase. As the Fermi energy increases, more-rapidly oscillating eigenstates become involved. To plot and thereby analyze this behavior, we do the following: Along the horizontal axis we use the energy of each of the levels of the ring with the adatoms at opposite sides. Thus, as we set the Fermi energy further to the right along this axis, more levels become filled. At each of these 52 discrete energies, we plot as the vertical coordinate the shift of the level when the adatoms are brought to nearby sites. It is these shifts, due to the "interference" of the proximate adatoms, which in the weak-V limit tend to scale as V4 (cf. Eqs. (11.15) and (11.16)), i.e. as a next-order effect after taking into account the V2 shifts due to adsorption. The sum of these shifts, for the occupied levels, is the pair interaction. ! Thus, what we have plotted is the integrand in Eq. (11.15), essentially the phase shift 11. Near the bottom of the band, the integrand is, as noted above, generally negative, but with increasing energy, it begins to oscillate in sign. (The closer the adatoms, the larger is the energy between sign changes.) In performing this analysis, the shifts alternate between the expected behavior and a much weaker shift of the uninteractive ring eigenstates. Thus, in Fig. 11.2 we combine pairs of shifts, plotting their sum vs. the average of their (unshifted) energies (from the case of adatoms at opposite sides). We now seek to show that these results offer a decent finite-size approximation of the quasi-continuous behavior considered in the previous section. For an infinitely long chain, one can derive the analytic expression (cf., e.g., Economou (1979) or Davison and Steslicka (1992) for background information, or Kalkstein and Soven (1971), with no intralayer hopping): -i
[_e + i ~ ~ 5 - ] "
(1117)
c , , . ( s ) - ~41- - - -_- - 7
inside the band (1~1 < I); outside the band ~/1 - e2 ~ i.sgn(e) ~/E2 - 1 and Go, is pure real (Ueba, 1980). For the 50-atom ring, the substrate Green's functions can be computed numerically as G,,,(e) =
-•0
cos(kna) ~ e + cos(ka) + i8
(11.18)
k
1 Actually,it is half of the interaction, since we have been neglecting the factor-of-2 spin degeneracy in this section.
596
T.L. Einstein
0.03
0.06 f
. . . . . . . . .
,, . . . . .
~
.
.
0.02 u~
o
0.01
o
o.oo
:
O.02 0.00
._
-o.01
g c)
-0.02
-0.02 -0.04
-0.03
X
X
-0.06 -
.0
-0.5 0.0 0.5 e n e r g y r e l a t i v e to b a n d c e n t e r
1.0
.0
-0.5 0.0 0.5 e n e r g y r e l a t i v e to b a n d c e n t e r
Fig. 11.2. Integrand used to compute pair interaction energy for adatoms at nearest-neighbor and next-nearest-neighbor sites on a ring. Solid curve: continuum limit, as described in w 11.2.2.2. x- shifts of pairs of eigenvalues vs. average of their unshifted energy for a ring of 50 atoms. See text for discussion.
setting the infinitesimal at a value of, say, 0.1. ~While these Green's functions have many secondary oscillations, their overall behavior is rather similar to the analytic infinite-length Green's functions of Eq. (11.17). In any case, inserting the analytic form into Eq. (11.15), we produce the integrand (without the factor of 2) and coplot it in Fig. 11.3. We see that the couple-dozen pairs of levels from the ring provides a decent accounting for the results of an infinite ring in a form that may be more transparent. Our exercise further supports the idea that the pair interaction is a delicate mix of the couplings to all the occupied levels (or at least the half of them which are symmetric with respect to the inversion about the midpoint of the adsorption sites). Thus, a discussion in terms of H O M O (highest occupied molecular orbital) and L U M O (lowest unoccupied molecular orbital), i.e. frontier orbitals (Hoffmann, 1963, 1988) will not capture all the physics of the problem. On the other hand, with increasing separation there are more oscillations in sign as a function of oF. In the limit of large separations, reminiscent of stationary-phase problems, the interaction energy will be dominated by the endpoint of the integration, namely the behavior at By., making a frontier-orbital approach appropriate, if one has some grasp of the long-range behavior of wavefunctions at this energy. (In the section on asymptotics, we shall explore this problem further.) More importantly, in the limit of small V, the shifts and hence the interaction are quite small except when 8F is close to e,. In the limit of w e a k chemisorption, then, the H O M O / L U M O viewpoint may well offer a fruitful perspective on pair interactions. Burdett and Ffissler (1990), for example, in modeling CO adsorption find the interaction is strong only when 8F is near a large H O M O - L U M O gap. Before closing, we mention, for those particularly interested, some details skirted above. In Fig. 11.3 only 24 pairs of levels are included. In addition, there 1 The size of 8 in this discussion should be large enough so that the spiked distribution due to the discrete levels is smoothed but not so large that it is completed washed out. This parameter broadens the levels, a common way to represent a large system by a much smaller one with a limited set of eigenenergies.
597
Interactions between adsorbate particles
1.5 1.0 0.5
0.5
,~
o:o
0:0
o
-0.5
-0.5
-1.0
,:,
1.0
1.5i -
|
. .0
.
.
.
-0.5 0.0 0.5 e n e r g y relative to band c e n t e r
1.5
-1.0
-0.5 0.0 0.5 e n e r g y relative to band c e n t e r
.0
Fig. 11.3. Off-diagonal Green's functions Gol (nearest-neighbor) and G02 (next-nearest-neighbor) for a ring of 50 atoms (setting B= 0.1) compared with the continuum form (solid curves), computed exactly. x: imaginary parts" 0" real parts. are pairs above and below the band, corresponding to what have been called "split-off" states (ES) and amount to localized levels outside the substrate band (where Im G vanishes). As one might guess from Eq. (11.12), they are the solutions e• of the equation - 13,,- V2(Re Goo(e~) +_Re Go,(~)) = 0
(11.19)
For the case of isolated adatoms at opposite ends of the chain, the + term is absent and the solutions ~;0 are doubly degenerate. The eigenenergies of the 52• matrices correspond virtually identically to the solutions of these equations, which use the quasicontinuum Green's functions. For 3D substrates, these states typically occur only for strong coupling (large V) but in 1D they are always present, formally due to divergent van Hove singularities in Re G at the band edge, physically because of the large number of states near the band edge in 1D models. From the figures in ES ~, one sees that E+ + e _ - 2~0 is positive. This initially counterintuitive result can be derived analytically or graphically from the generic form of the Green's functions. The fact that ~+ shifts down from E0 less than ~_ shifts up corresponds to the relative decrease in shifts in the levels as one gets farther from the band and ~,,. The split-off state involves fully in-phase hopping around the ring. Perhaps when the adatoms are close to each other, the electrons get somewhat concentrated in the region near the adatoms, so that they cannot take full advantage of the hopping all around the ring. In any case, this result leads to the spikey behavior with the unexpected sign near the band edge.
11.2.3. Multisite interactions 11.2.3.1. Three-adatom (trio) interactions In general there will be several adatoms in close proximity. Eq. (11.2) anticipated the possibility of multiadatom interactions. The expectation of ES is that overlayer
1 Bewaresome misleading analysis in wII.B.3 of ES.
598
T.L. Einstein
electronic energies are overwhelmingly dominated by nearest pair interactions. To evaluate multisite interactions, and thereby check this idea, it is straightforward (Einstein, 1979a,b) to enlarge the matrices needed to compute the phase shift in Eqs. (11.9) and (11.10). If the sites to which the adatoms bind are lth, mth, and nth nearest neighbors, we find EF
Er=,,,, - - 2 ~ Im ln(A)dlz- E , - E,, - E,
(11.20)
where u
A = 1 - V4 G,,2. (G~, G~m GZo.)- 2 V 6 G,,3,,Go, Go,. Go.
(11.21)
As indicated in w 11.2.1, Eq. (11.21) does not factor, making an explicit subtraction of pair energies necessary. There are two parts of this new interaction: (1) a new triangular path, represented by the G G G term; and (2) an "incompleted cubic" term, marked by the absence of V8 and V~2terms that would be present in E, + Em + E, if their logarithms were merged. Trial calculations using just the "triangle path", with moderate V appropriate to chemisorption, reproduce the full interaction at least qualitatively. Computations of trio energies using Eq. (11.20) suggest that their magnitudes are determined primarily by the two closest (strongest) pairs. In explicit comparisons (Einstein, 1979a,b) of E,m n -- E223, E225, E238, and E335, for typical V and •,,, for all possible substrate fillings, the first two have the strongest trio interaction energy. E223, which has a 3rd neighbor spacing as its third side, is somewhat the larger, and is nearly as strong as E 3. The other two are smaller by at least half an order of magnitude. With increasing adatom separation the trio energies fall off rapidly, much like the pair energies.
11.2.3.2. Complete overlayers While quartets and higher-order terms could be calculated, numerical noise problems from successive cancellations would become troublesome. Starting from the other extreme, one can easily show (Einstein, 1977) that the indirect interaction energy per adatom for a complete ( l x l ) adlayer of N, adatoms is El:
f
2 ~] f Im In I[1 - V2G..(~) G(k,,,E)/[1
/1;NIl
k,
I
-
V2Gaa(~) Goo(F_,)]IdF..,
-.~
(11.22)
2 2 ~Im In [1 - V2 G.,.,{ G(k,, ~ ) - G,,,,(I~)}] de
71;N,,
kll
-,Do
where the summation goes over the surface Brillouin zone (SBZ), containing N, (the number of adsorption sites) points. G(k,,e) can be computed analytically (Kalkstein and Soven, 1971), rather like a semi-infinite chain.
Interactions between adsorbate particles
599
For a real monolayer, direct interactions between the closely-spaced adsorbates are likely to produce an interaction energy quite different from that predicted from Eq. (11.22). Not only does the direct interaction make a great difference for individual pairs (Burke, 1976), but it often leads to the formation of two-dimensional adlayer bands which overshadow any indirect effects (Liebsch, 1978). Therefore, we focus on the c(2x2) overlayer. Since the real space unit cell area is doubled, the SBZ is halved, most naturally taking the form of an inscribed "diamond" (square rotated by 45~ Points outside the new SBZ get folded back in, giving doubling of the (highly blurred) two-dimensional band-structure. The upshot is that for a c(2x2) adlayer, G(kai,E) in Eq. (11.22) is replace by (Einstein, 1977, 1979a) and Na = Nil~2 +
-
(11.23)
Based on these ideas, one can compare (Einstein, 1977, 1979a) the indirect interaction energy (per adatom) for a full c(2• overlayer with an explicit sum over the pair energies for all pair configurations arising in a c(2• pattern ~ only the five shortest contribute significantly - - weighting them according to the number per adatom existing in the pattern: two for pairs along the < 10> and < 11 > mirror axes, four otherwise. Overall, this curve does a good job of reproducing the c(2• plot. Trio interactions can also be included in the sum and help make up differences between the overlayer calculation and the explicit sum. Their contribution generally is important only near energies corresponding to the Hartree-Fock bonding and antibonding resonances in the DOS. In short, multisite interaction energies are not too important in total overlayer energies, although they may play a role in other circumstances. The other way to approach dense monolayers is to invoke results from the theory of alloys (Ehrenreich and Schwartz, 1976). Perhaps the simplest such scheme is the average T-matrix approximation (ATA) (Korringa, 1958), which assumes that adatoms are randomly distributed over the lattice sites. Urbakh and Brodskii (1984, 1985) work out the formal expression for Ap(~) and apply it to Pt(ll I)-H (cf. w 11.4.2.). The next level of sophistication is CPA, in which the self-energy of the "effective medium" of the alloy is calculated self-consistently; an application by Persson and Ryberg (1981) was noted in w 11.2.1.
11.2.4. Coulombic effects: self-consistency and correlation, and other improvements The issue of self-consistency has pervaded most subsequent efforts to apply tightbinding methods to the pair problem. The inability to resolve this problem in a satisfactory way is one of the greatest difficulties in extending this approach to quantitative investigations. In the LCAO framework, since the electron orbitals are fixed at the outset, self-consistency is discussed in terms of the Friedel (1958) sum rule - - which in this case requires charge neutrality within some finite range of an adatom w rather than Poisson's equation (Appelbaum and Hamann 1976). Typically, e,, is adjusted (making it a derived rather than a free parameter) (Allan 1970, 1994). The energies of nearest neighbor(s) on the surface may also be altered, thereby inviting new surface states (Kalkstein and Soven, 1971; Allan and Lenglart, 1972). Sometimes off-diagonal Coulomb terms are also included in various ways
600
T.L. Einstein
(Rudnick and Stern, 1973; Leynaud and Allan, 1975), meaning that changes in charge on a site affect the potential of its neighbors. Generally neutrality is required either at each site or just in the surface cluster consisting of the adatom and its nearest neighbor(s), excluding any longer-range oscillations. The quantitative results are rarely compelling. The qualitative results (Einstein 1975, 1979a) are plausible. A second approach assumes that in a strongly chemisorbed system, the essence of the pair interaction lies in a surface molecule. A small cluster is treated carefully, gaining an improved description of local Coulomb effects at the expense of any background effects from the substrate. From studies of W2H and W3H2, for example, Grimley and Torrini (1973) conclude that H atoms at nearest neighbor sites on W(100) will be unstable, the repulsive energy being of order 200 meV. This method is not extended readily to more widely separated pairs, since the distance from the adatom to the edge of the cluster should presumably be at least as large as its distance from the other adatom. Since the "substrate" w a v e f u n c t i o n s - via which the pair interacts m are sensitive to the details of the cluster, matching conditions to the background must be adjusted carefully. Moreover, in cases where adatoms bond to a common substrate atom, some anomalous structure may arise which should not be generalized (ES; Einstein et al., 1990). The best hope for cluster approaches is to embed them in well-characterized semi-infinite substrates (Grimley, 1976). Grimley and Pisani (1974) have taken this approach for clusters containing single adatoms and calculated in a S C F - L C A O - M O scheme. The embedded cluster technique has indeed flourished (cf. NATO conference proceedings in Pacchioni et al. (1992)). Most of the applications are to monomer adsorption, but the dissociation of (gas) dimers is also often considered (e.g. Cremaschi and Whitten (1981), Madhavan and Whitten (1982)). As noted, it is hard to imagine applying the method to larger pair separations. Feibelman (1989a) provides a lucid critique of this approach, questioning typical choices of bases and treatment of background effects. Also, since correlation is typically considered only in the cluster region, he wonders how much of the adsorbate binding energy actually comes from allowing substrate correlations in the bonding region. Grimley and Walker (1969) observe that while sizeable charge transfer might take place during chemisorption, little more should happen as a function of the relative placement of the adatoms. If energies in simple models could be determined in some plausible way, the pair interaction should work out satisfactorily even if the single-adatom results are somewhat inadequate. Moreover, the pair interaction is a rather insensitive function of e,,, as suggested by Table 11.1 and shown more convincingly by Fig. 11 of ES. Sch6nhammer et al. (1975) studied carefully the correlation effects in indirect pair interactions. Using a (100) cubium substrate with parameters appropriate to H on Ni, Sch6nhammer (1975) had previously shown from a variational approach that the single adatom binding energy is roughly 1/3 stronger than in Hartree-Fock (although the two curves did have the same structureless shape as a function of EF). They find that this correlation energy, 1/4 the binding energy, roughly cancels out when the pair interaction energy is computed. Although this cancellation is reported to be less complete for other parameters, the qualitative behavior holds for V's of
Interactions between adsorbate particles
601
order the "critical hopping" (below which Hartree-Fock local moments arise). In addition to confirming the anisotropic, oscillatory behavior of the pair interaction, Schtinhammer et al. (1975) corroborate the roughly exponential fall-off with separation (for interadatom distances of order 1 to 4 lattice constants). The implication of this work is that correlation effects (in the form of careful treatments of the Anderson Coulomb term), while important for single adatom effects, can (to a reasonable approximation) be neglected in computing pair (and higher order) effects. Later studies discussed below (w 11.3.5) cast doubt on how well this result generalizes to models treating the d-band aspects of the substrate. Over the last decade or more, research in chemisorption theory has stressed generation of numerical results to fit quantitatively data from UV photoemission, ion neutralization spectroscopy, low-energy-electron-diffraction (LEED), and scanning tunneling microscopy (STM) experiments. The primary object has been to compute the spatial and energy distribution of the electron density near the surface region and to find exact locations of surface states. For these applications self-consistency (here in a Poisson's equation sense) is crucial. The first attempts to gauge the role of such effects considered the adsorption of single adatoms on jellium. In semiconductors it is difficult to propagate electrons from one adsorption site to another, from a physics viewpoint because the Fermi energy lies in the band gap, from a chemical perspective because electrons are relatively localized in covalent bonds. (Some implications are discussed in the next section.) Tosatti (1976) has considered the interaction between adatom pairs on Si(100)(2• assuming a short-range defect potential for the adatoms and linear response by the surface electrons. His pair interaction is always repulsive, oscillatory (in strength) with separation, but with an exponentially decaying envelope (due to trying to propagate electrons in the gap). Realistic slab calculations for transition and noble metals began appearing about a decade ago and are becoming more or less routine for flat surfaces. They are discussed at length in volume 2 of this handbook. Nonetheless, even today most total energy self-consistent calculations consider only a (1• overlayer, with the full symmetry of the substrate.
11.2.5. Lattice indirect interactions: phonons and elastic effects To check whether there were significant interactions mediated by phonon rather than electronic degrees of freedom of the substrate, Cunningham, Dobrzynski, and Maradudin (1973) studied the contribution to the free energy of the interaction between two identical adatoms via the substrate phonon field. In their model, the adatoms sit in the atop position on the (100) face of cubium. Results are computed as a function of the three dimensionless quantities: adatom mass over substrate mass, adatom-substrate coupling over substrate-substrate coupling, and inter-adatom separation R (in lattice constants). They find that the zero-point energy is invariably attractive and that it decreases monotonically in strength with R, going like R-7 for large R. The attraction is at most 10-4h (1) L (where O) L is the maximum phonon energy) or of order 10-6 eV, and thus nearly always negligible.
602
T.L. E i n s t e i n
Given these negative results, little further work was done on this problem. However, beginning half a decade later, considerable interest has been paid to elastic interactions on surfaces. When electronic interactions play a significant role, it is generally not just difficult but artificial to try to isolate elastic effects from other electronic effects. However, for non-metallic substrates, where there are no electrons near the Fermi energy, focus on elastic interactions may often be a fruitful perspective. Also, in the asymptotic region, the elastic interaction generally dominates for large enough separation. In this section we first give a chronological account of studies of this interaction between adatoms. We then discuss in more detail, via a few examples, the interplay between elastic and electronic effects on metal substrates, and why this perspective is often not fruitful at close range. Lau and Kohn (1977) investigate the long-range interaction between two adatoms due to classical elastic distortion of an isotropic semi-infinite substrate, finding: h
0,
_
_
_
, , ,
l
!
I
l
l l
l ~
/ /
#
I
x
,
.-xx\-\ I
...... ,...",,,..",,,. ",,,
I
" ".".,"./'1 I "',lil'J
\7\",,'.."'" " 77)""''
,,
z
z
z
1
1
1
1
~
L
l
~
/
\
l
\
,
\
.
\
(e) Fig. 11.4. Schematic of the origin of the elastic repulsion between like atoms on an elastically isotropic substrate. (a) Response of the substrate to a single atom. Here the displacement is taken to be away from the adsorbate, though it is more likely to be toward the adatom. (b) When two adatoms are present, substrate atoms between them cannot relax fully.
603
Interactions between adsorbate particles
E~I~,(R)_ 1 - o" A~At, 4x~t R 3 ' Aa = ~-" F~j . ( R j - R a ) J
(11.24)
where F~j is the force exerted by adatom a at Rj, ~ is the Poisson ratio, and IX the shear modulus. For identical atoms, this interaction is always repulsive, due to frustrated relaxation of substrate atoms between the two adatoms, as illustrated in Fig. 11.4. Taking ~ = 1/2 and IX --- 10 -3 atomic units as typical, they estimate this repulsion to be of order 0.1 eV at R = 10 a.u. --- 5 /~. (If the a d a t o m - s u b s t r a t e coupling on a triangular surface has the form-3~/g0 6, where R0 = I R j - Ral is the spacing between the adatom and one of its 3 substrate neighbors, then the "virial" (Stoneham, 1977) A = - 6 T a 2Ros.) For different adatoms, the elastic interaction can have either sign. Its R -3 decay is reminiscent of the dipole-dipole repulsion. Inserting reasonable numbers for Xe pairs on Au, Lau and Kohn find at R = 10 a.u. that
I.
Q. .-..-
\ \ \ \ \ \ ~
\\\\\X
,,\\.X,\~
1 1 1 1 1 1 1 1 1 1 T t t t l l l l l /
t!
r r i I I I I II I [
,tZTrrl
~a~3]~
t Y,ZX l I
'~
[. 1 I / / 7
LI.Z/7/
\.~\A A
T ,,-,,,.~,.,.',,, ",,,'X.\
,_-.~.-~i~....-j,/ ~--'~"~'?//
/ 1 I / ,/ I
///11111
,,.,-....-........-......,,....-,...-....-....-/,/ X,.X,"~'%"-" ' " - " / ' / / / , ' I
\-..~--~~ X ",.,'~"'~--~,
i i i { i I I I I i I ],I ~ \ \ k,',,", I I I I I I I I I I t { ~ ~,\\\\ \\
(b} Fig. 11.4 (continued). Caption opposite.
604
T.L. Einstein
the elastic repulsion is 0.53 meV, compared to their dipole repulsion of 1.1 meV, i.e., about half as large. On the other hand, at nearest neighbor sites it is three or four orders of magnitude greater than the phonon-mediated attraction just discussed; in essence Cunningham and coworkers' calculation (Cunningham et al., 1973) gives the leading quantum correction to the classical distortive effects. Varying the vertical position of the adatoms relative to the isotropic substrate, Maradudin and Wallis (1980) also find the R -3 decay but find that the interaction is attractive if the average distance below the surface is greater than R/2~. Stoneham (1977) shows that if the substrate or the adatom-substrate coupling is anisotropic, then the elastic interaction between like adatoms can again be attractive. He estimates that the magnitude of interaction of neighboring bridge-bonded H on W is of order 0.1 eV, large enough to account for some measured interactions without recourse to electronic effects. He also considers additional elastic effects due to clusters of adsorbates. Lau (1978) in turn considers anisotropic substrates w i t h hexagonal or cubic symmetry. Using Green's functions derived by Dobrzynski and Maradudin (1976) and by Portz and Maradudin (1977), he works out explicit formulas. For Xe pairs on graphite, separated by 5 ]k, he finds a repulsion of 0.18 meV. On Au (100), with pairs of Xe again 5 A apart, he finds an attraction of 0.30 meV along the cube axis and a repulsion of 1.73 meV at 45 ~ He expects this anisotropic behavior to be fairly general. While these energies are quite small, he expects elastic effects to become stronger and play a significant role in distortive phase transitions. Kappus (1978) rederives the previous results on isotropic and cubic substrates, finding again the possibility of homonuclear pair attractions on anisotropic substrates. Between clusters a repulsive barrier arises, proportional to the product of the areas of the clusters, even in directions in which the long-range interaction is attractive. Kappus (1980) extends this work to consider an anisotropic force dipole tensor, which enters the calculation of the virials, but restricts the substrate to be elastically isotropic, a reasonable approximation for W. Again there is the possibility of elastic attractions between like adatoms. The formalism is applied to explain the ordered p ( 2 x l ) phase of O on W(110) (Engel et al., 1975; Wang et al., 1978). He obtains "reasonable qualitative agreement" with the pair interactions used by Williams et al. (1978) in a Monte Carlo simulation of this system. However, since they do not lead to the p(2xl ) superstructure, Kappus (1981) generalizes the model to include a nearest neighbor interaction, an electric dipole repulsion caused by adatom dipoles normal to the surface, and another long-range part coming from elastic dipoles of nearest-neighbor pairs of adatoms. This third energy leads to multisite interactions. Nonetheless, with an E2 interaction, he cannot stabilize the p ( 2 x l ) superstructure; such an interaction could, of course, arise from the electronic indirect mechanism, from small anisotropy in the elastic constants, or from a breakdown in the continuum approximation. ~ 1 This system has proved quite challenging. Rikvold et al. (1984) used a model with E 1< 0, E2 > 0, E3, and trio interactions, and still found that an attractive E51% of E l could introduce pronounced first-order behavior at both low and high coverages.
Interactions between adsorbate particles
605
Theodorou (1979) proposes an intriguing approach to the overlayer structure of W( 110)-O. He noticed that on a rigid substrate the W - O - W angle of bridge-bonded O was 102.2 ~ rather than the ideal 90 ~ Presumably, then, these two W ' s would be drawn toward each other; there are two other W's, at the far ends of the "diamond" at the center of which the O sits, which are repelled by a lesser amount. From this perspective, he estimated energies per O of isolated atoms, the chain constituent of the p(2• the p(2xl) itself, and a full ( l x l ) to be 0.15 eV, 0.05 eV, 0.15 eV, and 0.29 eV, respectively. In terms of interactions, he essentially finds an attraction El = -0.10 eV which duly produces chains. A repulsion in a different direction keeps the chains apart. Unfortunately, some more distant (second-neighbor in some direction) interaction between chains is also repulsive, preventing the p ( 2 x l ) from forming. He speculates about what other interactions might overcome this repulsion, noting that the small work function change suggests that dipolar interactions are insignificant. Apparently no resolution of this problem was ever achieved and the paper seemingly has had little impact on research in adsorbate interactions, though perhaps it influenced thinking about strained superlattices in heterostructures (Tserbak et al., 1992). Tiersten et al. (1989) note that Kappus (1978) smoothly truncates 2D integrals over the surface Brillouin zone with a cutoff parameter of order the inverse lattice constant and that his interaction energies between adatoms separated by less than a few lattice spacings depends sensitively on this cutoff. Thus, they conclude that a lattice-dynamics analysis of the substrate is needed in the non-asymptotic range instead of the continuum elasticity approach. Working in a mixed representation (cf. just above Eq. 11.5) they find an expression for the pair interaction energy in terms of (Fourier-transformed) local force vectors associated with each adatom and a substrate propagator between the sites. This propagator they take to be essentially the inverse of the dynamical matrix. (In elasticity theory, the propagator is an angular-dependent term divided by the magnitude of the 2D wavevector; one then readily recovers the R-3 decay.) Tiersten et al. (1989) apply their formalism to As dimers on Si(100). They plot the interaction along the three principal directions, finding that it (1) can change sign with increasing R, (2) is highly anisotropic, (3) is rather small, less than 10 meV (often much less) once R >_ 8 ,~. They also look at interactions between H pairs on reconstructed W(100). Again they find that the interaction can be attractive or repulsive, that it depends on the direction, and is at most about 3 meV for the shortest R's, and becomes less than an meV quickly with increasing R. Presumably electronic effects are much larger for this case. In both cases the sign of the interaction at small separations can usually be understood in terms of the dominant forces on the substrate atoms or by simple arguments based on interference of the relaxations produced by the individual adatoms (cf. Fig. 11.4.). Later, Tiersten et al. (1991) consider Si(100)-O, finding generally similar qualitative features, but with larger magnitudes, around 50 meV at 4 ,~, but then falling quickly to less than 5 meV, then to less than a meV. In other words, when electronic interactions are present (on metals), they should dominate, but on semiconductors or ionic crystals, these could be the leading interaction. Recently Rickman and Srolovitz (1993) present a very general Green' s function
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T.L. E i n s t e i n
formalism for finding the elastic interaction between defects of spatial dimensionality D and multipole character m on a surface. Specifically, they tabulate results for four generic defects: a point force (D = 0, m = 0), an impurity adatom or island (D = 0, m = 1), a stress domain (D = 1, m = 0), and a step (D = 1, m = 1). Since defects in general involve more than the lowest-order multipole, the results apply for large lateral separation R. For point interactions (D = 0) between an m-pole and an n-pole, the interaction E(R) o~ R -(''+'~+~), reproducing R -3 for the interaction between adatoms. For linear defects, E(R) o~ R -(re+n), o r C 1 ln(R/a) + C2 for m = n = 0. Further comments related to steps are deferred to w 11.4.3. The preceding discussion assumes that one can neatly distinguish between electronic and elastic interactions. Such a distinction is generally possible at moderate-to-large separations between adatoms, but fails in the "near" region: in computing carefully the electronic interaction between adsorbates (Feibelman, 1989a), relaxations can play an important role. There is clear experimental evidence that adsorption can distort the substrate in the vicinity of the binding site, although the precise nature of the deformation may be difficult to determine. For example, for N i ( l l l ) p ( 2 x 2 ) - O Narusawa et al. (1982) measured, with high-energy ion scattering, outward displacements of about 0.15 ]k of the three Ni's to which each adatom binds (i.e. substantial buckling and overall relaxation); from LEED analysis, Vu Grimsby et al. (1990) note, in addition, lateral "twist" displacements of about 0.07 A. However, Schmidtke et al. (1994) find in a subsequent LEED analysis no twisting, minimal relaxation, but buckling of 0.09/~,. In a painstaking LEED survey of Ru(0001)-S, Pfntir's group finds progressively greater substrate distortions with structures of increasing coverage: for the p(2x2) there is slight buckling and outward relaxation, o f - 0 . 0 3 ~, (Jtirgens et al., 1994). In the (q-3-• symmetry forbids such buckling; the relaxation is still comparably minimal (JiJrgens et al., 1994). In the 1/2 ML c(4x2) phase, there is substantial (-0.2 /~,) row buckling (Schwennicke et al., 1994), Ru atoms bonded to two S's relaxing more than those bonded to one S. (Moreover, the S atoms occupy fcc and hcp sites with equal probability, but are shifted laterally from the high-symmetry 3-fold position by --0.16/~,!) Finally, in the (q-7-xq-7-) at 0.57 ML, there is even stronger dependence of the Ru relaxation on the S coordination: surface atoms with 3 S's relax 0.39 A more than those with a single S (although the overall relaxation is minimal) (Sklarek et al., 1995). (It is also noteworthy that in all cases the local chemistry is preserved in the sense that S - R u bond lengths do not change by more than 0.05 ./k!) Since these displacements are based on fits to data, accuracy depends on the insight and ingenuity of the experimentalist. Using Tensor LEED and scanning tunneling microscopy, Barbieri et al. (1994) investigate two of the four ordered overlayers of S on Re(0001) (Ogletree et al., 1991) and find a similar increase in surface distortions with increasing coverage. 1 Substantial displacements of surface atoms will certainly affect the electronic states nearby (and so the interaction energy) and
1 Einstein(1991) points out that several distinct trio interactions would be needed to account for these ordered phases; presumably some of these are related to the local distortions.
Interactions between adsorbate particles
607
evidently can depend on the separation between the adatoms. It is a futile exercise to sort out which portion of the interaction is elastic. As more specific systems are carefully documented, it will be interesting and important to look for trends in the evolution of buckling with coverage. 11.2.6. A s y m p t o t i c f o r m o f the indirect interaction b e t w e e n a t o m s a n d b e t w e e n steps
In this section we present more information than in w 11.2.2 about the nature of the indirect interaction between widely separated adsorbates. Our intention is to stress the general features and underlying physics while skirting explicit formulas, which can become quite complicated (Einstein 1973, 1978, 1979a; Lau and Kohn, 1978; Flores et al., 1979; Roelofs, 1980). From Eq. (11.16) we see that the asymptotic behavior hinges on the behavior of Go,(~) at large R, where R is the vector from site o to site n. While studying scattering in solids four decades ago, Koster (1954) recognized that with the competition of rapid oscillations, the solution required stationary-phase arguments. He uncovered much of the essence of our problem, finding that G,,,,(E) o,: R -I exp(i k(I;) 9R)
(11.25)
where k(e) is that wavevector along a constant-energy surface at which the velocity (viz. V~ke)is parallel to R, as illustrated in Fig. 11.5. Moreover, the proportionality constant varies inversely with the Gaussian curvature of the constant-energy surface at k. More generally, if Go,(e) o~ k -~ R-" e x p ( i k R ) , then integration by parts (Grimley, 1967) leads to the important result V4
--2
2
E, ----~--Re [G,,,(eF) Go,(~;F)]
(11.26)
and the interaction decays like R -(2re+l). For surfaces, one can show quite generally that m = 2, i.e. that G,,,(e) ,,,: k -~ R -2 e x p ( i k R ) (cf. the discussion in the paragraph after Eq. ( 11.28)) and E, ~ R25 cos(2kr.R, + r
(11.27)
if the interaction is isotropic. The complex quantity Go~ is independent of the separation and so leads to the phase factor ~; from Eq. (11.14), this factor is given explicitly by (Joyce et al., 1987) = arg(G,,)2 = 2 arg [l~F- e , , - V 2 Goo(eF)] -l
(11.28)
which vanishes when V2npo(eF) << e F - e,,- V2ReG00(eF), e.g. when the coupling is weak or the adatom level is far from the Fermi energy. Gumhalter and Brenig (1995a) emphasize that the phase factor only appears in nonlinear theories. In studying the static response function, Rudnick (1972) found essentially this result for jellium confined by an infinite barrier. He interpreted the sinusoidal
608
T.L. Einstein k 3
"
)
kx
-.3
-2
-1
0
1
2
,3
Fig. 11.5. lllustration of the wave vector which dominates the asymptotic interaction. The curves indicate constant-energy lines in the surface Brillouin zone, in the lower third of the band of a simple cubic crystal in the tight-binding model. The dots denote sites in real space. The dashed line shows the R connecting the origin with a particular site. For ~ =-2.1, the arrows show the velocity Vk~ of two candidates for this wave vector; it is not the k at which E(k) intersects R, but rather the lower one at which VkC is parallel to R, which enters Eq. (11.25). For isotropic systems, the contours become circles (as near the bottom of the band, as depicted for E = -2.8), and there is no distinction between the two candidates. When the Fermi curve lies on more than one "sheet", one must sum over the contributions from each k with VkE parallel to R.
variation as a Friedel oscillation of the screening charge around a point impurity. Similar behavior was also found by M o o r e (1976) and Flores et al. (1977a). The response at point n due to a disturbance at point o of the infinite-barrier system can be described in terms of the bulk responses from the disturbance at o and from a c o m p a r a b l e disturbance at the mirror image of o, with the opposite sign to p r o d u c e a node along the barrier (Flores et al., 1979). Then the leading R -~ contributions to G,,, cancel, leaving the next order, with coefficient R -2, to dominate. Lau and Kohn (1978) verify that a similar asymptotic interaction occurs for a jellium substrate even if the barrier is finite. Treating the adatom-substrate interaction in s e c o n d - o r d e r perturbation theory, they can separate the pair interaction energy from the adsorption energy of single atoms, analogous to what was done above for the tight-binding model. The R dependence of the pair interaction is given by d z k, elk' R G (k,),
( 11.29)
where G(k,) is a kernal which depends only on the substrate energy spectrum (here of the free electron form). After detailed analysis, they find that asymptotically the integral is dominated by a singularity in the fourth derivative of G (k,)12k,~ times a unit step function. Using the results for generalized functions given by Lighthill (1958), they reduce behavior to the form of Eq. (11.27).
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609
Lighthill's formulas can be applied more generally to find the asymptotic form of the Gon(e), particularly in the tight-binding model; there is no need to assume 2D isotropy, weak coupling, or large separation between eF and ea, viz. ionic bonding (Einstein, 1973, 1978, 1979a). (From another approach with different expansions, Le Boss6 et al. (1979) rederive most of Lau and Kohn's results without the latter restrictions.) The final expressions are rather cumbersome. An interesting qualitative aspect is that the use of surface rather than bulk Green's functions means that one must be somewhat cautious in applying Koster's approach. (Cf. Flores et al. (1979) for some details of the application to surfaces.) Only kll is a good quantum number (since the surface destroys crystal translation invariance in the k• direction), so we have a Fermi "loop" rather than a surface. For each kjl, however, the kernel of the surface Green's function can be rather simply related to those of bulk Green functions (Kalkstein and Soven, 1971). For the (100) face of a simple-cubic tight-binding model, it turns out that there are essentially two subbands, each with 2/3 the bulk bandwidth B and centered at B/3 from either band edge. Thus, in the central portion of the band, there are essentially two sheets to the Fermi surface, so that there are two special values of kll to be considered. For a more realistic model of the substrate, there might well be even more. On the other hand, near the bottom of the band, the tight-binding dispersion relation simplifies to the parabolic form; Lau and Kohn (1978) manipulate the Anderson model (in essence, applying the Schrieffer-Wolff (1966) transformation) to make contact between their free-electron calculations and the earlier tight-binding work. (See also Einstein (1978).) An R -5 interaction at large separations is clearly of rather academic interest. However, in the infinite-barrier model, prompted by Hjelmberg's (1978) explicit numerical calculations, Johansson (1979) and Johansson and Hjelmberg (1979) notice that in addition to this "far asymptotic" region, there is a region with bulk R -3 interaction occurring for much smaller values of R, still larger than k~:~ but smaller than the distance to the barrier, so that the image does not cancel the leading term due to the atomic charge. (Cf. comments after Eq. (11.28).) When R is a few lattice spacings (specific range dependent on the electronic charge density, i.e. kF, of the jellium), there typically is a crossover region between the bulk-like and far asymptotic limits, with a decay exponent varying continuously from 3 to 5. Le Boss6 et al. (1978) also find the R-3 decay, but do not report the transition or far asymptotic region and attribute (Le Boss6 et al., 1979) the lack of R-5 decay to other factors. Seemingly the latest word on this problem is Eguiluz and coworkers' numerical treatment (Equiluz et al., 1984), based on a Kohn-Sham self-consistent approach, of two charges in AI- and Na-like jellium. They recover Lau and Kohn' s result (Lau and Kohn, 1978) as the dominant result for contributions to the response function from wavevectors at least 2kF, but their calculations show that this weak oscillatory term is masked by a much larger, monotonically-decaying attractive interaction due to smaller wavevectors. (The singularity in the integrand at 2kF is numerically invisible.) When the charges are placed outside (inside) the jellium, the direct Coulomb repulsion overwhelms (roughly compensates) this attraction, which is presumably a manifestation of the polarization screening. In their range of study (which does not reach the "far asymptotic" regime), they also see the R -3 decay of
610
T.L. Einstein
the envelope of the oscillatory part. The oscillations are observable in the total interaction only when the charges are inside the jellium, and are strongest when the charges are near the surface, with initial oscillation amplitudes somewhat larger than 10 meV (larger for "Na" than "AI"). Although for charges outside jellium there is no observable evidence of long-range oscillatory interactions due to polarization of the substrate electron gas, the authors carefully note that their model does not allow for electron exchange coupling to the substrate expressed in Eq. (11.5) In the asymptotic regime, the lateral interaction may be analytically tractable but is generally insignificant. At short distances, the interaction is far more complicated, since it depends on all the occupied states and not just those at one point on the Fermi surface. By the distances that the asymptotic form dominates, the interaction is quite small and is often masked by other interactions decaying like R -3. Thus, a particularly significant result of Lau and Kohn is: if the indirect interaction is mediated by a surface state (assumed to be circularly symmetric), the singular nature of expression (11.29) appears in the first derivative of G(kll) at 2kF, and the prefactor of cos(2krR) in Eq. (11.27) becomes R-2! Of course kF is now associated with the cylindrical Fermi surface. The slower decay should in retrospect be not so surprising, since the curvature of the cylinder vanishes along the axis direction, so that the form derived by Koster would diverge. For the more general cases of a (2D) hexagonal or square tight-binding substrate, Volokitin (1979) and Braun and Medvedev (1989), respectively, also find asymptotic R-2 decay, the latter suggesting that such behavior might be seen on Re (0001). With the axis of the cylindrical Fermi surface parallel rather than perpendicular to the physical surface, as obtained by adsorption onto the edge of a semiinfinite square tight-binding net, Braun (1981) and Braun and Medvedev (1989) find R -4 and R-2 for the "surface" (edge) and "bulk" contributions, respectively, to the interaction energy; the physical analogue is Re (1010) in the [1210] direction. Lau and Kohn (1978) also consider a model in which the Fermi surface is defined by two exactly parallel planes spaced Ak apart in the x direction" they find Ex ~ x-~cos(Akx). Braun (1981) and, more explicitly, Braun and Medvedev (1989) illustrate this decay for the case of a tight-binding chain as the substrate. In rederiving these asymptotic behaviors, Flores et al. (1979) also find a fractional exponent for a conical Fermi surface. Lau and Kohn's idea of mediation of interactions by quasi-one-dimensional states with consequent x -~ decay has captured the imagination of many for years but has only very recently been applied to a physical system: Ni(110)-H (Bertel and Bischler 1994; Gumhalter and Brenig 1995). This speculative recent work is discussed at the end of w 11.4.3. Further analytic progress was achieved by Brodskii and Urbakh (1981). (For a more general theoretical review from their perspective, see Urbakh and Brodskii (1985).) They note that in the Lippmann-Schwinger integral equations underlying the formalism in w I 1.2.2.2, behavior is dominated by poles in the Green' s functions at the resonance energies of the closely-spaced and the infinitely-separated adatoms, as well as by the singularity in the energy spectrum of the substrate. Making a zero-range potential approximation, they recover the structure of the asymptotic form reported by Einstein (1978). They also obtain a somewhat similar expression
Interactions between adsorbate particles
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assuming a separable potential. More remarkably, they derive, with suitable approximations, analytic expressions for the "intermediate asymptote" regime, for smaller separations than the asymptotic regime. In trying to make contact with experiments, they consider both wide and narrow bulk bands, surface bands, and possibly different species of adatoms. Including the possibility of interactions between heterogeneous pairs, they produce a table with at least four different power-law decay exponents. Apparently independent of all the above work, Ebina and Kaburagi ( 1991 ) apply methods of Brovman and Kagan (1974) for finite-temperature Green's functions to study interactions on jellium substrates. By approximating surface electrons as two-dimensional objects, they implicitly focus on interactions mediated by surface states. From a step-like anomaly in the second-order susceptibility, they find a trio interaction dominated by wave vector kFf3-. In their calculations, there is also a contribution from wavevector 2kF. Seemingly this competition in the asymptotic regime reflects the two interaction terms in Eq. (11.21). While most of this chapter is devoted to individual adatoms on flat surfaces, it is worth mentioning some relevant results for vicinal surfaces, i.e. surfaces misoriented slightly from high symmetry directions. On semiconductors, the energetic interactions between steps, if noticeable, are repulsive. In contrast, on metals evidence is emerging that the interactions can be oscillatory in sign. We discuss this novel application further in the w 11.4.3.
11.3. Attempts to model real systems
11.3.1. Tight-binding, jellium, and asymptotic-ansatz The philosophy behind the single-band tight-binding calculations of ES is that the d-band is primarily responsible for the lateral indirect interactions. Burke (1976) raised doubts about the adequacy of this idea, even for refractory transition metals like W, by performing more realistic tight-binding calculations with a five-fold degenerate substrate band. His goal was to reproduce the pair data for transition metals on transition metals, gleaned from experiments performed by Tsong and coworkers (1973, 1975), Bassett (1975), and Graham and Ehrlich (1974) using field ion microscopy. Generalizing ES, Burke first showed how five-fold degenerate adatoms may be incorporated into the formation starting with Eq. (11.9) by an orbital-peeling matrix procedure. The idea was (1) to focus on one of a pair of nearby adatoms, (2) to remove it from the system orbital by orbital, and (3) to replant it infinitely far away (as though there had been originally five single-level adatoms rather than one at each site of the pair). In this procedure, the bulk is unspecified. In actual calculations, the substrate was the (100) or (II0) face of a semi-infinite bcc crystal, with adatoms imagined as the same element (viz. W) as the substrate and sitting in the otherwise vacant lattice sites above the surface. All diagonal matrix elements are set at the energy zero; the possibility of having to modify what amounts to % = 0 is discussed and dismissed, thereby neglecting
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T.L. Einstein
self-consistency corrections completely. Overlap is also excluded. Slater-Koster (1954) matrix elements between nearest and next-nearest sites are calculated in terms of the two sets of 3 d-d tight-binding parameters. Alas, these six values are simply scaled up from narrow band values (Pettifor, 1969), ignoring any details of hybridization with s-p electrons. To compute the substrate Green's functions, Burke combines the continued fraction approach with a scheme counting poles and zeros on the real energy axis. When the adatoms are separated by just a (bulk) nearest or next-nearest neighbor distance, a direct interaction between them can (and should) also be included. With no direct interaction, the results look similar to those of ES: oscillatory in sign, peaking in strength when eF is near Ea. Inclusion of the direct interaction makes a substantial difference for small interadatom separation, leading to an attraction at all eF due to the bonding between the adatoms themselves (Desjonqu~res and Spanjaard, 1993). Overall, Burke reconfirms that the pair interaction energy has roughly the same size as in ES and that it oscillates as a function of eF for fixed separation and as a function of R for fixed eF. In a rather cursory look at decay with separation, Burke finds much faster fall-off on the (110) face than the (100); there are no analytic results. Burke was disappointed to find the calculated pair binding of a nearest-neighbor dimer on the (110) surface to be nearly five times the experimental value of 0.3 eV (Tsong, 1973; Tsong et al., 1975; Bassett, 1975). Another difficulty is that subsequent work suggests that the adsorbed W sits in a "surface site" rather than a "vacant lattice site". Burke alleges this makes little difference but gives no supporting evidence; in light of the results of ES - - cf. Table 11.1, especially the difference between the two bridge configurations m this insensitivity is surprising. Burke suggests a number of sources of error, but aside from adding a Coulomb counter term or massaging parameters, it is not clear how to improve matters. His dissatisfaction with this approach was heightened when he could not explain the ordered phases of Ni(100)-O (Holloway and Hudson, 1974; Demuth and Rhodin, 1974); in this case the computed strength is typically much too small, of order 1-10 meV, to account for disordering temperatures. As discussed in w 11.2.6, Flores et al. (1977a), Lau and Kohn (1978), Le Boss6 et al. (1978, 1979), Johansson (1979), Johansson and Hjelmberg (1979), and Eguiluz et al. (1984) show that with a jellium substrate there are also ~ndirect interactions of substantial magnitude. The last group in fact explore the interaction between two protons on/in AI as a function of separation, for several distances from the surface. A major result of the latter four of these studies is that the ultimate [R-Scos(2kFR)] asymptotic regime is not reached till separations R so large that the interaction is negligible. At shorter spacings, the interaction goes first like R-3cos(2kFR), as in the bulk; for larger R, typically those of most interest, the decay exponent increases smoothly to the surface value of 5. Rogowska and Wojciechowski ( 1989, 1990) use these ideas to consider noble-metal adatoms on jellium, using the exponent 3 exclusively (for separations larger than suggested by Johansson and Hjelmberg (1979)), offering ways to compute the charge density, from which to get kF. While this approach may be reasonable for free-electron-like substrates, the eventual application to a W(110) substrate (Rogowska and Kolaczkiewicz 1992) seems questionable.
Interactions between adsorbate particles
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Proposing an alternative to the cluster-based picture of H - H interaction of e.g. Grimley and Torrini (1973), Flores et al. (1977b) consider the system Pt(111)-H. From a variety of experimental evidence, they argue that the H atoms sit in regions of high electron density, viz. center sites. They then invoke (without justification and rather implausibly) the asymptotic form of the interaction, Eq. (2.27), for five structures involving separations between one and two Pt nearest-neighbor spacings. Taking into account the coverage of the ordered phase and the idea that the assumed potential form must have a minimum at the pair spacing of the ordered structure, they deduce this spacing and the associated graphitic (2• overlayer, found subsequently by LEED (Van Hove et al., 1978). Noting that for most of the structures there are at least two identical nets with the particular choice of neighbor spacing, they argue for the existence of two different binding states, consistent with experiment (Christmann et al., 1976). Consistent with experiment again, these become identical at monolayer coverage, when both nets are complete (Burch, 1980).
11.3.2. Embedded cluster model Muscat (Muscat, 1985a; Muscat and Newns, 1981) was the first to allow explicitly for contributions of both free and d-like electrons in producing lateral interactions between adatoms, in his case H atoms. In his embedded cluster model, spheres are centered on the sites of the H adatoms as well as on several nearby metal atoms in the substrate. Within the latter muffin-tin spheres, he places self-consistent bulk band-structure potentials (Moruzzi et al., 1978). The spheres are then embedded in some model of a free-electron gas, usually infinite-barrier jellium. (In some later work, the jellium contribution is taken from effective medium theory (NOrskov, 1982; Nordlander and Holmstr6m, 1985), discussed below.) The d-wave contribution comes from the l = 2 solutions. Again, interaction energies are calculated from changes in all the one-electron energies. The technique was applied to a wide variety of late-transition and noble metal substrates. Pair interactions generally have the correct sign and order of magnitude to corroborate the energies deduced from Monte Carlo simulations of the experimental phase diagram (but were often off by factors of very roughly 3). In these calculations the distance d between the H proton and the jellium edge (taken as a plane half-way between the surface atoms and what would have been the next plane above the surface (Muscat, 1986)) is the only explicit adjustable parameter. By quoting the results for a few values of d, Muscat gives some idea of an intrinsic uncertainty in this approach. While the variation is not negligible, the qualitative and usually semi-quantitative results are not overly sensitive. More specifically, this method also evaluates interaction energies based on integrations over one-electron phase shifts. For a pair of adatoms, an explicit subtraction is required (in contrast to Eq. (2.15)). As an illustration, the phase shift for a single H muffin tin centered Zo from the infinite barrier is rl(e) = tan-'
1 -j,,(2kzo) cot 80 - no(2kzo)'
(11.30)
614
T.L. Einstein
where 50 is the phase shift of a single muffin tin in a free-electron gas. For two H atoms, the spherical Bessel function jo(2kzo) is replaced by jo(kR)- jo(k~[R2+4z2), similarly for the spherical Neumann function no(2kzo), and there is a second tan -l term in which these Bessel functions are added rather than subtracted. To calculate 8o one must stipulate the spherical potential in the muffin tin; Muscat and Newns (1981) choose a simple exponential with a prefactor set to produce a bound state just below the bottom of the sp-band of Ni. On jellium (at the separations of C sites on Ni( 111 ), neglecting any difference between "fcc" and "hcp" sites), they then find E~ = 900 meV, E2 = - 7 0 meV, and E 3 = - 3 5 meV. Since E2 < E3, these interactions would lead to a ( l x l ) rather than the experimentally observed graphitic (2x2). Next, they add a (hexagonal) cluster of (seven) spheres at the appropriate positions to represent the top layer of Ni. The effect is calculated using standard multiple-scattering (KKR) techniques, using only the l = 2 component of the Ni-centered spheres. To the relatively simple argument of tan -~ above, one subtracts summations over products of scattering matrix elements. In this case, for d = 0, they now find E~ = 450 meV, Ez = 2 meV, and E 3 = - 9 meV. (As a measure of sensitivity, for the largest Idl, d = -0.4, E2 = -10 meV and E 3 = - 1 9 meV. For Cu with d = 0, E~ = 600 meV, E2 = - 2 meV, and E 3 = 0 meV.) To calibrate these numbers, Bartelt et al. (1983) show with Monte Carlo simulations that the transition temperature at the saturation coverage of the overlayer (i.e. the correct number of adatoms to form a defect-free overlayer) is below 100 K, far below the value of 270 K determined in LEED experiments by Christmann et al. (1979). Extending earlier work (Muscat 1984b, 1985b), Muscat (1986) gives the most comprehensive results, treating the close-packed faces of seven substrates: Ti, Co, Ni, Cu, Ru, Rh, and Pd. He first shows that the fcc site is more favorable for H adsorption than the hcp site (by about 10 meV for Ni up to 137 meV for Ru), except for Cu. Most of this energy is due to one-electron contributions, computed as described with phase shifts. To assess Coulomb corrections due to changes in electron density at the adsite, an effective medium function is computed. This correction is roughly an order of magnitude smaller, around 10 meV. The principal goal was to evaluate the relative stabilities of the ground states of various possible ordered overlayers. In this regard, pairwise interactions alone, out to sixth neighbor (i.e. third neighbor for the Ni's or the same kind of 3-fold site), suffice. For Ni, Co, Ru, Rh, and Pd, the dominant interaction is an attractive sixth neighbor (producing (2x2) islands). An unsettling feature of the numbers is that the interactions do not tend to decay with increasing separation; these attractions have the largest magnitude, except for the very-short-range enormous El repulsion. For P d ( l l l ) - H , Muscat confesses substantial problems in comparison with experiment: his interactions are quite small, dominated by E 6 = - 1 0 meV, which would produce a p(2x2) rather than the observed graphitic (2x2). He also notes in comparison with EAM calculations by Foiles and Daw (1985) that he does not obtain subsurface occupation, which they found to be crucial. On the other hand, in reexamining the system Ni(l 11)-H with larger clusters, he basically reproduces his earlier values for E~, E2, and E 3 but now cites small E4 and E5 repulsions and a remarkable attraction E 6 - - 1 8 meV (at an H - H separation twice the substrate
Interactions between adsorbate particles
615
nearest-neighbor distance). He claims "excellent agreement with experiment" (Christmann et al., 1979). This conclusion provides an opportunity to warn the reader that there are so many degrees of freedom in these systems that one can easily be tempted into unwarranted enthusiasm. In this case, Muscat obtains not only the correct ordered state, a graphitic (2x2), but even a good estimate of the disordering temperature for the saturation coverage. However, a more detailed look at the experiment reveals a fairly broad (2x2) region in the phase diagram which disorders continuously to a disordered state. In a Monte Carlo simulation using Muscat's interactions, Roelofs et al. (1986) find, in contrast, a very narrow pure (2x2) region surrounded by very broad coexistence ("island") regions produced by the anomalously strong 6th-neighbor attraction. Roelofs (1982) and Nagai (1984) both wrote down sets of interactions based on fits to the whole phase diagram rather than on any microscopic computation; their sets had relatively weaker E6 interactions and stronger shorter-range repulsions. Another problem with the embedded cluster model is that it is expensive to extend the clusters, since the number of spheres grows rapidly with number of shells. Reduced symmetry in the clusters severely complicates the calculation, making it taxing to include local distortions. In general, there is no unambiguous way to find the parameter d nor to assess the accuracy. It is not clear what would happen for a transition metal with wider d-bands or for a more complicated adsorbate. In spite of these criticisms, I hasten to add that these calculations were the state-of-the-art in their time. They made several clear predictions and usually produced energies with sensible magnitudes.
Special cautionary case Fe(110)-H Muscat's (1984a) most extensive tabulation of trio energies is in his treatment of F e ( I I 0 ) - H . Experimental determination of the adsorption site was problematic. Adsorption was first thought to occur in the long-bridge site (based on LEED (Imbihl et al., 1982)) and then in the short-bridge site (based on EELS (Bar6 et al., 1981)), a conclusion consistent with Muscat's calculations. The system has an interesting phase diagram with a phase transition that was thought to be highly unusual. Painstaking calculations (Kinzel et al., 1982; Selke et al., 1983) using lattice gas models were performed to elucidate the system. Eventually, however, they were supplanted by the conclusion from detailed LEED work (Moritz et al., 1985) that H sits in the quasi-three-fold site and that the ordered phases observed in LEED have different real-space symmetry. 11.3.3. Effective medium theory and embedded atom method m semiempiricism The first of the semiempirical methods (NCrskov 1977), effective medium theory (EMT) begins with the self-consistent calculation of the function Ec.z(P) o f a n atom i, with nuclear charge Z i, in a homogeneous electron gas of density p. (This laborious calculation need be done only once. While this procedure to get Ec.z(p) is typical, NCrskov (e.g. 1993) mentions an alternative.) For non-noble-gas atoms, these functions have a simple shape with a single minimum at a value on the order
616
T.L. Einstein
of 0.1 bohr -3. In a solid each atom sits in the tails of the electronic charge density of its neighbors_ The total energy of asolid is then approximated, to first order, by the sum of Ec.z,(P) for all atoms i. Here p is the average over the atomic sphere. There are correction terms due to electrostatic effects (discussed in w 11.2.1) and to changes in one-electron energy sums, say between an adsorbed system and the same atoms before adsorption (essentially the topic of w 11.2.2). An early application was to adsorption on jellium (NCrskov and Lang, 1980), with later studies of transition metal substrates, reviewed by NCrskov (1994). Variants of this method are the quasi-atom approach (Stott and Zaremba, 1981) and the corrected effective medium theory (Raeker and DePristo, 1989, 1991). Applications of EMT to adsorbate-adsorbate interactions has centered on the issue of poisoning and promoting by preadsorption (NCrskov 1993, 1994). Most of this interaction is electrostatic and was discussed earlier in w 11.2.1. From another perspective, the role of rehybridization, i.e. altering of the chemical nature of the adsorption bond, has been stressed by Feibelman and Hamann (1984) and by MacLaren et al. (1987). These calculations focus on the change in the local density of states around metal atoms due to preadsorbed electropositive or electronegative atoms. Part of this change is electrostatic, but the rest, in EMT language, must be attributed to one-electron effects, i.e. the sort of covalent aspects discussed in w 11.2.2. There is a general result that emerges nicely in EMT (NCrskov 1993) that will be useful in later analyses. If one plots the cohesive energy of an fcc metal as a function of coordination number for fixed interatomic spacing, the curve does not decrease linearly as it would for a simple nearest-neighbor pair interaction model. Instead the decreasing curve has a positive second derivative. The effective pair interaction, defined as the derivative of this curve, is therefore enhanced for small coordination numbers (below 5) and diminished for larger such numbers (above 5 or 6), compared to (1/12) the bulk cohesive energy. Thus, interactions on surfaces should be considerably smaller than one would predict based on near-neighbor bond models gauged by bulk cohesive energy. The semi-empirical embedded atom method (EAM) (Daw et al., 1984, 1993; Foiles et al., 1986) has offered a relatively easy way to contend with the low-symmetry problems. In this approach, the cohesive energy is written
where the p"'s are spherically-averaged computed atomic electron densities, the prime on the summation indicates j = i is excluded, and U is the electrostatic Coulomb repulsion ZiZF2/IRol. The effective charge densities Z inserted into U are determined by the formula Z(R) = Z0(l + 13zR)exp(-o~fl). The embedding energy can be determined numerically by embedding an atom in a homogeneous background, as in effective medium theory (NCrskov 1982, 1994) or by using the "universal" binding curve of Rose et al. (1984)" E(a) = -Eh( 1 + a*) exp(-a*), a* - (a - ao)/)~
(11.32)
Interactions between adsorbate particles
617
where a 0 is the equilibrium separation. Typically, the parameters are adjusted to fit the bulk properties such as lattice constant, cohesive energy, and elastic constants (e.g. ~, can be obtained from the bulk modulus). For adsorption of one species on another, one can fit adsorption position and vibration frequencies (Voter, 1987). In the "glue model" (Ercolessi et al., 1986, 1988) (and for Voter-Chen (1987) potentials) one fits, in addition to F and U, the atomic density p (Tosatti and Ercolessi, 1991). The fact that fitting functions are not uniquely specified leaves the method vulnerable to criticism, but alternatively can be viewed as a strength in that one can tailor functions for specific applications and gauge uncertainties by use of different sets of functions. In contrast, in an earlier but similar scheme, Finnis and Sinclair (1984) took the functions F to be proportional to the negative square root (of p), as they would be in the lowest approximation to a tight-binding model. Since the numbers produced by EAM and similar calculations can be tuned somewhat arbitrarily, they are most useful in identifying trends and rough magnitudes that do not depend on the detailed choices. For high-symmetry systems more exacting band-structure techniques become feasible and offer more reliable information. (However, the flexibility of EAM can often lead the practitioner to unexpected structural revelations that might elude someone calculating with a scheme that depends on human ingenuity to determine the likely choices for equilibrium sites. For example, for P d ( l l l ) - H the existence of subsurface sites and their domination of the interactions needed to describe the phase diagram (Felter et al., 1986; Daw et al., 1987) were discovered "by accident" during dynamical simulations!) EAM is quite helpful in assessing the effects of coordination number on bonding. This theme underlies more exact work by Feibelman, to be discussed below. The driving program developed at Sandia-Livermore easily allows for substrate relaxations or for preventing the motion of any atom in any direction. On the other hand, since there is no Fermi surface in the method, EAM cannot describe any effect involving Friedel oscillations, such as the asymptotic form of lateral interactions. In EAM calculations of N i ( l l l ) - H and Pd(100)-H, Einstein et al. (1990) assessed the ability of EAM to predict lateral interactions. The origin of the interaction in this framework comes from the change in the argument of the embedding functions of the atoms in the cluster of atoms in the vicinity of the adatoms. Presuming the overlap of the adatoms is negligible and their atomic density decays fairly rapidly, the primary contribution to the interaction, in the EAM formulation, comes from substrate atoms "touching" both adatoms. Specifically, by expanding the embedding functions, we focus attention on the effect of a small increase in density due to a second adatom adding density to a substrate site (Foiles, 1985): 2
(11.33) J
To lowest order, the positive curvature of F(p) leads EAM to predict repulsive interactions, with their magnitude proportional to the number of shared substrate
618
T.L. Einstein
nearest-neighbors (except at the shortest separations, when direct interactions can overwhelm the physics). We will find a similar result below in a second-moment, tight-binding picture (cf. Eq. (11.37)). In first-principles calculations Feibelman (1988b, 1989a) found such a repulsion for AI-S and AI-Te dimers on AI (100) and for H-S on Rh(001). Furthermore, for trio interactions, EAM is rather insensitive to the configuration of the trimer: because the charge densities are spherical, the only dependence of the trio interaction on the angle (as opposed to the length of the legs) of the trimer comes from substrate relaxations. Such relaxations are particularly small for closepacked (111) fcc surfaces (cf. Wright et al. (1990), discussed near the end of w 11.3.5, for evidence that a (100) fcc surface can be expected to have significant relaxations.) Thus, for (111) fcc surfaces, in EAM one can to good approximation replace explicit treatment of trios (and higher-order multi-atom terms) by pair energies which depend on coordination (Fallis et al., 1995). While this approximation may be reasonable for the systems for which EAM works well, viz. late transition and noble metals, it is unlikely to be viable for most refractory transitionmetal systems (even their closest-packed (110) bcc faces), since angle-dependent bonding is important; cf. w 11.3.5. On the other hand, when viable it can be very helpful when doing simulations, and it highlights the idea that high-coordination atoms bind less strongly to another atom than low-coordination atoms. In the bond-saturation model (BSM), one posits that the cohesive energy of each atom depends only on its (nearest-neighbor) coordination. For the particular case of Pt( 111 ), Fallis et al. (1995) report that the bonding of adatoms can be characterized simply by a quadratic expression A + Bz + Cz 2. Here z, the coordination number within the adlayer, ranges from 1 to 6, A =-961.423 meV, B = 97.5456 meV, and C = -4.86116 meV. The positive value of B is a reflection of the positive curvature of the embedding function in EAM. The strength of the interaction between two adatoms with coordinations zt and z2 is then the average A + B(z~ + zz)/2 + C(z 2, + z~)/2. The same tactic can be applied more generally to problems in growth, where the total coordination rather than just that in the overlayer is considered. The dependence of bond strength on coordination was already discussed early in this section in the context of EMT and is reconsidered in w 11.3.6. For Ni(l 1 I)-H only the first-, second-, and third-nearest neighbors are above 1 meV, since only these involve shared substrate atoms. Their magnitudes are comparable to those found by Muscat (1984b, 1985b, 1986) but all are positive, consistent with behavior deduced from Monte Carlo fits of the phase diagram (Roelofs et al., 1986). We find a tiny attractive trio interaction for the smallest equilateral triangle of adatoms in the same kind of three-fold site, comparable in size to that found by Muscat but of the opposite sign. Overall, the signs of the interactions seem more reliable than Muscat's, and there are no anomalous attractions, but since the second-neighbor repulsion is less than 3/2 of the third, a p(2x I) overlayer is predicted instead of the observed graphitic (2• (or (2x2)-2H) (Christmann et al., 1979). Truong et al. (1989) extend EAM to a procedure called EDIM (embedded diatomics-in-molecules); they obtain magnitudes for the lateral interactions more consistent with expectations from experiment, but with the same
Interactions between adsorbate particles
619
sign as we found. There are a number of modifications, with no commentary on the effect of each. A likely possibility is the allowance, for Ni's in the top layer, of a different number of s-electrons from the bulk value. Since EAM successfully treated alloying at surfaces and phase transitions of one noble metal on another (Foiles 1987), we expected (Einstein et al., 1990) that late transition metals adsorbed on each other would be more accurately described in EAM. Wright et al. ( 1990)' s studies of Pt, Pd, and Ni on Pt(100) bear out this belief. We defer this discussion, as well as comparisons between EAM and tight-binding results, until the end of w 11.3.5. Another issue of concern for adsorbates is the large charge gradient near surfaces. For the reconstruction of Au(110), EAM predicts (Foiles 1987) a (lx3) pattern rather than the observed (lx2). To rectify this problem, Roelofs et al. (1990) include the leading correction from such gradients, using Daw's (1989) modification of EAM formalism. Moreover, to treat this system with Monte Carlo simulations, they decompose the interactions of Au atoms in the top layer, finding that not only are trios significant, so are "quartos" (i.e. the interaction energy of four surface atoms minus the constituent pairs and trios); even the close-packed "hexto" interaction has a strength -3.4 meV. To assess the role of the gradient contribution, I quote some numbers for pair interactions I computed in an early stage of this project before the corrections were implemented: for adatoms on neighboring rows, at the same position along the row or shifted by one unit (so somewhat diagonal), the interactions a r e - 1 0 and +17.6 meV, respectively, without the gradient term v s . - 2 . 6 and +12.3 meV with corrections. In short, the gradient corrections do not change the qualitative results but are important for quantitative assessments. More recently, Haftel (1993) proposed that many of the problems in applying EAM to surfaces could be cured by increasing the curvature of the embedding functions F(p), particularly on the low-density side of the bulk value. The impact of this procedure on pair interactions has not yet been explored.
11.3.4.Empiricalschemes To illustrate why EAM and related calculations are called semi-empirical rather than empirical, in spite of the several adjustable parameters, we present an example of a truly empirical scheme. To take advantage of the success of computationally intensive schemes such as FLAPW (Wimmer et al., 1981) to compute details of monolayer adsorption, Gollisch (1986) constructed an effective potential Ui, a generalization of the Morse form, with several parameters to be fit to the numerical "data":
U~=~"b~J[Q~176176176176 ~ ~ , j
(11.34)
The two global parameters s and It, on which the quality of the potential depends sensitively, adjust the exponents of competing terms. Three more parameters, a, b, and ~, adjust the scale and exponent of a separation-dependent interaction function Q, here a sum of two exponentials, introducing four more parameters. These seven
620
T.L. Einstein
parameters, computed from bulk properties, are tabulated for each element of interest. The off-diagonal a, b, and ~, (i.e. those for differing atoms) are computed as the geometric mean of those values for the two constituents. As a test of the accuracy of the numbers produced by this scheme, Roelofs and Bellon (1989) compute the resultant phase diagrams, using transfer-matrix finitesize scaling, for Cu and Au on W(110). In both cases the pair interactions for the three shortest separations are all attractive, indicating the formation at low temperatures of coexistence between a low-coverage lattice "gas" and a high-coverage lattice "liquid", both with ( l x l ) symmetry. Accepting Gollisch's pair values, they try to fit the experimentally determined phase boundary by tuning the multisite interactions. For W(110)-Cu the "quarto" interaction is negligible while they estimate that the (repulsive) trios underestimate the actual values by about a factor of two. For W(1 10)-Au they find that if they include the repulsive trio interactions computed by Gollisch (as well as the attractive pairs) then to fit the temperaturecoverage phase boundary would require a repulsive rather than the computed attractive quarto (with a magnitude at least a third smaller). 11.3.5. Field-ion microscopy, modern tight-binding, and more on semiempiricism
While field ion microscopy (FIM) has long been arguably the most direct and convincing way to see atoms on surfaces, only in the last decade or so have technological advances made it possible to accumulate enough data to contribute detailed quantitative information about the interaction between adsorbates. In the earlier years of this work, it was necessary to azimuthally average data in order to obtain tolerable statistics (Tsong, 1973). Casanova and Tsong (1980, 1982) plotted the pair interaction energy of Ir-Ir, of Ir-W, and of Si-Si on W(110) as a function solely of separation; hence the oscillating curve added to guide the eye actually misleads it: this plotting strategy might be satisfactory for physisorbed atoms (which are not amenable to FIM), but it obscures the anisotropy that we have seen to be ubiquitous and significant. Moreover, Watanabe and Ehrlich ( 1991 ) comment that such a plotting scheme could mislead one into thinking that a diffusing adatom could get trapped between two radial barriers in the potential, when in fact the adatom can skirt the repulsive sites because of the strong anisotropy of the interactions. In recent years it has become possible to accumulate enough data to assess interactions between pairs of adatoms at dozens of distinct separations, as best illustrated by an intensive set of experiments by Watanabe and Ehrlich (1989, 1991, 1992), Ehrlich and Watanabe (1991). This process can be eased by using two different atoms, one of which bonds more strongly than the other, so that one can study equilibration at a temperature at which only the more weakly bonded atom is mobile. ~ An additional advantage is that the stationary atom can be set near the This idea dates back at least to 1977, when Cowan and Tsong (1977) studied the interaction between a W adatom and a substitutional Re atom on W(110). Without benefit of a PC image digitizer, they found strong deviations in the site distribution from random (viz. the same measurement with no Re atom substituted).
Interactions between adsorbate particles
621
center of a facet, minimizing the number of "snapshots" with an adatom near the edge of the terrace, where fringe fields make the data questionable. Accordingly, Watanabe and Ehrlich (1989, 1991) fix a W or a Re atom near the center of a W(I 10) plane on a FIM tip and monitored the distribution of a mobile Pd atom. They observe that Pd is most frequently found at the nearest-neighbor position of the fixed adatom, but the nearby second and third neighbor sites are not populated. From 1638 observations of a Re-Pd pair equilibrated at 205 K, they deduce E1 = - 3 6 . 8 + 1.0 meV, E2 >- 45 meV, E 3 _>45 meV; from 1288 observations of a W - P d pair equilibrated at 225 K, they find, rather similarly, E~ = -50.4 + 8.0 meV, E2 -> 40 meV, E 3 2 40 meV. (Actually, these are free energies rather than energies; we neglect the distinction in quoting numbers. See w 11.4.2 for more comments on the analysis.) Along the close-packed [li-1] the interactions are attractive out to -10 A. The decay in strength is monotonic except for the second site (E4), which is considerably smaller, giving an oscillatory appearance to the decay_vs. R. Along the [001 ] direction the interactions tend to be repulsive, along the [110] beyond one spacing, they tend to be weakly attractive, in both cases with some exceptions. Overall, then, the interaction is highly anisotropic and oscillatory in sign, extending to large separations, with no simple pattern, i.e. consistent with the general picture presented in w 11.2.2. With further work they were able to observe pairs of identical adatoms, Ir-Ir (Watanabe and Ehrlich, 1989, 1992) and Re-Re (Watanabe and Ehrlich 1992) on W(110). From 2232 observations of the Ir pair equilibrated at 375 K, they find qualitatively similar behavior to the two heteropairs" E~ = - 8 6 + 2 meV, E2, E3 = 70 + 30 meV. In all directions there are oscillations in sign and non-monotonic decay in amplitude (see Fig. 11.6). From 3145 observations of the Re pair equilibrated at 390 K, they find different behavior in that the interaction is repulsive at all close spacings" E~ = +21.5 + 9.3 meV, E2, E3 > 70 meV. At larger R, the interactions again become attractive and are dependent on the orientation of the pair on the surface; the interaction is oscillatory and anisotropic. Watanabe and Ehrlich (1992) also try to assess the trio interaction. It is not feasible to measure this interaction directly because at the temperatures at which trimers dissociate, the liberated adatoms quickly move to the edge. From an Arrhenius plot of lifetimes of linear (straight) trimers, they deduce a dissociation energy, from which they subtract the diffusion barrier Ej to find the trimer binding energy. From this they subtract the pair energies of the three legs of the trimer to obtain an attractive trio interaction o f - 1 3 0 + 70 meV for the linear (L) configuration. On (110) bcc surfaces there are two other trimer configurations with two nearest-neighbor pairs (i.e. legs in the [111] direction)" nearly equilateral (P for "pointed"), and H20-like (O for "open" or bent). For Re the trio interaction is even more dramatic, stabilizing L and O trimers in spite of the short-range repulsions, with energies o f - 2 4 0 meV a n d - 2 1 0 meV, respectively (Fink and Ehrlich 1984b), suggesting trio attractions o f - 3 4 0 meV and at least-380 meV, respectively" the P trimer is unstable, so has a quite different trio energy. In both cases the adatoms are at neighboring sites and are comparable in size to the substrate atoms, so that direct interactions undoubtedly play a dominant role in these interactions, which are much stronger (relative to the constituent pairs) than expected from w 11.2.3. I.
622
T.L. Einstein
Fig. 11.6. (a) Distribution of separations between two lr atoms on W(110) at 375 K. Observations over the entire surface have been folded into a quadrant, and distant separations not plotted. (Along the bounding axes the number of observations is doubled.) (b) Distribution of separations between two noninteracting identical atoms on W(110). (c) From the ratio of these two distributions, the (free) energy of interaction between the Ir pair is computed and plotted as bars vs. R. Gray bars indicate repulsions, black bars attractions. Standard errors are shown at a few locations, based on statistical uncertainty. From Watanabe and Ehrlich (1992), with permission.
Interactions between adsorbate particles
623
Unfortunately, many kinds of atoms cannot be probed with FIM: FIM is largely limited to refractory transition metals. Scanning tunneling microscopy (STM) does not share this restriction; it has been used to examine a breathtaking range of systems (Gtintherodt and Wiesendanger, 1992). On the other hand, one cannot quench a whole (STM) sample nearly so quickly as one can an FIM tip ~ to get "snapshots" of rapidly evolving configurations, even if one has a low-temperature STM. If the STM scan time is not fast compared to the hop rate of the adatoms, the analysis is considerably more difficult. (Cf. Giesen-Seibert et al., 1993.) Much of the theoretical work in the 1980s on pair interactions was spurred by earlier FIM measurements of transition metals on (other) transition metals. In examining the dimer attraction for the 5-d series on W(110), Bassett (1975) measured a striking minimum for Re, with a rapid linear increase in attraction for lower Z and a slower increase for heavier Z. This observation was particularly intriguing because the adsorption energy and activation energies where largest in the middle of the series and so motivated Desjonqu6res and Spanjaard (1993) and coworkers to undertake several theoretical studies. Bourdin et al. (1985) propose a very simple analytical model. They claim that Burke's excessively large energies were due to his neglect of both core-core repulsions and electronic correlations (but cf. w 1 1.2.4), which they compute to second order in U/w, where U is the Hubbardlike intraatomic Coulomb repulsion and w the bandwidth. Bourdin et al. consider linear trimers at nearest-neighbor separation. They make several simplifying assumptions: (1) The substrate is rigid and the adatoms sit exactly at high-symmetry sites. (2) The core-core repulsion is the same for all the adsorbates. (3) The one-electron, "band" contribution comes solely from broadening of the adsorbate levels. Thus, the dominant interaction is directly between adsorbates rather than through the substrate. This ansatz is only reasonable at short spacings. (ES explicitly neglected these direct effects in their tight-binding calculations.) (4) The local density of states increases with coordination number (as one would expect from tight-binding theory) and is taken to be constant over an energy range (as for a 2D band). (5) The Coulomb integral U is the same for adatom, dimer, or trimer and independent of N,,, the number of d-electrons on the adatom. The band contribution to the dimer interaction energy E~ is
EI
b--
10(N,,/10) . , (1 - nN,~/10)J (w~- w**)
(~ ~.35)
where w~ - w.. is the increase in the width of the rectangular local density of states due to bringing the adatoms to neighboring sites. This contribution alone would produce the expected but incorrect result of maximum binding at N,, = 5. The additional contribution from correlation, E'~~ = - 90U2(wT ~- w2') [(N,,/10) (1 - N,,/I0)] 2
(11.36)
1 Watanabeand Ehrlich (1991) note that FIM tips can be cooled from a high equilibration temperature of 350 K to the imaging temperature of 80 K in under 5 seconds.
624
T.L. Einstein
is positive and tends to destabilize the dimer. They also consider trimer energies, adjusting U first to produce the experimental dimer pair energy. The estimated trio energies are (with a small exception) attractive and range from the same magnitude as the pair energy down to below an order of magnitude smaller. Oils et al. (1988) consider the magnetic contribution, which they find to be repulsive by a similar amount because the magnetic moment on the Re atoms decreases considerably when the dimer forms. Bourdin et al. (1987) find that on more open bcc surfaces (viz. W(211) and Ta(211)), on which the adatoms can increase their coordination number by additional bonding to subsurface atoms, the repulsive contributions decrease or disappear, and the maximum bonding of 5d dimers occurs near half filling. In this study the local density of states for the calculation of the band contribution comes from a continued-fraction expansion of the Green's function. Desjonqu6res et al. (1988) also apply their approach to compute potential energy curves for adsorption of gas dimers on bcc substrates. Desjonqu6res and Spanjaard (1993) present an appealing, simple argument stemming from this work that predicts that a repulsive interaction between adatoms close enough to share N common substrate bonding partners but too far apart to interact directly. When the (z-coordinated) adatoms are far apart, there are 2z substrate atoms which gain an extra bond to an adatom; their band energy is proportional to -(It + ~2)1/2, where ~t is the centered second moment of the density of states of an atom on the clean surface and 13 ": V. When adatoms share N host atoms, their band energy ,,,: -(It + 2132)!/2 while the energy of the other 2(z-N) substrate atoms is unchanged. Finally, at close range only 2z-N surface atoms are coupled to adatoms, N fewer than at large separation. Hence, in this second-moment picture, the pair energy E~N) E,N, ": - N4~t + 2132 - 2 ( z - N) 412 + 132 - N ~
+ 2z ~/l.t + 132 (11.37)
,,,: N(241a + ~2 _ 41.t + 2132 _ ~/~) > 0 This generic result was noted above for EAM in conjunction with Eq. (11.31 ). In a more sophisticated study, Dreyss6 et al. (1986) find similar results for W(110)-Re. Also considering only d-electrons, they consider the same three contributions, treating the one-electron energy using 5-fold degenerate tight-binding bands, the correlation energy using second-order perturbation theory (but with local atomic densities computed from their Green's functions), and the repulsion using Born-Mayer interactions. They also take some account of self-consistency by shifting atomic levels (cf. w 11.2.4.) They compute interaction energies for the three trimer configurations, L, P, and O, as well as the six shortest-separation pairs. For the pairs, including correlation energy with intra-atomic Coulomb integral U = 1.6 eV has a considerable effect, in most cases reversing the sign of the interaction; U is taken as just big enough to make the nearest-neighbor pair interaction repulsive, to reproduce experiment. Making use of an effective coordination number, they obtain for Re on W(110) the nearest-neighbor pair interaction
Interactions between adsorbate particles
625
E~ = - (242.8 + ~ - 3.35) Ebo~k All0.4
(11.38)
and for the L trimer interaction (lamentably called "trio"), i.e. trio plus 3 constituent pairs E(L)
=
- (2~/2.8 + ~ + 42.8 + 2~
_
5.04) ~bulk "-'cob / ~ / 1 0 . 4
(11.39)
Here ~ is related to the square of the interadatom hopping. With no direct interactions (~ = 0), both energies vanish. With direct interactions (~ = 1), they estimate E~ = - 1 . 5 eV and E(L) = - 2 . 9 eV. Additional correlation effects (since some are seemingly included in the total energies leading to the formulas) counteract these unphysically large numbers. The correlation contribution is about 10% of the band contribution and is most important for the half-filled band. If local charge neutrality is invoked (Bourdin et al., 1985), the P-trimer is favored, inconsistent with experiment (Fink and Ehrlich 1984a,b). This configuration is destabilized by correlation energies computed in first-order perturbation theory, stemming from charge transfer. Very recently Xu and Adams (1994, 1995) have developed a semiempirical scheme for treating bcc transition metals with minimal non-d bonding (i.e. with about half-filled d-bands). In the spirit of the Finnis-Sinclair (1984) method discussed after Eq. (11.32), they seek to approximate a tight-binding model. However, rather than using just a second moment (so square root) approximation to describe the d-band width, they follow Carlsson' s (1991) approach and also include third and fourth moments to describe the band shape, in particular three- and four-site contributions. (Recall that the nth moment of a nearest-neighbor tight-binding model counts paths that return to the origin after n nearest-neighbor steps.) For each metal (W, Mo, and V were studied), the model contains 10 adjustable parameters: one to weight each of three computed moment terms, five to characterize the pair potential (i.e. the U(Rij ) of Eq. (11.31)), and a pair of radial cut-offs for the pair potential and for the moment terms. These are determined by fitting to 12 bulk properties, 2 calculated energy differences with other lattice structures and 10 measured properties: sublimation energy, lattice constant, relaxed vacancy formation energy, 3 elastic constants, and 4 zone-edge phonon frequencies. The model differs from a similar one by Foiles (1992) by including the third-moment term and by using different fitting criteria, demanding in particular that W(100) and Mo(100) reconstruct. To test the model, Xu and Adams compute surface properties. The relaxations agree well with experiments and larger-scale computations, and the surface energies do not suffer the great underestimation well known to occur for EAM. On the other hand, the calculations require two orders of magnitude more CPU time than EAM. Xu and Adams (1995) recently applied their model to the pair interaction between W atoms on W(110). As observed in some previous studies of bcc (110) substrates (cf., e.g., Williams et al. (1978), Roelofs and Bellon (1989)), the pair of quasi-three-fold sites have lower energy (in this study by a mere 8.8 meV) than the bridge site between them, where the next layer would grow. The main findings are that the interactions are strongly anisotropic (consistent with the highly anisotropic
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band structure) and oscillatory in any particular direction. At large separations, the interaction is slightly repulsive. The authors attribute this effect to strain fields, which might well mask electronic effects at large separations; this notion could be confirmed with a calculation of interactions with the strains prohibited by hand ("frozen out"). In comparing with Watanabe and Ehrlich's (1989) data for Ir pairs on this surface, the authors find general agreement, particularly when one allows for the experimental error bars. A detailed comparison between calculated W pairs and measured Ir pairs seems risky, since Re pairs differ notably from Ir pairs on this substrate, and Re is closer to W on the periodic table. The authors fret that their computed interaction for nearest-neighbor W pairs along the (11 l) direction is too strongly attractive a t - 2 . 6 3 3 eV, while for Ir the measured attraction is j u s t - 0 . 0 8 2 eV. Recalling Burke's (1976) result that the strong attraction between neighboring W ' s is due to direct interaction, one would expect the Ir attraction to be at least somewhat weaker since Ir atoms are smaller than W's. Xu and Adams present a discussion of the signs of short-range interactions in terms of bond coordination numbers and changes in bond strength with bond length (again, with no frozen-lattice calculations to quantify the effects); such effects are discussed at length in the next subsection. For Pd(100)-H EAM calculations (Einstein et al., 1990) find that the minimum for H atoms to be slightly below the top Pd plane rather than slightly above. The magnitudes of the lateral interactions are more consistent with experiment, viz. 87 [94], 54, and -9 meV for the first-, second-, and third-neighbor interactions, E~, E 2, and E 3, respectively. (The bracketed value for Et is obtained from analysis of ordered overlayers. By symmetry, local distortions that plague the isolated pair are removed.) However, since E2 is more than E~, a p ( 2 x l ) ordering is predicted rather than the c(2x2) observed by Behm et al. (1980). (This problem as well as the too-low binding site may be due to use of rather primitive EAM functions which were readily available for Pd.) The smallest-area right-triangle configuration has a trio energy EaT = -25 meV; it plays no role in the balance between these two ordered states but does affect the phase diagram, as we will see shortly. It may not be a coincidence that the placing of H lower into the surface than in reality leads to more realistic binding energies: for Pd(l l 1)-H (Felter et al., 1986; Daw et al., 1987) as noted above, the interactions producing the ordering come from the subsurface H' s; those on top of the surface have little interaction, as for Ni(l 11). For comparison, Stauffer et al. (1990) have used a state-of-the-art tight-binding approach to present a wealth of information on H atoms near Pd(100). The H atoms are only allowed to sit in lattice planes of the substrate lattice, so the results for the center site of the top layer are the ones of most interest. Then E~, E 2, and E 3 are + 14, -182, and +41 meV, respectively. Removing the constituent pair interactions from their tabulated trimer energies, I find that EaT = - 3 2 meV and the "linear-triangle" configuration ELT = - 7 2 meV. It would be interesting to know how these numbers would change if the H's were moved slightly above the surface; since the dependence on layer index is not monotonic, there is no obvious interpolation. In comparison with our EAM numbers, the tight-binding EaT is quite similar but ELy is much bigger than expected even in crude calculations and certainly in EAM. Moreover,
Interactions between adsorbate particles
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the pair interactions are starkly different. While their pair energies do lead to the observed c(2x2) ordered phase, the enormous size of EJEi would produce a broad coexistence region of c(2x2) + "gas" that persists to a temperature close to Tc of the pure c(2x2) phase (Roelofs et al., 1986). Binder and Landau (1981) had in fact conjectured such regions (of more modest size) on the basis of Monte Carlo simulations. Such a stable coexistence region would presumably have been observed in experiment (Behm et al., 1980). It is my understanding that subsequent experimental investigation of the low-T, low-coverage region produced no evidence of islands, and so nothing was published. On the other hand, it is conceivable that the islands could be the stable phase at so low a temperature that the adatoms cannot diffuse adequately to achieve the equilibrium configuration. Motivated by FIM measurements, Wright et al. (1990) applied EAM to Pt, Pd, and Ni on (center sites of unreconstructed) Pt(001). Their main goal was to study whether, for small clusters of adatoms, linear chains or compact islands were more stable by computing the cluster binding energy, i.e. the difference between the total energy of the slab plus cluster of adatoms and the same number of adatoms isolated from each other. For Pt clusters, chains are preferable for clusters of 3 and 5 atoms; otherwise (up to 9 atoms) compact islands are favored, by a small but ever greater amount as size increases. These preferences are consistent with experiments by Schwoebel et al. (1989). For Pd the results are nearly the same (except for 5-atom clusters); in sharp contrast, for Ni the compact configurations of the clusters are usually not bound, so chains are favored by a considerable amount. (Presumably relatedly, Chen and Tsong (1991) find with FIM that Ir trimers form chains on It(100) but clusters on lr(l 11).) Wright et al. (1990) find that substrate relaxation is a key factor in these interactions. On a frozen slab, compact islands are always preferable for Pt and Pd, and by a more substantial margin for larger clusters; even for Ni, the compact shape is favored for several sizes. For each type of adatom and each size, the relaxation contribution (the difference between total energy and that with a rigid substrate) favors the island configuration. The essence of this difference is that most of the relaxation occurs in the top layer along the circumference of the cluster, which for given size is clearly longer for the chain shape. Specifically, around an isolated Pt adatom, the four substrates relax laterally outward by 0.18 ]k and upward by 0.08 A. For an atom next to the center of a 4-atom chain, these numbers roughly double to 0.39/~ and 0.13 ]k, respectively, while for an atom at the side of a 4-atom square, they scarcely increase, rising to 0.23 /~ and 0.09 ]k, respectively. Furthermore, the substrate atom at the center of the square cannot (by symmetry) relax laterally and sinks inward by 0.21 ]k. There is no discussion of the heights of the adatoms when close together compared to when isolated. If one allows for these relaxations, it becomes difficult to define the sorts of lateral interactions we have been discussing. Nearest neighbors near the center of a chain will have a different E~ from those near the end; moreover, the size of these interactions will depend on the length of the chain. Except for Ni, the energies associated with the relaxations are smaller by a factor of about 1/2 or 1/4 than the energies for a frozen substrate. While the relaxation issue becomes crucial when we try to distinguish between configurations, it may not be paramount for other
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properties such as phase boundaries. For frozen Pt (100), Wright et al. (1990) tabulate the E l to b e - 2 9 9 , - 2 6 3 , a n d - 6 4 meV for Pt, Pd, and Ni adatoms; E 2 to be +59, +4, and +97 meV, respectively, for the three kinds of adatoms; and ERX to be - 7 6 , - 1 2 , and -40 meV, respectively. Thus, the sign of the interaction does not depend on the kind of adatom. The small E~ attraction and large Ez repulsion for Ni are important factors in its preference for the chain configuration, in addition to the large relaxation difference between the two configurations. For weaker adsorbates than transition metals, the substrate relaxations should be less dramatic or important. For example, for H on Ni(111), discussed above, Einstein et al. (1990) found a negligible contribution from substrate relaxations: Ni atoms moved by roughly 0.01 A or less, and contributed 1/2 meV per adatom in a test calculation. Furthermore, when the adatoms are in the intermediate (and certainly in the asymptotic range), there should be little "cluster" shape dependence on the lateral energies or on the off-diagonal Green's function Go,. On the other hand, presumably the local coupling parameter would need to be recalibrated to take such effects into account, unless it were obtained semi-empirical from fits to adsorption energies of low-density clusters. More generally, for stronger adsorbates one must worry seriously about this problem, as discussed at the end of w 11.2.5.
11.3.6. Scattered-wave theory Feibelman (1989a) notes that scattering-theory methods are designed to take advantage of the rapid screening by metallic-substrate electrons of the potential associated with defects, specifically adatoms. Because of the screening, the wavefunctions related to the adatoms can be described as solutions to a scattering problem involving incident and scattered Bloch waves of the clean surface. If N basis orbitals are needed to describe the spatial region where the potential is unscreened, then one must numerically solve NxN sets of linear equations. This focus on orbitals in the adsorption region has philosophical similarities to the tight-binding picture explored at the outset: one isolates the changes due to adsorption from the otherwise perfect, semi-infinite substrate. However, here both the background substrate and the adsorption area are described with far greater sophistication, without recourse to simple model parameters. To date, though, only pairs of adatoms in the near region have been investigated, and there is no attempt to focus on G,,,, i.e. a propagator between the adsorption sites. Both adatoms are part of the same cluster, and the heritage is from cluster methods. Here, though, the adsorbate region is not so much a cluster to be embedded into an indented substrate but a scattering zone that perturbs the substrate B loch states. Specifically, this approach builds on the local-density-approximation (LDA) description of surface electronic structure (Lundqvist and March 1983), solving the Kohn-Sham (1965) energy-minimization problem, and uses state-of-the-art expressions for exchange-correlation potentials and (norm-conserving) pseudopotentials. The equations are solved self-consistently: from a guessed scattering potential, the electronic density is obtained from Dyson's equation, and an iteration-relaxation scheme (Johnson, 1988) is invoked.
Interactions between adsorbate particles
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Feibelman (1987a,b, 1988a,b, 1989a, 1990) applied this self-consistent, matrix Green's-function (MGF) scattering theory (Williams et al., 1982; Feibelman, 1987a) extensively to adsorption on AI (001). Most of these papers involved single adatoms, but Feibelman (1987b, 1988a,b) discusses AI-AI, AI-S, and AI-Te dimers. For AI dimers, Feibelman (1987b) finds that because of the direct bond between the adatoms, their bonds to the substrate weaken, consistent with Pauling's (1960) bond-orderbond-length correlation, and so they sit farther (by 0.16/~) from the substrate than a single AI. (The most dramatic consequence is that the diffusion barrier for dimers is lower than for monomers!) From naive counting of bonds, the strength of which are deduced from the bulk coherence energy, one might guess that El ---0.556 eV. From the EAM work cited earlier, and more directly from our summary of NCrskov's (1993) discussion of cohesive energy vs. coordination number, we would already guess this estimate to be much too high. Careful calculation shows the interaction to be j u s t - 0 . 0 7 eV. In addition to the vertical relaxation, this result includes lateral relaxation of each AI toward each other by 0.05/~ from the center site. If the separated A1 adatoms are not allowed to relax back toward the substrate, their pair attraction would b e - 0 . 1 8 eV. Feibelman does not specify how much the attraction would decrease if the neighboring adatoms were fixed in the high-symmetry positions, with lateral relaxation forbidden. Note also that the relaxations are of the adatoms relative to their isolated-adsorption positions. There is no discussion of any distortion of the substrate neighbors on this fcc (100) surface, which we just saw plays a major role in the EAM study of Ni, Pd, and Pt on Pt(100). (However, since Feibelman (1990) later shows that AI diffuses by an exchange mechanism on AI(100), such distortions presumably do occur.) In later papers, perhaps to keep computations manageable, Feibelman (1988a, 1989a) fixes adatoms at their ideal "isolated" positions and just computes the force acting on each member of the dimer, in addition to the pair energy, to monitor the sort of corrections which would enter if the second-order relaxations were allowed. Recapping, Feibelman's (1987b, 1989a) key physical idea is that E! is small because, as pairs of AI adatoms are separated, the rupturing of their direct interaction is partially compensated by an increase in their bonding to the substrate. This viewpoint provides a fuller picture than the usually reliable insight (Desjonqu/~res, 1980), cited in a footnote, that the (fractional) "disposition of adatoms toward lateral interaction" (i.e. IEaa/Easl, presumably) decreases as the strength of their bonds to the substrate increases. Stumpf (1993) finds rather similar results for AI( 111 ) using a different self-consistent total-energy calculation. His slab is 5 layers thick, with the top two (and adatoms) allowed to relax. He finds for AI dimers on hcp sites that E ! = - 0 . 5 8 eV, attractive again because of the low coordination of the separated adatoms; the AI adatoms again relaxed toward each other. Similarly, if one AI is replaced by Si, El = - 0 . 5 6 eV. Motivated by the discovery of Na-induced vacancy structures (Schmalz et al., 1991), and bearing on the preadsorption problem, he finds that there is an attraction (-0.21 eV) between an AI adsorbate and an Na in a surface vacancy. On the other hand, if the Na were in a step vacancy, there is a repulsion of 0.06 eV; this change is attributed to the stronger binding of the Na and especially the AI at the step, so that they are more stable in their isolated configuration.
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T.L. Einstein
Motivated by evidence of surface poisoning by chalcogens, Feibelman (1988) next considers the interaction of an AI adatom with an S or Te adatom, again at adjacent center sites on AI(001). The El repulsions are 0.25 eV and 0.22 eV, respectively. From the force calculation, he finds that the pairs would relax outward as for AI dimers, but away from each other. (With such relaxations, the repulsions would decrease somewhat.) He shows that these interactions cannot come from hard-core or charge-transfer (dipolar) mechanisms and describes his numerical findings in terms of bond-order-bond-length ideas. Feibelman (1988, 1989a) focuses on the two substrate Al's to which both adatoms bind. The bond to each adatom weakens when the other adatom is adjacent. Feibelman asks how the electrons rehybridize to optimize their energy. The chalcogens, with valence two, can simply shift away and strengthen their bond with the two farther substrate atoms; the A1 dimer, each with valence three, shift toward each other to form a direct bond, achieving an attraction rather than a repulsion. The situation is reminiscent of the simple argument sketched in the preceding section in Eq. (11.37), but the behavior of the A1 dimer would require an extension to include the direct coupling. To distinguish these cases a priori, without the reliable numerical output, would be a formidable task. The simple chemical arguments are really compelling only when joined with solid numerical evidence, as Feibelman often remarks. Feibelman ( 1991 ) reports the first application to dimer (viz. H-H and H-S) adsorption on a transition-metal substrate, Rh(001). The adatoms are placed in nearest-neighbor center sites, sharing two Rh neighbors, at the positions they would take if widely separated. Since the H atoms sit close to the surface (0.65/~ above the outer Rh plane), they are well screened and there is scant interaction between them. This result is consistent with the observation by Richter and Ho (1987) that the desorption energy of H from Rh(001 ) is independent of coverage up to 0.8 monolayers. In contrast, the S sits much farther out (at 1.47/~) and so is less well screened. Consequently there is an H-S El repulsion of 0.17 eV, consistent with the observation by Brand et al. (1988) that H on Rh(001) avoids regions where S has been preadsorbed. To make progress on surface problems, improved scattering theories are being developed. Feibelman (1992) notes that the MGF approach requires that the number of equations N that must be solved simultaneously is larger than one would suspect solely from the size of the scattering length, because the kinetic-energy component of the adsorption coupling is not limited by screening but depends of the choice of basis. Accordingly, Scheffler et a1.(1991) have developed an alternative scheme in which they include the full kinetic energy in the initial description of the electronic wavefunctions, but must consequently perform a taxing real-space integration. Feibelman (1992), in turn, has found a way to simplify the MGF method, eliminating much of the extra computation due to the kinetic-energy matrix. It is now becoming possible to compute total energies using scattering theories to assess interactions. In the bulk, Dederichs et al. (1991) use a KKR approach to study vacancy-vacancy interactions at nearest neighbor, and in some cases next nearest neighbor, sites in Cu, Ni, Ag, and Pd. (Vacancy-vacancy interactions are essentially the same problem as adatom-adatom interactions, but with fewer parameters. Yaniv (1981) approached the problem using an approach identical to that
Interactions between adsorbate particles
631
in ES. Feibelman (1989a) remarks that these are all examples of problems involving point defects.) The calculations are rather demanding, requiring, e.g., that they forgo the usual muffin tin description of charge density. This approach is being extended toward surface problems involving spin interactions (Dederichs et al., 1993). (Note that the indirect interaction between spin impurities via hyperfine coupling to the conduction electrons had earlier been considered: with an infinite-barrier jellium substrate, Gumbs and Glasser (1986) generalize the results of Lau and Kohn (1978). Zheng and Lin (1987) start from Kalkstein and Soven's (1971) tight-binding substrate, similar to w 11.2.2, and apply second-order perturbation theory.)
11.4. Implications of pair interactions 11.4.1. Ordered overlayers and their phase boundaries As noted late in the introduction and discussed in more detail in Chapter 13, the lateral interactions we have been discussing can lead to the formation of ordered superlattices of adatoms. Such ordered structure can be measured readily by diffraction techniques, especially LEED, and much of a temperature vs. coverage phase diagram can be mapped out. There are typically problems at low temperatures due to slow equilibration and at high coverages due to breakdown of the lattice-gas approximation. Also it is not readily possible to measure the phase boundary between a pure phase and an adjacent island phase (e.g. a coexistence regime of the pure phase and the ( I x I ) gas phase). It is generally very difficult to deduce uniquely lateral interactions by fitting to phase diagrams. The typical approach is to choose the minimum number of interactions necessary to produce the correct topography of the phase diagram, and then to adjust their sizes to mimic optimally the available boundaries; such boundaries can be computed accurately using Metropolis (equilibrium) Monte Carlo methods (Roelofs, 1980, 1982, 1995; Binder and Landau, 1989) or transfer-matrix finite-size scaling (Kogut, 1979; Kinzel and Schick, 1981 ; Rikvold et al., 1984; Bartelt et al., 1986; Roelofs et al., 1986; Rikvold et al., 1988; Roelofs and Bellon, 1989; Nightingale, 1990; Myshlyavtsev and Zhdanov, 1993). Other methods, particularly mean field, but also quasi-chemical and cluster-variation, are ill suited for two-dimensional computations, for which fluctuations play a far greater role than in three-dimensional systems. In fitting a phase diagram, it is important to consider the entire range of coverages, not just the saturation coverage of the ordered structure. This point was illustrated for Ni(111)-H near the end of w 11.3.2 and for Pd(100)-H in w 11.3.3. In the simple case of a c(2• one can get some idea of the effect of additional interactions from the case study of Ag(100)-CI by Hwang et al. (1988). In an earlier paper Taylor et al. (1985) found that the phase diagram of this system is rather well approximated by the hard-square model: a square lattice gas with nearest-neighbor exclusion (E~ = +oo). However, the critical coverage for ordering at 300 K was measured (by LEED) as 0.394 + 0.007 ML, higher than the 0.368 ML of the simple hard-square model. They concluded that a second-neighbor repulsion E 2 in the range 20-26 meV could account for the experimental result. Hwang et al. (1988)
632
T.L. Einstein
found that this critical coverage 0c did not change even if the sample temperature was increased to as much as 600 K. With only E2 (and infinite El) there should be substantial variation in 0c. The simplest explanation was an additional small E 3 repulsion. A plot of possible values of E 2 and E 3 was produced: if 0.387 < 0c < 0.401 ML, then E 3 -- 4 meV if E2 = 20 meV, and E3 --- 3.5 meV if E2 = 24 meV. If 0.386 < 0c < -0.401 ML, the range of possible interactions more than doubles for both E z and E 3. Near the other limit of complexity are studies of ordering of multiple phases on close-packed surfaces, with adsorption in both kinds of center sites. Such models invoking up to 5 pair interactions, are applied to N i ( l l l ) - O (Roelofs 1982), Ru(0001)-O (Piercy et al., 1992), Ru(0001)-H (Sandhoff et al., 1993), and Ru(0001)-S (Sandhoff, 1994). In the latter cases, trio energies are also included (3 distinct ones for Ru-O!). Nonetheless, the fit of Ru-S is not fully satisfactory in that (1) the disordering temperature of the ('~t-3-x'~-) was comparable to that of the c(2• rather than nearly twice as high, the experimental result (Sokolowski and Pfntir, 1995); (2) no hint of the observed complex defect structure on the high-cover side of the (q3-xq3-) is found. For the problem of catalytic poisoning on P t ( l l l ) , Collins et al. (1989) apply transfer-matrix and Monte Carlo techniques to a two-species (H and S) lattice gas model. They use successively more sophisticated models and are able to account for the dependence of the H coverage on S coverage for several temperatures. The role of trio interactions is included in a review of them not long ago by Einstein ( 1991 ); see also Roelofs (Chapter 13). The presence of such an interaction in a lattice gas Hamiltonian will break up-down symmetry in the associated Ising model. Accordingly, it is widely expected that such interactions will ipso facto produce gross asymmetries in the phase diagram. For a single trio interaction, this expectation is often misguided. The crucial aspect is not whether there is an asymmetry in the ground state energy but rather whether there is an asymmetry in the (lowest) excitation energy from the ground state, which leads to disordering and so determines the phase boundary. If there are several distinct trio interactions, however, asymmetry in the phase boundary is usually unavoidable. A second key idea is that if one wishes to gauge the size of the trio interaction from a fit to the skewed phase boundary, it is important that one include all such interactions of comparable magnitude in the fit, or else one is likely to strongly overestimate the physical size of this multisite interaction. This problem is discussed in the context of W(110)-O by Einstein (1979b).
11.4.2. Local correlations and effects on chemical potential Until recently, most experimental probes of ordering on surfaces provided only statistical averages of correlation functions, convoluted with some instrument response function. In diffraction measurements, one can measure (subject to deconvoluting this response function) the long-range order parameter below the transition and its fluctuations above it, or in a different limit, an ill-defined sum over short-range correlation functions (Bartelt et al., 1985). Vibrational probes similarly give information about long-range order parameter, but with a far shorter range instrument response. Only in the FIM experiments cited earlier is use made of a
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sequence of real-space atomic-scale images. In those experiments, there are typically just two adatoms, so that it is not hard to find the degeneracy (or configurational entropy) of each possible energy state and so to work backwards to the interaction energies, as noted by Meyer (1992). For STM "snapshots" there are too many adatoms for analytic insertion of degeneracies and regression to interaction energies. Instead, one must tune estimated values of these energies until the configurations generated in Monte Carlo simulations adequately reproduce the STM images, as he illustrates for Cu(110)-O (Kuk et al., 1990); similarly, Schuster et al. ( 1991 ) estimate values for seven distinct lateral interactions for Cu(110)-K. The full power of STM and FIM as quantitative probes of atomic positions is that they allow experimental observation of specific (not just combinations of) short-range correlation functions. Meyer (1992) was the first to publish a way to exploit this potentiality by measuring correlation functions at two (or more) different temperatures. As in most methods, one must still posit at the outset which interactions to include. Then, he shows how to extract directly from a large number of snapshots the interaction energy associated with a particular correlation; the error varies as C --1/2, where C is the adlayer specific heat. He further discusses how best to choose the difference between the two temperatures: too close and there will not be enough difference; too far apart, and there will be inadequate relationship between the two correlation functions. Meyer (1993) subsequently proposes a way to extract the interaction energy directly from a set of STM "snapshots" without collateral Monte Carlo simulations: the presumed energy (in terms of a model Hamiltonian) is evaluated for each configuration and for other configurations created by moving each adatom in turn according to allowed kinetics. The interaction energies are obtained by best satisfying a steady-state criterion. Neither of these schemes have yet been applied to actual experimental data. Adsorption or desorption data is another more indirect way to look at lateral interactions. For example, Urbakh and Brodskii (1984, 1985) apply their ATA expression for Ap(~) (cf. w 11.2.3.2) to data for the isosteric heat of adsorption (and the change in work function) for P t ( l l l ) - H , achieving good agreement with experiments (Christmann et al., 1976; Norton et al., 1982). Braun et al. (1980) compute these quantities for a system in which charge transfer dominates the interaction, W-Cs, and find good agreement with experiment (Bol'shov et al., 1977), at least until the coverage at which metallization occurs. More generally, the adsorption/desorption rate depends intimately on the overlayer chemical potential It, which in turn depends subtly on the interactions as the coverage varies. In considering the effect of lattice-gas repulsions on temperature-programmed desorption spectra, Payne et al. (1991) consider just this issue. Starting with the relation 5 ~g/'kBT q(O,T) = -~ kBT + kBT 2 3T
(11.40) o
for the isosteric heat of adsorption q, they conclude that computations using transfer matrices do not show the anomalous behavior found in calculations using approximate techniques. In recent years Kevan's group has devoted considerable effort to trying to extract interaction energies from adsorption and desorption data, as reviewed by
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Wei et al. (1994,1995). While some of this work uses the virial expansion or the quasi-chemical method, they, too, recognize transfer matrices as the method of choice. For example, they consider the desorption of CO, measured by time-resolved EELS (electron energy loss spectroscopy), from the three principal facets of Cu. On Cu(100) Wei et al. (1995) find a strong repulsive El, E2 = -2.8 meV, and E 3 = +1.1 meV (or E2 = -1.3 meV and E3 = -0.9 meV in their first report (Wei et al., 1994): E2 and E3 are correlated, and both pairs give comparable fits; both energies are much below ksT, impeding greater precision). On the other two faces El is also strongly repulsive (>> kBT); on (110) E2 = - 1 0 . 3 meV and E3 = +8.2 meV, while on (111) E2 = +9.2 meV, E 4 - +13.4 meV, but, surprisingly, E 3 > 69 meV. On P t ( l l l ) , the interactions between CO's (in atop sites) are purely repulsive, with E~ again effectively an exclusion and non-monotonic decay since E2 < E3 (Skelton et al., 1994). In order to produce sensible strengths, they must also include E4 (< E3); they discuss consequent difficulties in extracting three interactions from fits to isotherms and the inability to do transfer matrix computations that include all the E 4 interactions. Applying the random-phase approximation (RPA) in a tight-binding formalism to EELS, Brenig (1993) shows that for high-symmetry, low-mass-adsorbate overlayers the single-particle dispersion and the phonon dispersion decouple" one first determines the band structure of the excited vibrational states (assuming a localized, zero-bandwidth ground state) and then finds the vibrational frequencies using standard lattice dynamics. The scattering intensity is proportional to the resulting RPA susceptibility. As a corollary, he notes that when indirect couplings are strong, then translation invariance of the interadatom interactions is likely to be lost, causing the zone-center wave vector to vary with adsorbate concentration. Applying this formalism to EELS data (Voigtl~inder et al., 1989) of high-coverage (1.5 ML) H on Ni(110), he analyzes the two low-lying optical modes with in-plane polarization using up to third-neighbor force constants. He finds that the lateral interaction between H atoms at the three shortest separations differs strongly from the "bare" [direct] interaction and has a [local] maximum near the H-H second-neighbor separation (viz. the Ni nearest-neighbor spacing). While he can deduce much information about the interactions, e.g. evidence of multi-site interactions, he discusses the non-uniqueness of his fit and the consequences of various assumptions about the tight-binding-like force constants. Unfortunately (cf. first paragraph of w 11.3.5), the potential m assumed to be isotropic m is plotted as a continuous function of R. Kang and Weinberg (1994) review the kinetic modeling of surface rate processes in terms of four levels of sophistication" (1) In the Langmuir picture of adsorption and desorption, adatoms are assumed to be randomly distributed. (2) Neglecting lateral interactions, one can approach precursor-mediated adsorption and desorption from kinetic and statistical perspectives. (3) Lateral interactions (typically just nearest neighbors) can be included in Langmuirian and precursormediated processes using a quasi-chemical approximation. (4) For reliable results, one must, as noted repeatedly above, turn to a more exact method, in this case Monte Carlo simulations rather than transfer matrices; the review describes several applications, mostly with the generic lateral interactions as arbitrary input parameters. The effect of lateral interactions on diffusion has generated interest for quite
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some time (Bowker and King, 1978). Excellent reviews of adsorbate diffusion by Naumovets and Vedula (1985) and by Gomer (1990) provide a wealth of information. The dependence of diffusion on these interactions also comes through the chemical potential; specifically, the ratio of the chemical diffusion coefficient D to the jump diffusion Dj is the "thermodynamic driving force" (Gomer, 1994) O(kt/kT)/OOIT, where Dj is a complicated average that is essentially a frequency factor times an Arrhenius factor and is generally similar to the tracer diffusion D* of single adatoms. (Note that jump rates ought also to depend on lateral interactions, although this complication is typically neglected.) Using Monte Carlo simulations, Uebing and Gomer (1991) study the effects on the three diffusion constants of several generic sets of lateral interactions on a square lattice. Except for a case with E~ < 0 and E2 > 0, the fluctuation method and the Kubo-Green approach give similar results. Using the transfer-matrix technique to calculate It, Myshlyavtsev and Zhdanov (1993) consider similar problems on a rectangular lattice with anisotropic interactions. Tringides and Gomer (1992) show that lateral interactions could produce anomalous behavior in diffusion constants measured by laser-induced diffusion compared with those from fluctuations around equilibrium, in contrast to their similar behavior in the absence of such interactions.
11.4.3. Surface states on vicinal and reconstructed fcc(110) surfaces The same mechanisms which underlie the interaction between atoms chemisorbed on flat surfaces should also play a role in the interactions between steps on vicinal surfaces. For most semiconductors the interaction potential between steps, U(l), is repulsive and decays as l-2, where I denotes the distance between steps, as reviewed by Bartelt and Williams (Chapter 2). This form describes energetic interactions expected from both dipole-dipole (Voronkov, 1968) and elastic effects (Marchenko and Parshin, 1980). As noted in w 11.2.5, this result can be argued from a very general Green's function perspective (Rickman and Srolovitz, 1993). Poon et al. (1990) found such behavior in a study of steps on Si(100) using the Stillinger-Weber (1985) interatomic potential. Using EAM to study vicinal Au (100) and (110), Wolf and Jaszczak (1992) assess how well Marchenko and Parshin's expression for two interacting steps (or another expression (Srolovitz and Hirth, 1991) for a periodic array of steps, which differs by just a numerical factor of order one) accounts for the computed step-step repulsion. They find first that the amplitude Gel of the 1-2 decay, which depends on Poisson's ratio, Young's modulus, and components of the linear force densities or stress factors, is nearly independent of l for large I. The Gem's for steps on the two surfaces are 50% and 70% of the value deduced directly from the orientational dependence of the surface free energy. The discrepancy is attributed to: (1) the fact that the expression for Qlassumes isotropic continuum-elasticity theory, while the environment near the step is highly anisotropic; (2) the bulk elastic moduli in the formula should be replaced by local responses near the surface, which have not been computed; (3) the treatment should be based on a fully relaxed flat surface with unrelaxed steps, i.e., doing the calculation correctly would require Gordian unraveling reminiscent of the comments near the end of w 11.2.5. Extending EAM calculations to six of the late
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transition/noble fcc metals, Najafabadi and Srolovitz (1994) also found 1-2 repulsions for l > 3ao; inclusion of a higher-order 1-3 term improved the ~2 of the fit by over an order of magnitude. Simple continuum elastic theory is thus deemed to fail at small I because it neglects the discrete atomistic nature of steps and surfaces and because the elastic field of a step cannot be adequately described by a surface force dipole alone. Detailed comparison shows that modeling steps as in-surface-plane dipole line forces in an isotropic elastic medium predicts elastic fields qualitatively different from those simulated. In both studies, it is important to remember that the EAM calculations are incapable, ipso facto, of including long-range electronic interactions since there is no Fermi-surface singularity. Recently, microscopic probes of surface structure, particularly the scanning tunneling microscope (STM) and reflection electron microscope, have permitted detailed measurements of the configuration of steps on single crystal surfaces. Specifically, the terrace-width distribution function P(I) provides a sensitive probe of step-step interactions. The simple/-2potential describes inadequately the terracewidth distributions which Frohn et al. (1991) have measured on vicinal Cu(100) surfaces" Although P(l) for Cu(1,1,7) has the shape expected for a simple repulsive potential, the width and asymmetry of P(l) for Cu(l,l,19) suggests attractive interactions between steps. Similarly, Pai et al. (1994) have recently reported STM measurements of vicinal Ag(110) surfaces in which steps appear noninteracting for {/) = 22 ]k, repulsive for (l) = 30 ~, and attractive for (l) = 40 ]k. While attractive interactions may result from surface stress relaxation in the vicinity of steps (Jayaprakash et al., 1984) or from dipole-dipole interactions (Wolf and Villain, 1990) (if dominated by the in-plane orientation), the most likely explanation is an indirect interaction between steps mediated by substrate electron states which can produce attractions at some step separations (Frohn et al., 1991; Redfield and Zangwill, 1992). In terms of the formalism in w 11.2.2.2, we can imagine the relaxation of each atom along the step edge as producing a localized perturbation on the substrate analogous to the chemisorption bond. In this perspective, we view the/-2 repulsion as arising from a naive integration along one of the steps of an r -3 point-point repulsion, thereby approximating the steps as lines of independent points (Redfield and Zangwill, 1992), although the result is more general. At small separations, the r-dependence of indirect interactions is usually quite complicated" however, for the nearest-neighbor tight-binding model, the asymptotic regime for indirect interactions via bulk states is reached in -4 lattice spacings (Einstein, 1978). In this asymptotic limit, we saw in w 11.2.6 (cf. Eq. (11.27)) that the indirect interaction reduces to r-Pcos(2kFr), where kF is the Fermi wavevector with velocity pointing in the ~ direction, p = 5 for mediation by bulk states near a surface, and we have assumed the phase factor ~ is negligible. The integration along the step edge is complicated by the oscillatory factor. ~ Redfield and Zangwill (1992) point out that, given site-site interactions of the form r-Pcos(~cr), the interrow
1 Redfieldand Zangwill (1992) pointout that this summation procedure is strictly valid only in the (weak) limit, when the local perturbation due to each site is independent of its neighbors.
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interaction has the form r-"cos(K:r+5), with m = p - 1/2 and 5 = ~/4.1 F o r bulk electronic states, p = 5, so m = 9/2. As n o t e d in w 1 1.2.6, w h e n m e d i a t e d by [2D-isotropic] surface states, Lau and K o h n (1978) s h o w e d that the interaction d e c a y s like r -2 leading similarly to m = 3/2. T o d e c i d e which case is a p p r o p r i a t e for a particular substrate, o n e o b v i o u s l y m u s t k n o w s o m e t h i n g about the electronic structure o f the surface. O n the highly anisotropic (1 10) faces of noble metals, there a p p a r e n t l y are surface states that are p r o m i s i n g c a n d i d a t e s to m e d i a t e interactions in the [001 ] direction. H o w e v e r , these states 2 exist only (in a gap) near Y, (the intersection o f the [001 ] direction and the surface Brillouin zone b o u n d a r y ) , which w o u l d suggest m = 2 rather than m = 3/2. In M o n t e C a r l o ( M C ) simulations relying on a t e r r a c e - s t e p - k i n k ( T S K ) m o d e l o f surface structure, Pai et al. (1994) use a rather a d h o c potential e m b o d y i n g these ideas" it contains an oscillatory term at l > 6 lattice spacings and a r e p u l s i v e / - 2 - l i k e form at smaller step separations. While there is insufficient data to warrant c o n f i d e n c e in the specific potential, it is nonetheless noteworthy that this potential, with reasonable p a r a m e t e r s , can a c c o u n t for the distributions m e a s u r e d at three d i f f e r e n t (l). In s u m m a r y , vicinal A g ( 1 1 0 ) p r o v i d e s the first e v i d e n c e o f an indirect interaction m e d i a t e d by a surface state. It also illustrates that w h e n such effects occur, the l o n g - r a n g e interaction is by no m e a n s negligible. W e also note that Xu et al. (1996) are a p p l y i n g their m e t h o d using a m o d i f i e d fourth m o m e n t a p p r o x i m a t i o n to tight binding, discussed in the latter part o f w 1 1.3.5, to c o n s i d e r step interactions. T h e y fit their results to the form o f a m o n o t o n i c i n v e r s e - s q u a r e law repulsion plus an oscillatory term as discussed above, i n c l u d i n g ~, and obtain a r e m a r k a b l y consistent, if curious, set of results. Very recently, G u m h a l t e r and B r e n i g (1995) studied the s c r e e n i n g p r o p e r t i e s o f q u a s i - o n e - d i m e n s i o n a l states, such as may arise in the troughs o f r e c o n s t r u c t e d (110) fcc metals such as Ni and Cu (but not Ag) and c o n s i d e r e d h o w such states m i g h t m e d i a t e the indirect interaction b e t w e e n H atoms. T h e y derive analytic
The essence of the derivation is taking the leading term of ~ (x2+ y2)-p/2 COS(I(N/X2+ y2)dy to be o
i x /' cos(~(x + y2/2x))dx = x-(/'- I~!f(Ic~), wheref(~:,x) contains products of trigonometric functions and 1)
Fresnel integrals but has a simple asymptotic limit ~ cos ~ + 2 One state has been observed often for (110) late-transition/noble fcc metals, about 2 eV above E F (Bischler and Bertel, 1994). These states are probably too far from Er: to play an important role. However, Liu et al. (1984) calculated on Au(110) a second surface state just below EF, over a narrower range near Y, and there is some calculational evidence of a similar state on Ag (K.-M. Ho, private communication). Courths et al. (1984) reported such a state in an angle-resolved photoemission (ARUPS) study of Ag: it was found to be dispersionless at 0.1 eV below El: and sharply peaked in intensity at Y, seemingly vanishing by 20% of the distance to F. While the effect of steps and disorder are unclear, it is plausible that this state could be broadened or shifted to cross EF in some small region near Y.
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expressions for the consequent indirect interaction first from second-order perturbation theory (as in Lau and Kohn (1978)) and then with non-linear screening as in w 11.2.2, but with overlap included, both explicitly (Anderson and McMillan, 1967) and as a proportionality factor for the adatom-substrate eigenstate coupling (Gumhalter and Zlati6, 1980). The proportionality constant for the adatom-substrate coupling and the adatom energy ea are related by the Friedel sum rule (cf. w 11.2.4). The surface states in question run along the localized chain states in the closepacked troughs of (H-induced reconstructed) Ni and along the step edges of similar metals. When trying to parametrize the indirect interaction, the authors find that unless they assume a very slow decay corresponding to R -! in Eq. (11.27), the coupling parameter is unreasonably large. However, the specific fit to the potential in Brenig (1993) (cf. w 11.4.2) is unconvincing since that potential is: (1) represented as isotropic (while the interaction is manifestly highly anisotropic); (2) interpolated from interactions at three separations, all less than two lattice constants (and so far from the asymptotic regime); and (3) indicative of the total lateral interaction, including the direct contribution (which is likely non-negligible at the shortest separation). They and Bischler and Bertel (1993) (also Bertel and Bischler (1994)) suggest that this chain state is similar to the state seen in inverse photoemission by the latter pair. However, this particular state S~ is 6 eV above the Fermi ! energy, so presumably completely empty and hence inactive.
11.5. Discussion and conclusions Two decades ago at a Nobel Symposium (Lundqvist and Lundqvist, 1973) papers were presented on both the Kondo problem and the pair interaction, both cloaked in the Anderson model. In the conference summary, Anderson quotes Harry Suhl as saying "Like South America, the Kondo problem will always have a great future." Not only are such statements no longer "politically correct", in the meantime the Kondo model was instrumental in the formulation of the renormalization group (Wilson, 1975) and was solved exactly by Bethe ansatz methods (Andrei et al., 1983); even the two-impurity problem has been solved (Jones et al., 1989; Affleck and Ludwig, 1992). In contrast, consider what we have learned about the pair interaction. The only exact results relate to the asymptotic regime. Until recent evidence on vicinal Ag (110) of indirect interactions via surface states, these results proved of purely academic interest. There have also been exciting observations recently of standing
The main import of Bertel and Bischler's (1994) work is to show that a one-dimensional sp-derived state can exist on the surface. It is curious that the dispersion of the state is flat in the direction (in k-space) parallel to the chain (where one would expect considerable dispersion); Bertel (private communication) points out that this behavior arises because the state is antibonding in the top layer of Ni but bonding in the second layer (cf. the LCAO discussion in Bertel (1994)). Unfortunately,experimental complications have so far prevented measurements in the perpendicular direction (along which the dispersion should be flat if the states are in fact quasi-one-dimensional along the close-packed direction).
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waves on surfaces (Crommie et al., 1993; Hasegawa and Avouris, 1993); there may be some relationship between them and the propagator which transmits the indirect interaction. The Anderson picture developed earlier at length does provide a convenient way to conceptualize the physics of the interaction. The earlier model work gives a general feeling for the relative size of the interactions associated with adatoms in various configurations. On the other hand, it has been difficult to improve the model or to achieve quantitative accounting of actual experiments, although there certainly have been several attempts in this direction (Sulston et al., 1986; Dai et al., 1987; Zhang et al., 1990; Cong, 1994; Sun et al., 1994). The model seems more limited in dealing with the adsorption of single adatoms than in the subsequent interactions between pairs. Perhaps if there were a compelling way to evaluate adjustable parameters for the single-adatom case, one might make progress in this direction. Models leave out many important pieces of physics which are particularly important in the "near" regime. Local distortions, changes in bond strength with coordination, and rehybridization subtleties have been seen to play a vital role in this regime. With scattering theory methods, compelling results have been obtained in a few simple cases. At present, progress seems to be computationally limited. It seems that in the foreseeable future, advances will come from improvements in the code rather than more powerful computers. In this regime, which is certainly the most important from a practical or chemical perspective, it is not necessary or perhaps even fruitful to concentrate on the Green's function carrying the disturbance produced by one adatom to the site of the second. Once the adatoms separate sufficiently so that they neither couple directly nor interact strongly with the same substrate atoms, the perspective stressing the propagation of disturbances should be the most appropriate. The pair interactions of remarkably few physical systems have been computed successfully. More strikingly, in many cases where two different methods have been applied, inconsistent results are found. The case of Pd(100)-H was discussed above. Consider now the case of Pt(11 I)-CO, not an ideal prototypical adsorbate from a theory viewpoint due to the two active orbitals of CO and the complicated adsorption mechanism. (It is also an intermediate case energetically, with a heat of adsorption of 1-11/2 eV (Toyoshima and Somorjai, 1979) so neither in the perturbative regime nor in the strong-adsorption regime of, say, H, which has an adsorption bond strength 1-2 eV greater (Christmann, 1988) and forms bonding and antibonding states during adsorption (Einstein et al., 1980). Persson (1989) (also Persson et al. (1990)) assumes that pair interactions depend only on separation R. Explicitly, his pair interaction consists of a Pauli (hard-core) (contributing 262 meV to E~ and negligible for larger R) and an indirect term which is also repulsive and decays (isotropically and rapidly) monotonically" (1.3 eV) exp[-(0.8 ,~-l)-R]. The two constants are chosen so that (1) the binding energy at half coverage is 0.25 eV less than that at zero coverage, and (2) the frequency of the frustrated translation at the atop site (preferred by 60 meV over bridge) increases from 49 to 60 cm -~. With this model potential he performs (off-lattice) Monte Carlo simulations which apparently do well at accounting for the experimental phase diagram. Joyce et al.
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(1987) present a strikingly different picture, but also achieve good agreement with different experimental data! They separate the interactions into direct, indirect, and site (atop vs. bridge, high-symmetry positions only) contributions. The direct part is formulated in terms of gas-phase Lennard-Jones potentials. The indirect part is assumed to come from sp electrons and expressed in terms of the asymptotic form, even at short range! They believe that the adatom-substrate coupling occurs via the 2rt* orbital (3 eV above EF) rather than the 5~ orbital (7 eV below). Their interaction is not purely repulsive, but oscillates in sign. Nonetheless, the results apparently fit desorption energies at four different fractional coverages ranging from 1/3 to 2/3. Wong and Hoffmann (1991) applied extended Htickel theory to CO on Ni, Pd, and Pt(111). Unfortunately, they only report results for two coverages (1/3 and 1/2), so it is unclear what the size and the sign of the pair interactions are. Very recently Jennison et al. (1995b), using a promising technique described below, found that the C O - C O interaction on Pt(111) is repulsive and decays monotonically (to at least 3 lattice spacings), similar to Persson's (1989) result. However, their admolecules are placed only on bridge (not atop) sites (favored by --0.1 eV); while the decay is sensibly less rapid, their repulsions are somewhat too strong" E 3 = 25 meV vs. E3 = 16 meV for Persson's experimentally-calibrated potential. Finally, both sets of interactions differ from the non-monotonic repulsive decay deduced by (Skelton et al., 1994) (cf. w 11.4.2). Until pair interactions can be computed readily and reliably, our general picture and its evolution provide a useful template with which to confront indeterminate interactions needed to begin Monte Carlo simulations. We have a good idea of which configurations should have comparable size (Einstein, 1979b). We can use phase boundaries to estimate interactions. When subtleties exist (Bartelt et al., 1989), they may provide particularly valuable insight into the size of small interactions. In some cases semiempirical methods can help in gauging interactions, but these usually only give significant interactions in the near region and certainly fail by the asymptotic limit, since they lack Fermi surfaces; they are best for late-transition and noble metals. Generalized tight-binding models, including d-band degeneracy and correlation effects, have been useful for mid-transition metals. Very recently Cohen et al. (1994) proposed a general tight-binding total-energy scheme that improves on previous similar schemes by adjusting the arbitrary zero of energy to eliminate the need for pair potentials; like EAM, it is in a sense an elaborate interpolation scheme, since parameters are fit using first-principles calculations. It has many times as many fitting parameters as the fourth-moment approximation method discussed earlier (Xu and Adams, 1994). It has done better than EAM in accounting for surface energies of late-transition and noble metals. Perhaps it or a related method will allow calculation of far more accurate Green's functions and, ultimately, interaction energies. As this chapter was in its final stage, Jennison et al. (1995) communicated noteworthy advances in computational capabilities. With a new Gaussian-based local-density-approximation code for massively parallel computers that uses Feibelman's LCAO method discussed in w 11.3.6, they can treat systems (large clusters, molecules) and elements (transition metals, oxygen) that pose difficulties
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for plane-wave methods. With a three-layer 91-atom cluster of Pt, they reproduce well the details of ammonia adsorption on a seven-layer slab of Pt(111). The top layer is a hexagon 7 atoms across. Like the CO molecules mentioned above, pairs of ammonia repel each other at the close and intermediate separations that can be computed on this cluster, decaying roughly like R-3; the magnitude is somewhat greater. Specifically, they find for a pair of NH 3,s that E3 = 85 meV, which is much greater than the "through-space" dipolar repulsion, which they calculate to be 15 meV for two isolated ammonias. Multisite terms are relatively small: by comparing a compact cluster of seven molecules to a hexagonal ring of six, they find the effective E:~ drops to 75 meV, suggesting an attractive trio energy (for the associated equilateral triangles) o f - 1 0 meV. For the coadsorption case of CO at bridge sites and NO at hcp sites, there is a weak attraction at the largest computable separations. For both C O - C O and C O - N O (but not NH3-NH 3) the LDA is expected to overestimate the adsorption energies and so the interactions, consistent with the abovenoted difference from Persson's results; gradient corrections (Becke, 1988; Perdew et al., 1992) are expected (Jennison et al., 1995a) to temper this overestimate. All this work considers adatoms at or near stable sites in the holding potential. The effects of interactions on diffusion barriers, i.e. with one of the adatoms near a saddle point in the holding potential, has not yet been approached systematically. Typically some unconvincing assertion is made about this contribution, which in some cases may significantly affect the kinetics. The generic problem we have considered has broad ramifications. There are obvious extensions to defect interactions. Many analogous features occur in adsorption in electrochemical cells (Rikvold and Wieckowski, 1992). A more novel related situation is the oscillating interaction of magnetic sandwiches of varying thickness (Herman and Schrieffer, 1992; Stiles, 1993). Hopefully synergistic progress, lacking to date, will permit results from one of these problems to impact on others. In summary, there has been decent progress in understanding the general principles of lateral interactions but limited progress in achieving detailed quantitative understanding. Interest has been rekindled recently in looking for long-range effects mediated by surface or even quasi-one-dimensional states. After a decade's work, issues of correlations and self-consistency that seemed particularly troublesome earlier (Einstein, 1979a) can be dealt with, at least in simple systems in the near regime (Feibelman, 1989a). The major issue today is the role of local relaxations and hybridizing effects. In the near regime, we may well be on the verge of significant progress. In contrast, it seems that advances in treatment of the intermediate regime Will require some imaginative way to incorporate the results of careful calculations of the clean-surface (for the propagator) and single-adatom (for the coupling) problems into a general framework that recognizes that the interaction will perturb the single-adatom solution weakly at most. Acknowledgements
My work has been supported by NSF-MRG Grant DMR 91-03031. In the later stages I also benefitted from the hospitality of the IGV, Forschungszentrum Jtilich
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and the support of a Humboldt Foundation U.S. Senior Scientist Award. I have been truly fortunate to have had enlightening conversations with a large fraction of the authors, both theorists and experimentalists, cited in the references. I thank N.C. Bartelt and H. Pfntir for critical readings of drafts of the manuscript. J.B. Adams, L.W. Bruch, A.G. Eguiluz, M.C. Fallis, D.R. Jennison, S.D. Kevan, J.K. Nr and B.N.J. Persson provided comments on specific passages. I am particularly indebted to J.R. Schrieffer for having proposed a thesis problem of such enduring interest.
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Stumpf, R. and M. Scheffler, 1994, Phys. Rev. Lett. 72, 254; Comp. Phys. Commun. 79, 447. Sulston, K.W., S.G. Davison and W.K. Liu, 1986, Phys. Rev. B 33, 2263. Sun, Q., J. Xie and T. Zhang, 1994, Solid State Comm. 91,691. Taylor, D.E., E.D. Williams, R.L. Park, N.C. Bartelt and T.L. Einstein, 1985, Phys. Rev. B 32, 4653. Theodorou, G., 1979, Surface Sci. 81, 379. Tiersten, S.C., T.L. Reinecke and S.C. Ying, 1989, Phys. Rev. B 39, 12575. Tiersten, S.C., T.L. Reinecke and S.C. Ying, 1991, Phys. Rev. B 43, 12045. Tom~inek, D., S.G. Louie and C.T. Chan, 1986, Phys. Rev. Lett. 57, 2594. Tosatti, E., 1976, in: Proc. 13th Int. Conf. Semiconductors, ed. F.G. Fumi. Marves-North-Holland, Rome, p. 21. Tosatti, E. and F. Ercolessi, 1991, Mod. Phys. Lett. B 5, 413. Toyoshima, I. and G.A. Somorjai, 1979, Catal. Rev.-Sci. Eng. 19, 105. Tringides, M. and R. Gomer, 1992, Surface Sci. 265, 283. Truong, T.N. and D.G. Truhlar, 1990, J. Chem. Phys. 93, 2125. Truong, T.N., D.G. Truhlar and B.C. Garrett, 1989, J. Phys. Chem. 93, 8227. Tserbak, C., H.M. Polatoglou and G. Theodorou, 1992, Phys. Rev. B 45, 4327. Tsong, T.T., 1973, Phys. Rev. Lett. 31, 1207; Phys. Rev. B. 7, 4018.. Tsong, T.T., 1988, Rep. Prog. Phys. 51,759. Tsong, T.T., 1988, Surface Sci. Rep. 8, 127. Tsong, T.T., P. Cowan and G. Kellog, 1975, Thin Sol. Films 25, 97. Ueba, H., 1980, Phys. Status Solidi(b) 99, 763. Uebing, C. and R. Gomer, 1991, J. Chem. Phys. 95, 7626, 7636, 7641, 7648. Urbakh, A.M. and M.I. Brodskii, 1984, Poverkhnost' 1, 27. Urbakh, A.M. and M.I. Brodskii, 1985, Russ. J. Phys. Chem. 59, 671 (Zh. Fiz. Khim. 59, 1152). Van Hove, M.A., G. Ertl, K. Christmann, R.J. Behm and W.H. Weinberg, 1978, Sol. St. Commun. 28, 373. Van Hove, M.A., S.W. Wang, D.F. Ogletree and G.A. Somorjai, 1989, in: Advances in Quantum Chemistry, Vol. 20, ed. P.O. L6wdin. Academic, San Diego, p. 1. Vidali, G. and M.W. Cole, 1980, Phys. Rev. B 22, 4661; 1981, ibid. 23, 5649 (E). Vidali, G., G. lhm, H.-Y. Kim and M.W. Cole, 1991, Surf. Sci. Rep. 12, 133. Lifshitz, I.M., 1964, Usp. Fiz. Nauk 84, 617 [Sov. Phys. Usp. 7, 549]. Voigtltinder, B., S. Lehwald and H. Ibach, 1989, Surface Sci. 208, 113. Volokitin, A.I., 1979, Sov. Phys. Semicond. 13, 960 (Fiz. Tekh. Poluprovodn. 13, 1648). Voronkov, V.V., 1968, Sov. Phys.-Crystallogr. 12, 728. Voter, A.F., 1987, in: Modeling of Optical Thin Films, ed. M.R. Jacobson, SPIE 821,214; Voter and Chen (1987) also fit data from dimers. Voter, A.F. and S.P. Chen, 1987, Mater. Res. Soc. Symp. Proc. 82, 175. Vu Grimsby, D.T.Y.K. Wu and K.A.R. Mitchell, 1990, Surface Sci. 232, 51. Wang, G.-C., T.-M. Lu and M.G. Lagally, 1978, J. Chem. Phys. 69, 479. Watanabe, F. and G. Ehrlich, 1989, Phys. Rev. Lett. 62, 1146. Watanabe, F. and G. Ehrlich, 1991, J. Chem. Phys. 95, 6075. W atanabe, F. and G. Ehrlich, 1992, J. Chem. Phys. 96, 3191. Watson, P.R., 1987, J. Phys. Chem. Ref. Data 16, 953. Watson, P.R., 1990, J. Phys. Chem. Ref. Data 19, 85. Watson, P.R., 1992, J. Phys. Chem. Ref. Data 21, 123. Watson, P.R., M.A. Van Hove and K. Hermann, 1994, Atlas of Surface Structures, Vols. IA, IB (J. Phys. Chem. Ref. Data, Monograph 5, Am. Chem. Soc./Am. Inst. Phys., N.Y.) Wei, D.H., D.C. Skelton and S.D. Kevan, 1994, J. Vac. Sci. Tech. A 12, 2029. Wei, D.H., D.C. Skelton and S.D. Kevan, 1995, Surface Sci. 326, 167. Whitten, J.L., 1993, Chem. Phys. 177, 387.
650 Williams, A.R., P.J. Feibelman and N.D. Lang, 1982, Phys. Rev. B 26, 5433. Williams, E.D., S.L. Cunningham and W.H. Weinberg, 1978, J. Chem. Phys. 68, 4688. Wilson, K.G., 1975, Rev. Mod. Phys. 47, 773. Wimmer, E., H. Krakauer, M. Weinert and A.J. Freeman, 198 l, Phys. Rev. B 24, 864. Wolf, D. and J.A. Jaszczak, 1992, Surface Sci. 277, 301. Wolf, D.E. and J. Villain, 1990, Phys. Rev. B 41, 2434. Wong, Y.-T. and R. Hoffmann, 1991, J. Phys. Chem. 95, 859. Wright, A.F., M.S. Daw and C.Y. Fong, 1990, Phys. Rev. B 42, 9409. Xu, W. and J.B. Adams, 1994, Surface Sci. 301, 37 I. Xu, W. and J.B. Adams, 1995, Surface Sci. 339, 241. Xu, W., J.B. Adams and T.L. Einstein, 1996, Phys. Rev. B. 54, 2910. Yaniv, A., 1981, Phys. Rev. B 24, 7093. Zangwill, A., 1988, Physics at Surfaces. Cambridge University Press, Cambridge. Zhang, T., M.C. Kang and W.C. Lu, 1990, Phys. Stat. Sol. (a) 120, K41. Zheng, H. and D.L. Lin, 1987, Phys. Rev. B 36, 2204.
T.L. Einstein
Part IV
Defects and Phase Transitions at Surfaces
This Page Intentionally Left Blank
C H A P T E R 12
Atomic Scale Defects on Surfaces
M.C. T R I N G I D E S Department of Physics and Astronomy Iowa State University Ames, IA 50011, USA
Handbook of Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B. V. All rights reserved
653
Contents
12.1.
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.
Point defects
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
657
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
657
12.3.
12.2.1.
T h e r m o d y n a m i c s of point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
658
12.2.2.
Detection of point defects with diffraction
658
12.2.3.
Role of defects in surface relaxation
L i n e a r defects
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview 12.3.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.2.
663 663
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
665
12.3.1.3. S t r e s s i n d u c e d c h a n g e s on s t e p p e d s u r f a c e s . . . . . . . . . . . . . . . .
667
Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ovcrview
674
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
674
Clean surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
675
12.4. !. 1. M o s a i c s t r u c t u r c , strain, s t a c k i n g faults
675
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1.3. F a c c t t i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 i 684
12.4.2.1. S u r f a c e o v e r l a y e r c o n f i g u r a t i o n s . . . . . . . . . . . . . . . . . . . . . .
684
12.4.2.2. Q u a n t i t a t i v e m e a s u r e s of o v e r l a y e r m o r p h o l o g y
686
.............
12.4.2.3. O v e r l a y e r s w i t h ( I x 1) s y m m e t r y . . . . . . . . . . . . . . . . . . . . . .
688
12.4.2.4. O v e r l a y e r s w i t h s u p c r s t r u c t u r e p e r i o d i c i t y
693
T h e role of dcfects in phase transitions Overview
12.6.
677
T w o - d i m e n s i o n a l ovcrlayers on surfaces . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
12.4.2.5. O v e r l a y e r d o m a i n s t r u c t u r e s w i t h l o n g - r a n g e p e r i o d i c i t y 12.5.
672
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1.2. L a t e r a l and v e r t i c a l d i s o r d e r on c l e a n s u r f a c e s
12.4.2.
662
12.3.1.1. Q u a n t i t a t i v e m e a s u r e s o f step m o r p h o l o g y
T w o - d i m e n s i o n a l defects 12.4.1.
660 662
Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.2. T h e r m a l r o u g h e n i n g of v i c i n a i s u r f a c e s
12.4.
655
........
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
697 701
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
701
12.5.1.
T h e r m o d y n a m i c effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
701
12.5.2.
Kinetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
703
T h e Role of Defects in Crystal G r o w t h
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
704
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
704
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
708
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
709
Overview 12.7.
Epilogue
654
12.1. Introduction
The study of single crystal surfaces on the atomic scale has reached a sophisticated level of detailed characterization, with the invention of atom resolving imaging techniques, like scanning tunneling microscopy (Chen, 1993) and the use of data acquisition methods like diffraction lineshape analysis (Lagally, 1985). These techniques have revealed the presence of an unavoidable number of defects which interrupt the two-dimensional translational symmetry and chemical purity of the substrate, and can dramatically change the surface structure. If the ultimate goal is not only to describe but to control surface properties, it is necessary to understand the types of defects present, their mechanisms of formation and how they modify surface chemistry. Such defects are expected to exist at least as projections of defects already present in the bulk but, in addition, other types of defects exclusively characteristic of surfaces are found. The majority of the studies carried out so far have emphasized structural information on the real space configuration of the defects. It has also been realized, especially with probes that are sensitive to the electronic and structural characteristics (i.e., STM) that defects dramatically change non-structural properties (i.e., electronic structure, density of states, carrier transport). Any loss of surface symmetry will affect the available eigenstates, which in turn will affect both the static and dynamic response of the system to any external perturbation. Despite the importance of clarifying the relation between different types of defects and the electronic structure of the surface, the main focus of this chapter is to concentrate on the structural aspects of defects, i.e., the kinds of defects present on surfaces, their densities, techniques used to evaluate them quantitatively, and their role in controlling the equilibrium configuration of the surface. Defects can be classified according to different criteria. Most commonly, the dimensionality is used to categorize them: zero-dimensional or point defects (vacancies, interstitials, impurities), one-dimensional defects (steps, dislocations, etc.) two-dimensional defects (absorbed overlayers, facets, stacking faults, domains, etc.). Other criteria used, in connection to diffraction, are the extent to which they affect the long-range order in the crystal. If the atoms simply move by a small amount, but maintain on the average their equilibrium positions, they have the same relative phase difference, and long-range order is not affected (i.e., thermal disorder). This type of defect is usually referred to as a defect of the first kind (see w 1.6.2). If the damage of the crystal is so substantial (i.e., sputtering) that atoms have terminally moved out of their equilibrium positions and the phase difference between them is not fixed, the long-range periodicity is lost and all diffraction beams broaden. Such defects are referred to as defects of the second kind (see w 1.6.3). Finally, it is possible that atoms selectively occupy a fraction of the lattice 655
656
M.C. Tringides
sites (i.e., overlayers forming superstructures over the substrate lattice). A random phase relation exists between atoms in the overlayer and long-range order is lost but atoms maintain their lattice positions even at large distances. Diffraction beams broaden differently depending on the momentum transfer wave vector. Such defects are named defects of the third kind. Defects of the third kind are usually realized in surface overlayers as domains (see Fig. 1.25). For ordered overlayers with (m • n) unit cells where m and n are integers, a perfect "infinite" size domain is never attained and different types of domain walls separate equivalent domains. The presence of walls, their type and density, and limitations in the ultimate domain size realized are issues to be addressed. The mechanisms generating defects result, in general, from an interplay between kinetic and thermodynamic factors. The non-equilibrium nature of crystal growth processes allows a wide range of possibilities, where imperfect phases are stabilized over practically infinite time scales. Large energetic barriers separating the different phases lead to very slow kinetics towards the defect-free state and a finite number of defects still remains on the surface. Conversely, defects intentionally introduced on the surface affect both the static, equilibrium properties of the surface (reactivity, electronic structure, charge carrier transport) and time-dependent processes (ordering kinetics in overlayers, crystal growth). Two main techniques have been employed recently to study surface defects: quantitative diffraction analysis with several monochromatized probes (LEED, RHEED, ABS, X-ray scattering) and microscopy (STM, LEEM, SEM, TEM). The two techniques are complementary to each other. Microscopy provides local, direct imaging of the defects while diffraction is sensitive to indirect, statistical information ~tbout the overall defect distribution. Diffraction requires extra modeling to extract the defect configuration because the information is collected in reciprocal space and a non-trivial transformation, (sometimes non-unique because of the phase information lost in the scattering processes) is needed to recover the real space configuration of the defects. Microscopy provides clear, detailed, unambiguous information because it visualizes the defects, but if a small number of images are collected, then carrying out ensemble averages to determine collective parameters might be limited. Predictive power depends on sufficient statistical averaging over all the configurations present at the substrate. Ideally, a combination of the two techniques in the same experiment provides consistency in the results. This is especially the case when kinetic information is of interest in addition to the determination of the static defect structure. Microscopy is inherently slower because the higher S/N ratio needed to guarantee that the highest resolution is obtained at the expense of time resolution. Table 12.1 compares the relative advantages/disadvantages of microscopy vs. diffraction (Henzler et al., 1992). A + sign indicates that the method has advantages; a - sign indicates it has disadvantages but its application to collect useful information about a system is not ruled out. For statistical or time-dependent information diffraction is superior, while for extracting a direct and unambiguous image of the defects microscopy is better.
657
Atomic scale defects on su~. aces
Table 12.1 Comparison between diffraction vs. microscopy in their characterization of atomic scale surface defects. A +(-) sign denotes an advantage (disadvantage) of the technique. A (-) sign does not exclude the use of a technique, but the sensitivity to defects is lower than the sensitivity of the technique with a (+) sign
Microscopy General features Existence of defects Identification of defects superstructure steps point defects islands domains
+
Qualitative features Single defect shape of islands nucleation sites correlation of defects rare events
+
Diffraction
Quantitative features Lattice constants and strain Average values Size distributions steps terraces islands domains Kinetic features Changes during deposition Changes during heating annealing Reversible changes order disorder melting
+
12.2. Point defects Overview
Point defects are unavoidably present on surfaces because of their high configurational entropy. They can be observed directly with the STM or in the increase of the diffraction background. The highest sensitivity is obtained with the use of He-scattering because the scattering cross section is at least 10 times larger than the
M. C. Tringides
658
expected geometric cross section of the defect. Although the number of point defects can be easily identified, temperature dependent measurements to determine the energies of formation are limited.
12.2.1. Thermodynamics of point defects Point defects are produced thermodynamically on any surface. One can write the free energy H of the system at finite temperature T as
H= U- TS
(12.1)
where U is the internal energy and S the entropy. The formation of n isolated defects costs nAU energy, where AU is the energy of forming a defect, which is balanced
by the corresponding entropy increase AS =ksIn (N) (where ~ ~ is the usual combinatorial expression which gives the number of'ways of pos ing n defects in N possible lattice positions) so the change in free energy is AH(n) = nAU- ksTln "N" AH n By minimizing this expressions as a function of n, - 0 , the number of An de(ects produced spontaneously is obtained n =N e -aU/kT"
(12.2)
The number of point defects is always larger than this limit because other types of point defects like chemical impurities can be present, absorbed from the background gas or by diffusion from the bulk towards the surface.
12.2.2. Detection of point defects with diffraction Diffraction can be easily used to detect point defects. They are defects of the first kind, i.e., they do not affect the long-range order of the surface and the full-widthat-half-maximum (FWHM) of the diffraction spots is unchanged. Detects raise the background intensity. For uncorrelated defects (i.e., randomly positioned on the surface) which scatter with atomic scattering factorfD, different from the substrate scattering factor f,, it can be shown easily from Eq. (6.34) of Chapter 6
I(Q)=
N
+-Nf~ -
N
f, +-~fD
N (12.3)
+rN/(Nn n J Zij eikQ~r'-'i)]j1 L N /f" + -NfD The first term in the brackets does not depend on the wave vector Q and is usually
Atomic scale dejects on su~'aces
659
called the "background". The second term results in the usual ~5-function type summation, expected for an infinite lattice. The diffraction pattern is as sharp as for a perfect surface but with a higher background which can be used as a measure of the defect density. However, such measurements are difficult because even for large differences in the scattering factors between substrate and defect atoms (for example,fD = 2f~) the background is only a small fraction of the peak intensity (not more than 2-3%) for a defect density of a few percent. In practice, the absence of background in a diffraction pattern during routine substrate cleaning is used qualitatively to decide whether the surface is well-ordered. Inelastic scattering is not always completely eliminated from the measured background so that quantitative analysis should separate out its contribution..Increased background can also result from other types of disorder (i.e., thermal disorder which is discussed extensively in w 6.1.4) that remove intensity out of the Bragg peak. A convenient rule for the analysis of the background, based on the kinematic approximation (i.e., single scattering by top layer atoms) can be used for quick evaluation: the integral of the elastically diffracted intensity over the Brillouin zone is conserved, so any loss of intensity at the peak position should be found in the background. A higher sensitivity to point defects, especially for small adsorbate amounts, is possible with atomic beam scattering than with other diffraction techniques (Poelsema and Comsa, 1989). Extremely high single atom cross sections, comparable to the ones found in gas phase atom scattering have been measured in He-diffraction experiments and attributed to the long-range attractive part of the He atom interatomic potential. At very low coverages 0 << 1 scattering by defects removes elastic intensity at the Bragg peak into diffusive channels /
- 1 - 0oN for 0 << 1
(12.4)
where/c~ is the initial intensity and 1(0) the intensity after 0 amount of adsorbate has been deposited. N is the substrate density, 0 the adsorbate coverage and o the cross section. Equation (12.4) holds at low coverages when adsorbate overlap is excluded and the single atom cross section is extracted. Figure 12.1 (a) shows the decay of the peak intensity decay during the adsorption of CO on Pt(111). The initial slope at t = 0 can be used to extract the adsorbate cross section o = 120/~2. It is clear that coverages as low as 0 = 0.01 can be easily detected with this method. Implicit in the use of the peak intensity decay as a measure of the adsorbate density, is the invariance in the spot lineshape with coverage as shown in Fig. 12.1(b), which implies that the functional form relating the peak intensity to the adatom density is independent of the type and amount of adsorbate on the surface. For higher coverage (0 >> 0), Eq. (12.4) needs to be modified. If no interactions are present between the adatoms, one can show that I
--
=
e
-cON
(12.5)
li)
i.e., the intensity decays exponentially. Deviations from the exponential dependence can be used to identify the type of interactions between the adsorbate atoms.
660
M. C. Tringides
1'01
2.8
Q.6 ~
~
O.2 0
, 0 t
,-
CO"in"
~
IIio<0-01
\ 2
,
(a)
.
.
6
~11. ' = ~ - " -
~o
t I rain )
x cleonsurfnEe 1.0
A
~H i
0.75
I/Io : 032
,,CO I/I0=0.26
0.50
0.25 O"
795 ~
80~
(b)
80 5o"
1
Of-'-
Fig. 12.1. (a) He-scattering specular beam intensity decay during deposition of CO on Pt( 1 I I ) at the indicated conditions. The initial slope of the decay can be used to measure the single atom scattering cross-section (Poelsema and Comsa, 1989). (b) Specular beam intensity angular profiles for different deposited adsorbates (H, CO) showing identical lineshapes. This justifies the use of the peak intensity as the measure of the deposited adsorbate amount (Poelsema and Comsa, 1989).
For repulsive interactions, one expects the normalized curve !/1o to fall below the non-interactive one (since the atoms are kept further apart than for the case of random deposition minimizing the overlap in the cross section of different atoms) while for attractive interactions the normalized curve is above the non-interactive one (the atoms form clusters so the total cross section is less than the one expected for random deposition) in their interaction with the He atoms.
12.2.3. Role (?['defects in surface relaxation In special cases, point defects in the form of low level chemical impurities can have a drastic change in the relaxation of the substrate. This is more probable with semiconductor than metal surfaces. For the Si(001 ) and Si(110) surfaces, it has been shown (Ichninokawa et al., 1985)that very small amounts of Ni can change the types of reconstruction observed. What is very surprising is the minute amount sufficient to globally affect the surface over many lattice spacings. For Ni amounts
Atomic scale defects on su~. aces
661
barely detectable with Auger spectroscopy, (2xn) reconstructions with n = 5, 6, 7, 8, and 9 are formed. A combination of LEED and Auger analysis has shown a correlation between the amount of Ni and n, the higher the Ni amount, the smaller n. These reconstructions are formed after the sample is rapidly quenched from a high temperature to room temperature. If the sample is cooled slowly, then the superstructures are not observed, presumably because the Ni impurities diffuse to the bulk. It is believed that the ordered superstructures are formed from the ordering of vacancy defects (i.e., missing dimers on Si(001)) which are known to exist on the clean surface. The exact mechanism of how small amounts of impurities affect the position of at least 50 Si atoms is not well understood. The study of the ordering kinetics of the formation of the (2xn) structures is not consistent with the diffusive motion of Ni impurities to the bulk (Tringides et al., 1989). The presence of missing dimers on the Si(001) can be clearly seen with STM (Wolkow, 1992). The detailed atomic positions on this surface are shown in Fig. 12.2. Vacancies present on the terraces presumably order to form the (2• superstructures. It has also been demonstrated that the vacancies affect the details of the Si(001 ) dimerization into ( 2 x l ) superstructures to eliminate two Si dangling bonds, to be discussed extensively in the following section. It has been theoretically debated, however, whether the dimer axis is parallel to the surface plane or if it is buckled at a non-zero angle. With the STM, both non-buckled and buckled dimers have been observed, especially at low temperatures where transformation between the two configurations is minimal. The buckled ones are stabilized by the presence of the missing dimers shown in Fig. 12.2. The STM can also provide direct images of single adatoms, intentionally dosed
Fig. 12.2. Low temperature (T = 120 K) STM 110x110 ,~ image of the Si(100) surface showing a small fraction of vacancies. The surface reconstructs into the (2xl) ordered structure by pairing adjacent Si dangling bonds to form dimers. At the top right and bottom left corners buckled dimers are clearly resolved. (The horizontal band in the middle of the image is an artifact) (Wolkow, 1992).
662
M.C. Tringides
Fig. 12.3. STM images of an Si(111) (7x7) surface covered with a small amount of oxygen. The underlying substrate reconstruction remains intact and two types of oxygen atoms labeled D (dark) and B (bright) are observed. They corrcspond to different electronic bonding configurations of the adatom with the Si substrate (Avouris and Lyo, 1991).
on the surface (Avouris and Lyo, 1991 ). Figure 12.3 shows the Si( 1 I I )(7• surface after it is exposed to submonolayer amounts of oxygen (0 = 0.2). Both dark and bright sites (labeled D and B, respectively) are observed in the image, corresponding to single oxygen atoms bonded with different electronic configurations to the substrate. This clearly demonstrates that the STM is sensitive to combined electronic and structural effects. By analyzing the I-V characteristics of the tunneling current with the tip positioned at different points in the unit cell, the local density of states can be mapped out and the nature of the bond further identified. The bright sites correspond to the insertion of oxygen into the Si-Si bond while the dark sites correspond to the elimination of a dangling Si bond by the attachment of an oxygen atom.
12.3. Linear defects Overview
Stepped surfaces can be routinely characterized with diffraction from quantitative analysis of the split spots expected at destructive interference conditions. Lineshape
Atomic scale defects on su~. aces
663
analysis can provide step edge roughness, terrace size distributions, and step height multiplicity. Similar information can be extracted from statistical analysis of STM images. Changes in the step morphology as a function of temperature (towards rougher configurations) or as a function of the intrinsic stress anisotropy (i.e. single to double height step transitions on Si(001)) have been analyzed with STM and diffraction. As explained in Chapter 2, such changes of the step morphology can be used to identify the thermodynamic driving forces and the interactions controlling the step structure. Dislocations lack long-range periodicity so they can be characterized better with microscopy. STM images clearly show dislocations projected to the surface plane and the Burger vector can be fully determined from the projection parallel and normal to the plane.
12.3.1. Steps 12.3.1.1. Quantitative measures of step morphology Stepped surfaces are routinely prepared macroscopically on single crystals by polishing a few degrees off the high symmetry direction. Microscopically, they are present on spherically shaped sharp tips separating flat low index planes. The crystallographic direction of a vicinal plane in Fig. 1.16 can be easily identified with the stereographic projection, Fig. 1.19. They can be imaged directly and atomically resolved with field ion microscope. An ideal, "staircase" stepped surface, of equally spaced straight edge terraces, results in a diffraction pattern that can be characterized easily, as shown in w6.2.1.2. Split spots are produced at the out-of-phase conditions (Qllb = (2n+ 1)rt) where there is destructive interference between adjacent terraces. The periodicity in Qtt can be used to measure the step height b while the splitting of the spots AQjj is a measure of the inverse of the terrace width. This simple picture of a stepped surface as a "staircase", is far from true in real systems. Several sources of disorder can complicate the description of stepped surfaces as a sequence of straight-edge, equal-length terrace structures. Figure 12.4 shows an STM image of a stepped Si (001) 45 ~ [110], consisting of (001) terraces, with the misorientation direction at 45 ~ azimuth towards the [110] direction. The step edges are not straight but they meander around an average position. The degree of meandering can be measured microscopically from the density of kinks, and the size distribution of the linear segments separating kinks in the two directions normal and parallel to the step edge. It can also be described statistically in terms of the mean square fluctuation of the step position away from its nominal position G(r) = ((h(r) - h) 2)
(12.6)
where h(r) is the step position at distance r along the step edge. For rough steps G(r) is expected to be a rapidly changing function of r. (The 2-d analog of the mean square fluctuation function G(r) is used to describe 2-d interfaces. The relation of G(r) to the type of the interface morphology grown will be analyzed further for homoepitaxial growth in w 12.4.1.2, since the richness and variety of growth fronts is larger in 2-d than in l-d).
664
M. C Tringides
Fig. 12.4. STM image of a stepped Si(001) surface at misorientation angle 0 = 0.5 ~ in the [110] direction, imaged over a horizontal scale of 1000 ]k. Two types of domains and single step heights are present. Depending on the annealing temperature, the external stress or the vicinality, a transition to single type domain and double height steps is observed. The transition is driven by the stress energy anisotropy in the two types of domains. (Swartzentruber et al., 1990.) Another deviation from the ideal "staircase" structure is realized in practice, if the terrace length is not uniform. The step separation distribution can be used to characterize the extent of deviations from the idealized system. It is essentially determined by the nature of step-step interactions (in Chapter 2, a general formalism describing the thermodynamics of stepped surfaces is described. We will briefly summarize the main results and the reader can consult Chapter 2 for in-depth analysis). Interactions between steps are first driven by entropic considerations since adjacent steps cannot cross each other (otherwise the configurational entropy of a single step is reduced). This amounts in effect to a non-thermal repulsive interaction between steps, with topological origin and, therefore, leads to a universal step separation distribution. Any deviations of the experimentally determined separation distribution from this universal form, implies the presence of truly energetically driven step-step interactions (a narrower distribution, than the one determined by the entropy, signals the presence of attractive interactions and a broader distribution signals the presence of repulsive interactions (see Fig. 2.13)). The physical mechanism responsible for step-step interactions in most cases is stress. Atoms at a step are under stress to maintain their bulk positions despite the lower coordination number. Locally, they can relax outwards to minimize the stress present. The relaxation of atoms at one step can oppose the relaxation of atoms at a neighboring step which effectively produces repulsive step-step interactions (see w 2.4.3). Although the separation step distribution is exclusively controlled by the
665
Atomic scale defects on suq. aces
direct or effective step interactions, step meandering and the form of G(r) is also sensitive to step interactions since fluctuations of a meandering step are limited by its proximity to an adjacent step. Single height steps are, in general, expected on clean stepped surfaces because the energetic cost of multiple height steps depends exponentially on step height, but in several cases transitions from single to multiple heights, are observed either directly in STM images or from changes in the Qz periodicity with diffraction (i.e. case 3 of w 2.3.2 and stepped Si(001) discussed in w 12.3.1.3). A stepped surface, prepared initially by mechanical polishing, is essentially in a metastable state. It can be driven to a morphologically different configuration (with different step edge meandering, terrace size distribution, or step height multiplicity) by varying different thermodynamic variables. For example, temperature increase results in reversible transitions to rougher step configurations, stress anisotropy related to inequivalent terrace reconstructions can produce multiple step transitions and non-uniform terrace distributions, surface reconstruction and adsorption of foreign atoms can result in the phase separation of a uniformly stepped surface into regions of different vicinality. These transitions in the step structure can be characterized in terms of intrinsic step parameters (kink formation energy ~,c, step energy X.,t, step-step interaction w 1, intrinsic step anisotropy Ac = (c~-o2) where o~, (Y2 are the stress components on inequivalent terraces) or the externally applied variables (strain, adsorbed gas coverage, etc.).
12.3.1.2. Thermal roughening of vicinal surfaces It has been theoretically predicted that a clean surface can undergo a roughening transition because the entropy gain, resulting from the proliferation of disorder (i.e., kinks, adatoms, vacancies, steps, etc.), outweighs the cost in energy at sufficiently high temperatures. It is far easier to realize such transitions on vicinal surfaces, where the lower coordination number of step atoms lowers further the energy cost. A simplified Hamiltonian written in terms of the solid-on-solid model (SOS), captures the essential physics of the problem. The vicinal surface, shown schematically in Fig. 12.5, is described as an array of columns of height Aij at locations (i j) and the Hamiltonian is 2
W/
(12.7)
where ~,c, wt were defined in the previous section. A roughening transition at TR is predicted for this model which can be followed from changes in the form of the correlation function G(r), Eq. (12.6). For T < TR step meandering is limited over finite distances resulting in exponential decay of G(r) on r, but for T > TR G(r) diverges logarithmically with r
G(r) - c(T)lnr
(12.8)
with c(T), a temperature dependent coefficient which is related to the microscopic parameters Xc, and wt.
M. C. Tringides
666
A..
~= ~ ! ~ .6_
.
~/'C--~
,. , - - ' ~ t , - "~
m a o -.,.
,~,
i--- 3 V
~= 3
(a)
.... j _1J
(b)
(c) Fig. 12.5. (a) Schematic presentation of a stepped surface used for the SOS Hamiltonian Eq. (12.7). A,./denotes the deviation of atom j along step i from its nominal position. (b) Thermal roughening of the stepped surface with the generation of kinks along steps. (c) Further roughening of the surface with the formation of islands on top of terraces. (Conrad and Engel, 1994.)
Diffraction experiments can be used to detect the roughening transition (Kern, 1994). Originally deviation of the peak intensity decay from ideal D e b y e - W a l l e r behavior was used, which is much easier to measure experimentally. With the proliferation of disorder at TR, an additional transfer of diffracted intensity away from the peak is expected, producing a faster drop than the exponential decay predicted by the Debye-Waller effect. (However, as TR is approached, anharmonic effects become important with increasing vibrational amplitude and can lead to an unreliable estimate of TR). A better indicator of the roughening transition is the profile lineshape I(QI I, Q I ) of the fundamental beams. This lineshape changes dramatically from Lorentzian (for T < TR) to power law (T > TR) I(Q,,QI) -
c'(T)
Q[~z-~)
(12.9)
The roughness exponent x jumps discontinuously from zero (Lorenztian lineshape) to a finite, universal value at TR followed by a linear increase with ( T - TR). This linear increase reflects the increasing amount of disorder and fractal-like defect structures generated thermally, at smaller length scales, which transfer intensity to larger wave vectors. Maximum sensitivity is expected at the out-of-phase condition where destructive interference between different levels is maximum. Figure 12.6 shows the dramatic lineshape change at TR, obtained from He-scattering experiments on N i ( l l 5 ) , measured by the increasing F W H M of the specular beam. No
667
Atomic scale defects on su~. aces
i
I
o
0.8
o
0.6 N
",_0.4 O3
o~ e
0.2OQ 0
0
I
o
e
~
~
OQO I
200
I
9 I
400
I
T (K)
I
600
A I
9 I
800
I
'
I000
Fig. 12.6. Dramatic increase of the FWHM of the specular beam (measured as a fraction of the Brillouin zone) above the roughening transition TR, obtained wi_th He-scattering on Ni(115) at the out-of-phase condition. (O, 9 denote measurements along the [ 110] and [552] azimuths). No change in the FWHM is observed at the in-phase condition (A) (Conrad and Engel, 1994).
changes are observed at the in-phase condition. Detailed analysis of the temperature dependent prefactor c(T) can be used to extract the microscopic, energetic parameters describing the step (i.e., Xc, wt), as predicted for the statistical mechanics of the Hamiltonian equation (12.7). Several, stepped surfaces have been studied (Cu(l 13), Cu(I 15), Ni(113), Ni(115)) with He-scattering, which has clear advantages because of the high defect scattering cross section (w 12.2.2) and X-ray scattering. The results obtained with the two probes are in agreement with each other. Typically TR - 700 K for the surface with the lowest misorientation (TR decreases at higher vicinality) and the interaction parameters are Xc - 3 T R and w t ~ T R. Tabulations of all the experiments carried out so far and the corresponding energetic parameters are given by Kern (1994). 12.3.1.3. Stress induced changes on stepped surfaces The Si(001 ) stepped surface of Fig. 12.4 slightly misoriented in the [ 110] direction, shows that adjacent terraces have different dimerization directions, as a result of the Si crystallography. The Si lattice is built out of two interpenetrating fcc sublattices as seen in Fig 1.5. Each atom is bonded to two pairs of atoms in the other sublattice, with the bonds of each pair orthogonal to each other. Adjacent terraces, separated by single steps, expose atoms of different sublattices so that the dangling bonds that determine the dimerization direction are rotated by 90 ~ The local atomic configuration at Si(001 ) step edges is shown in Fig. 12.7 (Webb, 1994). Open circles denote atoms with dangling bonds and the size of the atom indicates the level of its location. Depending on the reconstruction of the upper terrace, there are two types
M. C. Tringides
668
(a)
('0)
s.
(e) A
(d)
(e)
Fig. 12.7. (a) Illustration of the different types of steps on vicinal Si(001) surfaces with the misorientation along the [II0] direction. (a) Overall presentation of the surface with different dimerization directions in adjacent terraces. The other figures show the local step bonding configuration. (b) For SA steps, the upper terrace has dimer rows parallel to the step edge. (c) Double steps DB separate identical terraces with the dimer rows normal to the step edges. (d) For bonded SB steps second layer atoms at the step (solid circles) have no dangling bonds. (e) For non-bonded SB steps second layer atoms at the step have one dangling bond. The size of the circle denotes the level of the atom. Open circles indicate atoms with dangling bonds and solid circles indicate atoms without. DB type steps are energetically less costly but depending on the misorientation angle or annealing temperatures SA, SB steps are found because of the intrinsic surface stress anisotropy (Webb, 1994).
of single height steps: SA-type steps Fig. 12.6(b) which run parallel to the d i m e r rows (i.e., essentially all the s e g m e n t s parallel to the nominal step direction on " s m o o t h " steps and the s e g m e n t s running normal to the nominal direction on " r o u g h " steps in Fig. 12.4); and SB steps Fig. 12.6(d,e) which run normal to the d i m e r rows (i.e., the s e g m e n t s parallel to the nominal direction on " r o u g h " steps in Fig. 12.4). Two different types of SB steps are present d e p e n d i n g w h e t h e r the atoms at the step edge are bonded Fig. 12.6(d) or n o n - b o n d e d Fig. 12.6(e) (second layer atoms on the top terrace have fully satisfied bonds for b o n d e d steps while for
Atomic scale defects on su~. "aces
669
non-bonded steps they have one dangling bond). Figure 12.6(c) shows double steps DB which have the same microscopic configuration as SB steps but separate terraces with the same (2• type reconstruction. Double steps with similar configuration as SA steps are theoretically predicted to have higher energy and have not been observed experimentally. Initial diffraction experiments have shown that at high vicinalities (0 > 2 ~ mostly D a steps are observed. This is clearly seen from the higher (1/2,0) superstructure intensity (than (0,1/2) intensity) which implies predominantly (2• type terraces. Double periodicity between successive out-of-phase conditions is measured along the reciprocal lattice rod, indicating double height steps (Saloner et al., 1987). Spot profile analysis of the split spots with HRLEED is used at the out-ofphase condition to confirm that there is a considerable amount of step meandering although it is not possible to resolve the detailed picture of the alternating degree of roughness, seen in Fig. 12.4. Contrary to the picture of a perfect staircase that implies split fundamental spots, finite intensity is present at the center, confirming that one of the domain types is more populated than the other, so the cancellation of intensity is incomplete at the center of the profile. As a function of temperature stepped surfaces of high vicinality, with Da steps, undergo transition to surfaces with SB single steps. The transformation depends both on temperature and vicinality with higher temperatures required at higher vicinalities to attain a given fraction of SB steps (De Miguel et al., 1991)). It is also evident that the large-scale domain morphology of stepped and singular Si(001) is correlated to energetic differences in the local atomic configurations of the reconstruction. From images like the one shown in Fig. 12.4, information about the energy cost of the different bonding configurations at steps is measured (Swartzentruber et al., 1990). It is straightforward to conclude that SA type steps must be energetically favored since they are more abundant. A quantitative description of the step roughness is shown in Fig. 12.8(a), with s denoting the separation between kinks and n the length of kinks (for "rough" steps, separation lengths s correspond to Sa-type steps and kink lengths n to SA-type steps). The size distributions P(s) and N(n) for both separation and length distributions can be measured accurately by statistical analysis of a large number of STM images. For non-interactive kinks P(s) is expected to be given simply by the product
P(s) = p( 1 - p)~-~
(12.10)
where p is the probability to form a kink and ( l - p ) to continue with a straight edge. Figure 12.8(a) shows that the experimentally determined P(s) follows exponential dependence on s as predicted by Eq. (12.10) which implies simply that the energy cost of a kink of length n is proportional to its length
E(n) - (Zs^ n + ~,~)
(12.11)
where ~'sA is the step energy per unit length for SA-type steps and Xc the kink formation energy. N(n), which is generated thermally at the annealing temperature T = 625 K should follow Boltzmann statistics
670
M.C. Tringides
0.6
I
"
0.4
P(s)
s
, \
~
SA
I -o, J l
~
SB
,
a-O C'-O G'~'~I
"
0.2
(a) 0.0
0
I
I
I
2
I
I ~I~,-4,-.-~
4
B 8 s (dirners)
L
1
I0
k
l
12
I0 ~
t
10 3
I
N(n)
I
3"
I0 a
I
-f
10
(b)
1
1
0
1
1
2
4
6
n (dimers)
8
I0
Fig. 12.8. Measurement of the energetic parameter ks^, k~ of the different types of Si(001) steps. (a) The separation distribution P(s) vs. s, follows an exponential dependence on s (Eq. 13.12) that proves the absence of kink interactions. (b) The kink length distribution N ( n ) vs. n, measured for the annealing temperature T = 625 K. Since there are no kink-kink interactions, the cost in energy of a kink of length n is simply E(n) = n ks^ + k~ and obeys Boltzmann statistics. By fitting the data to the form N ( n ) = No e --e~"vkr the step (per unit length) parameters ks^ = 0.04 eV X,~ = 0.06 eV are measured within 10% uncertainty. (Swartzentruber et al., 1990).
N(n) = No e -e~')/kr = No era'^ " +~)/kr
(12.12)
(The STM image is measured at room temperature but the kinetics are slow enough, as deduced from the diffusion activation energy E = 0.65 eV discussed in w 12.4.2.3, to identify the imaged configuration as a frozen-in high temperature state). The N ( n ) vs. n data shown in Fig. 12.8(b) in a semi-log plot follow Eq. (12.12). The slope of the linear segment with the heating temperature T = 625 K, can be used to extract the step energy (i.e., energy/length) ~,s^ = 0.04 eV/2a and the intercept can be related to the kink energy ),.c - 0.06 eV/2a, with 2a the spacing between dimers. Similar analysis on the "smooth" steps was used to measure the corresponding step energy for SB steps, ~.s, - 0.11 eV/2a. The experimental uncertainty in the measured values is 10%. These values are in qualitative agreement with theoretical calculations, using semi-empirical, tight binding energy models which predict ~sA = 0.02 eV/2a, X s , - 0.3 eV/2a (Chadi, 1987). The sum of the step energies of SA- and SB-type (both the experimentally measured and the theoretically estimated values) is greater than the calculated energy for double steps ~'D, = 0.05 eV/2a, so double
Atomic scale defects on su~. aces
671
steps DB should be present under all conditions and single steps should be absent. As explained before, this is only true at higher vicinities and lower temperatures; otherwise single steps are observed, which implies that there must be additional energetic factors stabilizing the step configuration. Controlled strain experiments have indicated that a stress related contribution to the energy controls the equilibrium configuration of the Si(001). Compressive strain ~ext is applied by bending the free end of a sample, with an anvil assembly attached to a precision linear motion feed through. It is found that even for strains as low as 0.3%, a singular Si(001) surface with initially equally populated ( 2 x l ) and (1• domains, transforms into one with mostly (2xl) domains, which is the same preferred type as in the transition on the stepped surface. The transformation is reversible with strain and it occurs faster at higher temperatures. However, the final amount of asymmetry between the two domain types is independent of temperature. This implies that the transformation is different from the usual type of entropically driven order-disorder transition. Instead, the amount of asymmetry is simply related to the mechanical stress energy of the system which only depends weakly on configurational entropy. As explained in Chapter 2, surfaces even under equilibrium, can have a non-zero intrinsic stress tensor defined by I ~9E ~ = A-~c~3nij
(12.13)
where n 0 is the surface strain tensor, (ij) label possible directions on the surface, E the energy per unit area, and A0 the area of the surface unit cell. The energy can be easily calculated as a function of the atomic coordinates, in a given structure, which can be differentiated according to Eq. (12.13) to measure o 0. The Si(001) surface terraces with (2• and (1• types of reconstruction have different stress components, o~, o2, respectively, as a result of the anisotropy in the dimerization. Tight binding calculations have estimated the tensors to be o~ = 0.035 eV/A 2, O 2 - - - - 0 . 0 3 5 eV/A 2, with the negative sign indicating compressive and the positive sign tensile stress. At the steps separating different domain types there is a discontinuity in the stress which results in a force field F(~ = + (ol - o2)
(12.14)
with the sign changing at the two types of reconstruction of the step edge. This intrinsically generated stress is located at steps but it acts like an externally applied strain affecting the position of atoms over large distances. When the force field is integrated over the whole surface it adds a positive term to the surface energy which increases monotonically with the domain size L, and therefore, favors domains of smaller size. Since smaller size generates more boundaries, there is higher energy cost in the increasing number of missing bonds, which opposes the reduction of the domain size. The equilibrium configuration of the Si(001) surface is a result of the balance between these two competing terms. Mathematically exact expressions can be written in terms of the size of one type of the domain L
M.C. Tringides
672
(assuming straight step edges) and the asymmetry in fractional occupation of the two types of domains (p = 0 corresponds to equal populations of ( 2 x l ) and (1• domains, while p = 1 corresponds to a surface entirely covered by one domain type), E is the overall surface energy per area
E(L,p) = ~)'% + )'% + ~1 ~:extP(O, - 0"2) - 2 (ol
2rt 12.15)
w h e r e F~ext is the externally applied strain, ~t is the bulk modulus and v the Poisson ratio. Minimizing this expression as a function of the two parameters L, p can explain the different types of transitions on the Si(001) surface. For singular or surfaces of low vicinality, even for zero external strain tex t = 0 , E is minimized for p = 0 and some characteristic length L0 which explains why equal mixtures of the two domain types and single step heights are formed despite the lower energy cost of DB steps. The transition on singular surfaces with non-zero F_,ext :g: 0 results in p ~: 0 since the dependence on p of the second term far outweighs the weaker dependence of the last term. On a vicinal surface, one has the constraint L = constant, so E is simply minimized as a function of only p. It is clear for small enough L (i.e., high vicinality) that the contribution of the last term cannot compensate for the large number of domain walls (i.e., the energy cost of the first term in Eq. (12.15)). Refinements of Eq. (12.15) are needed to account for the entropy contribution with temperature increase, the dependence of the stress anisotropy on step meandering (straight steps in the calculation of Eq. (12.15) were assumed) and direct step-step dipole interactions due to the local strain caused by the rebonding of the S~ step. Such refinements can explain changes in the surface vicinality from double to single steps at higher temperatures and justify why a continuous transition is observed contrary to the predicted first-order transition, based on Eq. (12.15) (Webb, 1994).
12.3.2. Dislocations Dislocations are found in bulk materials as a result of kinetic limitations during crystal growth. Sometimes they can be eliminated after prolonged annealing, but often the times necessary are impractical. The definitions of the various types of dislocations have been given in w 1.6.3 with Fig. 1.24 providing a good visualization of screw and edge dislocations with their corresponding Burger vectors. Surface sensitive techniques provide information about the projection of dislocations onto the surface plane. If the distortion observed in the 2-d image is measured along different planar directions, the 3-d structure of the dislocation can be reconstructed. Since dislocations are aperiodic and irregular defects, microscopy is superior to diffraction in identifying their detailed configuration.
673
Atomic scale defects on su#aces
(a)
B1 \
B2 &
A---~ A'--~
CI
7'
7" C'
C2
(b)
Side View h :i.1~ w
~
v
Fig. 12.9. (a) STM image of a Cu(ll 1) surface show_ing a dislocation at the center. By viewing the picture at a glancing angle along the A,A' arrows ([ 1 I0] direction) a shift of d/3 is observed where d = 2.21 ,~ is the spacing between rows. If the picture is viewed along (C1, C') ([ 101] direction) the shift is 2d/3. These shifts provide the components of the Burger vector projected onto the surface plane. (b) A line scan along the A direction through the dislocation core, showing a single step height change 2.1 ]k. This and the information deduced from figure (a) uniquely specify a Burger vector b = I/2 [ 101 ]. A more detailed examination of the shifts show the presence of two peaks D~, D2 which suggests that the dislocation can be wr_itten as the sum of two partial dislocations b = I/6 [ 112] + 1/6 [211 ] (Samsavar et al., 1990). An atomically resolved S T M image of a Cu( 1 1 1 ) surface prepared by sputtering and annealing with a dislocation is shown in Fig. 12.9 ( S a m s a v a r et al., 1990). F r o m images on a larger length scale a dislocation density of 109 dislocations/cm 2 is measured; this is much higher than the densities found on s e m i c o n d u c t o r surface_s. In Fig. 12.9(a) the A arrow is along the [1 10] and the CI arrow is along the [101] direction of the C u ( l 11) surface. In the middle of the image the dislocation causes distortions that involve in-plane and out-of-plane d i s p l a c e m e n t s of atoms. T h e spacing b e t w e e n successive rows on the (111) surface (for the three e q u i v a l e n t directions) is d - 2.21 A; careful viewing of the picture at a glancing angle along the A and C I arrows shows that after crossing the dislocation, the rows are shifted by constant amounts d/3 (for rows along the A) and 2d/3 (for rows along the C l) directions. There is no shift along B2. T h e s e shifts are the projections of the d i s l o c a t i o n ' s B u r g e r vector onto the two i n d e p e n d e n t directions. Line scans along the A direction are shown in Fig. 12.9(b) which can be used to deduce the vertical positions of the atoms. The difference in the position of the atoms in the flat regions
674
M.C. Tringides
away from the dislocation is h = 2.1 A, which corresponds to the single step height on the Cu(111) surface. The lateral and vertical information about the dislocation fully determines the Burger vector b = 1/2[ 101 ]. The vector forms an angle c~ = 35 ~ with respect to the surface normal and since the dislocation line is expected to be normal to the surface (for well-annealed dislocations) it follows that this is a mixed dislocation ( a = 0 corresponds to a screw and c~ = 90 to an edge dislocation). The transition region around the dislocation core is fairly wide, far larger than what is expected, because of finite tip size effects. It clearly shows two distinct peaks D1,D2 which can be identified as the cores of two partial dislocations summing up to the mixed dislocation we have identified before; i.e., b = 1/61112] + 1/6121 1]. The vertical heights for the two partial dislocations are 2/3h and 1/3h so they add up to the single step height. This can be further confirmed if the in-plane shifts around the two cores in the surface plane are measured for each partial dislocation along two independent directions (as done before) although the uncertainty is higher since smaller displacements are involved. It is common for a dislocation to split into two partial dislocations to reduce abrupt structural distortions and to lower the strain energy.
12.4. Two-dimensional defects Overview
Structural information about the surface morphology, for clean and absorbate-covered substrates, can be readily obtained with both diffraction and STM. By measuring the intensity distribution as a function of Q= the step height distribution is obtained, while intensity measurements as a function of QII provide the terrace size distribution. This information can be verified directly with STM images. For well-annealed surfaces the number of layers exposed and, therefore, the interface roughness is relatively small. However, defects common in the bulk (mosaics, twins, strain), can be projected onto the surface. For epitaxially grown surfaces, one has the ability to manipulate growth and attempt a variety of combinanons for film/substrate materials, but usually at the cost of increasing interface roughness and lateral disorder. For ordered overlayers, the local arrangement of the overlayer atoms within the unit cell is easily observed with diffraction, even in cases where there is no commensurate relation with the substrate. Such arrangements can be also observed with STM whenever the overlayer atoms can be clearly distinguished from the substrate ones. However, identifying the long-range arrangement (type of domains, domain size distribution, shape of the domains) is a more difficult challenge. It can be done with diffraction by analysing the superstructure intensity when interference between different domains can be ignored and with STM by imaging the domains directly (although in large scale images atomic resolution is lost). LEEM is the ideal technique to map the domain distribution, but only for length scales larger than 100 ~,, the resolution of the technique.
Atomic scale defects on su~. aces
675
The driving forces responsible for the morphology of the overlayer are fairly well understood. For ordered structures which involve the repeat of small superstructure unit cells (a few times the substrate one), short-range absorbate-absorbate interactions of different strength or sign (i.e., attractive or repulsive) are responsible. In several cases domains (which can be as long as 100-200 ]k) order in superperiodic arrangements over several ~tm. The ordering is driven by long-range stress mediated interactions that establish the superperiodic arrangement as the configuration of the minimum energy. The repeated domain size results from a competition between the stress energy that favors small domains and internal domain energy that favors large domains. Despite the abundance of techniques and experiments for the characterization of surface defect structure, quantitative information about the energetic parameters responsible for the defect formation is rather limited and can benefit from additional thermodynamic measurements. An additional complexity is that in several cases thermodynamic information might not be sufficient because defect structures form in kinetically limited, metastable configurations i.e. domain wall network in overlayer domains, interface roughness in epitaxially grown films, etc. In such cases, including all the microscopic processes with their individual kinetic rates is necessary to account for the observed defect distributions.
12.4. I. Clean surfaces 12.4.1.1. Mosaic structure, strain, and stacking faults The preparation of clean surfaces in vacuum is, in general, a lengthy process. Different cleaning recipes are used, depending on the chemical identity of the substrate: high temperature flashing for refractory metals, sputter and anneal cycles for soft metals, cleavage for semiconductors, etc. Clean surfaces can also be grown epitaxially by direct deposition of atoms after the surface has been partially smoothed out with the use of buffer layers. By manipulating the growth parameters (deposition rate, substrate temperature) the quality of the growing layer can be optimized. Regardless of the method of preparation, a residual amount of disorder is still present on the surface and needs to be characterized before overlayer adsorption. Several of the defect structures present are projections of 3-d defects that commonly exist in the bulk. As discussed in Chapter 6, different crystal grains terminate at the surface in a mosaic structure that can be easily described with diffraction. When electrons are used as the incident beam, all diffracted beams (hk) broaden with Qz, the normal component of the momentum transfer. Strain is another type of bulk defect that extends up to the surface and results in finite displacement of atoms from their equilibrium positions r,, = ro, + 8,, (Welkie and Lagally, 1982). For non-uniform strain, 8,, depends on the position of the nth atom and can be described in terms of a distribution of displacements P(8). Diffraction can be easily used to determine P(8). The resulting effects on the distribution of diffracted intensity are similar to the effects of thermal disorder, discussed in Chapter 6. The attenuation of a given (hk) beam is independent of Qz but it increases with Qll, for
M.C. Tringides
676
B.I~_ [I~
=8.88 0
0
=
0.06 .
elP
qD
-
0.0J,
0. 02
o. BB
~
4. BO
I
J
5. 0 0
I
B. 0 0
l
10
0B
t
12.00
Q,(k') Fig. 12.10. FWHM of specular beam profiles normalized to the width of the Brillouin zone vs. Qz for a Ag film grown on mica substrate. The dashed line is the instrument response function. Both the mosaic size distribution and the average value of strain (flea,,-- 0.025) are obtained (Welkie and Lagally 1982). in-plane 5, displacement. Figure 12.10 shows a set of measurements of the F W H M s as a function of Q~ for different (hk) beams of Ag films grown epitaxially on mica substrates. The slight increase of the F W H M s with Q, is accounted for by the mosaic structure of the film, which is determined from lineshape analysis of the (00) beam. By modeling the dependence of the F W H M s on Qz for the other beams, a small amount of non-uniform strain <5>/ao = 0.025 is measured in these films. Uniform strains can also be present with all atoms displaced by the same amount from their equilibrium positions so the surface is described by a different lattice constant a'o = ao + Aao. Unlike the case of non-uniform strain, no diffracted intensity is redistributed in reciprocal space and the intensity profiles are unchanged. Uniform strain can be easily measured from the shift in the beam positions, observed from the new exit angle 6=
z~a a,,
-
cos0, sin0, - sin0,,
A0,
(12.16)
where 0o is the incident angle, and 0, the exit angle. Crystals constructed by building up successive layers in regular sequences A B C A B C A B C A B C ..... can have faults in their stacking that propagate up to the surface. For example fcc crystals are better visualized from the close packing of spheres with the centers of the spheres lying above the center of the enclosed triangular region (see Fig. 1.3) and shifted laterally in successive layers. An extrinsic stacking fault is formed when the "wrong" layer (indicated by the symbol !) is added resulting in a sequence A B C A B C A B A ! C A B C ...... or an intrinsic stacking fault is formed when a layer is removed A B C A B ! A B C .... Partial dislocations are formed as boundaries to the faulted areas. Semiconductors (Si, Ge) with the diamond cubic structure (Fig 1.5) which consists of two interpenetrating fcc lattices have a similar stacking sequence to fcc lattices, with each stack consisting of two
Atomic scale defects on su~. "aces
677
Fig. 12. l I. (a) Top and (b) side schematic view of the Si(l 11) (7• reconstruction. Each triangular hall of the (7• unit mesh corresponds to a different stacking sequence, AaC (left hal0 and AaB (right hall). The size of the symbols indicate the level of the atom (Takayanagi et al., 1986).
layers. Stacking faults in the diamond cubic structure are similar to the ones observed for fcc lattices. A well-known stacking fault is formed in the Si(111)(7• reconstruction (Takayanagi et al., 1985) shown schematically in Fig. 12.11 (a) as a top view and Fig. 12.11(b) as a side view. Different size atoms correspond to different layers with the bigger size corresponding to the top layer atoms. Changes in two successive layers are involved in producing the 7• periodicity. The solid circles denote atoms in deeper, unreconstructed layers. Adatoms on the top are shown as large, heavily outlined circles (12 in each cell). Atoms in the first reconstructed layer hold inequivalent positions denoted B, C in each triangular subcell and are responsible for the change in the stacking sequence. In the level below there are two groups of 15 atoms at sites directly above the unreconstructed A sites and 9 dimers at the sides of the subcells (smaller open circles). The arrangement of atoms in each subcell follows a different sequence AaB, AaC. A different fault in the stacking of a crystal can result in the sequence: ABCABC !ACBACBACB ..... where two equivalent stackings meet at a twin boundary. Each of the two segments of the crystal involves a normal ordering and no extra atoms are added. Twin boundaries do not have as drastic effect as stacking faults in the properties of the crystal. When they are projected onto the surface they result in a network of domain walls, to be discussed later in the section on overlayer adsorption.
12.4.1.2. Lateral and vertical disorder on clean surfaces The residual amount of disorder in a well prepared surface can be characterized in terms of several parameters that describe the vertical and lateral distribution of
678
M. C. Tringides
atoms: the number of exposed levels n = -N, ( N - l ) .... n .... (N-1), N, the step height distribution, the coverage 0, at level n, the average terrace size at a given level and the full terrace size distribution at level n, P(L,). Vertical disorder is statistically described in terms of the mean square height fluctuations defined by G(r) - ((h(r) - h(r)) 2)
(12.17)
that measures the mean-square deviation of the interface height from its average value h(r). A flat surface has G(r) = 0 while a thermally roughened surface above the roughening transition is described with a diverging form for G(r) as r ~ oo. In all cases G(r) is a monotonic function of r, describing the higher probability for more levels to be occupied at larger distances. Several different forms for G(r) have been given in w 6.2.2. The interface width w is obtained from the asymptotic value of ~--(r) as r --~ oo since no correlations will survive and all levels have the same probability to be occupied N
w 2 --G(r) - (2N 1+ 1-----~y--' n2 ..N2
(12.18)
n=-N
For surfaces that facet in pyramid-like configurations the height fluctuations scale like G(r) -- r 2 since the inclination of the surface is constant and the height excursion is linearly proportional to the lateral position on the surface. For a small number of exposed levels a detailed description of the atom arrangement at each level is possible while for larger values of N a statistical description is more meaningful. Diffraction can be used to deduce the previously described parameters at each level by modeling the distribution of diffracted intensity in reciprocal space l(Qi j, Q~) (Horn von Hoegen et al., 1988). Figure 12.12 shows a sequence of specular beam profiles I(Qjj) measured at different values of Qz on a homoepitaxially grown Si(001), obtained with HRLEED. Similar measurements can be performed with other diffraction techniques. Real space information can be extracted from the diffracted profiles by modeling the scattering process, within the kinematic approximation, which assumes single scattering off the topmost atoms. Although multiple scattering should be carefully considered for incident radiation that scatters strongly (i.e., electrons, ions), minimal effects are expected when long-range positional correlations and not the local atomic positions are probed. The kinematically scattered intensity can be written in general as the sum of two distinct components, a narrow instrumentally limited one 8(Q,) and a diffuse one
L(Q~) I(Q~,Q:) - A(Q.) 5(Q~) + B(Q z) L(Q~t)
(12.19)
Both A(Qz), B ( Q z) are periodic functions of Qz with periodicity 2rt/b where b is the step height. The narrow component is present because the atoms, despite shifts in the scattering phase from different levels, occupy lattice sites. From the data of
679
Atomic scale dejects on su~. aces
"'-'-~'-_'"AL 1.9
1.7
1,6
1.5
t.4
lo1, -~'L
----" ,-10t,
0~
20",,
Qll Fig. 12.12. Specular beam profiles/(QII) vs. QIIfor different values of Q~ for a Si(100) substrate grown epitaxially, showing the characteristic two-component structure. The decay of the narrow component c)l the profile with Qz can be used to measure the interlace roughness and the FWHM of the diffuse component at the out-of-phase condition Q:b/n = (odd) can be used to measure the average terrace size (Horn von Hoegen et al., 1988). Fig. 12.12 it can be deduced that single height steps are present on the surface b = 1.36 ,~ - ao/4. Such step height determination is straightforward with diffraction, because periodic variation of the sharpness of the measured profile with Q~ can be easily observed, corresponding to in-phase/out-of-phase conditions. For the low-index surfaces of single component crystals, single height steps are most c o m m o n l y observed because the formation of multiple height steps is energetically more cc)stly. (As discussed in w 12.3.1.3, it is possible for stepped surfaces to form double steps at high vicinality or with externally applied strain.) Multiple height steps can be formed by cleavage, when several planes of atoms are removed macroscopically, or in crystals with stoichometrically different planes of atoms, i.e., Cu3Au(001). Planes with and without Au atoms alternate in bulk successive layers, while only planes with Au atoms terminate at the surface, which result in double height steps (Niehus and Achete, 1993). Multiple height steps introduce higher periodicities in the dependence of the diffraction lineshape on Qz, 2rt/mb with mb a multiple step height. The fundamental periodicity 2rt/b of a single step is still easily measurable
M. C Tringides
680
but introducing higher periodicities smears out the sharpness of the modulation. The interface thickness can be characterized by the mean square fluctuations in G(r). For discrete levels, its asymptotic value simplifies to n=N W2 -- Z
n20n
(12.20)
n=-N
For the experimental data shown in Fig. 12.12, a low value w = 0.55 was measured which implies that only a few layers (less than 3) are exposed even after a large number of layers have been deposited. This corresponds to almost perfect layer-bylayer growth of the Si which was independently verified in the same experiment, from sustained diffraction oscillations of the specular intensity vs. time. In general, the maximum of the oscillations corresponds to the completion of a layer while the minimum to half-completed layers. Diffraction oscillations have been observed in many other systems with other diffraction probes and have been used as an independent measure of the absolute coverage. In many systems the oscillations are progressively damped with the degree of damping measuring deviations from ideal layer-by-layer growth and a finite interface width. The coverage at each level was determined progressively, from the start of the growth up to the topmost level, by fitting the decay of the intensity oscillations with time. The interface width extracted from the decay of the oscillations is in excellent agreement with the measured width w = 0.55 obtained from the variation of the Bragg intensity with Q~ as discussed before. Information about the lateral length distribution can be obtained by analyzing l(Qi I) at the out-of-phase condition where there is maximum sensitivity to surface disorder. The diffuse component of the profile Eq. (12.19) determines the lateral size distribution. The average of the FWHM is proportional to the average lateral size . The exact form of the profile space (i.e., whether it is Gaussian, Lorenztian, etc.) can be used to measure the terrace size distribution P(L,). For example, a Lorenztian lineshape corresponds to an exponential size distribution. For the sharp interface (2-3 layers) as in the experiment of Fig. 12.12, = 16 atoms extracted from the FWHM is the average terrace size of the topmost level. For the statistical description of the clean surface, the function G(r) monotonically couples the lateral to the vertical distribution. It has been predicted theoretically (Family, 1990) that the short range functional form of G(r) is a power law r 2'~ for r < ~ (where ~ is a characteristic lateral length) while for larger distances it saturates to its asymptotic value w2, limited by the finite number of levels exposed. Smaller values of ~ correspond to the presence of higher frequency spatial components or equivalently short-range roughness in the interface. It is important to emphasize that accurate testing of this form of G(r) requires sampling the interface over large distances. This is necessary not only for a reliable evaluation of the averaging process, but to confirm the predicted power law since power laws are notorious in the requirement of covering sufficient decades in variation both for the independent and dependent variable.
Atomic scale defects on su~aces
681
The description of a growing interface requires specifying the vertical (i.e., width of the interface) and lateral (terrace length) distribution. Microscopically, fluctuations in the incident deposition rate can produce rougher interfaces and smaller terraces while surface diffusion has the opposite effect. A rate equation can be written for the rate of change of the interface height h(r,t) at position r and time t, in terms of the incident flux and a gradient term reflecting the type of diffusion current operating. Under most conditions, it has been shown theoretically that the vertical and lateral distributions do not grow independently, but they obey self-affine scaling, i.e. if the vertical scale increases by a factor b and the lateral scale changes by a different factor b y, the interface morphology remains invariant. The value of the exponent y is universal, i.e. it does not depend on the details of the system but simply on the type of diffusion current present. Such predictions are important because they guarantee that a few measurements of the surface morphology at one time fully determine the evolution of the morphology at later times. Diffraction and STM have verified such predictions on epitaxially grown or sputtered-deposited films (You et ai., 1993).
12.4.1.3. Facetting A surface with an initially symmetric configuration can undergo a facetting transition to a less symmetric configuration, consisting of planes of different orientation, typically arranged in a ridge-and-valley morphology as shown in Fig. 12.15(a). This phase transition is driven by the lowering of the surface energy of the facet planes with respect to the energy of the original orientation. It can also be induced by surface reconstruction or the presence of foreign atoms. It requires a large amount of mass transport so it is best accomplished by heating the surface to sufficiently high temperatures. As explained in w 2.2, facetting transitions can also be observed from an initially vicinal surface which phase separates into regions of different vicinalities: larger flat terraces of the nominal low index orientation separated by steeper regions of higher vicinality where steps "bunch" together so the overall orientation is unchanged. Such step "bunching" has been observed during the thermal annealing of vicinal Si(111). An example of a facetting transition driven by the presence of foreign atoms is shown in Fig. 2.3. An initially vicinal Ag(110) surface misoriented by 2 ~ in the [001] direction (so the steps run along the closepacked [110] direction) develops flat terraces up to 300 A (10 times the original size) separated by steeper step "bunches" after treatment with oxygen at 150 Langmuir for 160 min. This transition can be lifted with the addition of CO. It is explained by the gain in energy of the (110) terraces as oxygen chains of (3• periodicity are formed normal to the step edge. Facetting can be easily deduced from the diffraction pattern which shows additional diffraction spots. It can be explained as a superposition of the patterns produced by the different orientations present on the surface. For a surface with one orientation, the specular beam is routinely identified as the convergence center of all the other spots with increasing beam energy. For a facet surface each facet has its own specular beam. Each side of the hill and valley structure can be thought of as a finite "staircase" surface with the characteristic tilted reciprocal lattice shown
M. C. Tringides
682
(010)
9
9
9 9
9 9
9 9 9
9
~-,----------6a ,~
--I
9
9 9
9 9
9 9 9
9 9 9
9 9 9
9 9
9 9
9
9 9
9
9 9
9 9
9 9 9
9 9
9 9 9
9 9 9
// (100) 9 9 9
9 9
9 9
9 9
9 9
9
(a)
lO
flat area oo 1o
/
, 000
QJl
(b)
Fig. 12.13. (a) Schematic presentation of the TaC(110) facetting transition into a ridge-and-valley uniaxial (along the [1 I0] direction) structure, after heating the surlace to T= 1700 K. The height of the ridge is six atomic levels and the distance between the top of the ridges six lattice spacings (Zuo et al., 1993). (b) Reciprocal lattice rod schematic presentation of a surface that has faceted into a uniaxial ridge-and-wllley structure. Three sets of rods are present corresponding to scattering off the nominal surface direction and the two sides of the ridges. The intersection of the rods at points shown in 3-index notation (hkl) corresponds to the conditions of constructive interference for bulk scattering.
in Fig. 12.13(b). For symmetric facet morphologies the reciprocal lattice rods are tilted symmetrically around the nominal direction. This can be used to determine the orientation of the facet planes from a family of I(Q) vs. Qll profiles obtained at different values of Q:. Figure 12.14 shows such measurements for TaC(I 10) after the surface is heated to 1700 K (Zuo et al., 1993). The wave vectors corresponding to the centers of the additional peaks in the (Ql~, Q~) plane follow the tilted reciprocal lattice rod direction and can be used to deduce 45 ~ as the facet orientation. The periodicity length of the facetting, which measures the average separation of ridges, can be obtained fiom the presence of satellite spots next to the specular beam. The inverse of the satellite wave vector measures the periodicity length which is the standard method to detect regularities with diffraction. Their position does not change as Qz is varied, so they can be easily distinguished from the additional spots introduced by the intersection with the tilted reciprocal lattice rods. The facet orientation and the periodicity uniquely determine the height of the ridge-and-valley
Atomic scale dejects on sudaces
683
-,-,' @
0
~ ~ 5
N
~:
_
......
-.
.
.
"62
"" ::2-
.
5.50 N
,.~5.38
,
.0
,
i,,,
-0.8
i
-0.6
,
(
-0.4
,
i=,,
i
-0.2 -0.0 kalGlo
,]
0.2
,
i
0.4
,
i
0.6
,
Fig. 12.14. Diffraction profiles/(Qll) vs. QII obtained at different scattering conditions Q:d/rt for a faceted TaC(110) surface. By tracing the position of the facet spots on the (Qii, Q:,) plane, the orientation of the tilted reciprocal lattice rods in Fig. 12.13(b) and the direction of the facet planes ace determined. The satellite peaks close to the specular beam which do not move with Q: can be used to deduce the distance between the top of the ridges (Zuo et al., 1993).
structure, which is essentially the interface width w defined in Eq. (12.18). This can be measured independently as a consistency test, from the decay of the specular beam intensity vs. Q~. (Knowing the facet orientation, the distance between the ridge-and-valley structure and the interface width determines the facet morphology" this information can easily distinguish between triangular vs. trapezoidal geometry.) An analogous analysis can be followed for different facetting structures (i.e., the phase separation into larger flat regions and steeper regions of step "bunches") but the tilted reciprocal lattice rods are not symmetrically positioned with respect to the nominal direction. For the TaC(I 10) transition the lateral separation of the ridge tops was 19 ~ w_hich implies that the ridges height is six layers. The facetting was uniaxial in the [ 1 I0] direction, exposing [ 100] and [010] planes at valley walls. Long, uninterrupted structures observed in the direction orthogonal to the facetting direction with STM have confirmed the diffraction results.
684
M. C. Tringides
Fig. 12.15. Schematic presentation of a "mound" morphology which can be obtained during epitaxial growth for systems growing in 3-d mode. Cu on Cu(100) is an example where this "mound" morphology was observed with He-scattering for T = 160-200 K. Facetting structures have been observed during homoepitaxial growth. They produce a "mound" morphology, shown schematically in Fig. 12.15 with a linear relation between the mean square height fluctuations G(r) vs. r. This type of morphology has been observed during the growth of Cu on Cu(100). After depositing 5 0 - 1 0 0 ML additional diffraction spots are present in the diffraction pattern. They correspond to the formation of regularly spaced pyramid-like structures with sides in the (113) and (115) (direction depending on the substrate temperature T = 16()-20() K) (Ernst et al., 1994). Microscopically this is related to the presence of an additional energy barrier (to be discussed in w 12.5 and 12.6) at the steps that limits transport between terraces from higher to lower levels, thus promoting "mound" structures.
12.4.2. Two-dimensional overlayers on su~, "aces 12.4.2. I. Surface overlaver cot!figurations The arrangement of atoms in submonolayer surface overlayers has a well-defined periodicity which can differ from the periodicity of the substrate as discussed in Chapter 1. This periodicity can be described by comparing the size and orientation of the overlayer to the substrate unit mesh. Simple overlayer unit meshes have either ( Ix l) symmetry identical to the substrate one or they have unit meshes with unit vectors that can be expressed as integer multiple combinations of the substrate unit vectors (w 1.4.3). The former case is expected in systems grown homoepitaxially. Although f o r ( l • symmetry it is possible to have a shift of the atomic position of an overlayer with respect to a substrate atom, this shift is constant in all unit meshes so it does not change the fundamental symmetry, but it can change the structure and electronic properties. For example, the binding site on square fcc(100) substrates can be the four-fold hollow site (site c in Fig. 1.12), at the center of the substrate unit mesh, without affecting the ( I x l ) symmetry. No antiphase walls between overlayer domains are introduced in such ( I x l) systems. If however, there are inequivalent binding sites, then it is possible to have different types of ( l x l ) regions with atoms occupying inequivalent sites in each region, which can introduce domain boundaries between them. For example on bcc(110) substrates if the binding site is the triply-coordinated ("hourglass" type) site shown in Fig. 12.16(a)
685
Atomic scale defects on su~. "aces
ort
brldge
a
Top View
~
-Si Trimer
O
:Ag atom
b Fig. 12.16. (a) Superstructures formed by chemisorbed O on W(110) at different coverages: (2• (Ix2), (2x2). Each one is four-fold degenerate. The oxygen binding site is the triply coordinated one, with two equivalent positions lor an atom to occupy, which doubles the domain degeneracy. (b) Si(l 11 )(',~-3><~ff)-Ag superstructure. It has three-fold degeneracy since its unit mesh can be orientated in three different directions with respect to the substrate unit mesh with the origin of the structure in any one of the four corners of the mesh. The local bonding configuration involves the formation of Si trimers. there are two possible positions for the atom to o c c u p y (at the top and b o t t o m o f the " h o u r g l a s s " ) which are not c o n n e c t e d by a reciprocal lattice vector. This arrangem e n t will be discussed further on the W( 1 1 0 ) - W system. F o r fcc(l 11 ) substrates it is possible for a d e p o s i t e d atom to sit on sites that b e l o n g either to fcc or hcp stacking, which can also result in a distribution of i n e q u i v a l e n t ( l x l ) d o m a i n s . O v e r l a y e r s with unit vectors that can be written, as integral multiple c o m b i n a tions of the substrate unit vectors p r o d u c e additional, superstructure spots in the diffraction pattern. As described in C h a p t e r 6, the reciprocal lattice unit mesh can be d e d u c e d from the size and orientation of the superstructure spots with respect to the f u n d a m e n t a l spots. For e x a m p l e , W ( 1 1 0 ) - O can form several s u p e r s t r u c t u r e s d e p e n d i n g on c o v e r a g e i.e. (2• (2x2). T h e s e have larger unit m e s h e s than the substrate but still have mesh vectors with the same orientation as s h o w n in Fig.
M.C Tringides
686
12.16(a). Unit meshes with different orientation from the substrate ones can be illustrated with the system Ag/Si(111) which is also discussed in Chapter 3. It can order epitaxially in the (f3-x'~3)R30 structure shown in Fig. 12.16(b) where the unit mesh is larger than and oriented 30 ~ away from the substrate unit mesh. An ideal overlayer extending over distances larger than the coherence length of the instrument should produce instrumentally limited superstructure spots. This is rarely observed which implies the presence of finite size domains on the surface, with different types of antiphase relations between them. Domains can form either because of kinetic limitations (i.e., slow diffusion), thermodynamic factors (i.e., topology of the phase diagram), substrate imperfections, or long-range stabilizing forces (strain). Coincidence lattices (w 1.4.3) have unit vectors that can be expressed as linear combinations of the substrate unit vectors with rational numbers as coefficients. The overlayer and the substrate lattices coincide after an integer number of m displacements in one and n displacements in the other direction which implies that satellite spots should be observed close to the fundamental ones, with the inverse of the separation between the satellite and fundamental spots corresponding to the distance over which matching between the overlayer and substrate lattice is attained. Coincidence lattices can be easily formed with systems that have small amounts of lattice mismatch with the coincidence distance being proportional to the inverse of the lattice mismatch. A higher periodicity, observed with LEED on Ni(l()0), is named "(7• structure, because of the presence of additional spot at wave vectors 1/7 the size of the Brillouin zone. It forms during the oxidation of Ni(l()()) and can be explained in terms of coincidence between the overlayer NiO(100) lattice and the Ni(100) substrate meshes (Wang et al., 1993). Registry is attained between 7 Ni(100) and 6 NiO(100) unit cells that have approximately 6:7 ratios in their lattice constants. Incommensurate structures can form with the overlayer unit mesh having no epitaxial relation to the substrate unit mesh. They can still be expressed as linear combinations of the substrate unit vectors but irrational numbers are needed as coefficients. New superstructure spots are also present exclusive to the overlayer. Different types of domains are possible with a continuously varying phase relation between them since overlayer domains can "float" anywhere on the substrate with arbitrary orientation.
12.4.2.2. Quantitative measures of overlayer morphology The arrangement of atoms on a surface can be described in terms of the spatially dependent autocorrelation function (Eq. 1.19) that defines the relative probability of two atoms to be separated by distance R P(R) - J 9( r)9( r + R)dr
(12.21)
where p(r) denotes the occupation probability at site r. This quantity can be measured directly in diffraction experiments, since the diffracted intensity depends on the relative phase shift of the scattered wave from
Atomic scale defects on su~. aces
687
the two atoms. The autocorrelation function is simply the Fourier transform of the intensity. Depending on the overlayer symmetry either the autocorrelation function of the overlayer or a combination of the substrate/overlayer autocorrelation functions is obtained. For systems that have ( l x l ) periodicity only the fundamental beams are present which are sensitive to an effective autocorrelation function defined by all positional correlations of the two types of atoms: substrate-substrate, substrate-overlayer, overlayer-substrate, overlayer-overlayer. All possible pairs contribute intensity to the fundamental beams so a single profile measurement of one of the beams might not be sufficient to identify uniquely each separate autocorrelation function. In the cases where the symmetry is not ( l x l ) , a new periodicity is introduced, measured by the location of superstructure spots. It exclusively contains information about the overlayer. Within the kinematic approximation the autocorrelation function of the overlayer can be obtained by Fourier transforming the superstructure intensity
P(R)- ~
I(Q~) e -i~
dQj~
(12.22)
BZ
The substrate autocorrelation function can be convoluted out of the profiles of the fundamental beams since the overlayer is already determined from the superstructure spots. In heteroepitaxial systems there can be large differences between atomic scattering factors depending on the atomic number of each element. Light adsorbates, especially at high energies, have negligible contribution to the fundamental beam, so most often the analysis of the fundamental beam directly determines the substrate distribution. From the autocorrelation function other measures that characterize the overlayer morphology can be constructed. A clear visualization of the arrangement of the deposited atoms can be obtained in terms of the domain size distribution P(N). This is defined as the fraction of domains of a given size N. It can control the electronic and structural properties of the system; it can also reveal with appropriate modeling, the kinetic processes responsible for the final configuration. A full description of the overlayer configuration requires the use of another independent measure, the island separation distribution g(s), which is equivalent to the substrate size distribution. This is defined as the probability of any two islands to be separated by distance s. g(s) can be measured by diffraction since it determines the degree of interference between the scattering of two islands. For overlayers with ( l • symmetry, fundamental beams depend both on the combined substrate/overlayer size distributions which can be extracted by modeling the lineshape, as discussed before for the autocorrelation function. On the other hand, for superstructures one usually relies on the random phase approximation, which assumes that the measured superstructure profile fully determines the overlayer size distribution and there is no contribution from the substrate (this is equivalent to having g(s) a uniform distribution). Such approximation is justified at low coverage when the islands are separated by large enough distances that their relative phases average to zero; or at high coverage when degenerate superstructures of different domain types are present (i.e., ( l x 2 ) domains contribute intensity to the (0, 1/2) beam vs. ( 2 x l )
688
M. C. Tringides
domains that contribute intensity to the (1/2, 0) beam) to be discussed further in w 12.4.2.4. One domain type acts like a "sea" separating domains of the other type, which cancels out possible correlations in the island positions. In all cases it is far easier to extract the average domain size (usually from the inverse of the profile FWHM) than the full domain size distribution P(N). The previous measures have been emphasized historicallybased on the extensive use of diffraction techniques. With imaging techniques both domain size and domain separation distributions can be measured directly, as long as sufficiently large regions are imaged to ensure good statistical averaging. Thermodynamic or kinetic measurements however, are more difficult because imaging techniques require a minimum acquisition time to attain the highest resolution. However, additional detailed information is obtained about the domain morphology: island shapes, the microscopic structure of boundaries separating adjacent domains, positional correlations between the domains and the terrace boundaries are clearly visible. It is more difficult to determine the exact shape of domain walls separating coexisting domains with diffraction since it requires measuring the diffracted intensity at large wave vectors (i.e., small distances) where the S/N ratio is low. Imaging techniques (especially STM with the advantage of atomic resolution) have been indispensable in probing the overlayer morphology at short length scales.
12.4.2.3. Overlayers with ( l x l ) symmetry The homoepitaxial growth of Si on Si(001) has been studied with STM (Mo et al., 1991). The atomic structure of the stepped surface has been discussed in conjunction with Figs. 12.7 and 12.8. A typical image after growth is shown in Fig. 12.17 at low coverage 0 = 0.07. The condensation of atoms into islands can be described in terms of the classical picture of nucleation. Clusters of size larger than the critical size cluster i are stable because the gain in cohesive energy exceeds the cost in boundary energy. In a deposition experiment, the island density N for sizes larger than i reaches steady state when the rate of deposition is balanced by the rate of atom incorporation into already formed islands. At this point, no new islands are nucleated. It has been shown with simple scaling relations that the island density in this steady state regime is related to the diffusion coefficient by N-- D -i/i+2. By measuring the island density attained at different substrate temperatures T = 300-600K the diffusion activation energy E = 0.65 eV of Si/Si(100) was extracted from the image of Fig. 12.17. Two additional features can be seen in Fig 12.17. The nucleated islands have elongated shapes with high aspect ratios up to 1:15. The long direction runs normal to the dimer rows, as expected, since they are incorporated into the next layer which should have a 90 ~ rotation in dimer orientation. This anisotropic island shape is a result of kinetic limitations: the islands evolve and stabilize into more isotropic shapes by heating at 700 K and cooling back to room temperature. Interestingly, there is a large difference in island density in adjacent terraces. Especially close to the steps the island density is smaller and "denuded" zones are clearly visible with widths that depend on the type of terrace. Larger "denuded" zones have been observed on the terraces with dimer rows running perpendicular to the step edge. The presence of "denuded" zones around steps is a result of adatom diffusion
Atomic scale defects on su~. aces
689
Fig. 12.17. STM images of epitaxially grown Si islands on Si(100). The islands are anisotropic indicating different accommodation coefficients for an incoming atom to attach to the sides of the island. Denuded zones are visible close to the steps with the ones on terraces with dimer rows running normal to the step edge larger indicating appreciable diffusion anisotropy. Th direction of the dimer rows is indicated with a white bar. (Mo et al., 1991 ).
towards and capture into stable binding sites decorating steps. The large difference in the size of "denuded" zones points to highly anisotropic diffusion on the two types of terraces as intuitively expected from the channeled structure, a result of the formation of dimer rows. Atom diffusion along the rows is faster than diffusion normal to the rows, so in terraces with rows running normal to the steps, the atoms can reach the step faster. The measured activation energy E = 0.65 eV is the diffusion barrier along dimer rows, with the diffusion coefficient in the normal direction being at least a factor of 1000 lower. However, the diffusion anisotropy cannot explain the island shape anisotropy since it favors the motion of atoms parallel to the narrow side and would add material to produce more rounded shapes. The shape anisotropy must be a result of differences in the local bonding configurations around the island perimeter; i.e., the accommodation coefficient of an atom must be different, with the elongated side having a smaller probability to accommodate an atom than the narrow direction. Both the diffusion and the island accommodation anisotropy do not affect the scaling relation relating island density to diffusion rate, in the limit of low coverage, when the average island size is smaller than the island separation. A low density of divacancies (i.e., missing dimers) which has been routinely observed on Si(001) is not correlated to the nucleation process as can be seen from the identical island density observed at different annealing temperature sequence. The island separation is approximately random and implies a uniform island separation function g(s), except near the step edges where the "denuded" zones occur. Nucleation processes on metal substrates have been studied with the homoepitaxy of Pt on Pt(111) in a wide temperature range T - 200-700 K (Michely et al.,
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Fig. 12.18. (a-e) STM images of the homoepitaxial growth of Pt/Pt(l 1l ) over the temperature range T = 200-700 K indicating a temperature dependent island shape transition, successively from dendritic, to triangular, to hexagonal, to inverted triangular, and finally to hexagonal shapes. (f) The shape transition can be explained in terms of edge diffusion anisotropy for diffusion along the two types of steps shown A,B. The step microstructure is different for the two types. In A type steps the island edge atoms must move to top sites (which is energetically costly); while for B type steps, they must move to fcc hollow sites (which is energetically favorable). (Michely et al., 1993).
1993) shown in Fig. 12.18. Dramatic island shape changes have been observed as a function of substrate temperature as a result of changes in the relative importance of kinetic factors (i.e., different diffusion barriers) with temperature. At low temperatures T = 200 K dendritic island shapes Fig. 12.18(a) are observed while at the highest temperature compact, hexagonal shapes are formed Fig. 12.18(e~), (e2). For the in-between temperatures still different shapes are seen: compact triangles with vertices pointing in one direction at T = 400 K Fig. 12.18(b), hexagons at slightly higher temperature T = 455 K Fig. 12.18(c), compact triangles with vertices pointing in the opposite direction at higher temperatures T = 640 K Fig. 12.18(d). The differences in the island shape points to the importance of step edge diffusion during growth. At low temperatures there is practically no diffusion of an atom at the steps (diffusion barrier E = 0.5 eV) while diffusion of single atoms on the terraces is faster (E = 0.25 eV). The incoming atoms diffuse at T = 200 K towards already formed islands, but once captured, they are practically immobile producing the fractal-like island morphology. Step edge diffusion increases with temperature but the step edge barriers are different on different type steps shown in Fig. 12.21(f). Because of differences in the local bonding configuration, the edge diffusion barrier at A type steps (characterized by the difficulty of moving a step
Atomic scale defects on su~aces
691
edge atom towards an on-top substrate site) is different from the barrier along B type steps (characterized by the ease of moving step edge atoms towards a triply coordinated substrate site). As the temperature is increased, diffusion at one type step is faster, followed by an intermediate temperature where diffusion at the two step types becomes comparable, until at even higher temperatures diffusion along the other type step becomes faster. (This assumes that each process follows an activated Arrhenius form over the whole temperature range and has a different prefactor, so necessarily the two rates intersect at the crossover temperature). The shape transition can be understood in terms of the step edge diffusion anisotropy if it is realized that the edge with the slowest diffusion rate will have the faster growth rate. This is the 1-d analog of the well-known enhancement of crystal growth rate on rough surfaces discussed in w 12.6: the edge with the slower diffusion has an irregular shape with a large number of kinks, which serve as traps for new atoms to arrive and be captured at the step edge. The intermediate temperature corresponds to the crossover point when the two rates are the same, so the two types of steps are present in equal amounts resulting in hexagonal shapes. At even higher temperature diffusion at the other type step is faster so islands have inverted shapes. The role of slow step edge diffusion can be seen in the images of Fig. 12.19 showing Au islands deposited on Ru(100) at room temperatures (Hwang et al., 1991). Dendritic island shapes are formed, a result of negligible edge diffusion. Heating to higher temperatures produces more rounded but still not compact islands until at the highest temperature T = 1100 K, an interconnected network of compact islands is formed. The fractal dimensionality of the islands grown at room temperature was shown to be 1.7 cc~nsistent with diffusion-limited-aggregation models. The island nucleation process can be monitored with diffraction by using the profile shape to measure the island size distribution or the island size separation. It
Fig. 12.19. Dendritic Au islands deposited on Ru(001) substrate at room temperature. The formation of fractal like island shapes indicates the absence of step edge diffusion (Hwang et al., 1991).
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has already been discussed during the growth of Si on Si(001) Fig. 12.12 with H R L E E D (Horn van Hoegen et al., 1988), that the diffuse profile c o m p o n e n t of the fundamental beam can be used to measure the effective average island size (from the F W H M ) or the full island size distribution P ( L ) (from the functional form of the profile). In the data shown in Fig. 12.12 a Lorentzian lineshape was fitted to the diffuse component which is consistent with exponential functional forms for both the island size and the island separation distributions. In this case it can be shown that the inverse of the measured F W H M is the geometric mean of the average island size and the average island separation. Different fundamental beam lineshapes (i.e., not Lorenztian) can be used to deduce other size distributions for the substrate and the overlayer (i.e., not exponentials). In particular, for distributions with island separations following a sharply peaked distribution at some characteristic distance s, the fundamental beam profiles should show satellite peaks at wave vectors q -- l / s . It can be thought that the islands are effectively forming a new "superlattice" with a larger lattice constant, s. (This picture is only approximate since at island sizes comparable to their separation, the relative phase of neighboring islands has a wide variation that can smear out the necessary diffraction condition for constructive interference between the islands. This should affect the
reciprocal space
LEED spot profiles for
O:0.5
E : 71.5 eV
monolctyers
at
E : 9 8 . 6 eV
T:400K 00 - spot
"E
......
!
0
0.1
-0.2
!
-0.1
1
0
I
0.1
i
0.2
position on screen (in fractions of normal spot distance}
surface
Fig. 12.20. Study of the homoepitaxial growth of W/W(110) with LEED spot profile analysis. At the in-phase-condition sharp (00) profiles are observed (left) while at the out-of-phase condition satellite "ring" structure is present (middle) with the characteristic distance R o~s-~ the inverse of the island separation. Schematic representation of the corresponding reciprocal lattice rod structure (right) (Hahn et al., 1980).
Atomic scale dejects on su~. aces
693
satellite peak intensity but not the satellite positions which can still be visible with sufficient detector sensitivity). Figure 12.20 shows this characteristic "ring" structure around the specular beam, observed during the growth of W on W(110), at a coverage 0 = 0.5 in the temperature range T = 300-430 K (Hahn et al., 1980). The satellite "ring" structure is observed at energies corresponding to the out-of-phase condition (middle) while sharp profiles are observed at in-phase-condition (left). The reciprocal space structure is shown schematically as azimuthally symmetric cylinders surrounding the (00) rod (right), over a wide range in Qz values (the two in-phase-conditions are denoted with the 3-d notation (110), (220) since they are identical to Bragg conditions for bulk scattering). The inverse of the satellite spacing 1/s is a measure of the island separation which can be combined with the absolute coverage to estimate the island density N. The azimuthal symmetry of the "ring" is consistent with no preferred orientation in the nucleation centers and with circular island shapes. From the data shown in Fig. 12.20 the island density is measured to be lx 1014 islands/cm 2. When the profiles are measured at different substrate temperatures, the "ring" diameter decreases, which indicates larger separations between the islands and smaller island density, as expected from the increasing role of diffusion. Scaling relations are expected between the island diameter and the diffusion coefficient, as described before in the STM experiments of the growth of Si/Si(100) which can be used to extract the single atom diffusion barrier of W on W(110).
12.4.2.4. Overlayers with superstructure periodicity As discussed before W(110)-O has been well-studied to measure its equilibrium phase transitions (Lagally et al., 1980) and under non-equilibrium conditions to monitor the domain growth kinetics (Wu et al., 1989). It can be used as a prototype system to illustrate the types of disorder in systems forming domains and methods to characterize it. As shown schematically in Fig. 12.16(a) different superstructure domains are formed depending on the coverage and/or temperature range. The exact T-0 phase diagram, shown in Fig. 12.21 has been mapped out with LEED intensity vs. temperature measurements at fixed coverages, to determine the transition temperatures where the intensity drops to zero. At low coverages, partially formed ( 2 x l ) and (Ix2) regions coexist with a lattice gas (single isolated atoms); as the coverage approaches the exact stoichiometry of the (2x I ) at t3 = 0.5, four degenerate (2xl) domains form along the two close packed directions. With further increase in coverage, (2x2) and ( 2 x l ) domains coexist until the ideal coverage of the (2x2) phase 13 = 0.75. At even higher coverage there is coexistence between (2x2) domains in the "sea" of the ( l x l ) phase of isolated holes. Different types of walls separate the domains, which can differ in their length, atom density, shape, etc. Domain walls are important in extracting the domain size distribution because depending on their separation antiphase relations can average to zero or persist to produce non-zero interference terms. The phase between two domains separated by a distance s---~an be written A ~ - ~l" s--?.For s--having a sufficiently wide distribution A~ can average to zero and the random phase approximation can be employed to quantitatively analyze the diffraction pattern. This condition is easily satisfied at
694
M. C. Tringides
800
p(IXl)
I
700
g
L.G.
w
p(2 X I1 + p(2X 2)
w
p[ 2X 9 p(~x
w500
40C 0
l
p(2 Xl) ,~ Q2
*
L G_ =
o',COvERAGE
0
'o'8'
Fig. 12.21. Temperature-coverage W(110)-O phase diagram obtained from the analysis of LEED superstructure intensities as a function of temperature. Depending on the temperature and coverage different ordered phases are formed. The phases are degenerate so a network of antiphase walls separates domains of different type (Lagally et al., 1980). low coverage, with few (2• domains randomly positioned on the surface. It is also satisfied at high coverage if all types of domains are present with equal probability. Both the (2• and the (2• domains are four-fold degenerate. (For each close packed direction there are two sublattices to be occupied by overlayer atoms in forming the (2• for a total of four domain types and there are four sublattices for the holes to order in forming the (2• At coverages with all eight domains present there is a "sea" of domains of different type separating domains that contribute intensity to the same superstructure spot, to randomize the phase. However, at exactly 0 = 0.75 only (2• domains are present and the different domains are in contact without a "sea" to randomize the phase. Thus, at this coverage the interference term will have a negative contribution to the ( 1/2, 1/2) diffraction spot. In this case it is more difficult to extract information about the domain size distribution and the density of domain walls. The overlayer morphology of the W ( 1 1 0 ) - O described above from diffraction experiments has been supported with direct STM imaging at different oxygen coverages (Johnson et al., 1993). Figure 12.22 shows a sequence of images obtained at increasing coverage reproducing with remarkable clarity the unit meshes shown in Fig. 12.16(a). At low coverage 0 = 0.2 rows of the (2• 1) phase are visible as dark depressions in Fig. 12.22(a). Close to the stoichiometric coverage 0 = 0.5 the surface is covered with the different types of (2x I ) domains (Fig. 12.22(b)). At even higher coverage ( 2 x l ) and (2• domains coexist (Fig. 12.22(c)). The domains have a compact morphology with a straight network of separating walls. They have average sizes of less than 50 ]k. Even when heated to high temperatures T - 1750 K they do not exceed 75 ]k wide. A closer examination of Fig. 12.22(a) shows that the spacing between the two domains indicated by arrows is not simply an integer multiple of twice the size of the unit cell a = 3.14 ]k. There is a shift of 0.25 A from
Atomic scale defects on su~. aces
695
Fig. 12.22. STM images of W(I 10)-O overlayer showing the formation of the different superstructures corresponding to Figs. 12.16(a), 12.21 and the domain walls separating them. (a) At low coverage 0 = 0.2 ML (2xl) islands and lattice gas (i.e. isolated atoms on the bare substrate) are present with the island rows shifted by an additional 0.25/~ indicating the triply coordinated as the binding site. (b) At 0 = 0.5, the four types of (2xl) are present. (c) At even higher coverage (2• and (2x2) domains coexist. Typical size of a domain of one type is 50 ]k (Johnson et al., 1993). the expected value. This shift points to an additional type of domain degeneracy, which results from two inequivalent choices for the binding site (w 12.4.2. I) shown in Fig. 12.16(a) as the triply-coordinated site in the "hourglass". This doubles the types of domains that can be present at a given coverage and justifies further the use of the random phase approximation on this system. For the W ( I 10)-O, the overlayer unit meshes of the various structures have less symmetry than the substrate because of their larger size. However, they maintain the same orientation with respect to the substrate. Lower overall s y m m e t r y is present with overlayers that have unit meshes rotated with respect to the substrate mesh. Fig. 12.16(b) shows the real space mesh for the S i ( l l 1)(q-3xq3-)R30-Ag superstructure. The unit mesh is rotated 30 ~ with respect to the substrate mesh.
696
M.C. Tringides
There are four equivalent choices for adsorption of the Ag atoms at the corners of the unit mesh, but only three symmetry equivalent orientations of the unit mesh. For domains of one orientation, the other two types serve as the intervening "sea" to randomize the phase and to justify neglecting the interference term in the analysis of diffraction profiles. Rotational domains of continuous variation in the unit mesh orientation of the overlayer with respect to the substrate can be present during the growth of an overlayer on a lattice different from the substrate one. Ag crystallites grown on Si(111) is an example (Gotoh and Ino, 1983). Any azimuthal dependence is averaged out resulting in symmetric "rings" of intensity around the fundamental diffraction beams. A domain structure of very high degeneracy is formed during the Si(111 )(7x7) reconstruction. The local arrangement of atoms within the unit cell has been discussed before as an example of a stacking fault. The long-range morphology of the reconstructed surface is shown in the LEEM image of Fig. 12.23. Triangular shaped reconstructed domains growing normal to steps are visible on the vicinal surface. Each triangle contains a network of domain boundaries separating degenerate (7x7) domains (Bauer and Telieps, 1985). The formation of overlayer domains on stepped surfaces can have dramatic effects on their size distribution (Wang and Lu, 1982). Symmetry is lost on stepped surfaces since the direction parallel to the step edge ceases to be equivalent to the direction normal to the step edge. This can affect the population of domains with ordering wave vectors that are correlated to the step direction. One example has already been discussed, the stepped Si(001) surface. In this case, strain oi" high
Fig. 12.23. LEEM images of the Si(111) (7x7) reconstruction showing its nucleation and growth after a quench from T= 1450 K. The triangular domains are reconstructed regions nucleated at step edges. Each domain contains domain walls separating degenerate (7x7) type of domains (Bauer and Telieps, 1985).
Atomic scale dejects on su~aces
697
vicinality can cause an asymmetry between the two types of domains, favoring the one with dimer rows parallel to the step. High vicinality with step edge parallel to the [110] direction can affect the ( ~ - x ~ - ) R 4 5 clean W(001) reconstruction which is also discussed in Chapters 3 and 14. This reconstruction has been shown to be an order-disorder transition, similar to a Jahn-Teller transition. The displacement of atoms into the zigzag pattern splits the degeneracy of a set of electronic states at the Fermi level, into bonding and antibonding ones, so the energy of the reconstructed is lower than the energy of the unreconstructed surface (Fu et al., 1985). Two possible types of domains can be present, rotated by 90 ~ from each other. In one domain, the microscopic displacement of the atoms is along the [ 110] direction (i.e., parallel to the steps) and in the other type, the displacement is in the [ 110] direction (i.e., normal to the steps). On singular surfaces, both domain types are present in equal amounts. On the vicinal surfaces of large misorientation (i.e., 3.3 ~ domains with atom displacements perpendicular to the step edges occupy a larger fraction of the surface. (Step meandering is also possible which can effect the distribution of the two domain types. There is an inhibition zone around the step edge, where the reconstruction is excluded, equal to approximately 20% the terrace width).
12.4.2.5. Overlayer domain structures with long-range periodicity The overlayer structures we have described so far do not have strong correlations in their positions. Domains nucleate more or less randomly on the surface with the only requirement being that their total area be consistent with the overlayer coverage. Even for systems showing satellite "ring" structures in diffraction, no explicit interactions between domains are needed to produce the "ring". The "ring" results from the formation of denuded zones around the island perimeters as atoms are captured efficiently in the surrounding region which imposes an implicit correlation in the island positions. The driving forces responsible for the domain formation are adsorbate-adsorbate interactions, (of either sign but short range), but they cannot induce long-range ordering in the arrangement of the domains. Contrary to the majority of cases, few overlayers have long-range correlations in their structure. Such domain morphologies, require the presence of stabilizing, long-range interactions. Stress relaxation is most commonly the reason for these long-range interactions, especially if there is large lattice mismatch between the overlayer and substrate. Long-range dipole-dipole inverse power law (l/r") interactions can also induce long-range domain periodicity, but to a lesser degree. The separation distribution g(s) is sharply peaked at the characteristic "lattice constant" s of the superperiodicity. Such 2-d "superlattices" of domain boundaries can be observed easily with the STM as a regularly spaced network of spacing s or with diffraction as a sequence of strong satellite peaks, separated by 2~/s (not a single weak satellite "ring" as for the case of denuded zone formation around islands). Long-range periodicities have been observed in the Cu( 110)(2x I )-O reconstruction (Kern et al., 1991 ). The (2x 1) ordering observed at low oxygen coverages is an added row structure of long C u - O strings in the [001 ] direction, held together by strong short-range attractive interactions. At higher coverages the reconstruction is different producing a c(6x2) LEED
M. C Tringides
698
Fig. 12.24. (a) Long-range periodicity of the Cu( 110)(2xl )-O reconstruction showing a stripe pattern of period 86 ~ at oxygen coverage 0 = 0.26. The ratio of the reconstructed domain size to the domain separation is close to 1/3, which for low coverage is predicted to be universal. (b) He-scattering diffraction pattern off the Cu( I I 0)(2xl )-O with satellite peaks of periodicity and intensity consistent with the image of a periodic island grating shown in (a) (Kern et al., 1991).
pattern. Figure 12.24(a) shows regularly spaced features separated by D - 86 ]k at an oxygen coverage e - 0.26 after the surface is annealed at T - 550 K. The length L of the strips in the [ 110] direction depends on coverage. The domain superperiodicity extends over macroscopic distances without interruption over step edges as seen in Fig. 12.24(a). The size of the domains is stabilized as a result of attractive interactions between strings and repulsive long-range elastic interactions between strips to relieve the stress generated by the reconstruction. A generic form for the interaction energy, in terms of two competing terms, can be written (similar to the one used for vicinal Si(001 ) surfaces, Eq. (12.15)) 2c 2c2 (.L+D / F - L + ~D - L~+ D In ~, 2~a sin(rt8 )
(12.23)
where c i is the energy cost in boundary free energy, c2 is a function of the elastic constants and the stress present in the C u - O domains, and e the coverage. The first term is the cost in boundary energy that decreases with larger domain size (L --+ ~ )
699
Atomic scale defects on su~. aces
while the second is the strain energy which favors smaller sizes L. The equilibrium value of L can be obtained by minimizing the free energy i)F/i)L = 0. For low 0 < 0.4 a universal value for the ratio L/D = 1/3 is obtained in good agreement with experimental observations. Long-range correlations in domain morphology have also been observed in the A u ( 1 1 1 ) ( 2 2 x ~ -) reconstruction (Harten, 1985) as discussed in Chapter 3. Ideally the terminated surface should follow the fcc stacking sequence, with atoms occupying the hollow site (out of two choices) which is consistent with the correct sequence. The other half of unoccupied hollow sites should correspond to hcp stacking. The reconstruction involves the gradual displacement of atoms from fcc to hcp sites along the [ 110] direction so the lateral spacing of the atoms increases from 2.75 to 2.87 A, a 4% expansion. Twenty-two top layer atoms correspond to 23 substrate ones with the unit cell 63 ~ = 23x2.75 ,~. This lateral top layer expansion produces lateral and vertical mo_dulations in the atom positions" two ridges of height 0.3 ~ are formed along the [121] direction as the atoms are pushed outwards to accommodate the lateral shift. As can be seen in the STM image in Fig. 12.25(a) (Barth et al., 1990), these ridges appear as two closely spaced bright regions separated by dark (mostly fcc type) on the outside and smaller dark (mostly hcp type) in-between regions. The expansion is uniaxial along the [110] direction but since the surface has three-fold symmetry, there are three equivalent [110] directions and correspondingly three equivalent [121 ] directions. The reconstruction direction changes over a distance of 280 A, from one of the orientations to an equivalent one, rotated by 120 ~ producing a characteristic herringbone type structure. The driving force of this long-range periodicity is similar to the stress mediated transitions that drive the double to single height step transition on Si(001) and the long-range correlations observed in the Cu(110)p(2xl)-O reconstruction. The cost in energy in forming walls between equivalent domains is compensated by the lowering of the strain energy (that is present in the (22• reconstruction because of the atom displacement). The competition stabilizes the size of the domains to a characteristic size. Herringbone structures of different orientation can form on larger areas because of the three equivalent [121] directions, as can be seen in large-scale STM images that include several terraces. The interaction between surface defects (i.e., steps) and the Au( 111 ) reconstruction shows some unusual features. The ridges prefer to run parallel to the step direction so pairs of bright lines form U-type connections close to the steps. Each bright ridge can be thought of as a dislocation since the atoms on either side are displaced by non-lattice vectors. The ridges provide preferred nucleation sites during epitaxial growth, illustrating how surface defects can promote better growth. This possibility has been demonstrated during the epitaxial growth of Ni on the reconstructed A u ( l l l ) substrates (Chambliss et al., 1991). As shown in Fig. 12.25(b) Ni islands nucleate only at the "elbows" of the brigh_t ridges where they changoe direction by 120 ~ The islands are lined up along the [121] direction spaced 140 A (which is half the herringbone periodicity), since nucleation is possible at both "elbows", one of which bulges in and the other bulges out. The spacing of the islands in the orthogonal [110] direction is simply the unit cell size (63 A) because m
700
M. C. Tringides
Fig. 12.25. (a) Herringbone reconstruction of the clean Au( 111 ) surface after sputtering and annealing at T = 600~ The reconstruction is unaxial along the < 110> direction. Wide dark regions correspond to fcc stacking, narrow dark regions correspond to hcp stacking and the bright regions correspond to the ridges tbrmed between fcc and hcp stacking. The change in the reconstruction orientation is a result of the 3-1old symmetry of the surface and has periodicity of 280 A (Barth et ai., 1990). (b) Nucleation of Ni islands on the herringbone Au(l 11 ) surface. They form a regularly spaced network because the islands nucleate at the bulging-out elbows of the herringbone structure (Chambliss et al., 1991). islands nucleate only at the " e l b o w s " of one of the two ridges. D e t a i l e d crystallographic analysis shows that the ridge w h e r e Ni nucleates has a n o n - z e r o B u r g e r vector, while for the other ridge the B u r g e r vector is zero. Such a highly r e g u l a r n e t w o r k of Ni islands extends over the whole terrace; it can be different on adjacent terraces, since it follows the orientation of the reconstruction. S T M i m a g e s show Ni islands s u p e r d o m a i n s rotated by 120 d e g r e e s on different terraces. The nucleation of Ni islands in such spectacular o r d e r e d a r r a n g e m e n t s s u g g e s t s that Ni
Atomic scale defects on su~. aces
701
diffusion on Au(111 ) is fast and the sticking coefficient at the "elbow" sites is unity, because a constant island density is formed for all Ni coverages. In the same study, an entirely different island arrangement was observed during the growth of A u / A u ( l l l ) , proving that epitaxy is highly complex and system specific. No regular network of Au islands is observed, most likely because of the low sticking probability of Au atoms at "elbow" sites. This is consistent with the large choice of binding sites for Au (fcc vs. hcp) that is responsible for the reconstruction in the first place. The presence of dislocations at the "elbows" does not substantially increase the sticking probability as for Ni atoms.
12.5. The role of defects in phase transitions
Overview Defects can drastically change the degree of transformation, speed and the nature of a phase transition. The thermodynamic (i.e., how the phase transformation depends on temperature at equilibrium) and the kinetic (i.e., how the phase transformation is completed as a function of time) behavior has been studied mostly with diffraction techniques because of the necessary statistical average. A new length scale is introduced, the average spacing between defects, which in both cases limits the domain size attained in the ordered phase.
12.5.1. Thermodynamic effects In Chapter 14 different types of phase transitions in 2-d systems have been presented. A general condition for truly singular behavior as the phase boundary is approached is for the system to be infinite. Phase transitions in finite systems are smeared out because the development of long-range order is limited by the size of the system. In addition to finite size other forms of disorder (impurities, distribution in interaction energies, deviations from stoichiometry, etc.) can modify the expected singular behavior. Disorder introduces an additional finite length scale (the distance between impurities, finite width of domain walls, etc.) that limits the attainment of long-range order. Phase transitions of any order are affected. Figure 12.26 shows the typical effects expected close to a second order transition (Kleban, 1985). The critical temperature is shifted towards lower temperatures and the divergence of a typical thermodynamic parameter (compressibility, correlation length, etc.) is rounded off. For finite size effects, a quantitative theory of the temperature shift and the smearing of the transition has been described in Appendix C of Chapter 13. Briefly, the free energy becomes a function of the scaled variable L/~ so as T --) Tc, and ~ diverges so that the ratio L/~ --) 0. A scaling form has been used to describe the dependence of the free energy on L/~ that can been used theoretically to deduce the critical exponents from the dependence of the roundingoff effects on L. It has not been possible to apply this analysis to real experimental systems since it is not easy to vary the substrate size in situ under ultra high vacuum
702
M.C. Tringides
I
II
~Lelu
P" IrI-P
i I I
'/ xl
! I
~ i_-'Iv
T. (L)J'~ITc ---t
I"--- ~L-X I I
= L-J,'u
Fig. 12.26. Finite size effects at a second order phase transition. The transition moves to lower temperatures and the divergence of the appropriate thermodynamic quantity is smeared out. The changes can be described in terms of finite size scaling theory.
conditions. Ex situ results on surfaces with different average terrace lengths (i.e., stepped surfaces, sputtered surfaces, etc.) is difficult because of the lack of absolute calibration standards in temperature and coverage. For a given average terrace size, finite size effects have been routinely taken into account in extracting critical exponents and in identifying the corresponding universality class. However, finite size limitations are important also away from the phase boundary without the need of the sophisticated finite size scaling analysis. The case of the W ( 1 1 0 ) - O phase diagram is a good illustration (Lagally et al., 1980). As seen in Fig. 12.21, the T-0 phase diagram consists of regions with different superstructures: (2xl), (2x2), ( l x l ) , etc. When the oxygen atoms are initially deposited in a disordered state, it was found that for a temperature T within the ordered region, the size of the domains formed was limited to sizes much smaller than the ones expected thermodynamically. The average domain size, obtained by annealing at T is 3-4 times smaller than the size obtained after annealing at a higher temperature followed by cooling at T, indicating also the importance of kinetic effects. The average domain size follows a 0 ~/2 dependence on coverage, indicating that the system cannot evolve to its lowest energy state of a single-phase domain which should be independent of coverage. The kinetic limitation must be related to the finite terrace size (estimated to be around 200 A) and the existence of a step edge barrier that suppresses the motion of atoms between terraces. Such barriers will be discussed further in w 12.6 in the context of epitaxial growth. Each terrace acts like a thermodynamically independent system with a fixed coverage determined by the initial deposition (which for a given terrace can deviate from the nominal coverage because of statistical fluctuations). This can explain why smaller domains are grown at lower coverage. By heating to higher temperatures, the step edge barrier can be overcome and the domains can grow as large as the terraces.
Atomic scale defects on surfaces
703
12.5.2. Kinetic effects The role of disorder in the evolution of a system to a new equilibrium state has been systematically studied by measuring the time dependence of the domain size (Tringides, 1994). For disorder-free systems it has been theoretically predicted and has been verified on several surface overlayers that the growth kinetics under the highly non-equilibrium conditions during time evolution towards a new state, follow two basic laws" (i) the average domain size grows like a power law in time L - A(T)t x, with x a universal growth exponent (ii) the domain size distribution P(N, t) is time invariant i.e., it is fully determined by knowing simply the value of the average size at some time t. Such experiments are easily carried out with diffraction by measuring the non-equilibrium structure factors S(q, t) as a function of wave vector q and time t. Figure 12.27 shows results for the ordering of W(I 10)-O at 0 = 0.5 towards the single phase ( 2 x l ) structure an initially disordered phase (Wu et al., 1989). The measured lineshapes sharpen with time, as the domains grow larger. However, when plotted in the scaled form (S(q,t)/Sma x vs. q / F W H M ) with S(q) the lineshape and F W H M the full-width-at-half-maximum, all the data collapse onto a universal curve that verifies that growth is self-similar and the domain size distribution is time-invariant. Both growth laws are modified in the presence of disorder. Finite size effects lead to saturation and slowing down when the domains become a considerable fraction of the terrace size (40%); for terraces of size larger than 200 /~, this still allows the verification of the power law prediction, since at least one decade in length evolution is available to extract reliable growth exponents. Disorder effects in the form of added impurities (quenched or annealed), distribution in the adatom-adatom interactions, and deviations from stoichiometry have been studied both theoretically and experimentally. Monte Carlo simulations on lattice gas models with a fraction of impurities added have shown that, independent of whether impurities are mobile or not, power law growth slows down after the characteristic time necessary for the 1
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Fig. 12.27. Scaling of the superstructure (1/2,0) intensity profiles during the formation of the (2• ordered phase out of an initial random configuration for W(110)--Oat 0 = 0.5. This shows that for surfaces without impurities the domain size distribution is time invariant (Wu et al., 1989).
M.C. Tringides
704
0% 0.5% 2% 5% 10% 20% 50%
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Fig. 12.28. The results of Monte Carlo simulations showing how the presence of impurities decreases the growth law exponent for the average domain size L = A(0tx. For the case of no impurities (0%) the classical exponent x = 1/2 is expected while as the impurity amount increases, x decreases monotonically to zero (Mouritsen and Shah, 1989). domain size to become comparable to the distance between defects as shown in Fig. 12.28 (Mouritsen and Shah, 1989). This result is further supported by the linear dependence of the domain size, obtained at fixed time, on the inverse of the square root of the impurity concentration. In analogy with quench impurities, annealed impurities are found to segregate at the domain boundaries pinning additional domain evolution. The random-field-Ising-model (RFIM) has been the prototype system used to understand the role of disorder in modifying equilibrium phase transitions. The RFIM has also been used to understand the role of disorder in non-equilibrium kinetics for a system quenched from high to low temperature (Gawlinski et al., 1985). The power law growth for the normal Ising model is replaced by a logarithmic time dependence and the self-similarity in the domain size distribution breaks down because the larger domains are pinned to a m a x i m u m size. Experiments were carried out on W ( 2 1 1 ) - O with H R L E E D by intentionally varying the amount of Nz impurities which are known to be immobile at the temperatures used (Zuo et al., 1988). For the pure system the domain size growth exponent is x = 1/2 and the self-similarity of the domain size distribution was initially verified. With increasing impurity amount the growth follows a logarithmic time dependence as expected for the RFIM and the scaled structure factor lineshapes
S(q,t) change with time, indicating a non-invariant distribution P(N,t). S(o,t)
~2.6. The role of defects in crystal growth
Overview" The presence of defects on surfaces can provide a large number of stable binding sites to accelerate crystal growth. The dramatic increase in crystal growth rates out of solution or during epitaxial deposition has been shown on thermally
Atomic scale defects on su~. aces
705
roughened surfaces. Non-thermal ways to increase the growth rate include the formation of screw dislocations initially or seeding the surface with small amount of impurities that act like surfactants promoting layer-by-layer growth. The importance of defects and impurities to promote crystal growth can be seen from a simple comparison of the dependence of crystal growth rates on the orientation of the substrate. The growth of close packed planes is the rate limiting step in the process, since they grow at rates which are slower by several orders of magnitude than the growth rates of open planes. This is clearly related to the lower coordination number of an atom in an open plane and the availability of unsatisfied bonds to accommodate the incoming atoms into stable sites. By forming defects of any dimensionality (i.e., vacancies, steps, dislocations, islands, etc.) on closed packed faces one can mimic the reduced coordination number of open planes. The easiest pathway to increase the number of defects is to raise the temperature. As discussed in w 12.3.1.2 for stepped surfaces, the increase in the configuration entropy of a surface with temperature far compensates the energy cost, so the surface roughens above the roughening transition. In the context of crystal growth, great interest was generated to understand the experimental and theoretical properties of the roughening transition (Burton and Cabrera, 1949). More specifically it was important to determine the transition temperature TR in terms of the microscopic interaction parameters and to characterize the sharpness of the transition in terms of the diverging thermodynamic quantity. Experimental measurements have been carried out to characterize the transition for close-packed and stepped surfaces, in terms of the diverging interface width Eq. (12.8) which can be measured from the power law decay of the substrate diffraction beam lineshape Eq. (12.9). TR increases with decreasing vicinality of the surface, as discussed in w 12.3.1.2, because the values of the energetic parameters of the Hamiltonian equation (Eq. 12.7) )~,, w~ are 2-3 times larger than the values on stepped surfaces. The transition signals the spontaneous formation of steps at zero free energy cost which implies that the nucleation barrier for island formation is also reduced to zero. The statistical mechanics of the roughening transition have been described before for a thermally annealed stepped surface (i.e., constant number of atoms), so in this short section we will concentrate on how roughening modifies the crystal growth rate in the presence of an incoming flux of atoms (i.e., a finite chemical potential). It is common practice to grow crystals at higher temperatures so it is important to understand how the enhancement in the growth is related to the surface roughness. Monte Carlo methods have been used to model the growth process in Ising-type models that include the minimum number of the relevant, physical parameters; i.e., the chemical potential specifies the incoming flux and the nearest neighbor Ising interaction controls the bonding strength of an atom adsorbed on the surface (Weeks and Gilmer, 1979). The binding energy of an adsorbed atom determines its re-evaporation rate back to the vapor. After a sufficient transient time has elapsed, a steady-state is attained with the newly arrived atoms incorporated into the already grown structures, and the surface is growing at a constant rate. The growth rate is defined as the net number of layers added to the crystal in unit time. Figure 12.29 shows a plot of the growth rate (normalized to the incoming atom flux)
M. C. Tringides
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Fig. 12.29. Monte Carlo simulations showing how the crystal growth rate (i.e., the number of new layers added to the crystal per unit time) depends on the chemical potential Ala (i.e., supersaturation) for different substrate temperatures. The top curve corresponds to T > TR, where TRis the temperature of the roughening transition, and shows that there is a linear relation between growth rate and supersaturation, a result of the increased thermal roughness on the substrate. For T < TR (two lower curves) the growth rate requires a minimum supersaturation because the surface is relatively flat and the nucleation barrier to condense new islands is finite (Weeks and Gilmer, 1985).
vs. the ratio of the chemical potential to the substrate temperature. Different curves correspond to different substrate temperatures and the importance of the roughening temperature is clearly seen. For T < TR there is a finite plateau region indicating a zero growth rate because no islands have nucleated yet to promote the capture of the incoming atoms. The top curve corresponds to temperature T > TR when islands and steps are spontaneously formed so the incoming atoms can be easily incorporated into the surface. There is a linear relation between the crystal growth rate and the chemical potential. Growth rates at T < T R can be increased if defects are externally introduced onto the substrate. Figure 12.30 shows how a screw dislocation can provide a large number of strongly bound sites for incoming atoms to decorate step sites projected onto the surface (Keller and HOche, 1987). A characteristic spiral is formed around the screw dislocation. The growth rates for T > TR are not affected since defects can proliferate thermally but substantially reduced plateaus are observed for T < TR than the plateaus shown in Fig. 12.29. The importance of maintaining a rough substrate during growth has been recently demonstrated in molecular beam epitaxy (MBE) experiments. Although M B E growth rates are extremely slow relative to the ones for conventional growth out of a melt, there are clear potential advantages for combining epitaxially different materials into sharp, atomically controlled layers. As discussed before, the goal is to grow new materials with interfaces in a layer-by-layer fashion that is expected to increase the stability of the grown film. The oscillations in diffraction intensity during growth, have been used to decide the quality of the grown film. It is widely accepted
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Fig. 12.30. Electron microscope image of NaCI growth from the vapor phase showing the importance of dislocations in crystal growth. The steps can be easily imaged when they are decorated with Au. The evolution of the spiral dislocation can be followed after different NaCI amounts have been deposited. In the left figure two spirals of the same sense of rotation merge into each other and in the right figure two spirals of opposite sense of rotation produce concentric circular steps (Keller and HOche, 1987).
that layer-by-layer growth requires a minimum substrate temperature, so the incoming atoms have sufficient diffusion lengths to join nucleated islands and complete a layer before the next layer has nucleated. Typical diffusion barriers on metallic and semiconducting substrates are 0.2 eV and 0.6 eV, respectively, which implies that layer-by-layer growth should be absent at substrate temperatures T < 80 K (for metals) and T < 200 K (for semiconductors). Quite surprisingly, diffraction intensity oscillations have been observed on both metallic and semiconducting substrates at temperatures where no thermal diffusion is expected. Although the detailed mechanism responsible for the oscillations is still unclear, it has been realized that a critical factor promoting layer-by-layer growth is interlayer diffusion; i.e., the transfer of atoms from a higher to a lower level (Kunkel et al., 1990). At low temperatures a large number of small and ramified islands are formed which aids interlayer diffusion: the deposited atoms have diffusion lengths larger than the small island size so they can reach the step edge and move to the lower level before they have a chance to nucleate new islands. This explains why He-diffraction oscillations are observed for T < 200 K and T > 450 K for Pt/Pt(l 11 ), but are absent in the in-between range. Interlayer diffusion is present at low temperatures because of the small island size and at high temperatures because of the increasing probability to cross the step edge barrier. The seemingly paradoxical result that at low temperatures a constantly roughened layer produces a smooth film is understood, if it is recognized that good quality layer-by-layer films should be grown with minimum vertical disorder (by reducing the interface width) even at the cost of maximizing the amount of lateral disorder at the top layer. Roughness is equally important for conventional or epitaxial growth but the underlying mechanism is
M. C. Tringides
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Fig. 12.31. Diffraction intensity drop vs. time plots, during epitaxial growth of Ag/Ag(11 I). No oscillations are observed when a clean substrate is used because the step edge barrier limits interlayer diffusion. When a small amount of Sb is dosed on the surface (0 = 0.2 ML) diffraction intensity oscillations are observed, indicating layer-by-layer growth, most likely because the step edge barrier is rcduced. Such experiments demonstrate the importance of impurities (i.e., surfactants) in promoting the growth of smooth films (van der Vegt et al., 1992). different. For conventional growth, the large number of available free sites, in a roughened substrate, accommodates atoms easily into stable low energy structures. For MBE growth, the fractal-like low temperature morphology provides alternative kinetic pathways for the non-equilibrium, smooth film structures to grow layer-bylayer growth. It is also well-known that crystal growth is improved with the addition of a small amount of impurities that act like stable nucleation sites of the deposited atoms to stick to. Similar impurity effects have been observed during crystal growth, by introducing small amounts of foreign elements, named surfactants. Deposition on systems that normally grow rough with the simultaneous occupation of many levels at the interface (3-d mode) can be manipulated with the addition of surfactants to grow layer-by-layer. A g / A g ( l l l ) grows in 3-d mode despite the rapid terrace diffusion because of the presence of a step edge barrier that limits interlayer diffusion. By introducing a small amount of Sb (0 = 0.2 ML) the growth mode is changed from 3-d to layer-by-layer, as can be seen from the strong oscillations observed in X-ray scattering, Fig. 12.31, which are present only on a substrate dosed with the surfactant (van der Vegt et al., 1992). The exact mechanism responsible for the change in the growth mode has not been fully identified. However, it is clear that most of the Sb floats to the surface, as successive layers are formed, so it plays the same role during the entire deposition time. Sb has the net effect of minimizing the difference between the step edge barrier E~ and terrace diffusion barrier Et, but it is not clear if this is realized because E, is reduced o r E t is increased.
12.7. Epilogue A description of the structure of different kinds of defects on single crystal surfaces
Atomic scale defects on su~. aces
709
has been presented based mainly on the quantitative analysis of diffraction and STM images. The types of defects presented is by no means exhaustive and several interesting topics on defect structures have not been discussed (i.e., domain wall configuration in incommensurate structures, dislocation networks, correlated roughness in ultrathin films, etc.). The main emphasis of the chapter was on the structural characteristics of defects. Discussion of their effects on the electronic properties of the interface has been minimal. This is partially due to the limited amount of combined experimental work relating directly the surface electronic structure to the defect configurations. Such experiments require the combination of structural and spectroscopic techniques to be carried out in situ on the same substrate, which is instrumentally far more demanding. Potentially, the STM can be used to provide such information because the technique is sensitive to both the electronic structure and surface morphology. However, the measured quantities depend on both features and decoupling each contribution is not easy. In the current article there was more emphasis on the static defect structures (which can be produced either thermodynamically or kinetically in low temperature metastable states) than time-dependent effects. Topics relating the dynamics of defects and changes in defect distribution have not been included. For example, the diffusion of defects either on the surface or into the bulk can change the defect arrangement and the way surface properties are modified; on the microscopic time scale the coupling of defects to the vibrational motion of the crystal can change the phonon dynamics. Structural information about surface defects is readily available. It is clear from the information presented that defects of large variety and complexity are unavoidably present on crystal surfaces. The current experimental probes sensitive to atomic scale defects (STM, diffraction) can provide complementarily a clear picture of the defect configuration. The knowledge of the type, number, and defect distribution is essential to extend the picture of a surface, presented in the other chapters of the volume as a perfect, highly symmetric arrangement of atoms, into more realistic, non-ideal structures found in nature. A cknowledgements
I would like to thank Susan Eisner for the efficient and skilled typing of the manuscript and Kevin Cook for his careful preparation of the figures. I would like to thank Bill Unertl, the editor of the volume, for his valuable input in the form and substance of the manuscript. Ames Laboratory is operated by the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.
References Avouris, Ph. and I.W. Lyo, 1991, Surf. Sci. 242, 1. Barth, J.V., H. Brune, G. Ertl and R.J. Behm, 1990, Phys. Rev. B 42, 9307. Bauer, E. and W. Telieps, 1985, Surf. Sci. 162, 163. Burton W.K. and N. Cabrera, 1949, Disc. Farad. Soc. 5, 33
710
M. C. Tringides
Chadi, D.J., 1987, Phys. Rev. Lett. 59, 1691. Chambliss, D.D., R.J. Wilson and S. Chiang, 1991, Phys. Rev. Lett. 66, 1721. C.J. Chen, 1993, Introduction of Scanning Microscopy. Oxford University Press. Conrad, E.H. and T. Engel, 1994, Surf. Sci. 299, 398. De Miguel, J.J., C.E. Aumann, R. Kariotis and M.G. Lagally, 1991, Phys. Rev. Lett. 67, 2830. Ernst, H.J., F. Fabre, R. Folkets and J. Lapujoulade, 1994 Phys. Rev. Lett. 72, 112 Family, F., 1990, Physica A 168, 561. Fu, C.L., A.J. Freeman, E. Wimmer and M. Weinert, 1985, Phys. Rev. Lett. 54, 2261 Gawlinski, G.T., S. Kumar, M. Grant, J.D. Gunton and K. Kaski, 1985, Phys. Rev. B 32, 1575. Gotoh, Y. and S. lno, 1983, Thin Solid Films 109, 255 Hahn, P., J. Clabes and M. Henzler, 1980, J. Appl. Phys. 51, 2079. Harten, V., A.M. Lahee, J.P. Toennies and Ch. Wtill, 1985, Phys. Rev. Lett. 54, 2619. Henzler, M., M. Horn, V. Hoegen and U. Kohler, 1992, Advances in Solid State Physics, Vol. 32, ed. U. R/Sssler. Viewweg, Braunschweig/Wiesbaden. Hwang, R.Q., J. Schr6der, S. Gunther and R.J. Behm, 1991, Phys. Rev. Lett. 67, 3279. Horn, M., U. Gotter and M. Henzler, 1988, J. Vac. Sci. Tech. B 6, 727. lchninokawa, T., H. Ampo and S. Miura, 1985, Phys. Rev. B 31, 5183. Johnson, K.E., R.J. Wilson and S. Chiang, 1993, Phys. Rev. Lett. 71, 1055. Keller, K.W. and H. Htiche, 1987, Electron Microscopy in Solid State Physics, eds. H. Bethge and J. Heyendreich. Elsevier, Amsterdam. Kern, K., 1994, The Chemical Physics of Solid Surfaces, Vol. 7: Phase Transitions and Adsorbate Restructuring at Metal Surfaces, eds. D.A. King and D.P. Woodruff. Elsevier, Amsterdam. Kern, K., H. Niehus, A. Schart, P. Zeppenfeld, J. George and G. Comsa, 1991, Phys. Rev. Lett. 67, 855. Kleban, P. 1984, Chemistry and Physics of Solid Surfaces V, eds. R. Vanselow and R. Howe. Springer Vcrlag, Berlin. Kunkcl, R., B. Poelsema, L.K. Vcrheij and G. Comsa, 1990, Phys. Rev. Lett. 65, 733. Lagally, M.G., 1985, Methods of Experimental Physics, Vol. 22. Solid State Physics: Surfaces, eds. R.L. Park and M.G. Lagally, Academic Press, New York. Lagally, M.G., T.M. Lu and G.C. Wang, 1980, Ordering in Two-Dimensions, ed. S. Sinha. Elsevier, Amsterdam. Michely, T., M. Hohage, M. Bott and G. Comsa, 1993, Phys. Rev. Lett. 70, 3943. Mo, Y.W., J. Kleiner, M.B. Webb and M.G. Lagally, 1991, Phys. Rev. Lett. 66, 1998. Mouritsen, O.G. and P.J. Shah, 1989, Phys. Rev. B 40, 11445. Niehus, H. and C. Achete, 1993, Surf. Sci., 289, 19 Poelsema, B. and G. Comsa, 1989, Scattering of Thermal Energy Atoms from Disordered ,Surfaces. Springer Verlag, Berlin. Saloner, D., J.A. Martin, M.C. Tringides, D.E. Savage, C.E. Aumann and M.G. Lagaily, 1987, J. Appl. Phys. 61, 2884. Samsavar, A., E.S. Hirschom, T. Miller, F.M. Leibsle, J.A. Eades and T.C. Chiang, 1990, Phys. Rev. Lett. 65, 1607. Swartzentruber, B.S., Y.W. Mo, R. Kariotis, M.G. Lagally and M.B. Webb, 1990, Phys. Rev. Lett. 65, 1913. Takayanagi, K., Y. Tanishiro, S. Takahashi and M. Takahashi, 1985, Surf. Sci. 164, 367. Tringides, M.C., 1994, The Chemical Physics of Solid Surfaces, Vol. 7: Phase Transitions and Adsorbate Restructuring, eds. D.A. King and D.P. Woodruff. Elsevier, Amsterdam. Tringides, M.C., J.G. Luscombe and M.G. Lagally, 1989, Phys. Rev. B 39, 9377. Vanderbilt, D., O.L. Alerhand, R.D. Meade and J.D. Joannopoulos, 1989, J. Vac. Sci. Technol. B 7, 1013. van der Vegt, H.A., J.M.C. Thornton, H.M. van Pinxtesen, M. Lohmeir and E. Vlieg, 1992, Phys. Rev. Lett. 68, 3335. Wang, G.C. and T.M. Lu, 1982, Surf. Sci. 122, L635.
Atomic scale defects on su~. aces
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This Page Intentionally Left Blank
CHAPTER 13
Phase Transitions and Kinetics of Ordering L.D. ROELOFS Physics Department Haverford College Haverford, PA 19041, USA
Handbook o.[Su~. ace Science Volume 1, edited by W.N. Unertl
9 1996 Elsevier Science B.V. All rights reserved
713
Contents
13.1.
Introduction 13.1.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.2. Phase transition p h e n o m e n a at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2.1. C l e a n s u r f a c e s
13.2.
13.4.
717
13.1.2.2. A d s o r b a t e s y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
717
Surface kinetic p h e n o m e n a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
718
13.1.4.
Basic phase transition ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
718
13.1.4.1. T h e l s i n g m o d e l in 2-d
719
. . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.4.2. T r a n s i t i o n o r d e r and the free e n e r g y . . . . . . . . . . . . . . . . . . . .
720
13.1.4.3. T h e L a n d a u rules
723
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A d s o r b a t e phase d i a g r a m s and the lattice-gas analogy The lattice-gas analogy
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
725 725
13.2.2.
Particle-vacancy s y m m e t r y and trio interactions . . . . . . . . . . . . . . . . . . . .
13.2.3.
C h e m i s o r p t i o n and physisorption . . . . . . . . . . . . . . . . . . . . . . . . . . . .
729
13.2.4.
Phases of m o r e c o m p l e x s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . .
730
Universality and classification of transitions
. . . . . . . . . . . . . . . . . . . . . . . . . .
728
736
13.3.1. Critical e x p o n e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
736
13.3.2.
Universality classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
746
13.3.2.1. E x p o n e n t v a l u e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
747
13.3.2.2. M a g n e t i c m o d e l s
750
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
754
13.4. !. Reconstruction of metallic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
755
13.4.1.1. R e c o n s t r u c t i o n r e s u l t i n g in i n c r e a s e d s u r f a c e p a c k i n g d e n s i t y . . . . . .
755
13.4.1.2. D i s p l a c i v e r e c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . .
757
13.4.1.3. M i s s i n g row r e c o n s t r u c t i o n s
759
. . . . . . . . . . . . . . . . . . . . . . . .
13.4.1.4. A d l a y e r - i n d u c e d r e c o n s t r u c t i o n 13.4.2.
13.4.3.
13.4.4. 13.5.
717
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.3.
13.2.1.
13.3.
716 717
Reconstruction of s e m i c o n d u c t o r surfaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
764
! 3.4.2.1. ( 111 ) S u r f a c e s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
765
13.4.2.2. (001) S u r f a c e s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
766
Adsorption effects on reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
768
13.4.3.1. M e t a l l i c s u r f a c e s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
768
13.4.3.2. S e m i c o n d u c t o r s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . .
770
Diffusion in reconstruction systems
771
. . . . . . . . . . . . . . . . . . . . . . . . . .
O r d e r i n g kinetics at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1.
760
772
Theoretical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
773
13.5.1.1. G e n e r a l f r a m e w o r k
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
775
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
778
13.5.1.2. N u c l e a t i o n
714
13.5.1.3. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
779
13.5.1.4. C o n s e r v a t i o n conditions
780
..........................
13.5.2. Results on late-time ordering from experiment and simulation 13.5.2.1. Verification of scaling in e x p e r i m e n t and theory
............ .............
780 782
13.5.2.2. Finite-size effects and other limitations on e x p e r i m e n t and simulation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.3. Ordering in coexistence regions
............................
783 784
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
787
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
788
Appendix A: Physically allowed phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Critical point singularities and modern theory of critical phenomena
793 794
..........
B. 1.
Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
795
B.2.
Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
799
Appendix C: Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
802 803
C.2.
Length scales
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
803
C.3. C.4.
Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
805 805
715
13.1. Introduction This chapter introduces the partially-related subjects of surface phase transitions and the kinetics of phase development and relates them to the broader context of two-dimensional (2-d)critical phenomena. In this introductory section the context of work in this subfield is discussed, the scope and organization of the chapter are presented, and references are given for related topics not covered in detail. The section concludes with a brief discussion of the 2-d Ising model whose known solution can be related via the lattice-gas analogy to surface phase transitions and which provides something of a paradigm for 2-d phase transitions and kinetic behavior more generally.
13.1.1. Overview Surface phase transitions and kinetic phenomena have applications ranging from the elegant theory of 2-d critical phenomena to the technologically important problems of surface reactions, diffusion and catalysis. As such, this area of surface science has been an active one since the origin of the field and has attracted the attention of an array of investigators of extraordinarily broad emphases. The area is thus sc~mewhat difficult to encompass in a brief treatment, and some aspects will c)f necessity be left to lengthier presentations or review articles. Closely related material is contained in Chapters 2, 3, 10, and 12 of this Handbook. The phase transitions exhibited by lattice gas models and the real systems they describe can be related to those occurring in simple models for 2-d magnetism. Lee and Yang (1952) first drew attention to the direct analogy between a simple attractive-interaction lattice gas model and the 2-d lsing model for ferromagnetism. Later the relevance of these ideas to experimental surface physics was noted and emphasized by several workers including Estrup (1969), Park (1969) and Doyen et al. (1975). These direct correspondences between magnetic and surface phase transition systems are presented in w 13.2. Beginning at about the same time and continuing into the 1980s, the general theory of phase transitions advanced tremendously with the advent of new caiculational techniques and theoretical constructs. The importance of the divergence of the length scale of the fluctuations that occur at continuous phase transitions was realized and exploited via scaling. The concept of universality, which allowed the grouping of disparate phase transition systems into a small number of universality classes, was broadly tested and confirmed. Members of a universality class display identical mathematical behavior in the vicinity of critical points. Universality and classification are the subject of w 13.3 with focus on their implications for surface phase transitions and the experimental tests that have so far been conducted. 716
Phase transitions and kinetics of ordering
717
The following subsections will briefly enumerate surface phenomena encompassed by these topics, identify where in this chapter or elsewhere in this volume detailed treatment is to be f o u n d - or, when no further coverage is attempted, review articles to be consulted, and present the organization of the chapter.
13.1.2. Phase transition phenomena at surfaces This subsection lists phase transition phenomena that have been observed in clean and adsorbate-covered metal and semiconductor surface systems.
13.1.2.1. Clean surfaces Surface reconstruction: The strong perturbation of a surface on the electronic structure of a metal or semiconductor drives rearrangements of surface layer atoms away from the simple, bulk-termination structure for many systems. Investigation of some of these systems, notably the Si(l 11) and W(100) reconstructions, has been one of the central thrusts of surface science. Reconstruction is discussed in w 13.4; see also Chapter 3. Surface roughening: In some cases, depending on both the material and the orientation of the surface, the thermodynamic equilibrium form of a surface may depart from simple planarity at a lower temperature than that at which the material melts. This phenomenon has been extensively investigated for a wide variety of metal and semiconductor systems both theoretically and experimentally. This and related phenomena such as the orderliness of steps on vicinal surfaces have been treated in Chapter 2. Surface melting: The process of bulk melting seems to initiate at particular crystalline surfaces for at least some materials at temperatures as much as 10% below the melting point, T,,,. Liquid-like ~ layers whose thickness increases as T,,, is approached have been observed using ion scattering and quasi-elastic neutron scattering, on (usually the more open) crystalline faces of several metals and semiconductors. The intriguing connections between surface melting and roughening have been noted by Lapujoulade and Salanon ( 1991 ). Since this phenomenon is fundamentally a bulk process B indeed it has also been called surface-induced melting B it will not be further discussed in this chapter. References and a detailed discussion are given by van der Veen (1991). Surface magnetism: A surface layer may exhibit magnetic properties differing in isotropy, symmetry or Curie temperature from the bulk it terminates. Bona fide 2-d magnetic phase transitions are then realizable, but since this is more directly a surface electronic effect than a consequence of surface structure, these phenomena are not treated in detail in this volume. The review by Falicov et al. (1990) may be consulted, and see also the chapter by Kirschner in Vol. 2 of this series.
13.1.2.2. Adsorbate systems Addition of a foreign species to a surface introduces other possible occurrences of phase development and transitions.
1 The phases cannot be true liquids since contact with the underlying solid induces partial order.
718
L.D. Roelof,;
Adsorbate ordering: Order of ( l x l ) or higher periodicity may occur within the adlayer, leading to the possibility of critical points and transitions between phases of differing symmetry. Phase diagrams ~may be determined in the (T,0) and/or (T, bt) planes and the character of transitions between phases may be ascertained and critical exponents determined. These matters are discussed in {} 13.2 and 13.3 and for physisorption systems by Suzanne and Gay in Chapter 10. Adsorbate-induced reconstruction: Adsorption typically results in displacements of nearby substrate atoms. In some cases these adsorption forces can couple to surface reconstruction modes leading to the possibilities that adsorption may induce or enhance or eliminate reconstructions of the clean surface.
13.1.3. Surface kinetic phenomena The possibility of phase changes occurring on surfaces clearly implies the opportunity of examining systems away from thermodynamic equilibrium and their approach there to, i.e. the kinetics of phase development. This chapter will focus primarily on this topic as it relates to the phase transition behavior of surfaces. The topics of significance include the development of order over time in systems which have been subjected to rapid variation of external conditions forcing the system across a first-order phase boundary, usually a temperature quench or an 'upquench'; and critical slowing down in the vicinity of continuous transitions. These topics are discussed in w 13.5. One might reasonably expect simple diffusion on surfaces to be relatively unimportant to questions of long-range order. There are, however, several situations where it in fact can play a significant role. These include: cases of reconstruction requiring mass motion of surface-layer atoms; and development of long-range order in adsorbate systems under conditions of fixed 0. Some recent developments in the study of this subsidiary topic are gathered in w 13.4.4. The purpose of this volume is primarily pedagogical, and its limitations include incompleteness of citation and (lack ol) depth of coverage. Section 13.7 comprises a brief bibliography that lists several important review articles and other sources pertinent to the subject. Readers interested in a particular topic in greater detail and more complete handling of the literature are urged to consult the sources given there.
13.1.4. Basic phase transition ideas In order to illustrate the phenomena typical of first- and second-order phase transitions in 2-d systems and to introduce the necessary vocabulary and notation it is useful to begin by considering the 2-d Ising model and its solution. Readers familiar with the general theory of phase transitions can skip this section and proceed to w 13.2.
1 A phase diagram is a 'map' showingthe phases that occur for a given systemand the nature and locations of the boundaries between them in a space defined by the relevant thermodynamic variables. Examples are given throughout this chapter.
Phase transitions and kinetics of ordering
719
13.1.4.1. The Ising model in 2-d The simplest 2-d model for a ferromagnetic system is the Ising model based on a lattice {i} of magnetic moments, s~, which can orient themselves in just two directions, s~ = +1. These moments are taken to interact via separation-dependent exchange constants, Jk--the subscript k denotes a neighbor relationship, Ji is the interaction energy for a nearest neighbor pair, J2 for a second-neighbor pair, etc. If these interactions favor alignment of neighboring moments we have the possibility of a ferromagnetic ground state. A Hamiltonian including the effect of an external magnetic field, h, may be written
H , - J, Z
s,s +
Z
s;s + ... + h Z s,
(ij) 2
(t.'/') i
i
where the index (ij)k on a sum denotes that the sum is to be taken over all kth-neighbor pairs on the lattice. This model was proposed by Ising and Lenz in the 1920s and solved for the case of a 1-d lattice. The 2-d case proved to be intractable and resisted exact solution until Onsager obtained the partition function Z,(T,h)- y j exp [-H,(is,i~)/kT]
(13.2a)
and the associated (magnetic) Gibbs free energy Gl(T,h) - - k T log Z i
(13.2b)
but for the case of h - 0 and nearest-neighbor interactions only. (The history of the Ising model is discussed by Brush (1967).) A bona fide phase transition is a change in symmetry of the material. This usually involves a change in the translational or magnetic order, the degree of which may be quantified by an order parameter. In the case of a ferromagnetic system the useful order parameter is the spontaneous magnetization, N
1 OG M(T,h) - ~ y j s~- Oh
(13.3a)
i=1
where N ~is the number of sites on the lattice. (The second equality is noted in most statistical mechanics texts, or it may be derived straightforwardly from Eqs.( 13. I and 13.2a,b.) If M ~ 0 for a sample of macroscopic size, then somehow the spins are 'communicating' across the entire sample. This characterizes what is meant by the concept of long-range order (LRO). Note that the presence of LRO implies an alteration of the symmetry of the system, in this case a loss of u p - d o w n symmetry in the spin orientation.
I The effect on transitions of the size of the system is discussed more fully in Appendix C. Here N is being used simply to normalize the order parameter conveniently.
L.D. Roeh?f;~
720
D e t e r m i n i n g M requires a differentiation with respect to h and O n s a g e r had d e t e r m i n e d Z~ only for h = 0. H o w e v e r Y a n g (1952) was able to e x t e n d O n s a g e r ' s t e c h n i q u e to d e t e r m i n e M(T, h=0) directly and two solutions are f o u n d
sin421~ji
-
/
1
w h e r e 13 -k-~T" M• vary from +l in the fully o r d e r e d state at low t e m p e r a t u r e to 0 a b o v e the critical t e m p e r a t u r e T c given by
sinh42~Ji = 1 ~ kBTc --- 2.2692 IJiI.
(13.4)
N e x t we discuss the global phase d i a g r a m implied by these results.
13.1.4.2. Transition order and the free energy The (T,h) and (T,M) phase d i a g r a m s of the model are given in Fig. 13. I. In panel (a) the bold dashed line along the h=0 axis denotes a first-order transition ~. It is c o n v e n t i o n a l to denote first-order phase b o u n d a r i e s via d a s h e d lines. The first-order line is t e r m i n a t e d by an asterisk d e n o t i n g the critical point at T c. Note that if one sets h = () and varies T through the critical point, the m a g n e t i z a t i o n will be o b s e r v e d to vary along one of the two curves M+(T) or M_(T). T h o u g h the b e h a v i o r is highly n o n a n a l y t i c at T = T c, see Eqs. (13.3 and 13.4), the m a g n e t i z a t i o n is c o n t i n u o u s so that this is a s e c o n d - o r d e r transition. Isolated s e c o n d - o r d e r transition points are usually d e n o t e d by asterisks; when they o c c u r along an e x t e n d e d curve a bold solid line is used. The model is in a f e r r o m a g n e t i c phase for h = 0 and T < To. For T > Tc the s y s t e m is p a r a m a g n e t i c . If h ~ 0, M :~ 0 for all T and there are no phase transitions. The k n o w n d e p e n d e n c e of the m a g n e t i z a t i o n on T and h allows the phase d i a g r a m to be presented also in terms of T and M as in the lower panel of Fig. 13.1. T h e shaded region b o u n d e d by the M• curves is inaccessible u n d e r normal e q u i l i b r i u m conditions, as e x p e c t e d for a f e r r o m a g n e t i c system. As first discussed in the influential t e x t b o o k of L a n d a u and Lifshitz (1969) the order o f a phase transition in a particular system is d e t e r m i n e d by the ' d e p e n d e n c e '
1 Concerning the order of a transition, phase transitions are always characterized by nonanalyticity of thermodynamic variables (see Appendix B), but come in two distinguishable varieties. One finds first-order transitions across which all thermodynamic densities (see Appendix A for a discussion of the distinction between thermodynamic fields and densities) are discontinuous - - in the present case the equilibrium magnetization jumps between M+ and M as one crosses the bold dashed line. (Nonequilibrium behavior, usually in the form of hysteresis is often observed.) In a second-order or continuous transition on the other hand, all densities vary continuously through the transition, but with still a singularity, a slope discontinuity, which thus manifests itself in the derivative of next higher order. The terminology is attributed to Ehrenfest.
721
Phase transitions and kinetics o.fordering
TII
(a)
2.0 -~
|
1.0 .I
TT
(b)
iiiiiiiiiiiii!iiiiiiiii!iii!i!iiiiiiiiiii!i!!ii~!iii!iiiiiii!i!i!i!iii!i!i!iii!iiii!ii!i!iiii ii!iiii!iiiii!ii!ii!iiiiiiiii.iiiiiiiiiii!ii!iiiiiiiiUiiiiii!!i iiiii!iiiiiiiiii!iiii i iiiilili iiii!!iii iiii iiiiiiiiiiiiiiiiiiiiiii i !iiiiii
-I .(
0
.0
Fig. 13.1. Phase diagram of the Ising model, Eq. (13.1). (a) (T,h) phase diagram; (b) (T,M) phase diagram. of its free e n e r g y on the o r d e r p a r a m e t e r a s s o c i a t e d with the p h a s e transition, t N e a r a c o n t i n u o u s p h a s e transition, M is small in m a g n i t u d e so that G ( T , h ) can be e x p a n d e d a b o u t M = 0. T h e t e r m s w h i c h a p p e a r in this e x p a n s i o n can be d e d u c e d f r o m the s y m m e t r y of the s y s t e m . F o r e x a m p l e the free e n e r g y e x p a n s i o n appropriate to the 2-d Ising m o d e l is G(T,h;M) - hM + a(T) M 2 + b(T)M 4
(13.5)
1 T and h are the canonical variables on which G, the magnetic Gibbs free energy in this case, depends. Considering the dependence G(M) means that for each value of M the overall partition function sum is restricted to include only those states having that value of M. Strictly speaking this theory should apply only close to the critical point since we want to be able to describe G adequately as a low order expansion in terms of M.
L.D. Roelofv
722
2nd - order
1st - o r d e r
F
M
M
F
F
F
M
Fig. 13.2. Free energy variation associated with first-order (left panels) and second-order (fight panels) phase transitions. The absence, for h = 0, of a term linear in M can be understood by noting that if all spins in the system are overturned G should not be affected. Analysis of other terms requires use of group theory and is treated in Landau and Lifshitz. Persson (1991) displays the determination of the Landau expansion for several models with relevance to surfaces. Equation (13.5) encompasses both types of transitions occurring in the 2-d Ising model. Figure 13.2 displays characteristic variations of G(T,h;M) for the two cases. The panels on the right show G(T,h=O;M) for the continuous case. For a < 0 (top panel) there are two equilibrium solutions with nonzero order p a r a m e t e r - this represents the situation for T < Tc. For a > 0 (bottom panel) there is only a single solution with M = 0 thus corresponding to T > Tc. As a passes through zero (middle) panel, M moves continuously between these two possibilities; and one sees in addition that the nature of the variation is such that we should expect large fluctuation of M in the vicinity of Tc - - t h a t is to say, M can vary
Phase transitions and kinetics of ordering
723
significantly with little cost in free energy. These large fluctuations, even more than the continuous variation of M, are the hallmark of a second-order phase transition. The panels on the left depict a first-order transition for T < Tc so that a has been taken to be negative. Now as h varies through zero at fixed temperature, the equilibrium value of M must move discontinuously from one minimum to the other. Note as well that this picture provides an explanation for the hysteresis often observed accompanying first-order transitions. In the lower panel, the ball describing the system has been left in the right-hand minimum, having been prevented from moving to the minimum on the left, obviously now the equilibrium situation, by the free energy barrier that will only be surmounted when h is large enough to overwhelm the effect of the term aM 2.
13.1.4.3. The Landau rules Landau and Lifshitz (1969) j also deduce from the free energy formalism several rules governing whether a given phase transition may be continuous. The rules are based only on the symmetry of the order parameters. It is important to emphasize at the beginning of this discussion that any transition can turn out to be first order. This is because a phase of virtually any s y m m e t r y can be stabilized by appropriate attractive interactions, and in that case disordering it should require a latent heat. The Landau rules only identify those transitions which have the possibility of being, on account of their symmetry, continuous. It is equally important to emphasize that the Landau free energy formulation does not account for fluctuations, i.e. the fact that important configurations of a system near its transition may be more aptly described via an order parameter which varies spatially. Therefore the 'rules' are approximate 2, and in fact some counterexamples have been found among 2-d phase transition systems to Landau's second rule, The rules, nonetheless, are still valuable in that the first and third rules seem to be valid even in 2-d and for the conceptualization of phase transitions from which they arise. The latter leads to the important notion of Landau classification, discussed in w 13.3.2. Unfortunately the Landau rules were originally formulated and are usually still presented in the rather esoteric language of group representation theory 3. Their translations into the language of surface science are, however, quite accessible and this section motivates and states them in that language and gives some examples. We will suppose that the variable being changed to drive this system through its transition is the temperature, T, but these rules apply equally well to transitions driven by varying say the chemical potential.
! This material is given in Chapter XIV, w 145 of the third edition of that text. 2 Unfortunately,the degree to which they are likely to break down increases as spatial dimensionality decreases, since fluctuations become more significant in lower dimension systems. 3 For example the second rule is stated as follows in Landau and Lifshitz (1969): "...the symmetric cube [r 3] of the representation F in question must not contain the unit representation...". Their explanation, is however, more approachable than that of most others.
724
L.D. Roelofr
The first Landau rule states that continuous transitions happen only between states whose symmetry groups lie in a group/subgroup relationship. (By group we mean the set of symmetry operations, translations, rotations and worse, of an ordered phase.) To understand this rule consider an example based on a made-up system. Suppose we have an adsorption system of rectangular symmetry, with lattice constants say, a and b, ~n the x- and y-directions respectively, and that 3 phases occur thereon: two ordered, a ( I x 3 ) and a ( l x 2 ) ; and one disordered and hence of ( l x l ) symmetry ~. Next consider the set of s y m m e t r y operations of each. The ( l x l ) phase has the largest group of operations that bring it into itself, including in particular translations by distance b in the y-direction. The (1• and (1• phases have smaller groups, since they do not contain the above-mentioned translation, for example. (They do, however, include translations by 2b and 3b respectively in the y-direction.) The groups of symmetry operations for the ( I x 2 ) and (1• phases are both obviously subgroups of that of the (1• According to the first Landau rule then there can be continuous transitions between the ( 1• 1) and either the ( 2 x l ) or the ( 3 x l ) . But it is clear that the (1• is not a subgroup of the (1• 2 and vice versa; and the first Landau rule prohibits continuous transitions directly between them. The second Landau rule, though properly stated in a more complicated manner, essentially states that continuous transitions are impossible if there are third-order terms in the free energy expansion. This is the rule that is known to be violated in some 2-d systems, most notably the 3-state Potts model (to be specified in w ! 3.3.2.2), so we will say relatively little except to motivate it. Suppose one included a third-order term c(T)M 3 in Eq. (13.5), and now imagine again the phase transition sequence presented in the right-hand panels of Fig. 13.2. In this case as a(T) passes through 0 and becomes positive, marking the disordering in the absence of an M :~ term, there will now be a new well on either the positive or negative side of M = 0. The system will remain in this minimum until a(T) becomes sufficiently positive to raise it above the one at M = 0. At that point the system will 'fall' (discontinuously) to M = 0 or exhibit hysteresis. Thus the transition must be first-order, unless by some odd coincidence b(T) happens to vanish exactly when a(T) does. The third Landau rule, sometimes called the Lifshitz criterion, has to do with higher-order periodicities and incommensurate phases. It can be phrased as follows. If the Q-vector of an ordered phase is not at a high symmetry point in the surface Brillouin zone (see below for more specificity) then there are only two possibilities for the disordering transition of the phase: either the phase becomes incommensurate, meaning that the Q-vector begins to move continuously with T before it 9
A
A
/k
,
.
A
I For a discussion of the nomenclature of surface phases, see Chapter 1. Some readers might argue with the assignment of p(lxl) symmetry to a disordered phase. This is in fact the correct assignment, since from the phase transition point of view, there has been no breaking of the substrate symmetry. The average occupancy of any regular sublattice of the lattice is the same as that of any equivalent one. 2 Thatof the (1 x2) has the translation by 2b and the other does not, so the group for the (Ix2) phase cannot be a subgroup of that of the (lx3); and likewise in reverse for the translation by 3b.
725
P h a s e transitions a n d k i n e t i c s o . f o r d e r i n g
broadens and diminishes in intensity to signal the disordering; or the transition must be first order. An example of an application of this rule is the disordering transition of the famed (7x7) reconstruction (see Fig. 6.5) of the Si(l 11) surface, whose beautiful diffraction pattern is one of the must sees of surface science. (McRae (1983) has nice photographs.) The Q-vectors of the ordered phase occur at the seventh-order positions, not high symmetry points in the hexagonal surface Brillouin zone of this (111) surface. Study of this transition has an interesting history (see w 13.4.2.1), but it is now known that the phase does not become incommensurate and that the transition is first-order, as expected from the third rule. We next discuss surface realizations of these general phase transition ideas.
13.2. Adsorbate phase diagrams and the lattice-gas analogy In accounting for the phase behavior of adsorbate systems, the principle degrees of freedom describe the occupation of a regular lattice of adsorption sites. (See Chapter 9 for a discussion of how these sites are identified experimentally.) When potential complications like surface inhomogeneities (steps, defects, impurities, etc.), reconstruction, dissociation (in the case of molecular adsorption), etc. may be ignored, it is natural to describe the cohesive energy ~ of the adlayer in terms of occupation variables, n, e {0,1 }, where {i} indexes the lattice of (assumed equivalent) binding sites for the adspecies in question. (A specific energy expression in terms of ni's will be given in Eq. (13.6).) The resulting approach, known as the lattice gas model 2, is uniquely useful for surface systems and has been widely applied. (Obviously, however, the model cannot and does not encompass some of the degrees of freedom possessed by atoms bound to surfaces, e.g. the vibrations of adatoms within their binding wells or their relaxation within the wells in response to interactions with other adatoms, and so must be applied with caution. Moreover the substrate may become more substantially involved. Some obvious limitations are discussed near the end of w 13.2.2.)
13.2.1. The lattice-gas analogy The lattice gas Hamiltonian is written in terms of the n; defined above as
H, o-e, Z , , , j (ij) ,
Z",", + ... (ij),.
Z",
(13.6)
i
1 By cohesive energy one means the change in energy, at T= 0 (i.e. no kinetic energy or zero point energy), that occurs when the originally distant and noninteracting adparticles are brought to and adsorbed on the surface. 2 'Gas' does not imply anything about the density of the phases to be described. The model can describe systems ranging from fully condensed ( Ix 1) layers to extremely dilute low-coverage systems.
L.D. Roeh?]:~
726
where Ei is an interaction energy for an ith neighbor pair of occupied sites and ijk was defined following Eq. ( 13.1 ). (E0 is a site binding energy which is often dropped from the treatment by shifting the zero of the chemical potential appropriately.) The similarities between HLG and HI (Eq. (13.1)) are obvious and the exact analogy first noted by Lee and Yang (1952) is seen through the following transformation
si ~ 2 n i - 1 J~ ~
(13.7)
l/4 Ei
h ----)- (p + ~j) where
la,, y__, giEi
(13.8)
i
with g, representing the number of ith neighbor pairs that occur per site on the lattice in question. Thus the phase behavior of a simple lattice gas with an attractive nearest neighbor interaction can be deduced from that of the Ising model; the (T,p) phase diagram is depicted in Fig. 13.3a. The dependence of the coverage 0 on T and la can be obtained from the transcription of the magnetization, N
M~~
1
~(2ni-
1)-20-
1
(13.9)
i= I
and in turn allows deduction of the (T,0) phase diagram of the attractive-nearestneighbor lattice gas shown in Fig. 13.3b. In surface nomenclature, one is dealing with an adsorbed p( 1• 1) phase ~. Some minor confusion is occasionally generated by the fact that the lattice gas analogy implicitly carries one between the canonical and grand canonical partition functions. In the case of magnetic systems one calculates the canonical partition function, Z, and from that the magnetic Gibbs free energy G(T,h) = E - T S - hM; while for simple lattice gas systems one wants the grand partition function, ~ and the associated grand potentiafl, f2((,p) = E - T S - p N = -kBT log ~. From experience with 3-d phase transitions, one is accustomed to the terms gas, liquid, f l u i d and solid for phases of differing density and character. These terms, though occasionally used in the surface phase transition literature, are not particularly meaningful since the distinguishing features of 2-d lattice gas phases are rather
1 Adsorbedp(lxl) phases seem rather uncommon in the literature, but this may be due to the fact that LEED (see Chapter 7) is not conveniently sensitive to overlayers with periodicity similar to the bulk on which they are adsorbed. Several cases of metal on metal epitaxy do occur and Kolaczkiewicz and Bauer (1985) have developed a novel technique based on work function variation to detect the boundaries of the coexistence region in this case. 2 See, for example, Chapter 23 of Morse (1969).
727
Phase transitions and kinetics of ordering
T
(a)
0.50
dilute
i
phase
0.25
dense phase
0.00
-2.0
"1
T
J (b)
dilute gas , ~1 ~
lattice'liquid' ,
ii••••!i•i•!i••!i!i•i!i••ii•i••ii!i!i•!i!iiiiii!iii !•i!i•i!i•i••iiii••iii!c i•iiiiii i ii!!i!iiiiiiiiiiiii!iiiiiii iiiiiiiiiiiiiiiiiiiii i ii!•i•i•iiii•iiiiiii!iiiiiii i.i•ivu::n sso o 2s I ~ . . " . : . i i i
0.00
0.0
0.5
1.0
o
Fig. 13.3. Transcription of lsing model phase diagram into the corresponding attractive nearest-neighbor-lattice-gas phase diagram. (a) (T,I.t) phase diagram showing first-order boundary terminated by
a critical point indicated by the asterisk. (b) (7",| phase diagram showing coexistence region occurring lor T < T,:. dissimilar to those of the 3-d situation. This subject is dealt with more completely in w 13.2.4. In the meantime, it is worthwhile noting that most ~ lattice-gas phases are 'solid' from the point of view that continuous motion of particles is hindered by the lattice potential provided by the underlying solid surface. Secondly, even in 3-d systems the only real distinction between 'gas' and 'liquid' phase is a matter of density, the two phases becoming indistinguishable above the liquid-gas critical point. The umbrella term 'fluid' therefore is sometimes used to cover both cases. The situation in the simple lattice gas phase diagram of Fig. 13.3 is similar in that 'fluid' phases of differing density are separated by a first-order discontinuity below
1 Incommensuratephases are an important exception. See Chapter 10 and w 13.2.3.
L.D. Roelq[:s
728 1.0
0.8
Au/W(110) To= 1130 K
I-
0.6
0
0.2
O.t, O (ML}
0.6
O.B
1.0
Fig. 13.4. (T,0) phase diagram of W(110)-Au as obtained via work function studies, after Kolaczkiewicz and Bauer (1985). The boundary encloses ap(l • coexistence region. The phase diagram is incomplete for experimental reasons for coverages just above the critical coverage.
a critical point, but become indistinguishable above i t - hence the occasional use of the terminology 'lattice gas' and 'lattice liquid' for phases of this sort. An example of a surface system exhibiting a phase diagram topologically similar to that of Fig. 13.3 is W(110)-Au, a p ( l • phase, whose phase diagram, obtained by Kolaczkiewicz and Bauer (1985), is shown as Fig, 13.4.
13.2.2. Particle-vacancy symmetry and trio interactions HLC gives rise to a phase diagram symmetric about 0 = l/z, as can be seen by applying the particle-vacancy transcription n; --) ( 1 - n;) in Eq. (13.6). One expects, however, at least for chemisorption systems (see Chapter 9 and w 13.2.3), that the full energy of an adlayer system will not be describable just with pairwise interactions as in Eq. (13.6). Trio interaction terms, adding to the Hamiltonian terms of the form
H, = E(~I ~.~ n;njnk (ijk)•
where (ijk)p represents a trio of sites of some particular arrangement. In some cases even higher-order terms may also be required. Trio and higher order terms are not
Phase transitions and kinetics of ordering
729
invariant under particle-vacancy interchange and therefore constitute one possible explanation for lack of symmetry about half-monolayer coverage, as seen for example for W(110)-Au in Fig. 13.4. Anomalously large trio and higher order interactions are needed to explain the pronounced asymmetry for W(110)-Au and have been attributed by Roelofs and Bellon (1989) to strong relaxation effects of the adsorbate atoms in their binding hollows as a function of local environment. This is obviously a non-lattice gas effect and illustrates a limitation of the model as applied to surface systems. See also Persson (1991) for extensions of this non-lattice gas effect to other systems. The lattice gas model cannot in simple form deal with many other real-world situations including: quantum effects as proposed for H on various substrates by Christmann et al. (1979) and Hsu et al. ( 1991 ); the effect of surface steps and other defects, particularly when these degrees of freedom are not fixed; nonregistered binding as often occurs in physisorption systems (see w 13.2.3); etc. Before proceeding to phase diagrams containing phases of more complicated symmetry a digression on two particular limiting cases of possible experimental regimes will be helpful.
13.2.3. Chemisorption and physisorption In discussing surface phase transition phenomena, it is important to bear in mind that the phases on a surface are, at least in principle, in thermodynamic contact with a 3-d gas phase of the adsorbate species. The effect of this contact is determined mostly by the strength of bonding of the adspecies to the substrate. In the case of a physically-adsorbed (or physisorbed) species (see Chapter 15) the bonding is weak ( 10 meV order of magnitude) relative to the energies of mutual interactions between the adsorbates, i.e., the Ei's of Eq. (13.6). In the physisorption limit, the adlayer and 3-d gas rapidly come to equilibrium with respect to particle exchange such that the chemical potentials of the two phases become equal. The experimentalist therefore can control the chemical potential of the adsorbed layer by controlling the pressure of the surrounding gas. (The chemical potential is simply related to the pressure for ideal gases. See Reif (1965), for example.) The (T,I.t) phase diagram can thus be directly measured, and, given a method for determining coverage on the surface, the (T,0) phase diagram may also be obtained. See for example the careful and complete X-ray diffraction study of Xe physisorbed on graphite by Hong et al. (1989)'. In the chemisorption limit on the other hand, the binding of the adsorbate particle to the surface is strong relative to the Ei's (2-5 eV order of magnitude), so that at the energy scale at which the surface phase transitions occur, the rate of desorption from the surface is so low as to be negligible. If the experimentalist has achieved ultra-high vacuum, the rate of adsorption is also slow relative to experimental time scales so that the coverage is effectively fixed. In this limit only the
1 Otherexamples are given in Chapter 10.
730
L.D. Roeh?]:v
(T,0) phase diagram can be conveniently measured; the corresponding (T,~) diagram must be inferred from the former using the known properties of phase diagrams, as will be discussed in w 13.2.4. It is noteworthy that the ability to fix and control 0 provides convenient access to the coexistence regions, e.g. Fig. 13.3. Diffusion limitations lead to interesting kinetic behavior in coexistence regions. The equilibrium state within such a region is a fully segregated coexistence of the two phases on either side of the relevant first-order line, in the present case a dilute 'gas' and a dense p ( l x l ) phase (containing a dilute gas of vacancies). For most experimental initial conditions full equilibrium is difficult to attain due to the fact that as clusters of adparticles grow they become increasingly immobile. This immobility is exacerbated by surface defects in many cases. Eventually most systems stop short of full segregation, so that the often-encountered characterization, island phase, is an accurate description of the character of the phase. The development of order in coexistence regions will be discussed further in w 13.5. An important structural concept related to the physisorption/chemisorption distinction is registry, which can be defined to be the degree to which adatoms reside at the minima of the binding potential provided for an isolated adatom by a clean surface. Obviously, if adatom-adatom interactions are significant, one would expect the adatoms to deviate from these minima. A useful generalization is that, in the absence of reconstruction, chemisorbed species tend to be well-registered, while in the case of physisorption this is less likely to be the case, though there are exceptions. In any event, lack of full registry does not necessarily rule out the use of the lattice gas approach. However, the interaction energies used in this case must reflect the (spacial) relaxation of adatoms within their binding sites (and also that of nearby substrate atoms as well!). Furthermore, these relaxations depend strongly on local coordination, so that higher-order interactions are induced (see Roelofs and Bellon, 1989).
13.2.4. Phases of more complex symmetry Many adlayer phases whose symmetry differs from that of the clean substrate occur on surfaces. The simplest may be obtained by applying the lattice gas transformation, Eq. (13.7) to the Ising antiferromagnet, whose Hamiltonian may be written exactly as in Eq. (13.1), but with the sign of Jl positive so as to encourage neighboring spins to point in opposite directions. The solution to the antiferromagnetic Ising model (on a square lattice) is straightforward because the following transcription carries one between the two models s, ~ - s , s,--) si J!---) - J!
h ~---~h~
for i ~ Sl for i e
S2
(13.10)
Phase transitions and kinetics of ordering
731
Ca) T c
~ ~ ~
. . . . . .
/
t;t;t; .
"h c
.
0
§ § § § § §
2 2 2 2 2 2
4 4 4 4 4 4
7 7 7 7 7 7
2 2 2 2 2 2
4 4 4 4 4 4
7 7 7 7 7 7
2 2 2 2 2 2
4 4 4 4 4 4
7 7 7 7 7 7
2 2 2 2 2 2
4 4 4 4 4 4
7 7 7 7 7 7
\ .
he
h
Fig. 13.5. (a) (T,h) phase diagram of the anti-ferromagnetic (AFM) Ising model. The ordering in each region is indicated by the arrays of + and -. The solid line is a second-order phase boundary separating a region of c(2• symmetry from the paramagnetic p(l• phase. (b) Phase diagram of the AFM ising model in (T,h,h~) space. Viewed in the (T,h~,0) plane the phase diagram is identical to that of the ferromagnetic Ising model, Fig. 13.1a. where S~ and S 2 denote the two interpenetrating sublattices of sites defined by c(2x2) order and where h, is the staggered field, i.e. an external antiferromagnetic field ~. The transcription, Eq. (13.10), preserves the Hamiltonian so that Onsager's solution also covers this case. The (T,h) phase diagram of the Ising antiferromagnet with J2 (and all higher-order interactions) set to 0 is given in Fig. 13.5. Because of the applicability of transformation Eq. (13.10), we know that Tc is again given by Eq. (13.4). The phase diagrams are, however, quite different because of the interchange of h and h~. h no longer forces the system into a paramagnetic phase, but only decreases the transition temperature symmetrically in both directions and does not alter the fundamental
1 Obviouslyone of those theoretical artifices somewhat difficult to realize experimentally.
732
L.D. Roelofs
nature of the transition. Thus, along the entire solid curve in Fig. 13.5, one has a continuous phase transition at which the order parameter appropriate to this transition, M s = -~
(13.11)
si i
~
i~
S2
goes to zero with temperature variation similar to that of Eq. (13.3). (Note that Eq. (13.11) differs from (13.2) because of the different symmetry of the phase being described. M~ is the appropriate order parameter since it varies from +1 in the two possible fully ordered states to 0 above the transition.) The relation between ferro- and antiferromagnetic Ising model phase diagrams is clarified in Fig. 13.5b, which shows the dramatic effect of turning on the staggered field, h~ (analogous to a magnetic field in the ferromagnetic model) in the antiferromagnetic model. For h~ ~: 0 the system has c(2x2) symmetry for all T and there are no phase transitions. It is worth emphasizing what precisely is meant by s y m m e t r y and its connection to long r a n g e o r d e r (LRO) in this context. Genuine phase transitions, marked either by discontinuities or nonanalytic variation of measurable quantities occur only in the thermodynamic limit, i.e. in the limit of infinite system size', c(2x2) symmetry then means that the M~ order parameter defined in Eq. (13.1 1) is non-zero when the sums are carried out over the entire infinite system. There are two common misapprehensions. One is that the order must be perfect. It need not be; in an infinite system an extremely small imbalance in the terms in Eq. (13.11) constitutes long-range c(2• order 2. Another error is to assume that local c(2x2) coordination constitutes c(2x2) symmetry. That condition is neither sufficient, since a part of the system not seen may lead to cancellation of the order, nor even necessary (!!!). Consider for example the low-coverage (0.14 < 0< 0.18) phase discussed by Bartelt et al. (1989), which exhibits long-range c(2x2) order, but whose local coordination is p(2x2). The widespread use of LEED (see Chapter 7) in surface physics is a two-edged sword in this respect. On the positive side, LEED allows convenient detection of the symmetry of metallic and semiconducting surface systems. On the negative side, the limited resolution,--100/~, of commercial instruments hinders distinguishing between long- and short-range order. The lattice gas analog of the Ising ferromagnet has a (T,t.t) phase diagram identical with the (T,h) phase diagram of Fig. 13.5a, but with the factor of 4 reduction in Tc resulting from Eq. 13.7b). The phase that occurs inside the continuous transition boundary is of c(2x2) character, of course, with saturation coverage 0 = 0.5, existing over the range 0.37 < 0 < 0.63 at low temperature.
! The variationwith system size N as N approaches infinity has been investigatedand employedto improve the accuracy of calculations based on noninfinite systems. This area of investigation is termed.#nite size scaling and is discussed briefly in Appendix C. 2 One might say, however, that such a phase is 'not well-ordered' despite the fact that it does exhibit long-range order.
Phase transitions and kinetics of ordering
733
The lattice-gas order parameter analogous to M~ of Eq. (13.11) can be defined as the staggered coverage
0s -- N
ni i
~
ni
(13.12)
i
0~ is the density conjugate to the field g~ in a thermodynamic framework. 0~ takes on values of +1 in the two fully-ordered degenerate c(2x2) ground states, and goes to zero in disordered or other p ( l x l ) symmetry states. Lattice-gas order parameters are usually not mentioned in the literature on surface phase transitions because, unlike the magnetic order parameter of Eq. (13.2), there is no convenient way to measure them directly. Instead, since 0, is pl'oportional to the Q{, = ( a ~, ) component of the kinematic diffraction amplitude t, 1
A(Q) - ~ ~_~ nj e i~
(13.13)
J
where {rj} are the lattice positions, one can measure the square of 0, by measuring the c(2x2) component of the kinematic diffraction intensity,
l~atQ) = A'(Q)A(Q)
(13.14)
That is, N2
I(Q") - - 4 0~
(13.15)
I(Q) can be measured via X-ray or electron diffraction (see Chapter 7): in the former case one typically needs a high intensity source and a high-Z adsorbate to achieve adequate signal; in the latter a simple LEED apparatus suffices, but coherence limitations and multiple scattering limit the resolution and complicate the analysis respectively. This exhausts phases possible in models including only nearest-neighbor interactions on a square lattice, but further complexity is possible if longer-range interactions are present. Consider the addition of an attractive second-neighbor interaction, E2, to the above picture. At low temperature, such an interaction obviously tends to stabilize clusters of c(2x2) coordination, even at low overall coverage. The result is addition of low- and high-coverage coexistence regions as shown in Fig. 13.6a. One may accordingly deduce the (T,g) phase diagram given in
1 The value of Q0 assumes a square lattice of lattice constant a.
734
L.D. Roelo.l':v (=)
Dilute Gas
Dense Gas
Coexistence ~ DiluteGas+c(2x2) 0.0
Coexistence DenseGas+c(2x2) 0.5
1.0
(b)
TT
dilute
~
(lxl)
,,,,
~
I/ /
~ -1.0
dense (lxl)
I 0
Fig. 13.6. Phase diagram of a lattice gas with E~ repulsive and E2 = -!/2 El. (a) (T,O) plane. (b) (T,l.t,l.t.~) space. The dots represent tricritical points, so named because three critical lines meet there as indicated by the addition of the la.~axis on the low coverage side. Fig. 13.6b by noting that coexistence regions indicate coverage discontinuities, i.e. first-order transitions as in the p ( l • lattice gas (recall Fig. 13.3.) These are, as usual denoted by the dashed lines in Fig. 13.6b. Above the coexistence regions the nature of the transitions is not changed and these are denoted as solid lines in the (T, la) phase diagram. The point at which a line of continuous transitions becomes first-order is called a tricritical point, because in the expanded space including a staggered chemical potential, l.t~ (which favors one c(2• sublattice over the other in analogy with the staggered magnetic field, h0, one sees in Fig. 13.6b that the point is actually the junction of three critical lines. Figure 13.6 provides a good starting point for discussing the rules that physically allowable phase diagrams must obey. This matter is discussed in Appendix A.
Phase transitions and kinetics t~ordering
735
TIK]T
25i I[10-11A1
3~176 I
20- .._._ -'-""...~.~//0.t.0 ~0.37 ~0.32
I F'*., z~176 F '4. ,so ~ ~. F R,=,rz~i lOOj--
25
0
J
J I
J
0 (o)
t
0.S
i
i
_8=0.S0
is ~ / ~ ~ o . z s ,o 5 !
I
I___
G
0
1.0
J
100 (b)
I ~ ~
1~,..__
200 T [K] 300
Fig. 13.7. Phase diagram of H/Pd(001) as presented in Fig. 15 of Binder and Landau (1981). Panel (a) shows the experimental data (the crosses) of Behm et al (1980) and the calculated phase boundary (dashed curve) which best fits the data. Panel (b) shows the raw LEED intensity data of Behm et al (1980) from which the phase boundary points were deduced. Binder and Landau note that the discrepancy between the experimental points and the calculated boundaries is due to a mingling of long- and short-range order in the experiment by a combination of finite size effects and instrumental limitations. Binder and Landau (1976) first discussed phase diagrams of this sort and later made application (Binder and Landau (1981)) to the system Pd(100)-H, whose phase diagram had been measured by Behm et al. (1980) (see Fig. 13.7). The comparison between theory and experiment is somewhat problematic because of the limitations of LEED as previously noted. Structures of p(2x2) character are also a common occurrence in square lattice surface systems; see for example the study of Ni(001)-Se by Bak et al. (1985). One might naively expect to be able to force the occurrence of this phase with a repulsive E 2 (Binder and Landau, 1976), but in fact this is not sufficient (Binder and Landau, 1980). The p(2x2) phase can be considered to be made up of double-spaced rows of adatoms running in the x- and y-d~rect~ons. Clearly, however, as shown in Fig. 13.8, with only E, and E 2 repulsions, individual rows can slide either in the x-direction or in the y-d~rectson (but not both s~multaneously) without changing the energy. Random slipping of that sort yields a phase with LRO of either ( l x 2 ) or (2x I ) symmetry. This mechanism may be the explanation of the streaky diffraction pattern observed for Ni(100)-S by Oed et al. (1990). One may continue by considering adlayers on surfaces of different symmetry: the centered-rectangular lattices often found on bcc(110) surfaces; the rectangular systems on fcc(110) surfaces and the triangular and hexagonal lattices on the f c c ( l l l ) and the basal plane of the hcp crystal structure. Phases of periodicity different from that of the substrate abound also on these surfaces; some will be noted in subsequent sections. A
A
,
A
,
.
.
736
L.D. Roeloj':v
I 0
0 0
I0
t~:
0 I
0 0
I 0
0 0
0
0 0
0 I
0 0
I
0
0
0
~I~OOQO
0 I
0 0
~O~l
0
0
0
I
0%~
0
0
I
0
0
0
I
0
0
0
0
Fig. 13.8. The p(2x2) phase formed by El and E2 repulsions only, is unstable with respect to slipping of rows in either direction. The interactions do not couple adjacent rows.
13.3. Universality and classification of transitions All continuous 2-d phase transitions can be grouped into a small number of so-called universality classes. The members of each class display identical behavior near their respective critical points. These classes, the simple magnetic models which name them, and the means by which such identifications are made are described in this section. The similarities of behavior are usually described through critical exponents which characterize the singular behavior (see Appendix B) of various thermodynamic and ordering parameters. These exponents are defined and their known values given in this section. The experimental determination of critical exponents is illustrated via discussion of the investigation of O/Ru(001) by Pfntir and Piercy (1989). The section concludes with a discussion of the real-world hindrances that complicate critical exponent measurements in surface systems. 13.3. I. Critical exponents In the vicinity of a second-order phase transition the free energy of an infinitely large system is mathematically singular at the critical point, (T c, h = 0) for a magnetic system, or (To, l.t0) for a lattice gas system. (The origin of the singularity is discussed in Appendix B.) One then expects all thermodynamic and ordering properties to vary singularly in the vicinity of the critical point, the nonanalyticities being characterized by the critical exponents. Measurements of these exponents are of interest as tests of theory of critical phenomena and to determine the classification of a given transition. It is conventional and convenient to use variables centered on the critical point to characterize these singularities. The reduced temperature, t=
T-L L
(13.16)
Phase transitions and kinetics of ordering
737
and h itself which vanishes at the critical point are the standard choices for m a g n e t i c systems. In the case of lattice gas systems, we use t as a b o v e and a reduced c h e m i c a l potential ,-..,
la - ~t - ~t0
(13.17)
It is n e c e s s a r y to pause at this point for a brief notational aside. Section 13.5 of this chapter is c o n c e r n e d with the time d e v e l o p m e n t of order, so that one also needs a s y m b o l to denote time. "t" will be used to represent time in that section, so that reduced t e m p e r a t u r e and time will be distinguished only via use of a script font for the latter. (This should not lead to intolerable confusion; "t" occurs only in this section and "t" only in w 13.5. The c o n v e n t i o n a l critical e x p o n e n t definitions for a m a g n e t i c system are as follows. M e a s u r a b l e quantity Specific heat* O r d e r p a r a m e t e r vs. temp. Susceptibility O r d e r p a r a m e t e r vs. conj. field Correlation function Correlation length (~ defined via
E x p o n e n t definition C h -- Iti-'x m -- (-t) 13 Z " Itl-V m• ,-- +_lhl~/~' F(r) - r -n ~ -- Itl-~ Fr(r) -" e -r/~
Conditions h = 0 t < 0, h = 0 h - 0 t= 0 t= h = 0 h = 0 h = 0, t ~ 0)
(13.18) (13.19) (13.20) ( 13.21 ) (13.22) (13.23) (13.24)
*The quantity is denoted Ch to denote the specific heat measured at constant field strength in a magnetic system. The lattice gas analog is C measured under conditions of constant ordering field (usually 0, of course), not order parameter. (Fr(r), the r e d u c e d correlation f u n c t i o n , is the correlation function with contributions due to L R O subtracted out. It is defined for spin and lattice gas s y s t e m s below.) In the case of Eqs. (13.18, 13.20, 13.23) and (13.24) the functional form of the variation is similar above and below To, but with different amplitudes, or leading constants. In lattice gas systems one m e a s u r e s analogous quantities to d e t e r m i n e the c o r r e s p o n d i n g exponents. The specific heat is u n c h a n g e d so that the e x p o n e n t c~ can be d e t e r m i n e d via c a l o r i m e t r i c m e a s u r e m e n t s of C , (fixed c h e m i c a l potential, i.e., pressure) for adsorption on high-surface area materials ~ - see for e x a m p l e Bretz (1977), T e j w a n i et al. (1980) or Z h a n g et al. (1986). H o w e v e r , it is more useful to focus on the correlation function 1
F(r) - ~ ~ n(r~) n(r + r,)
(13.25)
i
1 Calorimetry is not feasible for standard crystalline surfaces because of the difficulty of extracting the signal due to the surface from that of the bulk of the substrate.
L.D. Roelof~
738
for three reasons: 1-" m o r e directly m a n i f e s t s the physics of critical p h e n o m e n a (see A p p e n d i x B for a discussion of this c o n n e c t i o n and the related subjects of scaling and r e n o r m a l i z a t i o n ) ; because all the above critical e x p o n e n t s can be e x t r a c t e d f r o m m e a s u r e m e n t s of I"; and b e c a u s e F(r), or at least its F o u r i e r t r a n s f o r m , 1
I(Q) = ~ ~_~ I-'(rj) e ; ~ r,
(13.26)
J can be c o n v e n i e n t l y m e a s u r e d for m a n y adsorption s y s t e m s using electron or X-ray diffraction. I(Q) is called the structure f a c t o r and can be shown to be identical to the k i n e m a t i c diffraction intensity ~ defined in Eqs. (13.13 and 13.14). The detailed form of F differs according to the s y s t e m in which the p h a s e transition occurs, but two key aspects are g e n e r a l i z a b l e to all critical points and allow d e t e r m i n a t i o n of the critical exponents. T h e s e are the signature of L R O c o n t a i n e d in the l a r g e - r limit of F(r); and shorter range variation (for t ~ 0) that describes the fluctuations. T h e s e two aspects are transparently d i s t i n g u i s h a b l e in the m a g n e t i c (Ising) case where the analogous s p i n - s p i n correlation function 1 1-'")(r) - N y-' s(ri) s(r + ri)
(13.27)
i
varies as sketched in Fig. 13.9 for t < 0, t = 0 and t > 0. Below Tc, F~r~(r) tends for large r to a constant value of M 2 due to the L R O that occurs in that regime. Even in the context of LRO, however, the system exhibits fluctuating regions of spins o r i e n t e d in the direction opposite to that of the overall order. T h e s e regions range in size up to a value of ~(T) and therefore at shorter distances ~t)(r) displays an e x p o n e n t i a l decay 2 so that, for T < T c, the overall functional form is s o m e t h i n g like G~t~(r) =
m 2+
D_e -r/~ + (shorter range corrections)
(13.28)
with D_ being a T - d e p e n d e n t constant. To focus on the fluctuations one defines a reduced correlation function by subtracting out the contribution due to L R O
1 Multiplescattering in LEED (see Chapter 6) hinders to some extent the extraction of the simple kinematic intensity. However, the phenomena of interest with respect to phase transitions typically occur over a rather limited range of Q-vectors (in the natural units of ~c~) since they concern relatively long-range correlations. The multiple-scattering-induced variations in the diffracted intensity, on the other hand, typically vary more slowly with Q since they arise from interference between beams that have interacted with several surface atoms in a fairly small region. 2 The decay is exponential because starting from within one of the "out-of-phase" regions, the probability that a "mistake" occurs, taking the phase back to that in which the LRO is occurring, is a (temperaturedependent, of course) constant if the interactions are short-ranged so that adjacent bonds do not behave in correlated fashion. As in the case of nuclear physics, constant decay probability results in exponential decay (there in time, here in space) of a population, in this case, the members of the fluctuating, out-of-phase domain.
739
Phase transitions and kinetics of ordering
1.0
(a)
0.8"
H
s._ v
0.6
\.\ ~
0.4
0.2
0.0
-9. . , 0
,
-
20
I(0)
,
.
|
40
.
60
|
.
80
, 100
(b)
~ _ _ ~ __~
Trrl2 t,~0 !
0
Q
Fig. 13.9. Schematic Ising model" (a) correlation function for t < O, t > 0 and t = O; and (b) its Fourier transform in the t < 0 case. I(Q) is the kinematic diffraction intensity as measured in a scattering experiment, for example. From I(Q)one can determine m2, ~ and Z; the temperature dependence of these quantities give the critical exponents [3, v and 7 respectively.
l-'l/~(r) = l-'Ct)(r) - M 2
(13.29)
The (2d) Fourier transform of Eq. (13.28) gives the beam profile in reciprocal space - - the L R O contribution b e c o m e s a 8-function at Q = 0 (and the other 2d reciprocal lattice points) and the decaying exponential b e c o m e s a Lorentzian of half width at half m a x i m u m w = 1/~ centered at Q = 0, /U)(Q,t) = M2(t) 8(Q) +
Z(t)
1 + QZ ~2
(13.30)
as sketched schematically in Fig. 13.9b. (The amplitude of the Lorentzian term has been denoted by Z(t), the susceptibility as in Eq. (13.20); this identification is established below.) The second term in Eq. (13.30) is often termed the diffuse intensity contribution or the critical scattering. By m e a s u r i n g l~t)(Q) for temperatures in the vicinity of T c and resolving the T - d e p e n d e n c e of m 2 and ~, the exponents [3 and v can be determined.
740
L.D. Roelo.[2~
For t > 0 LRO is absent but short range correlations persist, giving again (to leading order) an exponentially decaying F (t), G(t)(r) = D+ e -re" + (shorter range corrections)
(13.31)
The hallmark of a critical point is the divergence of the correlation length ~ that occurs upon approaching it from either direction in T. At criticality the system exhibits domains or fluctuations of order on all length scales and F can no longer exhibit exponential decay. Instead the correlations decay more slowly, via the power law of Eq. (13.22). (For the 2-d Ising model rl takes the value 1/4.) One also finds in the large-r limit at criticality that 1-" becomes isotropic, i.e. having circular symmetry, and thus the corresponding dependence in reciprocal space becomes I(Q) ... I r-n eiQr rdr -- f rZ-neiQ(," dr
(13.32)
1 Q2-q
(Methods for determining the asymptotic dependence of Fourier transforms of singular functions are given in Lighthill (1958).) Equation (13.32) offers, at least in principle, the prospect of determining the exponent 11. Tracy and McCoy (1975) have demonstrated that in practice there are great difficulties originating from the sensitivity to variations of t away from 0. Before placing these ideas in the lattice-gas context we return to the appearance of the susceptibility as the amplitude of the diffuse intensity contribution in Eq. OM (13.30). To measure X;(t) - --~-, one does not actually need to measure the magnetization or even to apply a field. ~Z is available from the correlation function via the fluctuation-dissipation theorem (see, e.g. Reif (1965) or by twice differdhtiating the free energy, since the definition of )(; and Eq. (13.3a) imply that 1
= k ~ ~ F,.(R)
(l 3.33a)
R
This identification is not hard to prove since the definition of X and Eq. (13.3a) imply that 1 32G - N ~)2h
(13.33b)
Equation (13.33a) then follows via use of Eqs. (13.2a,b). Thus )(; is obtained from the Q = 0 value of the structure factor.
Phase transitions and kinetics of ordering
1
Z = , _ ~ (I(Q = O) -
741
(13.34)
m 2)
i.e. the intensity at the center of the diffuse part of the diffraction beam as also shown in Fig. 13.9. Finally, as first noted by Fisher and Langer (1968), and as applied to lattice gas systems by Bartelt et al. (1985), even the specific heat exponent, o~, can be extracted from the correlation function. Integrating I(Q) over an extended range about the point at which critical scattering is occurring is a way of probing the short-range correlations. Like the energy, these must vary like Q,,,~x
I~(Q) = f I(Q) d2a -- C• l-~ +fit)
(13.35)
o
where C• are constants pertaining to the variation above and below the transition temperature and fit) is a smooth analytic function. The choice of QmaxiS not critical and quality of the data may be optimized by adjusting it between the limits of the resolution of the instrument, AQ, and the entire surface Brillouin zone (see Chapter I). A small value for Qmaxreduces signal magnitude (decreasing the signal-to-noise ratio). Increasing Qm~,xincreases signal magnitude, but as the limit of the full surface Brillouin zone is approached the amplitudes, C• of the singularity in Eq. (13.35) must tend to zero since the kinematic intensity integrated over a full Brillouin zone is a conserved quantity. A typical choice is to integrate over 2-5% of the surface Brillouin zone. This method of critical exponent measurement has become popular in the literature. It has been used, for example by Clark et al. (1986) for the reconstruction transition of A u ( l l 0 ) (see w 13.4.1.3), and by Pfntir and Piercy (1989) for the adsorbate disordering transition of Ru(001)-O (see below). All these ideas are readily transposed to the cases of multiple-spaced phases seen in lattice-gas systems. Shifting from the p( l xl ) order of the 2-d Ising model and the attractive lattice gas simply shifts the location of the wavevector at which critical scattering is manifested within the surface Brillouin zone of the substrate. In the case of the c(2x2) lattice gas, the LRO 8-function and accompanying critical scattering occur at Q0 = -a'a -
as previously noted. Since measurements proceed in
diffraction space this sin~ple'shift is no obstacle, and is in fact beneficial, since in most cases there is no substrate contribution to the diffraction signal in the vicinity of the critical wavevector, whereas at the integer-order positions, that contribution would typically dominate. (The overlayer also contributes a 8-function intensity proportional to 0 at the Ghk' S, where h and k are integers.) The real-space correlation function, Eq. (13.25), for such systems manifests the long-range, multiply-spaced order, if it exists, by settling down to a regular oscillation in the limit of large R. Figure 13.10a shows this situation schematically for t < 0. The large-R limit tends to a non-zero average proportional to 0. At shorter ranges one sees the decay of
L.D. Roelo.l:~
742
.................
................I ................
..... ~... ....... ........
zr' . . . . . . .
,..~
(a)
[(o)
4es2
.I.
!
~/a
Q
(b)
Fig. 13.10. Schematic correlation function (panel a) and diffracted intensity (panel b) for a lattice gas phase of double-spaced order for t < O. Solid lines denote 8-functions. correlations due to fluctuations, again in the context of the oscillations due to the multiple-spaced order. Analysis, however, proceeds in diffraction space where the total kinematic scattering intensity for t < 0 resembles the schematic sketch of Fig. 13.10b, so that abstracting away the trivial oscillations and eliminating the noncritical averaged contribution proportional to 0, requires no special effort, only the shift to the appropriate location, Q0, in diffraction space. The first diffraction-based measurements of the critical exponents of a surface phase transition were accomplished by Horn et al. (1978) in the physisorption system Kr/Graphite using X-ray diffraction. This early work was done using a high surface area form of graphite as a substrate to obtain sufficient signal strength, at the cost of rather significant finite-size limitations (see Appendix C). Later work using synchrotron X-ray sources has achieved sub-monolayer sensitivity; the reader is referred to Chapter l0 for a more complete treatment. The extraction of the critical exponents for a typical chemisorption system via LEED was first attempted by Roelofs et al. (1981) for Ni(111)-O and has since been
Phase transitionsand kineticsof ordering
743
accomplished for several other systems. The most recent example is the analysis of the order-disorder transition of a p(2• oxygen overlayer on Ru(001) by Pfniir and Piercy (1989) and this case is more instructive of the state-of-the-art. The approach used is to fit measured diffraction beam profiles about Q = Q0 to a function of the form
l(Q,t) = (m2(t) 5(Q _ Qo) + 1 + ( Q~(t) - Q,,)2 ~2 ] o T(Q)
(13.36)
where ~ denotes a 2-d convolution and T(Q) is the instrument response function (see w 7.3). T(Q) can be measured by scanning the beam in question under conditions of nearly perfect order (usually a carefully annealed, low-temperature configuration). The fit determines the functions M2(t), ~(t) and ~2(t). lin t may be obtained by directly integrating the measured profile j, or that reconstructed from the fitting procedure the latter approach giving somewhat better control over the parameter Omax in Eq. (13.35) since otherwise T(Q) contributes to the integration. Obtaining critical exponents from t-dependent data is an underdetermined problem due to the effect of the limitations of finite-size rounding (Appendix C), the lack of prior knowledge of Tc, and the fact that the power law forms of Eqs. (13.18-23) only give the lowest-order singular dependence. This last implies that one would want to use data only quite close to (the as yet unknown) To, while the first concern prevents closer approach to Tc than the point at which ~ becomes comparable to L, the linear dimension of defect-free regions on the surface 2. (This highlights the crucial importance of surface quality for exponent measurements. Step-free - - and mostly defect-free J regions of size on the order of L -~ 500 A are necessary to approach Tc within Itl < 0.01.) The approach used by Pfniar and Piercy (1989), which seems to have been quite successful, is as follows. Determine J using measurements of the T-dependence of the diffracted intensity well below the transition ~ and divide out the D e b y e - W a l l e r factor 3, assuming typically that the latter is independent of Q in the immediate vicinity of Qo; Resolve the beam profiles from the instrument response function via parametrized fits based on Eq. (13.36); Integrate the profiles over 2 - 5 % of the surface Brillouin zone to obtain lint(T); Determine Tc as the inflection point of the variation of lint(T) and use that value for all subsequent analysis; - E x a m i n e log-log plots of the variation of lint, M 2, etc. versus t to ascertain the temperature range in which the data is not seriously influenced either by finite-size -
-
-
-
1 Or equivalently, in the case of electron diffraction, by using a Faradaycup with an aperture of the requisite size.
2 Finite-sizescaling can be used to partially overcome this limitation if L can be determined, varied and controlled at least crudely. See Appendix C. 3 The Debye-Wallerfactor characterizes a non-lattice-gas effect, the loss of coherent diffraction intensity due to the uncorrelated, thermally-induced vibrations of the adatoms within their binding hollows. See Webb and Lagally (1973) for a definitive treatment.
744
L.D. Roeh~;~
effects (see Appendix C) close to Tc or by corrections to scaling away from Tc, this being the interval through which linearity is maintained; Perform least squares fits to determine the exponents using those ranges of temperature. In performing the fits it is useful to use all known constraints (and to test the effect of relaxing them!). For example, in the fit of lin t o n e expects the amplitudes C+ and C_ of Eq. (13.35) to be equal for 2-d Ising and Potts models (see following subsections for definition of Potts models). In the case studied by Pfn~r and Piercy (1989), this procedure yielded exponent values within 10% of the expected results; the most impressive verification of universality to date. (The specific values will be discussed and compared to those of the corresponding magnetic model in the following subsection.) An alternative approach to structure factor data analysis that determines Tc and the exponents y and v simultaneously is based on the notion of scaling. The origin of scaling is discussed in more detail in Appendix B and its implication, among others, is that in the vicinity of the critical point the diffuse intensity, defined in Eq. (13.30), can actually be written as a function of a single variable -
x - q~
(13.37)
where ~ is defined in Eq. (13.28) and Figs. 13.9 and 13.10 and q = Iql = IQ -Qol. The specific form of the diffuse intensity is Id(q,T) = Itl-g S•
(13.38)
The functions X§ and X_ are appropriate respectively for T > Tc and T < Tc respectively and are called scaled structure factors. An analysis based on Eq. (13.38) proceeds as follows. Begin by eliminating from the total diffracted intensity the Debye-Waller factor and the fi-function part due to LRO, and taking account of the instrument response function as in Eq. (13.36). Then one 'guesses' values for T~, y and v, and, based on those values and the variation of ~ with t in Eq. (13.23), makes plots of the function -
-
ItF Id (q,T) vs. x
(13.39)
as in Eq. (13.37) for all measured profiles either above or below the assumed value o f T c. Then one adjusts the values of T~, y and v until the plots are well superimposed. Assuming success has been achieved in superimposing the measured structure factors for a substantial temperature range above (or below) the critical point, one has demonstrated scaling and simultaneously determined the values of the exponents y and v as well as the form of the interesting scaling function X§ (or X_(x)). From these it is possible to extract the exponent 1] since X,(x) goes asymptotically to 0 like -
X•
---x 'a-2
(13.40)
for large x (see Tracy and McCoy, 1975). (As T --~ T~, the range over which the
745
Phase transitions and kinetics of ordering
, i,
3.0
I-
i , i , I ; I ;I
' I i I;
I'
I ; I '
2.0
CO
o
1.0
i I
0
o.s89
~._
~~___ o.sTs
I
I
i-
o
[-,
-
~o.ss,
~---~o.s55 i
,
0
I
2
,
i
i~
4
,
I
,
I
,
6
~
I
,
I
8
~
.
o.sqos ~
I0
| I~
-t-" k
(a)
5"0 L
4.5 .-.
1
I
I
' I
"
I'
I
-
!
4.0
-
t~
_o o
3.53.02.52o -I.75 .
-
.
.
-I.50
.
.
.
-I.25
.
.
.
-I.00
-0.75
-0,50
-0.25
3xlO - 7
Ioql o ( k )
(b) Fig. 13.11. Demonstration of scaling of the structure factor as calculated via Monte Carlo simulation of a lattice gas model for the order-disorder transition of a (ffx~J3-)R30 ~ phase by Bartelt et al. (1987). A lattice of linear dimension 60 sites and periodic boundary conditions was used. Panel (a) displays the superposition of structure factors for various temperatures above T,:, each plot being labeled via its temperature in units of E~. The value of T,: used in this plot was 0.338 Ej. Panel (b) shows an attempt to extract the value of the exponent q from the diffuse intensity for the profile with T= 0.3485 E~, the profile closest to T,: not strongly affected by the finite size of the lattice used. See text for further discussion.
746
L.D. Roeh?]:v
variation is controlled by the leading dependence specified in Eq. (13.40) becomes larger and larger.) This method has not yet been applied to experimental data, but has been tested via simulations in which the structure factor is obtained via Monte Carlo calculations (Bartelt et al., 1987). The results of this analysis of the order-disorder transition of a (f3-x'~-)R30 phase on a triangular lattice ~is reproduced in Fig. 13.11. This mode of analysis obviously cannot eliminate the influences of finite size (see Appendix C). In the case of the analysis displayed in Fig. 13.11, profiles at temperatures within 2% of Tc could not be made to scale. A second breakdown occurs at large q (and therefore for smaller x's as t increases in magnitude) and reflects the lattice structure of the substrate. The curves diverge from the common scaling function when q has reached halfway from Q0 to Q = (0,0) where a strong intensity contribution due to the substrate is centered. (The point Q0 at which ordering and critical scattering is that which characterizes ('4"3-• order.) Figure 13.11 b shows an attempt to extract the value of 1"1 using the profile closest to Tc that scales - - thus avoiding the complication of finite-size effects (see Appendix C) - - from the variation of ld VS. q which should go as indicated by Eq. (13.32). The expected value ofrl is 4/15 =0.2666... (see w 13.3.2). In Fig. 13.1 lb a trend toward larger 1"1 and thus better consistency with the expected value is found for data ranges restricted to larger values of q. However, a considerable discrepancy still remains at t = 0.02, this for a lattice size of linear dimension L = 60, which indicates the difficulty of reliably extracting a value for rl from systems of modest size.
13.3.2. Universality classes As noted above, continuous phase transitions can be classified into a rather smaller number of universality classes within which the critical exponents and certain amplitude ratios agree. This categorization (often termed Landau classification after the originator m see Schick ( 1981 ) and the textbook treatment in Landau and Lifshitz (1969)) depends on the spatial dimensionality of the lattice on which the transition occurs and the symmetry of the order-parameter whose variation defines the transition. The analysis of the symmetry of the order parameter is most conveniently done using group theory as generally applied in condensed matter physics. The discussion given here will of necessity be rather sketchy; the treatments by Schick (1981) and Persson (1992) can be consulted for further detail. Briefly, group theoretical analysis of the symmetry of the order parameter allows one to write the free energy near a continuous phase transition as an expansion in terms of the order parameter, which may have more than one component. Such an expansion, Eq. (13.5), has already been presented for the single-component-order-parameter Ising model in w 13.1.4.2. The key idea of universality is that in the vicinity of a critical point, the physics is
1 A triangular lattice can be realized in surface systems when the adsorbate binds to the hollow on a hexagonal surface such as the basal plane of graphite, or in atop binding on fcc(l I 1) and basal plane of hop crystals.
Phase transitions and kinetics ~'ordering
747
controlled by the very long-range correlations described by the divergence of ~; that therefore the local details of the system become irrelevant; and thus that the nature of the transition is controlled by more global considerations like the number of components in the order parameter and how they interact, as indicated by the free energy expansion. Stated in the language of Landau theory then, systems whose free energy expansion in terms of the order parameter are similar, have asymptotically identical critical behavior. Thus, for example, although one cannot make a direct analogy between the 2-d, square-lattice Ising model and say an attractive-interaction, lattice gas on a triangular lattice, the two systems are expected to have identical critical exponents because they agree as to dimensionality of space and symmetry of order parameter. A rather interesting illustration of classification is provided by Tejwani et al. (1980) who measured, via calorimetry, the specific heat singularity of a monolayer of He adsorbed on clean graphite and on Kr-plated graphite. Although the He layer is structurally identical in the two cases, its array of binding sites is altered from triangular (on the clean graphite) to honeycomb on the Kr-coated graphite with the result that symmetry group of the order parameter changes from that associated with the 3-state Potts model (see w 13.3.2.2) to that of the Ising model. The measurements confirmed the expected shift in value of or.
13.3.2.1. Exponent values Table 13.1 presents the exponent values of the known 2-d universality classes along with relevant surface realizations. The latter are discussed in w 13.3.2.2 where the magnetic models that name the classes are described. Table 13.1 requires some comment. The table presents the values of the exponents defined in w 13.3.1 as well as those of Yh and Yt, the leading renormalization eigenvalues, which characterize the behavior of the system near its critical point when examined on varying length scales. These eigenvalues determine the asymptotic decay of all correlation functions at T,.; Appendix B gives the details including the derivations of expressions for the critical exponents in terms of Yh and Yr. The case of the XY model with cubic anisotropy is clearly a bit different than the others in that for some of the exponents a range of values is given. This indicates that these exponents are expected to vary continuously (bUt still satisfy the scaling relations given in Appendix B) with the strength of the anisotropy, a parameter involved in the definition of the model. One therefore says that this model displays nonuniversal behavior. As demonstrated by Jos6 et al. (1977), the nature of the transition varies between the limiting cases of the Ising model and that of a Kosterlitz-Thouless transition (see w 13.3.2.2) whose critical behavior was analyzed by Kosterlitz and Thouless (1973) and Kosterlitz (1974). A calculation by Hu and Ying (1987) suggests the exponents have values rather close to those of the 2-d lsing model except in the vicinity of vanishing anisotropy where pure KosterlitzThouless behavior should occur. One should therefore expect that measured critical exponents of systems classified as XY models will often have values close to those of the Ising model. The critical exponents deriving from classical mean field theory (see any statistical mechanics textbook m e.g. Reif (1965) where the theory is called
748
L.D. Roelof~ Table 13.1 Two-dimensional universality classes, exponent values and realization
Exponent
f~ 8 ll v Yt .Vh Example of realization: Disordering of...
Ising
3-State Potts
4-State Potts
XY with cubic anisotropy
l st-order
Mean field
0* 1/8 7/14 15 1/4
1/3 1/9 13/9 14 4/15 5/6 6/5 28/15
2/3 1/12 7/6 15 1/4 2/3 3/2 15/8
[0,-co**] [1/8, oo] [7/4,00] 15 1/4
1 0 1 oo 0 1/2
0 1/2
[1,0] 15/8
2
(~3x73-3) R30 on triangular lattice
p(2x2) on triangular lattice
(Ix2) on square lattice; or (2x2) on centered-rectan gular lattice
(Ix2) on triangular lattice
I 1
15/8 c(2x2) on square or rectangular lattice; or p( I xl ) on any lattice
[1,oo]
1
3 0 1/2
2 Long-range attractive interactions on any lattice.
*Whcn cx -->0 the remaining singularity is logarithmic since for small x, Itl-x goes like I - x In Itl +... **A negative specific heat exponent implies a cusp rather than a divergence. When the exponent goes to - ,,,, the cusp becomes an essential singularity. (For a discussion of the classification of singularities see any text covering complex analysis or mathematical methods more generally; Wong (1991) is a good choice.) The essential singularity in the specific heat of the XY model is, tor all practical purposes, invisible, since the singular function and all of its derivatives vanish at the phase transition. (The function exp[-l/Itl], for example, has an essential singularity at t = 0.) See, for example the simulation study of Tobochnik and ('hester (1979). Fortunately the transition is apparent in other observables.
C u r i e - W e i s s t h e o r y , or B r o u t ( 1 9 6 5 ) h a v e b e e n i n c l u d e d in T a b l e 13. I. M e a n field t h e o r y ( M F T ) is not c o n s i s t e n t with the s c a l i n g h y p o t h e s i s and t h e r e f o r e v a l u e s o f yj, and y, h a v e not b e e n g i v e n . M F T can be and o f t e n is a p p l i e d to lattice gas m o d e l s w i t h s h o r t - r a n g e i n t e r a c t i o n s as a first t h e o r e t i c a l a t t e m p t (see S c h i c k ( 1 9 8 1 ) for a c o n v e n i e n t p r e s c r i p t i o n for c a r r y i n g o u t the c a l c u l a t i o n s ) ; but as c a n be seen f r o m T a b l e 13.1, the t h e o r y is not s u c c e s s f u l in d e s c r i b i n g p h a s e t r a n s i t i o n s in unodels with s h o r t r a n g e i n t e r a c t i o n s . M F T not o n l y fails to g i v e an a c c u r a t e a c c o u n t o f critical e x p o n e n t s , but also in s o m e c a s e s , n o t a b l y that o f the 3 - s t a t e Potts m o d e l , fails to p r e d i c t the o r d e r o f a t r a n s i t i o n p r o p e r l y . M o r e o v e r , as s h o w n by B i n d e r and L a n d a u ( 1 9 8 0 ) , p h a s e b o u n d a r y l o c a t i o n s for 2-d lattice gas m o d e l s c a l c u l a t e d via M F T m a y be i n a c c u r a t e by f a c t o r s o f 2 or 3. T h e f a i l u r e s o f M F T are d u e to its n e g l e c t o f f l u c t u a t i o n s ; the v a l u e o f e a c h d e g r e e o f f r e e d o m is set e q u a l to its s e l f - c o n s i s t e n t l y c a l c u l a t e d t h e r m a l a v e r a g e . In the c a s e o f l o n g - r a n g e a t t r a c tions, this is a less s e r i o u s a p p r o x i m a t i o n , s i n c e the s y s t e m is less a f f e c t e d by its o w n s h o r t r a n g e f l u c t u a t i o n s . I n d e e d F i s h e r et al. ( 1 9 7 2 ) f o u n d that the c r i t i c a l e x p o n e n t s for a m a g n e t i c s y s t e m with i n t e r a c t i o n v a r y i n g like s i -r3 s t- are i d e n t i c a l to
749
Phase transitions and kinetics of ordering
the mean field values given above. Translating to the lattice-gas equivalent we 1
would expect that a system exhibiting long-range ~ attractions should have mean field critical exponents. There is one important universality class not directly represented in Table 13.1, that of the Heisenberg model which can support two different forms of anisotropy. (This model and its surface realizations will be described in more detail in w 13.3.2.2.) The model is not included in the table since the best currently available theoretical evidence suggests that the model displays only either Ising-type transitions or first-order transitions. (Recent experimental work has, however, raised some important questions concerning this conclusion, as will also be discussed in w 13.3.2.2.) One therefore expects to find surface realizations of first-order phase boundaries not terminated by a critical point as in Fig. 13.3. Given this expectation, the exponents associated with a scaling treatment of first-order transitions due to Fisher and Berker (1982) have been included in Table 13.1. Note that both renorrealization eigenvalues have assumed values equal to the dimensionality of the lattice, d = 2 in this case; this being the diagnostic characteristic of the renormalization fixed point associated with a first-order transition. The exponent values are as might be e x p e c t e d only o~ and v require comment. With o~- 1 the energy, which varies as in Eq. (13.35) has no singularity upon approach to the transition. (The discontinuity in E at the first-order transition is not manifested by the critical dE exponent values.) With C-~---~ the variations in Eqs. (13.18) and (13.35) are ordinarily consistent. Obviously, however, this relation breaks down when c~ --~ 1, since the derivative of a constant vanishes and the value of ~ = 1 does not therefore imply a power-law specific heat divergence as in Eq. ( 1 3 . 1 8 ) f o r first order transitions. Rather the latent heat of the transition is exhibited as a ~i-function in C. (For systems of finite size the ~5-function is broadened thus allowing confusion with power-law growth near a critical transition. In that case one might extract an effective value of o~ from data, but there is no theoretical reason to expect that value to be related to the oc - 1 of Table 13.1. Finite size effects on first-order transitions are discussed further in Appendix C.) The value v - I/2 suggests a divergence of the correlation length, something one might not expect at a first-order transition. Fisher and Berker attribute the apparent divergence of ~ to the LRO possessed by both phases in the vicinity of a first-order transition. This interpretation, however, may be inadequate as indicated by the simulation study of the order-disorder transition of a p(2x2) phase on a honeycomb lattice of Bartelt et al. (1987). In this work apparent critical scattering and divergence of the correlation length with v~ff = 0.55+0.15 were found to be associated with an apparently first-order transition.
1 Dipolar repulsions go like r--3, but cause non (lxl) ordering (see Roelofs and Kriebel, 1987). Any order-disorder transitions that occur in that context would display the critical behavior appropriate to the relevant order parameter symmetry.
L.D. Roelo.f~
750
Some elaboration beyond this simple scheme is necessary to account for the effect of other r e l e v a n t operators. See Appendix B. 13.3.2.2. M a g n e t i c m o d e l s In this subsection the magnetic models which name and characterize the universality classes included in Table 13.1 are defined and described. All the models are for ferromagnetic behavior and can be defined on a 2-d square lattice using nearest neighbor interactions only. The description will in each case only specify the nature of the degree of freedom associated with each site and the form of the interaction, J, between neighboring 'spins'. The Hamiltonian of the 2-d Ising model was given in w 13.1.4 and therefore need not be repeated here.
(i) The Potts m o d e l s : The q-state Potts models are generalizations of the 2-d Ising model, which can be taken to be the case of q = 2. In the q-state Potts model each spin may be found to be oriented in q possible directions and interacts with its neighbors via a Kroneker ~5-function interaction.
Jq_po,t.,(ni, rt]) - - J 8,,,.,,j
( 13.41 )
The n;'s represent the degrees of freedom associated with each site and take on integer values {1, 2, ... q} to denote the q possible orientations; J is a positive constant for the ferromagnetic case. It is important to distinguish the standard Potts model defined above from the p l a n a r Ports or clock model in which the basic degree of freedom is also taken to be a vector of q possible distinct orientiations, but whose basic interaction is in the form of a dot product Jq cl,,ck(si, si) = - J si " Si
(13.42)
(The 3-state Potts and 3-state clock models are identical up to a shift of the energy zero, which has no physical significance, but for higher q the differences are non-trivial. Betts (1964) noted that the 4-state clock model can be reduced to two superimposed lsing models, and so has the critical behavior of that universality class.) Fc)r q > 4, the q-state Potts models all are known to exhibit first-order transitions. Wu (1982) in a review devoted just to the Potts model gives references detailing this work and the relations between the Potts and other models which were exploited to determine the critical exponents given in Table 13.1. He does not, however, note that the exponents for the q = 3 case are known from the solution by Baxter (1980) of the hard hexagon lattice gas model. See also Huse (1984). (ii) X Y model: In the XY model the basic degree of freedom is taken to be a spin that can be oriented in any direction in the plane. Hence the alternative name of p l a n a r model. The interaction is taken to be of dot product form Jxv(Si, SJ) = - J si " Si
where J again denotes a positive constant.
(13.43)
Phase transitions and kinetics of ordering
751
The XY model as usually discussed includes anisotropies that favor certain orientations, relative to crystalline axes, over others. One expects to encounter 2-, 3-, 4- and 6-fold anisotropies in real materials, so that the XY Hamiltonian is usually written including a single-site term of the form, Hp_..,m., - - h , ~ cos(p0,)
(13.44)
for p-fold anisotropy, where 0i represents the angle spin sg makes with say the x-axis. hp is a positive constant representing the strength of the anisotropy. Jose et al (1977) give a complete account of the behavior expected for the cases p = 2,3,4,6. In the case of p = 2 and 3 the perturbation represented by the anisotropy is relevant so that the critical behavior is governed by anisotropy. One then finds Ising behavior for the p = 2 case and 3-state Potts behavior for the p = 3 case. For p = 4 or 6 one has more interesting situations. We will focus here on the p = 4 case, that of 'cubic' anisotropy, since that is the only case that appears in Table 13.1. (It should be noted, however, that the 6-fold case has relevance to the theory of 2-d melting, see Halperin and Nelson (1978).) No exact solution for the XY model with cubic anisotropy is available, but various approximate calculations have determined the essence of phase diagram and the critical behavior of the model. The phase diagram is given in Fig. 13.12a. We begin with the situation for h4 = 0. In the absence of anisotropy there exist spin wave excitations whose energy vanishes in the limit of long wavelength. Under these circumstances the model cannot develop rigorous long-range ferromagnetic order at any finite temperature. Rather the correlation function, the analog of Eq. (13.27), displays power-law decay F(r) ~ r -n with I"1 increasing from 0 at T = 0, linearly with temperature. (See Kosterlitz and Thouless, 1973.) A power-law decaying correlation function is associated with critical behavior ~ see Eq. (13.22) and thus we must regard the h4-axis as a line of critical points with varying critical exponents, since 1"1is varying. Such a l-d locus of critical points is called a critical line and is often denoted by a line of asterisks as in Fig. 13.12a. According to spin wave theory the line would continue to arbitrarily high temperatures. Kosterlitz and Thouless (1973), however, pointed out that the h4 model supports another sort of excitation, the vortex, at sufficiently high temperature. Their analysis established that below a particular temperature, vortices and anti vortices ~ swirls in the opposite direction ~ occur in bound pairs leaving the correlation function still in the form of Eq. (13.27). However, at the so-called Kosterlitz-Thouless point (labeled 'K-T' in Fig. 13.12a) at which point 1"1has reached the value of 1/4, the vortices and anti vortices unbind resulting in a more rapid (exponential) drop off of correlations and thus constituting a transition to a paramagnetic phase. This occurs at TK_T----0.8 J. F o r h 4 :g: 0 a gap occurs in the spin-wave spectrum so that long-range magnetic order is possible at low temperature. The regions above and below the h 4 axis below TK_Tare thus ferromagnetic. (The phase diagram is symmetric about h 4 = 0 since a sign change in h4 is equivalent to a 45 ~ rotation, leading to order in one of the 45 ~ 135 ~.... directions rather than along the axes.) In the limit of large anisotropy the spins are constrained to point in just 4 directions. We thus have the 4-state clock
752
L.D. Roeh?f~
XY Model w/cubic anisotropy Y
x
(a)
+oo
,
~u
4 clock~ (Ising) h4
+
. K-T T/J
N
9 9,
~
Heisenberg Models
corner-cube anisotropy
face-centered anisotropy (b)
Fig. 13.12. Other spin models. (a) The XY model with 4-fold or cubic anisotropy. The upper panel shows the basic spin degree of freedom associated with each site as a moment of fixed magnitude able to point in any direction in the x-y plane, but with the anisotropy of 4-fold symmetry favoring certain directions over others. The lower panel shows the phase diagram of the model. See text for explanation. (b) The Heisenberg models. Preferred orientations of spins in the Heisenberg Model with corner-cube anisotropy are shown on the left; and on the right, similarly for face-centered anisotropy.
Phase transitions and kinetics of ordering
753
model in this limit, which is known to have Ising exponents. The disordering transition for large h a thus differs markedly from that at h4 = 0 and the character of the disordering transition is expected to vary continuously along the curve connecting those limits. Thus we have another critical 'line' (not straight in this case). Hu and Ying (1987) have performed a careful simulation study of this 'non-universal' behavior along this critical line, verifying the approximate character of the phase diagram shown in Fig. 13.12a and proposing forms for the variation of the various exponents with h a. For example the correlation length exponent defined in Eq. (13.23) is found to vary like v = [I - exp(-ch4)] -j
(13.45)
where c is found to have the approximate value of 3.6 when h 4 is expressed in units of J. Experimental verification of this fascinating theoretical panoply has been elusive. There has thus been some attention paid to the surface realizations the XY model, whose disordering transitions are expected to be located at varying positions along the curving critical lines. Successful measurements of critical exponents of such transitions would be useful checks of these predictions of non-universal critical behavior. The varying surface atom displacements that occur in the surface reconstruction of W(001 ) are subject to a Hamiltonian similar in symmetry to that of the XY-model so that this system, particularly when the addition of adsorbates is used to vary the Hamiltonian parameters, constitutes another possible experimental test. This matter will be explored in greater depth in w 13.4. (iii) T h e H e i s e n b e r g m o d e l s : The Heisenberg model is the extension of the Ising and XY cases to classical spins that can point in any direction in 3-d space ~. We consider the model here only with classical spin degrees of freedom, located on 2-d lattices and with ferromagnetic interactions. (A 2-d, quantum, antiferromagnetic version of the model is of some possible interest in the theory of high temperature superconductivity.) The interaction between neighboring spins is again taken to be in the form of a dot product, Jttei~(si , Si) = - J si " si
(13.46)
where J is again a positive exchange constant tending to produce ferromagnetic behavior. Again, because of long-wavelength, low-energy excitations, long-range order is possible only when anisotropies are present to create gaps in the spin-wave spectrum. Crystal fields give anisotropies of various symmetries. Uniaxial anisotropies
1 Ironically, Heisenberg proposed this model because of the failure of the 1-d Ising model to develop long-range order as determined from Ising's solution of this case. The Heisenberg model supports an even greater multiplicity of spin waves at long wave-length even than the XY-model and so is even less able than the XY-model to develop long-range order on either 1- or 2-d lattices.
754
L.D. Roelofs
lead either to Ising or XY behavior depending on whether the spin orientations along a particular axis are enhanced or suppressed. We focus here on the cases of corner-cube- and face-centered-anisotropy depicted in Fig. 13.12b, which result in models with new surface realizations. These Heisenberg models have not been solved exactly either, except that in the cornercubic case (CCH) and in the limit of large anistropy (where one has just the 8 allowed orientations) the model is identical to 3 decoupled, superimposed Ising models (one each for the x-, y- and z-directions) and so should have that critical behavior. Landau classification (see Schick, 1981) places the order-disorder transition of a p(2x2) state on a honeycomb lattice into the CCH universality class, and so there is the possibility of surface realizations. However, Landau classification is based only on the symmetry of the order parameter and interactions, and is not a quantitatively exact Hamiltonian transcription. The surface realizations might have effective Hamiltonians with s m a l l - but symmetry r e s p e c t i n g - corrections to Eq. (13.46). Grest and Widom (1981) studied the effect of small alterations of the dot product interaction of Eq. (13.46) and found that the transition becomes first-order for one type of variation and remains continuous and Ising-like for its converse. Their study did not, however, encompass all possible symmetry preserving variations of the interaction and therefore should not be considered exhaustive. Thus the most one can say at this point is that, for systems classified into the CCH universality class, either Ising critical behavior ~or first-order transitions could be expected. The face-centered case (FCH) also has surface realizations, most particularly the disordering of a p ( 2 x l ) structure on a triangular lattice as occurs for example for oxygen adsorbed on Ru(001) at 0 = 1/2. Renormalization group studies by Nienhuis et al. (1983) suggest that the transitions of the model will always be first-order in 2-d. The experimental study of the disordering of Ru(001 )p(2• ) - O by Pfntir and Piercy (1990), however, suggests that the transition is continuous and has critical exponents in rather good agreement with those of the 3-state Potts model. This apparent conflict has not been resolved at the time of this writing, but further theoretical work on the nature of the transitions of the FCH case appears to be warranted.
13.4. S u r f a c e r e c o n s t r u c t i o n
A surface is said to have reconstructed when the atoms in the surface layer and possibly those in nearby underlying layers, spontaneously, or in response to the presence of an adsorbate, shift from their expected (bulk-termination) positions in such a way as to alter the 2-d periodicity of the surface 2. Because of this symmetry
l Schick (1981 b) proposed this as explanation for the observation of Ising critical exponents by Roelofs
et al. ( 1981) for the disordering of p(2x2) oxygen on Ni(111). 2 Theoretical aspects of the driving forces for surface reconstruction are discussed in Chapter 3; experimental aspects are considered in Chapters 7, 8 and 9.
Phase transitions and kinetics of ordering
755
change, reconstruction phenomena are conveniently studied with diffractive methods, especially LEED (see Chapter 7) and X-ray diffraction. In many cases reconstruction is found to be influenced by temperature and one of the guiding themes of the field has been the elucidation of the nature of the observed transitions and the detailed character of the phases involved. New experimental techniques were often developed or were first applied to address long-standing controversies, and the subtle driving forces continually challenged the accuracy of theoretical treatments of the electronic energy in the surface region. The presence of a surface usually causes alteration of equilibrium atomic positions in the perpendicular direction as well. This phenomenon is termed surface relaxation, and is observed to extend several atomic layers into the crystal. Since no symmetry changes or phase transitions are involved in surface relaxation, it is not considered further in this chapter. It would be very satisfying to offer a unified explanation for reconstruction. However, the mechanisms involved are as diverse as the types of bonding that occur in condensed matter physics, and the best one can do is to offer a categorization, perhaps along the lines of the organization of this section.
13.4. I. Reconstruction of metallic surfaces Reconstruction has been observed on many metallic surfaces. Estrup (1984) gives a useful review. Observed reconstructions can be classified into a few categories, based on the character of the atomic movement involved, and the apparent driving force. The discussion in the following subsections is similarly organized.
13.4. I. I. Reconstruction resulting in increased surface packing density The more 'open' surfaces of highly coordinated metals, including AI and the noble metals, tend to reconstruct in such a way as to increase the packing density of the surface layer. On the (100) surfaces of Au, Pt and Ir a pseudo-hexagonal layer, whose spacings are not too dissimilar from that of the close-packed (111) planes, forms over the second layer which largely retains its square symmetry. (See also Chapter 3.) The case of Au(100) is typical. Recent definitive studies include: high resolution LEED by Liew and Wang (1990); scanning tunneling microscopy by Binnig et al. (1984); transmission electron microscopy by Yamazaki et al. (1988); and X-ray scattering by Zehner et al. (1991). An approximate (lx5) periodicity, as shown in the sketch in Fig. 13.13, is found. The atomic spacing in a close-packed (111 ) plane of Au is 2.88 ~. This is also the nearest neighbor distance in the unreconstructed (100) plane, but note that a reconstruction of (Ix5) periodicity - - the periodicity in this direction is evident in Fig. 1 3 . 1 3 b - gives a nearly hexagonal layer with nearest-neighbor spacings of 2.88 ~ and 2.804/~. This reconstruction is evidently driven by the improved coordination offered by the pseudo-hexagonal structure. The precise measurements of Liew and Wang (1990) indicate that there also occurs a slight compression in the other direction as well, such that 29 Au atoms occur in 28 spacings of the crystal in the (110) direction. This reduction of spacing below
756
L.D. Roel~?[;~
Fig. 13.13. Pseudohexagonal reconstruction of Au(001) and similar surfaces, after Yamazaki, et al. (1988). White circles are top-layer Au atoms, successively darker shading indicates layers further into the crystal. (a) depicts the reconstruction on a defect free surface. The overlayer is a distorted hexagon rotated by an angle from the surface crystalline axes [ 10] and [01 ]. (b) shows the effect of step edges in the [ 10] direction on the reconstruction, which is aligned to the step direction to achieve hexagonal coordination at the step edge (see dashed outline in lower portion of figure). The repeat distance along the step edge is about 28a; the sketch shows the registry of the reconstructed top layer at positions separated by half the repeat distance in that direction.
what occurs in the undistorted c l o s e - p a c k e d plane of Au probably results f r o m the fact that e v e n in the p s e u d o - c l o s e - p a c k e d a r r a n g e m e n t the Au atoms are not so fully c o o r d i n a t e d as they ' w a n t ' to be, and therefore respond by s h o r t e n i n g s o m e o f the bond distances. Finally, when the surface is relatively free of steps, the p s e u d o - h e x agonal top layer can further lower its e n e r g y via a slight rotation, as shown in Fig. 13.13a. T h e rotation angle o~ is d e t e r m i n e d by Y a m a z a k i et al. (1988) to be about 0.7 ~.
Phase transitions and kinetics ~" ordering
757
Both this rotation and the tendency of the reconstructed first plane to lock in to high-order integer multiples of the spacings established by the unreconstructed bulk are common features of overlayers that would prefer incommensurate spacings. This subject is covered in more detail in Chapter 10. Finally one should note that the areal density of atoms in the reconstructed first plane is 24% greater than that of a (100) surface. Thus the development of the reconstruction must involve large-scale motion on the surface (see w 13.4.4) and surface steps and other extended defects doubtless play a significant role. This is consistent with the observation that good order does not develop unless the material is heated above 500 K. Further detail has emerged from the X-ray diffraction studies of Zehner et al. (1991), who report a slight rotation of the pseudo-hexagonal structure for T < 970 K, an unrotated reconstruction in the range 970 K < T < 1170 K, and a disordered surface layer for all T > 1170 K up to the melting point.
13.4.1.2. Displacive reconstruction Displacive reconstruction refers to a situation in which the stable surface phase is of different symmetry than the bulk termination, but in a manner that does not involve atomic migration. A prototypical case is that of W(001) which undergoes a reconstruction to a phase characterized by c(2x2) symmetry in which zig-zag rows of atoms develop as shown in Fig. 13.14. This reconstruction, originally thought to be an adlayer phase, was first recognized as such by Debe and King (1977) and Felter, Barker and Estrup (1977). Its detailed structural character was first determined by Debe and King (1978). In this case there is no gross change in surface coordination; rather the effect appears to be driven by covalent bonding considerations primarily involving atoms in the top two planes, Singh and Krakauer (1988) display bending-charge-density plots in their ab initio electronic total energy calculations which clearly show
Fig. 13.14. Displacive reconstruction of the W(001) surface. The small arrows on two atoms in t_he top row emphasizes that the direction of motion is along the (or equivalently the <11> direction) primarily in the surface plane. The scale of the motion is exaggerated by about a factor of 2 to improve visibility.
758
L.D. Roeh~.#;
Fig. 13.15. The c(q7--• reconstruction of Mo(001) seen at glancing incidence looking along the zig-zag rows. These rows are similar to those which occur in the W(001) reconstruction (see Fig. 13.14), but here after each triple of zig-zag rows, there is a row of undisplaced atoms. additional interactions, particularly between a surface atom and the four nearest atoms in the layer beneath. The origin of this extra charge is mostly the 'dangling bonds' associated with presence of the surface. In the case of the W(001) surface, the reconstruction 'disappears' at a Tc of roughly 220 K'. The nature of the phase above Tc was initially controversial, with some workers suggesting that the ( l x l ) periodicity represented an ordered surface of bulk-termination character and others arguing for an interpretation in which the loss of periodicity was accounted for through a loss of long-range order in the displacements, whose magnitude and local character remain similar to that below To. This latter view has recently prevailed according to evidence from: LEED (Pendry et al., 1988); ion scattering (Stensgaard et al., 1989); X-ray diffraction (Robinson et al., 1989); and core-level-shift spectroscopy (Jupille et al., 1989). This evidence along with the observation that the phase transition appears to be continuous allows its classification into the universality class of the x - y model with cubic anisotropy (see w 13.3.2). Displacement magnitudes are small, on the order of 0.2 ]k in the case of W(001 ), and reconstruction energies are in the I0 meV range. These situations therefore represent stringent tests of electronic structure/total energy calculations. State-ofthe-art approaches (see Singh and Krakauer, 1988; Wang and Weber, 1987; Fu and Freeman, 1988) successfully account for the structure of the phase, and produce reasonable (order of magnitude) values for the reconstruction energy. However, detailed simulations of the disordering of the surface layer (see Roelofs et al., 1989) based on such calculations, reveal quantitative discrepancies as large as a factor of two in transition temperature. The (001) surface of molybdenum undergoes a related reconstruction, though in this case the wavevector of the reconstruction is found to be shifted slightly from the c(2x2) positions. This reconstruction was first reported by Felter et al. (1977)
1 The'unreconstructiontemperature' may appeal"to be substantiallyhigheron surfaces with closely spaced steps. (See Wendelken and Wang, 1985.)
Phase transitions and kinetics o.l'ordering
759
and was originally thought to be incommensurate. More recent studies (see Daley et al., 1993; Smilgies et al., 1993) establish that the phase actually is commensurate with c(~--x~-7--) periodicity, as shown in Fig. 13.15. The mechanism underlying this reconstruction has not been definitively settled. The theoretical accounts of C.Z. Wang et al. (1988) and X.W. Wang et al. (1988), which articulated a mechanism based on Fermi surface nesting successfully accounted for incommensurate ordering. However, a picture based on short-range interactions (see Roelofs and Foiles, 1993), seems promising for explaining a commensurate phase. Unidirectional displacive reconstruction is also possible; Itchkawitz e t a l . (1992) have reported that the top layer of K(110) shifts laterally along the [110] direction by about 0.25 ,/k. (See w 13.4.1.4 for a similar reconstruction induced by hydrogen adsorption.) 13.4.1.3. Missing row reconstructions
The (110) surfaces of several fcc noble metals are observed to reconstruct to the missing row phase shown in Fig. 13.16. This reconstruction occurs spontaneously for Ir, Pt and Au and can be induced by low-coverage alkali adsorption for Ag, Pd and Rh. (See Chapter 3 and Ho and Bohnen (1987) for experimental references.) This reconstruction is not driven simply by coordination, since there is no net change in the total number of missing first neighbors when it occurs. Thus more subtle many-body effects must be examined, and in the author's study, see Roelofs et al. (1990), of the energetics of this transition in the case of the Au(l 10) surface,
Fig. 13.16.The missing row reconstruction that occurs on fcc(110) noble metal surfaces. (a) top view; (b) side view.
760
L.D. Roelo.f~
the dominant driving force is found to be a reduction in the many-body or glue ~part of the interaction, which depends in a highly nonlinear way on coordination. These reconstructed phases appear to undergo disordering transitions at sufficiently high temperatures, but there has been some controversy over whether the disordered phase remains f l a t - but with the 1/2 layer of 'extra' rows disordered or whether the surface roughens. In the former case the transition should occur in the universality class of the 2-d Ising model as pointed out by Per Bak (1979). The first experimental study, carried out via LEED at relatively low resolution by Campuzano et al. (1985) seemed to confirm the 2-d Ising classification. Villain and Vilfan (1988) first raised the roughening possibility on theoretical grounds and noted that in that case one would not expect simple Ising behavior. Recent higherresolution studies of the Au(110) transition at 735 K, via X-ray diffraction - - see Keane et al. (1991), and the corresponding transition on Pt(110) which occurs at 960 K and has been investigated by Zuo et al. (1990) using high-resolution LEED indicate that the transition is of simple order-disorder (and thus) Ising character. The X-ray diffraction study of Keane et al. (1991) also reveals that the Au surface at least does eventually roughen as well, some 50 K above the disordering phase transition. The missing-row character of these phases was initially considered to be a problematic explanation of their apparent double-spaced periodicities, because it was thought that the extensive mass motion required for their ordering would be prevented by diffusion limitations. Many structural probes were therefore applied in attempts to shed light on this question, but the debate was only definitively settled by scanning tunneling microscopy in one of the earliest uses of that technique for metal surfaces (see Binnig et al., 1983). It was also pointed out by Campuzano et al. (1985) that the occurrence of steps initially on the surface as a result of sample preparation processes would, to some degree, mitigate the difficulty of ordering via diffusion. The significance of diffusion in non-displacive reconstruction is treated in more detail in w 13.4.4.
13.4.1.4. Adlayer-induced reconstruction The reconstruction modes considered in w 13.4.1-3 and others can also be induced by adsorption on surfaces which, when clean, do not reconstruct. There is a vast and growing set of known instances of this behavior; here we give some examples of cases that are relatively well understood at this time. It is worth noting parenthetically that in some cases these induced reconstructions are kineticaily hindered, and are not reversible. The induction by alkali adsorption of the missing-row reconstruction of w 13.4.3 has already been mentioned. The fact that coverages as low as 0.1 monolayers of K, for example, can induce the missing-row reconstruction on Ag(110) and the high electropositivity of the alkalis encouraged the hypothesis that charge donation into surface states drives the reconstruction. This long-range mechanism was supported
1 The term is due to Tosatti; see Garofalo et al. (1987).
Phase transitions and kinetics ~tf ordering
761
A
<
<100>
~
T
--
~ <110>
Fig. 13.17. The missing row reconstruction of Cu(l 10) induced by H adsorption (after Hayden et al., 1990). H atoms are shown as small solid circles in their probably binding positions, the trigonal sites exposed by the reconstruction.
by electronic-structure calculations of Fu and Ho (1989). In contrast, a local mechanism, in which the increased binding energy of the alkali adatom on the reconstructed surface tips the delicate balance (see w 13.4.1.3) was proposed by Jacobsen and Nerskov (1988). Recent experiment work, a scanning tunneling study of Cu( 110)-K by Schuster et al. ( 1991 ), reveals that even single isolated K adatoms induce the missing-row structure in small surrounding regions. This evidence strongly favors the local mechanism point-of-view. Hydrogen adsorption near monolayer coverage also produces the missing-row reconstruction of Cu(110) (see Hayden et al., 1990). Again, it would seem that the reconstruction is driven by the greater binding energy of the H-atoms in the trigonal sites that become exposed in the missing row reconstruction, as shown in Fig. 13.17 (see Christmann, 1988). Another, superficially similar, but actually quite different, reconstruction occurs on the Cu(110) surface, induced this time by the adsorption of oxygen at around 0 = 0.5. Again, rows are missing, but run in the orthogonal direction i.e. [ 100]. Ertl (1967) first deduced that the observed (2• periodicity was in fact due to a reconstruction of the surface, but again, recent scanning tunneling microscopy studies (see Coulman et al., 1990), were helpful in confirming the picture, which is shown in Fig. 13.18. In this case, it would appear that the reconstruction is driven by the unique propensity of Cu and O to form long, l-d strings. (See Jacobsen and NOrskov (1990) for a straightforward theoretical account of this behavior, which also occurs for O/Cu(001 ) according to Jensen et al. (1991 ).) These strings interact weakly with one another on the surface, ordering in the orthogonal direction to produce the reconstruction. The formation of this phase involves diffusion in an interesting way and is discussed in w 13.4.4. Pd(110), upon which the missing-row reconstruction can be formed via alkali adsorption, appears not to undergo this transition under hydrogen exposure. Instead, H-adsorption for 0 > 0.7 appears to cause a row-pairing reconstruction, in which adjacent, close-packed rows approach one another in pairs as shown in
762
L.D. Roelo.]:~
Fig. 13.18. Structure of the O-induced (2xl) reconstruction of Cu(110). The O atoms are denoted by the smaller solid circles; larger circles denote Cu atoms, successive darker shading indicates deeper layers of Cu. (a) top view. (b) side view, along rows. Positions of the O atoms are sketched roughly as determined by Robinson et al. (1990). Fig. 13.19. For Ni, which is immediately above Pd in the periodic table the situation is less clear. The Ni(110) surface also develops ( l x 2 ) periodicity under H-exposure, but the available experimental evidence, at this writing is insufficient to decide between the missing-row and pairing row candidate structures, although the weight of the evidence perhaps favors the latter. See Baumberger et al. (1986), for example, and Kellogg (1988) for the contrary view. A very different reconstruction category is exemplified by W ( 1 1 0 ) - H . Since tungsten is bcc, the latter surface features, long, flat-bottomed adsorption sites for H binding I as indicated schematically in Fig. 13.20a. Chung et al. (1986) suggest that if the binding well is truly flat-bottomed, then there should be an increase in H-binding energy if the entire top substrate_layer shifts uniformly in the [110] d i r e c t i o n - or the symmetrically-equivalent [110] direction, of c o u r s e - as shown in Fig. 13.20b. The driving force for the shift is that in the asymmetric well the energy of the lowest quantum state for a H-atom will be lower than in the flat-bottomed well where the first two levels are approximately degenerate. Thus we have a (H-coverage-dependent) ( l x l ) reconstruction, which manifests itself in LEED, as
1 The same hourglass adsorption hollows, but for metallic adsorbates figured in the discussion of asymmetrical phase diagrams in w 13.2.2.
Phase transitions and kinetics of ordering
763
Fig. 13.19. The H-induced, row-pairing reconstruction of Pd(110). (a) The clean surface, the large circles denoting Pd atoms, with second layer atoms shaded. (b) The reconstruction whose structure has been obtained by Rieder et al. (1983) and Demuth (1977). The smaller circles denote the binding sites for hydrogen. The inequivalent binding sites atop second layer atoms denoted by less darkly shaded small circles are opened up by the reconstruction, doubtless contributing thereby to the driving force of the reconstruction. observed by Chung et al. (1986), as an alteration in the point group symmetry of the diffraction pattern with onset at a critical coverage. Substitutional adsorption is also considered, technically at least, to be adsorbateinduced reconstruction, although one could perhaps also file it under surface alloy formation. For examp)_e K adsorption on AI(I 11) at room temperature results in a LEED pattern (,~-x~3)R30 symmetry and has recently found by Stampfl et al. (1992) to arise from substitution of the surface layer AI atoms with K. Other curious adsorbate-driven reconstructions abound. A rather noteworthy example occurs on the (001) surface of Ni and possibly Cu as well. This reconstruction has a pinwheel character and was first described by Onuferko et al. (1979) for the case of Ni( 100)-C. The structure was identified principally through its symmetry which is p4g, and is depicted in Fig. 13.21. This system, as well as N i ( 0 0 1 ) - O and Ni(001)-S, have been investigated by means of lattice dynamics by Rahman and Ibach (1985), who find that the observed reconstruction can be attributed, not to forces exerted directly by the C adlayer, but to an indirect effect, the weakening of the interactions between Ni atoms in the first and second layer of the crystal. Ying (1986) subsequently pointed out that C-Ni interactions may contribute significantly to the reconstruction - - thus allowing the assumption of a smaller reduction of the interlayer Ni force constants - - if one includes the interactions beyond first neighbor and if the adsorbate layer has (the observed) c(2x2) symme-
764
L.D. Roeh?]:~
Fig. 13.20. The H-induced top-layer-shift reconstruction of W(110) as determined by Chung et al. (1986). W atoms are denoted by the large circles, top-layer atoms are shaded and second-layer atoms are white. The H adatoms are the smaller black circles. (a) At 1/2 monolayer the adlayer order is (2xl) and the substrate unreconstructed. The plot on the right shows the binding potential for the H adatoms as a function of position within the long hollows. (b) At 3/4 monolayer the adlayer displays non-primitive (2• order, the top layer of the substrate is shifted relative to underlying layers and the adsorption well for the hydrogen is no longer symmetric and flat-bottomed. try. Although O and S do not induce reconstruction of Ni(001), they similarly alter the force constants of the surface and subsurface layers so as to soften vibrational modes of the same character (see Lehwald et al., 1985). Klink et al (1993) have recently been imaged this phase of N i ( 0 0 1 ) - C via scanning tunneling microscopy, confirmed the structure of Fig. 13.21, and found that the displacement magnitudes are of order 0.55 ]k. Chorkendorff and Rasmussen (1991) have suggested a similar interpretation of an observed H-induced reconstruction of Cu(001 ). The disordering transitions of these adsorbate-driven reconstruction systems have not yet been studied in detail or theoretically modeled.
13.4.2. Reconstruction of semiconductor surfaces Because of the directionality of covalent bonding, the surface reconstructions of semiconducting materials differ considerably in structural character from those typical in metallic systems. One finds large unit cells and often the involvement of several layers, so that the structural complexity can be considerable. The (7x7) reconstruction of the Si(111) surface is the most famous example with its unit cell consisting of 49 atomic sites in the surface layer and extending 4 layers into the surface, so that one has of order 200 atoms nontrivially displaced from bulk
Phase transitions and kinetics of ordering
765
Fig. 13.21. The C-induced reconstruction of Ni(001). Clean-surface positions of the first layer Ni atoms are the dashed circles and the reconstructed positions are the large solid, but unshaded circles. The C adatoms are denoted by the smaller shaded circles. termination positions. In fact, just solving this surface structure can be said to have preoccupied surface science for the 26 years from the discovery of the periodicity by Farnsworth et al. (1959), shortly after it became possible to prepare and maintain the clean surface, to the final solution of the structure by Takayanagi et al. (1985) using transmission electron diffraction. (Haneman (1987) provides a review of the study of silicon surfaces.) The structure of semiconductor surfaces is covered by Duke in Chapter 6 so that the present section will focus more briefly on what is known concerning the phase transitions of these surfaces. 13.4.2.1. ( I I I ) Surfaces
As noted above, the (111) surface of silicon exhibits a (7x7) reconstruction. Ge(111 ) reconstructs with c(2x8) periodicity; the local structure bears some resemblance to that of S i ( l l l ) 1. Both surfaces return to ( l x l ) periodicity upon sufficient heating, that of Si( 111 ) at Tc = 1130 K (see Bennett and Webb, 1981), and that of Ge(l I 1) at 573 K (see Phaneuf and Webb, 1985). In both cases the third Landau rule (see w 13.1.4.3) predicts a first-order transition unless the Q-vector of the phase moves continuously in the surface Brillouin zone while broadening. (No such m o v e m e n t has been reported; the beams are simply seen to decrease in intensity. See Bennett and Webb (1981), for example.) Nonetheless, the transition in the case of S i ( l l l ) seemed in L E E D measurements to be continuous, so much so that Bennett and
1 Althoughthe surfaces of the compound semiconductors, including GaAs for example, also reconstruct, they will not be covered in this chapter. Duke in Chapter 6 discusses their room temperature equilibrium structures in some detail and their transitions share many features with the elemental surfaces.
766
L.D. Roelo.f~
Webb (1981) measured effective critical exponents, while noting that their failure to observe beam broadening and/or critical scattering was inconsistent with the assumption of a continuous transition. The resolution of this apparent paradox came shortly after Bennett and Webb (1981) conducted their LEED study ~ and involved the effect of surface steps. Beautiful experiments by Osakabe et al. (1980) (grazing incidence electron microscopy) and Telieps and Bauer (1985) (low-energy electron microscopy or LEEM), established that step edges induce, or couple favorably to, the reconstruction, so that some (7• order remains on the surface above the Tc for a flat surface and then gradually decreases in extent as T is increased further. Thus the loss of order appears to be continuous unless there are no steps at all. The transition is also found to be reversible with nucleation of (7• order occurring at step edges and then growing across the surface as the temperature is decreased. Thus the transition is first-order, and Landau's third rule has apparently survived the test. The disordering of the G e ( l l l ) reconstruction is also first-order for similar reasons, but in this case, perhaps because of differences in the coupling between the ordered phase and surface steps, that was more immediately apparent in LEED. See Phaneuf and Webb (1985).
13.4.2.2. (001) Surfaces The (001) surfaces of the common semiconductors also reconstruct, most of them by forming rows of dimers on the surface. See Chapter 6 for a picture of the structures. The most significant contribution to the reconstruction energetics comes from the formation of the dimer rows. Because Si has the diamond crystal structure, the (001) surface is 2-fold, not 4-fold symmetric, and the dimer rows form only in one direction for a given surface plane. (For the next plane down or up, however, which occur if there are steps on the surface, the dimerized rows will run in the perpendicular directions so that most experimental studies see both directions.) Chapter 6 gives more of the structural details. Based on the fact that on a given terrace this (2• phase has two possible ground states, a single component order parameter analogous to the magnetization of the Ising model suffices to describe it in the infinite system. Therefore, if the transition is continuous, we expect it to proceed via an Ising-type transition with the associated critical exponents. (Of course any transition can also turn out to be first order according to the Landau rules; see w 13.1.4.3.) This system, however, contained several further surprises, which illustrate nicely how the complexities of real systems can undermine the usefulness of simple phase transition models. First, lower temperature studies have revealed that further ordering can occur from the (2• The literature on this point is complicated, Haneman (1987) offers a summary, but low temperature scanning-tunneling-microscopy studies (see Wolkow, 1992), establish that the lowest energy phase of the surface actually has c(4• symmetry. It seems that the dimers buckle, i.e. one atom of the dimer moves
i In fact, the observations of Osakabe et al. (1980), to be discussed following, are mentioned by Bennett and Webb in a 'Note added in proof'.
Phase transitions and kinetics of ordering
767
in the +z-direction and other in the-z-direction, and they interact with one another to order into a structure in which the dimer angle alternates. Saxena et al. (1985) analyzed the implications for the phase transitions of this reconstruction. Despite the seeming complexity of the unit cell of this reconstruction, there are again actually only two ground states, for a given choice of (2• ground state. Thus from the symmetry point of view, we can again expect an Ising transition between the c(4x2) and (2xl) phases. The (2xl) phase should then be considered as a disordered version of the c(4• phase with some of the dimer buckled in one orientation and the rest in the other, but with no long-range order in those orientations. The further disordering of the (already partially disordered) (2x 1) to a (1 x 1) surface would still be classified as an Ising transition and so one might wish to measure the critical exponents. However, there is still another complication. Recent discoveries indicate that flat Si(001) surfaces should not be taken for granted. It was noted above that on successive planes of the (001) surface the ( 2 x l ) reconstruction forms in perpendicular directions. Alerhand et al. (1988) first pointed out on theoretical grounds that steps on the surface, by allowing the reconstruction to occur in the two different directions, would work to relieve the anisotropic surface stress produced by the uniaxial reconstruction. Thus the surface if prepared without miscut would be expected to rearrange itself into a regular array of up and down steps with reconstruction in alternating directions on adjacent terraces. An elegant experimental study of the effect of uniaxial strain applied to surfaces of single crystal Si on the population of reconstruction directions by Men et al. (1988) confirmed the basic idea of the coupling between surface stress and direction of reconstruction. However, no study has succeeded in observing the generation of steps on previously flat surfaces. This failure has recently been overcome by simultaneous theoretical, Tersoff and Pehlke (1992), and experimental, Tromp and Reuter (1992), work showing that the stress may also be relieved, and with lesser kinetic hindrance than imposed via creation of new steps, on nearly flat surfaces through the development of long wavelength undulations in the few existing steps on the surface. What effect does this introduction of the third dimension have on reconstruction phase transitions? The 2-d models that lack the new degrees of freedom associated with 3-d become inadequate to the full description of equilibrium. However, it should be noted that the widths of terraces between the wavy steps seen by Tromp and Reuter are in the 1000-2000 A range. Thus one could still find ordering of the (2x l) over regions that large and perhaps one could still have rather sharp disordering transitions limited only by rather insignificant finite size effects. It seems quite certain that this reasoning would apply to the disordering of the c(4x2) phase into the (2x l) and temperatures well below room temperature. However, that reasoning may not apply for the disordering transition of the (2xl) phase, since that occurs at higher temperature where the character of the undulations may have changed, but more particularly because of the coupling between the step excitation and the direction of the (2x l) ordering. This last point deserves further experimental and theoretical investigation.
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L.D. Roeh?]:~
13.4.3. Adsorption effects on reconstruction Combining reconstruction and adsorption leads to interesting possible behaviors that go beyond those encompassed by the simple models that work well for these phenomena separately. One might then reasonably ask why one wants to study these more complicated situations. The answer is probably that adsorption represents one way of controllably perturbing a reconstruction system, an entree that can help determine the nature of complicated reconstructions. In other cases, one is more interested in the binding and reactive properties of adsorption on a given surface, and that surface just happens to reconstruct. (This seems to be commonly the case for adsorption on semiconducting surfaces.) From either viewpoint, many cases have now been investigated and we consider briefly in the following the effect of adsorption on reconstruction of some metallic and semiconducting surfaces. 13.4.3.1. Metallic surfaces The effect of adsorption on metallic reconstructions is highly dependent on the nature of the reconstruction. A few examples will serve to illustrate. A fairly complete theory for the effect of adsorption on the displacive systems of w 13.4.1.2 has been given by Roelofs et al. (1986). In these cases, the effect of adsorption upon the reconstruction is determined in large part by the binding site of the adsorbate, irrespective of the details of the interaction between the adsorbate and the nearby displaced substrate atoms. (The theory does not cover the case of long-range mechanisms as in the charge donation model for adsorbate-induced missing-row reconstructions, as described in w 13.4.1.4.) If the adsorption occurs into the atop site (labelled 'A' in Fig. 13.22) there is little effect on reconstruction in this case the principle interaction is with a single substrate atom. If the adatom binds in the centered site ('C' in Fig. 13.22), as for O or N on W(001), adsorption tends to oppose the reconstruction, since whatever the interaction between adsor-
Fig. 13.22. The effect of adsorption on the W(001) reconstruction. 'A' and 'C' denote sites for atop and centered adsorption respectively. 'B~' and 'B2' denote bridge sites rendered inequivalent by the reconstruction. Adsorption into either results in a rotation of the direction of displacements to lie along the axes of the surface as indicated by the two top-layer atoms adjacent to site B 1.
Phase transitions and kinetics of ordering
769
Fig. 13.23. Structure of 0.5 monolayer of H on W(001 ) as determined by Griffiths et al. (1981 ). The clean surface displacements (surface-layer W atoms are denoted by large unshaded circles) rotate around to be parallel to thex-axis, because of their attraction to the adsorbed H atoms (small black circles). The corresponding structure with displacements in the y-direction is degenerate.
bates is, it will in most cases favor equal displacements of the four nearest surfacelayer atoms, either toward, or away from the binding position. Since the reconstruction puts these four atoms at different spacing from the centered site, the two forces are bound to oppose one another. Thus, both O and N are observed to lift the W(001 ) reconstruction at relatively low coverages. For bridge-site binding, one has the opposite situation; reconstruction is enhanced by adsorption (references given in Roelofs et al. (1986)). Hydrogen is prototypical of this case and has been observed to have two important effects on the reconstruction: the reconstruction is enhanced, i.e. the disordering transition temperature is increased with increasing H c o v e r a g e - the full phase diagram is given by Barker and Estrup (1981); and the direction of the displacements is altered to be along the axes as shown in Fig. 13.23 (see Griffiths et al., 1981 ). This occurs because in bridge site adsorption the adatom interacts most significantly with two substrate atoms. Whether that interaction results in a net attractive force component in the surface plane (adsorption site B~ in Fig. 13.22) or a net repulsive force ~ (site B2) the reconstruction is favored by adsorption. Since alkali adsorption can induce the missing row reconstruction on some otherwise unreconstructed fcc(110) noble metals, it is interesting to consider their effect on those surfaces that spontaneously reconstruction. H~iberle et al. (1989) have studied the effect of K or Cs adsorption on the (1• reconstructed Au(110) surface. They find that adsorption at low coverage alters the reconstruction to (1• with 3 rows missing, one in the second layer, between successive remaining first-layer rows. At higher coverages alkali adsorption lifts the multilayer reconstruction. H~iberle, and Gustafsson (1988), for example, have investigated 1/2 monolayer of K on
1 A net repulsive force component in the surface plane acting on the neighboring surface layer atoms is possible if there is significant interaction between the adatom and the second layer substrate atoms beneath the bridge site.
770
L.D. Roeh?]:~
Au(110) using medium energy ion scattering (MEIS) and find that the equilibrium structure, which displays c(2x2) symmetry, consists of a flat (110) surface whose top layer is a K/Au alloy of c(2x2) periodicity I. The explanation for the change in behavior is that because they have donated charge to the surface, the K adatoms, which adsorb in the missing-row troughs at low coverage, repel one another strongly. When they are forced by increasing coverage into close proximity, it is advantageous for the structure to be rearranged such that Au atoms screen the K adatoms from one another; hence the alloy top-layer. The structure seen in MEIS is consistent with a first principles total energy calculation by Ho et al. (1989). Effective medium theory (see Jacobsen and NCrskov, 1988) appears to incorrectly favor a differing structure, a c(2x2) K layer atop a flat Au(110) surface. 13.4.3.2. Semiconductor surfaces Semiconducting surfaces are more reactive than most metals because of the localized and dangling orbitals (broken bonds) found in their surfaces. Therefore in some cases adsorption of say 02 on Si surfaces will lead not to subtle influences on the reconstructions, but rather to the growth of reacted areas on the surfaces, in this case forming a thin film of SIO2. Rather large doses of H2, on the other hand, are required to affect the reconstructions of the (001) and (111) surfaces, the former being eventually undone at a coverage of 0 = 2, while the (7x7) reconstruction retains its periodicity up to saturation coverage. (See Haneman (1988) for references.) The adsorption of metals on semiconducting surfaces often leads to rather complicated phase diagrams in which further study is needed to determine whether the observed order is due to adsorption or reconstruction degrees of freedom or both. See for example the rather complete study of Si(l I l ) - A u by Feidenhans'! et al. (1990) and the more recent report of further complexities of Shibata et al. (1992). For several other metallic adsorbates on the (7x7), e.g. Pb (see Ganz et al. (1991) and Ag (Ding et al. (1991) list experimental references and give a theoretical account ), one finds ('4-5x4-5-)R30 order at coverages around 1/3 monolayer. This phase may coexist with the (7x7) reconstruction elsewhere on the surface as occurs for example in the system Si(ll l ) - G a (see Zegenhagen et al., 1989). Likewise adsorption of Ag on G e ( l l l ) , at least at elevated temperatures, gives _ p h a s e diagram with coexistence between its reconstructed phase and a (',/-3-x'43)R30 overlayer. See Busch and Henzler (1990). In these cases the (7x7) reconstruction is lifted by the adlayer and the q3- order is primarily in the adlayer. In other cases metallic adsorption, rather than undoing the reconstruction, induces one of completely different character from that of the clean surface. In the system S(111)-Na, for example, a 3• pattern is seen to coexist with the (7x7) reconstruction and was found via tunneling microscopy by Jeon et al. (1992) to be of missing row character. The only generalization to make is that generalization is difficult, although there are similarities (noted by Dev et al., 1988) between metallic adsorbates in the same families on the periodic table.
1 The K atoms being larger protrude further from the surface than the Au atoms in the topmost plane.
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13.4.4. Diffusion in reconstruction systems
The necessity of long-range mass motion in nondisplacive reconstruction has resulted in continuing interest in the problem of self-diffusion on reconstruction surfaces. Consider, for example, the (2xl) "missing-row" phase induced on Cu(110) by a half monolayer of oxygen (see w 13.4.1.4), whose reconstruction mechanism was elucidated by Coulman et al. (1990) using scanning tunneling microscopy. When oxygen is added to clean C u ( l l 0 ) the phase actually forms by means of 'evaporation' of Cu atoms from terrace edges. These atoms then diffuse across the surface until they, together with the oxygen adatoms, find a growing "island" of the (2x l) phase and form the rows of which the structure consists. (Thus Coulman et al. (1990) correctly rename the reconstruction to be of the "added-row" type.) Other cases where diffusion plays a critical role in surface reconstruction will be discussed below. The study of diffusion on surfaces has a long history ~ and full coverage would be out-of-place in this context. However, the reader interested in exploring the fundamental bases of the field is referred to the classical treatment of stochastic problems by Chandrasekhar (1943), to Bonzel (1975) who introduces the application to surfaces and to Banavar et al. (1981), who present a full theoretical account. Atomic-scale investigations of diffusion have historically been most readily conducted via Field Ion Microscopy (see Chapter 8); Ehrlich (1994) gives a useful history and summary. For the clean missing row reconstruction systems, gradual ordering is observed from a variety of starting situations at or slightly above room temperature; see, for example, Ferrer and Bonzel (1982). To investigate this perhaps surprising degree of mobility, the same total cohesive energy methods which were used by, for example Garofalo et al. (1987) and Roelofs et al. (1990), to elucidate the driving force of the transition and reproduce the phase transition behavior, can be extended to investigate the energy barriers for simple diffusive moves of Au atoms on otherwise perfect Au(1 10) surfaces, both flat and reconstructed. Figure 13.24 shows examples of atomic moves that might be significant on a flat surface. Note that some of the 'moves' are c o n c e r t e d 2 and involve both the Au 'adatom' and an atom within the surface layer. To investigate diffusion barriers one calculates the total cohesive energy as the atoms are moved in small steps along a lowest-energy pathway (determined by minimizing the energy, allowing all other atomic positions to relax in response to the moving adatom). Not surprisingly, one finds, using the Embedded Atom Method (see Daw, 1989) to model the total cohesive energy, that
I Properlygoing back all the way to Einstein's theory of Brownian motion. 2 Concertedmovementsare significant in other surface diffusion situations. Feibelman(1990) determined, via total electronic energy calculations, that for AI(001)-AI concerted movements involve an energetic barrier about 1/3 as high as simple single atom moves. Experimental support for the significance of concerted movements is also available in some systems. Tsong and Chen (1991) present evidence for Ir(001)-Ir and Ir(ll0)-lr; Kellogg (1992) shows that concerted moves are also important in the non-homogeneous situation of Pt adatoms on various surfaces of Ni.
772
L.D. Roelo.[:~
Fig. 13.24. Some basic diffusion moves on an fcc(l 10) surface. The moves on the right side of the figure are direct; those on the left are concerted.
single atom diffusion along the rows is relatively free; the barrier height or activation energy is Eact = 0.27 eV. (These results are reported in Roelofs et al. (1991).) For single atom diffusion across the row, on the other hand, one finds Eact = 1.16 eV, a barrier that would imply very slow diffusion in that direction around room temperature (kTroom---0.025 eV). Consideration of the concerted mode of movement across the rows lowers the barrier to Eac t - - 0 . 3 5 eW providing a pathway which circumvents the direct limitation and allows for rapid motion in that direction as well. In light of these calculations, the spontaneous development of good reconstructed order at temperatures around 400 K is not surprising. One expects that similar situations obtain in the other systems that reconstruct in this fashion. These results may also provide some insight into homogeneous diffusion in nonreconstructing surfaces as well, where the process must play an important role in the annealing process, etc. Diffusion also plays a key role in other surface kinetic phenomena which are the subject of the next section.
13.5. Ordering kinetics at surfaces This chapter has presented a summary of equilibrium phase transition behavior at surfaces, but has not yet considered the time development of order, a nonequilibrium phenomena. In this section we present the theoretical picture that has been developed using approximate analytic approaches and the confirmation of those ideas via simulation studies of lattice gas models and experimental investigation of chemisorption systems. It would also be well to note that kinetic phenomena pervade surface science and in their breadth deserve eventually more complete treatment, perhaps a full volume
Phase transitions and kinetics of ordering
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later in this series. Because they are not intrinsically related to phase ordering, I have chosen not to cover some important aspects of surface kinetics including: adsorption and desorption kinetics, diffusion (but see w 13.4.4), reaction kinetics and oscillation, and the kinetics of epitaxial growth.
13.5.1. Theoretical introduction We consider the following situation. A system I possessing at least two distinct phases is allowed to come to equilibrium in one of its two phases. Then we abruptly alter the external conditions to be consistent with the other (or another) phase and observe the time development of order in the new phase. We expect the system, initially far from the new equilibrium to move toward it, but how do we characterize this development, how is it initiated, at what rate does it proceed, what universal patterns of development can be identified and what key variables determine which pattern is followed? We will pursue these questions in the following subsections, illustrating with some recent simulation results 2 for a particular model system and verifying the expected results with experimental and other simulation-based studies in w 13.5.2. Our illustrative system is a 2-d square lattice gas model whose interactions are suitable for the formation of a p(2x2) phase: El --->+oo (nearest-neighbor exclusion); E2 = 100 K; E~ = - 3 0 0 K; and E4 = 50 K. (All energies are given in the form of temperature equivalents. These interaction energies were not chosen to accurately model any particular system, but the resulting phase diagram is similar to that of Ni(001)-Se. See Bak et al. (1985).) The (T,~t) and (T,0) phase diagrams of this model are shown in Fig. 13.25. In later subsections we will discuss the development of p(2x2) order in this model under two different sets of conditions meant to simulate the typical experimental situation in, respectively, physisorption systems and chemisorption systems (see w 13.2.3). In both cases one begins with the system in equilibrium under one set of conditions, and then effectively instantaneously alters one system parameter or another in order to establish an initial out-of-equilibrium state. The experiment commences at that time, and one monitors the approach of the system to equilibrium at the new conditions. Specifically, for a system of physisorption character, imagine beginning with a clean substrate being held at a temperature at which p(2• order can be stable, and introducing to the chamber at time t = 0 the adsorbate, in gaseous form, at constant pressure which serves to fix the chemical potential ~t, at a value conducive to p(2x2) order. Thus one is instantaneously changing the experimental condition as shown via the bold arrow in Fig. 13.25a and one can follow the subsequent development
1 We assume the system is very large and discuss in w 13.5.2 the effects of size limitations. 2 The illustrative calculations -- both equilibrium (the phase diagram) and kinetic - - included in this section are original research doneby Jennifer Blue and the author, and have not been published elsewhere. Support via a CUR (Council on Undergraduate Research) Undergraduate Fellowship is gratefully acknowledged.
L.D. Roelof,;
774
400 disorder
C(2X2)
300 O0
9
[-200 p(2x2)
0
100
dv
_
0
Orv
-800
-600
-400
-200
0
l.t [K]
(a) 400
300
200-~
j ~v i ,,I ,2x,,i i i.. i i !
100
0
I
.~
0.0
0'.1
0.2
0.3
0.4
0.5
(b) Fig. 13.25. Phase diagrams for the lattice gas model used to simulate the kinetics of p(2• phase development, as determined by Monte Carlo simulation on a (48:<48) lattice with periodic boundary conditions. (a) (T,l.t) phase diagram. Solid dots denote continuous phase transitions; open dots, points on first-order boundaries. At the latter 0 is discontinuous and so these points open into coexistence regions as seen in (b) (T,0) phase diagram. The plotted points were obtained via simulation. The curves connecting them were drawn in to guide the eye; they have no further significance. Both diagrams also show via the bold lines (with arrows indicating the direction of the change) the discontinuous changes of condition after which the development of order is followed.
of the e x p e c t e d p(2• order. Or, for the c h e m i s o r p t i o n case, one can c o n s i d e r an abrupt c h a n g e in t e m p e r a t u r e while the c o v e r a g e 0 is fixed. To illustrate that situation in the p ( 2 • model system w 13.5.3 will present our study of e q u i l i b r a t i o n f o l l o w i n g a j u m p from a purely disordered s y s t e m with r a n d o m o c c u p a t i o n of sites
Phase transitionsand kinetics af ordering
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into a region where one expects an ordered phase to develop. Specifically we initiated the simulation runs by randomly occupying sites up to a coverage 0 = 0.10 monolayers. (This corresponds to the T ---) ,,o limit at this coverage.) We run the simulation program, however, at T = 100 K, thus simulating instantaneous cooling, as denoted by the bold arrow in Fig. 13.25b, of the system to a temperature at which clusters of the p(2x2) phase would be expected to develop I. The simulation models the hopping movement of the adatoms (not allowing any to leave the surface) and monitors the growth of islands of p(2x2).
13.5.1.1. General framework It is natural to suppose that the new phase begins to develop locally at small length scales and simultaneously at many points on the sample. Where and exactly when is unpredictable because of the randomness inherent in large statistical systems. The initiation of the development of order in a new phase is termed nucleation, and this phenomena, though not the central issue of this section, is interesting in itself and will be discussed briefly w 13.5.1.2 following. After the appearance of many small nuclei, each grows larger and we can identify domains of the developing order. This pattern of development of order is referred to in the surface literature as nucleation and growth. It is natural to attempt quantitative characterization by defining a time dependent domain size distribution, P(R,t), where R is a linear measure of the size of a domain. (One practical definition for R is where ~/A is the area of a developing domain.) One then hopes to obtain experimental measurements of and theories predicting the nature of P. From P can be obtained a measure of the average or typical domain size r(t), the basic quantity needed for theorizing. A moment's consideration, however, suggests that P itself is not the most experimentally convenient quantity, since the only practical way of characterizing the degree of order in a large system is with diffraction, which is sensitive to order and averages over a macroscopic area of a surface. The width of diffraction beams reflects the size of the coherent scattering regions producing them (see Chapter 7), so that the structure factor defined in Eq. (13.26) and its moments
~lQIml(Q)d2a Q,,, -
~ I(Q)d2Q
(13.46)
provide another characterization of the extent and range of the developing order. (The integrals in Eq. (13.46) are taken just over a conveniently chosen region surrounding the Q-vector Qo which characterizes the development of order.) A useful estimate of the average size of the ordered regions is
1 The upquench (see w13.5.2) is a popularexperimental alternative. In that procedurethe initial adsorption takes place at sufficiently low temperature that the adatoms are immobile and hence remain in their random arrival sites. At t = 0 the system is heated as rapidly as achievable to the temperature at which the kinetic behavior is to be investigated. This approach gives the same initial state as the T ---)oo limit of the simulation
776
L.D. Roelof~
1
r(t) = xiQ2
(13.47)
It is necessary to pause at this point for a brief notational aside. "t" has been used in the above to represent time and that notation will be used throughout this section. Note, h o w e v e r that "t" has also been used in w 13.3 to denote the reduced temperature (see Eq. (13.16). The notation conventions in the literature are sufficiently strict that the author has elected, with apologies, dual use of that character, distinguishing only via use of a script font for time. So, let us assume we have a domain of the new phase, a p(2x2) cluster in our example, of size r growing in a region that is otherwise in the form of the old phase, a dilute gas of adatoms in our example. Following nucleation we can expect exponential increase in r as can be seen as follows. The new phase has lower free energy per unit area than the old phase by some amount determined by the details of the system and the specific way the external parameters were changed. Simple rate assumptions suggest that the rate of change in any quantity is determined by the dependence of the free energy, G, on that quantity so that, for example 8r
8t
8G - -c'--
8r
(13.48)
where C' is some constant determined by microscopic rates of diffusion or adsorption. (The negative sign indicates that the development is in the direction of decreasing G.) For the growing domain G = - C " A where A is the area and C" is a constant related to free energy difference between the new and old phases. Relating A to r gives 8G 6r
-
C'"r
(13.49)
where C ' " is another constant. Combining Eqs. (13.48) and (13.49) in terms of yet one more positive constant, C, gives ~r - - = C r ~ r(e) = r,,ec' 6t
(13.50)
which establishes the result, an explosive development of order. Obviously, for many systems, including our p(2x2) example, the rapid rate of growth suggested by Eq. (13.50) cannot be sustained indefinitely, because different growing domains will soon come into contact, eliminating the free energy difference that had been driving the development of order. What then transpires is a bit more subtle. Growth continues and in this regime is called late-time growth. A simple argument, originally due to Lifshitz (1962) establishes that the rate of increase of r is now controlled by the local curvature, ~c, of the boundaries between domains. In the lattice gas context this notion can be understood as follows. Then adatoms at the edge of a cluster are the least well bound on average as they have lower coordination that their fellows in the cluster interior.
Phase transitions and kinetics of ordering
777
Fig. 13.26. Lattice-gas depiction of the Lifshitz mechanism by which clusters with the larger radius o1" curvature tend to grow at the expense of smaller ones. The dots represent adsorption sites on a square lattice and the large circles adatoms in regions of local p(2• order, solid atoms are in one sublattice and the shaded in another, that is surrounded by the larger cluster. The larger cluster will tend to grow because it has more attractive binding locations in the boundary (5 solid smaller circles) than does the smaller cluster, which has 4 (denoted by the smaller shaded circles). Thus, random moves by atoms across the boundary will result in net growth of the surrounding cluster over time at the expense of the enclosed cluster.
The larger the c u r v a t u r e , the more w e a k l y b o u n d a t o m s there are per unit length of the p e r i p h e r y . C o n s i d e r Fig. 13.26 which s h o w s two d o m a i n s one larger than the o t h e r and t e n d i n g to e n c l o s e it. Note that b e c a u s e of the c u r v a t u r e of the b o u n d a r y b e t w e e n the d o m a i n s there are m o r e f a v o r a b l e binding sites for a particle to join the larger d o m a i n than the smaller, and so the f o r m e r g r o w s at the e x p e n s e of the latter. F o l l o w i n g Lifshitz we can m a k e this a r g u m e n t quantitative. T h e a b o v e r e a s o n i n g e s t a b l i s h e s that the d e p e n d e n c e of the free e n e r g y on the size of the cluster is linear in the a v e r a g e curvature, ~: 5G
- C"K:
(13.51)
5r with C " again s o m e positive constant d e p e n d e n t on the details of the situation. N e x t we use Eq. (13.48) again, and relate K: to the d o m a i n size via K: = C'/r and find that 5r 5t
-
C r
(13.52)
w h e r e C is a constant. T h e solution to Eq. (13.52) is readily seen to be
r(t) ~: ~]t
(13.53)
so that g r o w t h in this r e g i m e while slower, c o n t i n u e s . T h e s e a r g u m e n t s were later f o r m u l a t e d in f i e l d - t h e o r e t i c form by Allen and C a h n (1979) with the s a m e final
778
L.D. RoelR]:,~
result so that this pattern has come to be known as the Lifshitz-Allen-Cahn (LAC) growth law. This basic result is thought to be generally applicable in 2-d systems, except when the growth of the phase must occur at fixed values of the order parameter, which matter will be taken up below in w 13.5.1.4. In the meantime we note that Eq. (13.53) has been broadly verified via simulation and experiment (some of this evidence will be summarized in w 13.5.2) and turn first to a brief discussion of nucleation, the process that initiates phase development.
13.5.1.2. Nucleation Conditions may occur under which the equilibrium phase will grow if established above some critical size, r c, but will 'evaporate' if smaller. The mechanism underlying this behavior is that edge adatoms are less tightly bound than surrounded adatoms and so are more likely to desorb; a small cluster has relatively more edge atoms and so is less stable. The development (nucleation) of a sufficiently large nucleus for the new phase eventually occurs as a result of fluctuations. This process may occur for example in physisorption systems and can be illustrated using the p(2x2) lattice gas model described in w 13.5.1. Our simulation thereof models the adsorption to and desorption from of adatoms the surface using the simplest assumptions concerning the dynamics. (Relative sticking probability and desorption likelihood are determined by the local binding energy of an adatom at the site, that being determined by the lattice gas interaction energies given in w 13.5.1.) This might be an appropriate scenario for physisorption systems. While simulating the time development of the system the program continuously monitors the size of all the adsorbed clusters. Figure 13.27 plots the percentage G of p(2x2) clusters ~ that eventually grow to fill the (48x48 with periodic boundaries) lattice as a function of their size N. The data was obtained by averaging over 12 independent runs, all of which eventually concluded upon reaching full p(2x2) order. One concludes from these results that if one defines the critical nucleus to be that associated with 50% probability of growth to phase completion, that at T = 100 K 2, Nc -- 20 adatoms. The clusters are relatively compact so that this translates to a linear size of approximately rc -- 4 or 5 lattice constants. That nucleation is strongly temperature dependent is also evident from Fig. 13.27, which also gives corresponding results at T = 90 K where Nc is reduced to about 15 adatoms. Quantitative variations within this behavior will clearly occur depending on system details. These results are therefore more useful as illustrative of lattice gas nucleation phenomena than as predictive for any particular system. We return next to the later-time development of order.
1 A cluster was defined via E 3 connectivity and clusters of sizes varying in size from 3 adatoms to 50 adatoms were monitored by the program. Plotting the island size distribution vs. time allowed easy assessment of whether a given island grew to fill the system or not, thus enabling us to obtain the histogram in Fig. 13.25. 2 Thisbehavior probably scales with the maximumtemperature at which the phase is stable, 200 K in this case.
Phase transitions and kinetics of ordering
779
100
-~ v-~
m, a m B r o s i a
~a
00
45
o []
4)
A
0
v
(3
O0
0
4O
$
"--~] 0
o0 j 10
9 I 20
9
310
9
i 40
'
u 50
9
60
Fig. 13.27. Nucleation in the fixed-~ (physisorption) case for the model. Open squares (filled diamonds) represent the percentage of clusters of size N atoms that eventually grew to full p(2• order at T = 100 K (T- 90 K) and ~t =-550 K.
13.5.1.3. Scaling It can be hypothesized that in the late-time regime there is really only one important length scale, that being the typical domain size, r(t). If this is the case then we should expect to be able to relate the domain size distributions, or equivalently the structure factors, at different times by a simple rescaling of the variables, viz. P(R,t) - P(x)/r(t)
(13.54)
where
x - R/r(t)
(~3.55)
or ..,.,
I(Q,t) = ly/Q2(t)
(13.56)
where
y - IQI{Q2(t)
(13.57)
Note that in each case we have scaled the relevant coordinate by the dimensionally consistent measure of the typical domain size, and defined a scaling function in terms of that variable, specifically, P(x) and l(y). The additional dividing factors in Eqs.(13.54) and (13.56) are needed essentially for normalization purposes. In Eq. (13.54) for example, as r increases, there are fewer domains in proportion to be counted in the distribution. Dividing by r(r takes account of this trivial correction. This scaling idea, like all others, despite its obvious appeal, must be tested via experiment and simulation before being accepted. Strong evidence of both sorts will be presented in w 13.5.2. Thus Eqs. (13.54-57) can be said to represent another important universal aspect of domain kinetics.
L.D. Roelof~
780
13.5.1.4. Conservation conditions It was noted in w 13.5.1.1 that the simple Lishitz-Allen-Cahn growth law of Eq. (13.53) does not hold under conditions of fixed order parameter, i.e. situations in which the microscopic processes leading to order conserve the global value of the order parameter. An example will help clarify this idea. Suppose that one has a lattice gas with attractive interactions leading to a (1• condensation phase diagram like that of Fig. 13.3, and suppose further that the one is in the chemisorption limit so that the coverage is fixed. Let the specific experiment be a quench from the disordered high temperature state into the coexistence region. Equilibration in this case will require the development of a single large cluster containing almost all of the adatoms, while the remainder of the surface will be sparsely occupied forming the dilute gas phase. But note, however, that for the system to order the adatoms in clusters must diffuse to the growing cluster. The order parameter in this case is the coverage and the basic movement by which the system moves toward equilibrium is the movement of an adatom from one site to another, which obviously then conserves the value of the order parameter. The treatment of this case was first provided by Lifshitz and Slyozov (1961), who established that the requirement of diffusion slows the late-stage growth to r(t) ~ fl/3
(13.58)
It is important to emphasize that constrained coverage is not the same thing as a conserved order parameter in most cases; the coverage is the order parameter only in the case of the ( l x l ) phase.
13.5.2. Results on late-time ordering from experiment and simulation As noted in w 13.5.1, there is strong evidence for the Lifshitz-Allen-Cahn growth law, Eq. (13.53) and for scaling; we present a sampling in this section. Beginning with the growth law, consider Fig. 13.28 (their Fig. 3) from Wang et al. (1991) which displays the ordering behavior of a (2x l) oxygen overlayer on W(112). In this experiment the peak intensity of the extra spots was used to probe the growth of the order; it is a measure of r(t) 2. (The assumption that the peak intensity measures r 2 requires that the intensity diffracted by the largest (2x l) islands is not resolved by the instrument.) The exponent obtained by the fitting lines to the log-log plots is 2n - 1.01 +_0.02. The exponent is independent of temperature and consistent with Eq. (13.53). It should be noted that Wang et al. use the novel technique of 'up-quenching' mentioned in w 13.5. I. Instead of equilibrating to a high-temperature disordered phase and then rapidly cooling the sample into the ordered phase, they adsorb the oxygen at low temperatures (<200 K) where it forms an immobile, disordered layer. The sample is then rapidly heated to a higher temperature at which the O adatoms become mobile and the development of order is monitored. Late-time simulation results show that other measures of order display the same growth behavior. Figure 13.29 (their Fig. 2) is from a study by Fogelby and
781
Phase transitions and kinetics o.[ ordering 100! W(112)(2x 1 )-O -m z iii I-1 z 10 9 v ,< iii n Q Lu N 2 10- " ,<
(1/2 O) BEAM O=0.5ML
/
rr
o z
& A
523K 503K 9 493K 9 478K + 468K
/
/
/
10+3 . . . . . 10 0
, 10 1
,
9
l
9 llllll
I
9
l
I 9 Ill|
10 2
9
9
9
9 I l ll
10 4
10 3
T I M E (sec.)
Fig. 13.28. Experimental results for the development of ( 2 x l ) o r d e r for W(112)-O from Wang et al. (1991). The solid lines are power law fits with exponents 1.01+0.02.
8:~/+21
1000
I
I
4- + 4. * ~.P `4.44.4#
+ 4.
1
++"+
4.+*'4-+++
1OO 4- 4- 4- 4-
~/X.E-
4 . + * # + + + + J
+ + + + +il 4-4-+4.4-1~ 4-
+-
4. 4+-
o
,1 +- ' t
+-
4-
+
4-+ k ' 2 l t ) ; T = O
4-'4-4.
10
1
++,t,
4-
+
( t ) ; T = 0.24
+. 4-
~. ~"
4- 4-
4- 4-+
L(t)
;T=O
4- 4 4 - 4 + + +
11
4-
lO
4-
4.
+
4.
1
100
1000
t
Fig. 13.29. Lifshitz-Allen-Cahn convergence in the simulation by Fogedby and Mouritsen (1988) of convergence to a (2xl) ordered lattice gas phase. All the calculated properties show the same power law behavior. AE(t) is the excess energy (relative to the final equilibrium value), k2(t) is the second moment of the structure factor (defined identically to Q2 in Eq. (13.46) and L(t) is the square root of the scattering intensity at the center of the peak. Temperatures are given in units of 1: - T/T~ where T,: is the ordering temperature of the (2xl) phase.
782
L.D. Roelof~
Mouritsen ( 1 9 8 8 ) o f a lattice gas model which orders into a (2• phase on a square lattice. The quantities plotted, all of which show time development consistent with Eq. (13.53) include parameters related to the energy of the system and its degree of order. (See caption for definitions.)
13.5.2.1. Verification of scaling in experiment and theory The notion of scaling, introduced in w 13.5.1.3, has also been amply demonstrated via experiment and computer simulation. Representative results from the same sources referred to in the preceding subsection are given here. First, for the experimental situation, see Fig. 13.30 which displays the scaled and normalized scattered intensity for the W ( 1 1 2 ) - O system, as obtained by Wang et al. (1991). In this plot the diffracted intensities at different times, scaled by their peak intensities, are plotted versus k/k, where k denotes the full width of that profile at half maximum, k-I functions as another measure of the domain size in the manner of Eq. (13.47), so this plot constitutes a verification of the relation Eq. (13.56). On the simulation side consider Fig. 13.31 from Fogelby and Mouritsen (1988), for the same system as gave rise to Fig. 13.29. The F 2 referred to in that figure is defined identically to the of Eq. (13.56) and their x is identical to the y defined in Eq. (13.57). The matching of the curves at different times is impressive. The figure also displays a Iogmlog plot of the same function, establishing that the tail of F 2, i.e. /, which probes the short-range structure of the growing domains, has a power-law fall-off of the form m
1.2 ra
(b) 1.0
A !
[11T1 SCAN
0.8
10 sec.
9 70 sec. 9
130 sec.
9 210 sec. 9 480 sec.
0.6
+
2235 sec.
x
H.T. annl.
0.4
0.0
_
-3
~
-2
m.
-1
0
1
2
3
k/k
Fig. 13.30. Scaling in the growth o1" (2• order ofW(112)-O. For t < 240 s (after which the growth saturates due to finite size limitations) the normalized diffraction profiles can be superimposed when
plotted vs. the scaled wavevector.
Phase transitions and kinetics r
783
20
-1
0
I
!
I~
x
1 I
-1
=,
F2(x) | 15 t
log lO F2 (x) -0
i=,
--1 101 .
-2 -3 --4
0
1
2
x
3
Fig. 13.31. Scaling of diffraction profiles in a simulation study. (The system is that described in Fig. 13.28.) See text for definitions of quantities plotted.
F2(x) -- x -~
(13.59)
with o3 = 3. ~ 13.5.2.2. Finite-size effects and other limitations on experiment and simulation Both s i m u l a t i o n - b a s e d and e x p e r i m e n t a l attempts to study surface kinetic b e h a v i o r are adversely affected ( s o m e w h a t differently, of course) by r e a l - w o r l d non-idealizations. This section briefly outlines the most significant of these limitations. U n d e r the heading of finite-size effects, one notes that real surfaces are neither infinite in extent nor of perfect quality, and likewise that c o m p u t e r m e m o r y allows storage only of finite-size site lattices and that s i m u l a t i o n - r u n lengths scale like the n u m b e r of lattice sites. In the case of simulation, one o b v i o u s l y cannot follow the o r d e r i n g b e y o n d the size of the region being simulated. One hopes and a t t e m p t s to test w h e t h e r the size is adequate to have reached the a s y m p t o t i c time d e p e n d e n c e . In e x p e r i m e n t s , real system size may be limited by steps on the surface or other defects. The saturation of g r o w t h e v i d e n t in Fig. 13.28 at large times is likely due to such size limitations. (Note that in the case of e x p e r i m e n t , one has the additional difficulty, that such limitations are m o r e difficult to accurately c h a r a c t e r i z e than in a s i m u l a t i o n w h e r e they are p r o g r a m m e d in and thus k n o w n exactly.) On the other
1 Theoretically one expects co = d+l where d is the spatial dimension. The analysis that gave this result is due to Ohta, Jasnow and Kawasaki (1982), and is rather involved. The starting point is an equation of motion for the movement of interfaces or domain walls between regions of order in differing phases. The velocity is taken to be proportional to the curvature of the domain wall in consistency with the physics portrayed in Fig. 13.26.
784
L.D. Roelof,~
hand, in experiment one can generally achieve better statistical quality of results, since a diffraction study of a macroscopic surface implicitly averages the ordering behavior over a large number of hopefully equivalent regions. Simulators must laboriously repeat their runs, averaging until the statistical fluctuations are sufficiently reduced. Surface point defects which couple differentially to the phases being examined will not only play havoc with nucleation, but also may ultimately influence the development of order in a region of size determined by a complex tradeoff between its preference for one phase versus the forces driving the ordering, and thus contribute another sort of finite-size limitation. One could attempt to characterize these effects by intentionally varying the density of such defects/impurities and looking for resultant influence on the ordering. The author is unaware of any such studies, the understandable tendency being to simply try to obtain the best possible surface and study only it. Another important sort of limitation is the fact that the local dynamics may significantly affect kinetic phase development. The LAC theory is not based on any particular assumptions concerning the local movements of adatoms underlying the development of order, except that it is assumed that the local dynamics do not conserve the order parameter (see w 13.5.1.4). One might think that in most cases, it would be fairly simple to determine whether hypothesized dynamics, e.g. shortrange hops to unoccupied sites, do or do not conserve the value of the order parameter. However, the matter is more subtle than that. Fogedby and Mouritsen (1988), for example, note that restricting the atoms to hopping between NN sites results in a saturation of growth in the development of (2xl) order on square lattices, while allowing NNN hopping as well allows growth to continue consistent with the expected growth law, Eq. (13.53). For most experimental systems one does not know what rules govern the local movements of atoms, although for this and for many other reasons one would very much like to. Even if one does have some information about the local dynamics, resulting limitations, which depend on the preferred local structure of domain walls, are very difficult to anticipate. Perhaps the only way to deal with this problem is by combining experiment and simulation, using the latter to correlate observed patterns of development with particular assumptions concerning the local dynamics. 13.5.3. Ordering in coexistence regions
Chemisorbed layers obey dynamics in which the coverage is fixed ~allowing access to regions of multiple phase coexistence. In the lattice gas model introduced in w 13.5.1, for example, there are regions in the (T,0) phase diagram (Fig. 13.25b) where the p(2• phase coexists with a dilute disordered phase and with the c(2• phase. It is not surprising that phase development occurs differently in such regions,
I Since once an atom binds with covalent strength to the surface, it does not leave during typical experimental durations
Phase transitions and kinetics of ordering
785
due to the need for more than one phase to develop and for the phases to dissociate spatially in order to attain a final situation satisfying the lever rule. As an illustration of these differences we present in this section model calculations for phase development in the p(2x2)-disorder coexistence region in Fig. 13.25b. Due to the coverage constraint, the behavior found in this setting is quite different from the nucleation and growth scenario of w 13.5.1.2. As the adatoms hop about the surface one sees first the rapid development of many small islands of 3 or 4 adatoms. The subsequent dynamics, however, are not those of random growth or decrease with a definable critical nucleus size. Rather, the larger islands gradually grow at the expense of the smaller ones, a survival o f the largest dynamic, until there is only one island left on the lattice 1. 9 The mechanism underlying this development can be simply understood as follows. Imagine a single p(2x2) cluster of adatoms in equilibrium with a low-density gas phase at the temperature at which the study was done (100 K) in a finite-size system (48x48 sites in our case). The associated equilibrium 'gas' coverage, 0g, will depend on the cluster size N, because a smaller cluster will have relatively more less-well-bound adatoms on its periphery than will a larger cluster. Since the number of 'corners' on the cluster should scale with the number of 'edge' atoms like I /N[N 0g ~ 1/,~-
(13.59)
at least for N not too small. This hypothesis can also be tested by simulation, by running to equilibrium with a particular total number, N t, o f adatoms present and then monitoring the size of the resulting single equilibrium cluster,
1 Our simulation cell is (48x48) in size. This corresponds roughly to typical experimental length scales for chemisorption systems, which are inevitably limited by steps and defects remaining on the surface, even after much care in sample preparation.
L.D. Roe~of,;
786
o.oo8
0.006
=
[] rl
o.oo2
o.ooo
9
0.0
I
"
0.1
I
0.2
"
I
0.3
0.4
1/qN Fig. 13.32. Simulation results for the p(2x2) lattice gas situation showing that 0~(N), the dilute 2-d gas coverage in equilibrium with clusters of size N, varies like 1A/N-.The calculations were carried out on a (48x48) lattice with periodic boundary conditions at fixed coverage according to the protocol discussed in the text. dN2 dt
o~ - ~tN2 (0g(N2) - 0g(N,))
(13.61)
Progress toward the final, single-cluster state is thus slow when the two are similar in size. Assuming that N2 < N, (by definition), we substitute for 0g from Eq. (13.59) obtaining dN2 dt
C, + C 2
4N2 Nt
(13.62)
where C~ and C 2 a r e positive constants. We can solve Eq. (13.62) approximately for t < t(), where t()is the time at which N2 --+ 0, because in that regime N~ is approximately constant. The approximate solution is 2
N 2 =_C , ( t - to)--5 C2
QCT~)
(t,,-
t) 3'2
(13.63)
which can also be straightforwardly verified by simulation. So in the final stages of evaporation of N2 one expects a linear rate of decay with time.
Phase transitions and kinetics of ordering
787
System size effects are important, however, if one takes into account the diffusion time scale. Recall that diffusion is necessary to maintain equilibrium between distant clusters. The time dependence for normal gas-phase diffusion goes like ~ so that the linear time dependence of Eq. (13.63) is only achievable in situation where the diffusion constant is sufficiently large such that the diffusion time between the clusters is not large relative to the rate at which atoms add to or leave from the clusters.
Acknowledgements The author is grateful to many present and past collaborators for sharing their insights concerning this field of research with him. One cannot mention them all, but Ted Einstein, Robert Park, Norman Bartelt, Peder Estrup, See Chen Ying, Steve Foiles, Murray Daw, Ernst Bauer, Jtirgen Behm and Peter Kleban deserve special thanks. Bill Unertl's suggestions for improving the manuscript were very helpful. The author also thanks Gwo-Ching Wang for providing Figs. 13.28 and 13.30, and Ole Mouritsen for Figs. 13.29 and 13.31. Bibliography
The present treatment of the subject of surface phase transitions cannot, by its nature provide full coverage of the literature, nor treat all subjects encompassed in full depth. The following annotated bibliography, which gives pertinent review articles and conference proceedings, may help supplement these failings. The list is ,,iven in reverse chronological order with an (arbitrarily chosen) cutoff at 1980. King and Woodruff (1994) - - Phase Transitions and Adsorbate Restructuring of Metal Surfaces. Vol. VII in Elscvier's series Chemical Physics ~?fSolid Surfaces and Heterogeneous Catalysis includes several important reviews. See especially Tringides' chapter on kinetics. Several surface phase transition systems are also covered in detail in other chapters. B.N.J. Pcrsson (1992) - - Ordered structures and phase transitions in adsorbed layers ( 135 pages) is a very complete and well-written review of the part of this field relating particularly to adsorption systems in equilibrium. Einstein ( 1991) J Multisite lateral interactions and their consequences (8 pages) reviews the determination of adatom interactions in chemisorption systems and the effect thereof on adlayer phase diagrams focusing especially on that of trio interactions. Two fairly recent NATO workshop proceedings include papers by many of the key workers in these fields: Phase Transitions in Surfiwe Films 2, Taub et al. (1991); and Kinetics of Ordering and Growth at Surfaces, Lagally (1989). For an introduction to kinetic behavior at surfaces see especially the contribution in the latter by O. Mouritsen. The series of international conferences on the Structure of Surfaces which have been occurring every third year have featured many interesting papers on the topic of surface phase transitions. Of special note is Part VII of the second of these, edited by van der Veen and Van Hove (1988). Andy Zangwill's (1988) excellent graduate-level textbook has useful chapters (5 and 11) on surface phase transition phenomena. Bauer (1987) - - Phase Transitions on Single-Crystal Surfaces and in Chemisorbed Layers (64 pages) is an exceptionally complete survey of phenomena and contains as well a solid section on theoretical background and a brief survey of useful experimental methods.
L.D. Roelo.l's
788
lnglesfield (1985) - - Reconstructions and Relaxations on Metal Surfaces (52 pages) reviews the theory and experimental situation with respect to metallic reconstruction. Estrup (1984) - - Reconstruction of Metal Surfaces (25 pages) is a useful survey of known phenomena occurring on metallic surfaces. Kehr and Binder ( 1 9 8 4 ) - Simulation of Diffusion in Lattice Gases and Related Kinetic Phenomena (42 pages) offers a useful theoretical background of nucleation and the kinetics of phase development in 2-d systems. Roelofs and Estrup (1983) ~ Two-Dimensional Phases in Chemisorption Systems (23 pages) focuses primarily on what can go wrong in the hunt for a chemisorption system in which the critical behavior of the simple magnetic models of w 13.3.2 is realized without major complications. Roelofs (1982a) ~ Order in Two Dimensions (25 pages) is a very brief introduction to the theoretical background for 2-d phase transitions. The topic of the effects of substrate disorder is discussed more completely there than in the present treatment. Einstein ( 1 9 8 2 ) ~ Critical Phenomena of Chemisorbed Overlayers (30 pages) surveys and discusses the measurement of critical exponents in chemisorption systems, mostly from a theoretical perspective. Roelofs (1982b) - - Monte Carlo Simulations of Chemisorbed Overlayers (31 pages) treats the determination of lattice gas phase diagrams via simulation techniques and offers some comparison with experimentally measured phase diagrams of chemisorption systems. Bauer (1980) ~ Chemisorbed Phases (49 pages) is a very complete survey of chemisorption systems exhibiting phase transition phenomena. The conference proceedings volume entitled, Ordering in Two Dimensions, Sinha (1980) is completely devoted to the subject of surface phase transitions and provides a useful picture of the state of the field in the late 70s.
References Alerhand, O.L., D. Vanderbiit, R.D. Meade and J.D. Joannopoulos, 1988, Phys. Rev. Lett. 61, 1973. Allen, S.M. and J.W. Cahn, 1979, Acta. Metall. 27, 1085. Bak, P., P. Kleban, W.N. Unerti, J. Ochab, G. Akinci, N.C. Bartelt and T.L. Einstein, 1985, Phys. Rev. Lctt. 54, 1539. Bak, P., 1979, Sol. St. Comm. 32, 581. Barker, R. A. and P. J. Estrup, 1981, J. Chem. Phys. 74, 1442. Bartelt, N.C., T.L. Einstein and L.D. Roelofs, 1985, Phys. Rev. B32, 2993. Bartelt, N.C., T.L. Einstein and L.D. Roelofs, 1987, Phys. Rev. B35, 1776. Barteit, N.C., T.L. Einstein and L.D. Roelofs, 1987, Phys. Rev. B35, 6786. Bartclt, N.C., L.D. Roelofs and T.L. Einstein, 1989, Surface Sci. 221, L750. Bauer, E., 1980, in: Phase Transitions in Surface Films, eds. J.G. Dash and J. Ruvalds. Plenum, New York, p. 267. Bauer, E., 1987, in: Topics in Current Physics, eds. W. Schommers and P. von Blanckenhagen. SpringerV erlag, Berlin, p. 115 Baumberger, M., W. Stocker and K.H. Rieder, 1986, Appl. Phys. A 41, 151. Barber, M.N., 1983, in: Phase Transitions and Critical Phenomena 8, eds. C. Domb and J.L. Lebowitz. Academic Press, London, 145. Baxter, R.J., 1980, J. Phys. A 13, L61. Behm, R.J., K. Christmann and G. Ertl, 1980, Surface Sci. 99, 320. Bennett, P.A. and M.B. Webb, 1981, Surface Sci. 104, 74. Betts, D.D., 1964, Can. J. Phys. 42, 1564, Binder, K. and D.P. Landau, 1976, Surface Sci. 61,577. Binder, K. and D.P. Landau, 1980, Phys. Rev. B21, 1941.
Phase transitions and kinetics of ordering
789
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Phase transitions and kinetics of ordering
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Appendix A: Physically allowed phase diagrams As originally emphasized by Griffiths and Wheeler (1970), simple geometric and topological considerations applied to the discontinuities and/or symmetry changes associated with phase transitions provide important constraints on phase diagram structure. These rules are more readily discussed in a representation in which the independent variables are taken to be the thermodynamic fields, like T, It, ~t.`in the present context, as opposed to densities, like S, O, | While there are formal intricacies (see Fisher, 1970) distinguishing fields and densities is conceptually straightforward; fields are driving forces and densities are the order parameters that develop in response to them (see also w 2.1 in Chapter 2.) One begins by envisioning the space defined by a set of fields chosen such that all observed phases have a corresponding driving force; including T as the field that generates the disordered phase. For the c(2x2) lattice gas with attractive E 2 the space is three-dimensional and based on the fields (T,~t,la.`) as in Fig. 13.6b. Let N be the number of necessary fields in the general case. Phase transitions in the space may take two forms: changes of symmetry and discontinuities of densities. Consider the latter first. A discontinuity, say in 0, cannot exist at just an isolated point. Rather, it will extend over an N-I dimensional surface (two-dimensional in the case of Fig. 13.6b). On geometric grounds this surface can only be terminated in a few ways. - One may reach a physical limit like T ---> 0 or la --->+ "'. - The surface may intersect with another such surface giving an N-2 dimensional surface of triple points. The dashed line in Fig. 13.6b is a line (really a curve) of triple points along which the two c(2x2) phases (note the degeneracy of the c(2x2) phase) coexist with the dilute ( l x l ) phase. The curves extending away from the It, plane are the loci of continuous, liquid-gas type condensation transitions (as in Fig. 13.3) occurring on just one c(2x2) sublattice. - The discontinuity can go continuously to zero along an N-2 dimensional bounding surface. (These are the one-dimensional curves shown as solid lines in Fig. 13.6b.). Anywhere along that N-2 dimensional surface densities will be observed to vary singularly (but continuously), i.e. a derivative of the quantity will diverge, as a remnant of the discontinuity. Outside the bounding surface all thermodynamic densities should vary analytically. These observations amount to stringent limits on phase diagram topology. A second and perhaps related consideration that constrains the structure of phase diagrams is that one cannot change symmetry without passing through a phase boundary. The phase bounded by higher temperature curve in Fig. 13.6b is labeled c(2x2) and represents a region of broken symmetry, in which the state assumed by the system has lower symmetry than its Hamiltonian. Since such symmetry breaking requires passage through a singularity, there can be no path to a region of differing symmetry that does not pass through a phase boundary. (We exclude paths leaving the p.` = 0 plane, since turning on a staggered chemical potential drives c(2x2) symmetry and the phase is no longer one of spontaneously broken symmetry.) Thus, phase boundaries delimiting regions of differing symmetry cannot just start or stop in 'mid air' as occasionally seen in the literature.
L.D. Roeh?fr
794
Appendix B: Critical point singularities and the modern theory of critical phenomena Section 13.3 provides the necessary nomenclature for experimental studies of surface phase transitions but does not attempt to deal with the rich theoretical structure that underpins the modern perspective on critical phenomena. In this Appendix we briefly sketch the classical theory of phase transitions and the modern (beginning in about the 60s) refinements stemming from the realization of the significance of critical fluctuations. The classical theory of phase transitions is based on the mean field approximation and is covered in most statistical mechanics texts: Brout's (1965) treatment is excellent. For understanding the modern theory one may consult various review articles including Fisher (1974), Kadanoff (1976), Wegner (1976) and the less technical article by Wilson (1979) or the textbooks that have recently appeared: Goldenfeld (1992) and Uzunov (1993). In the interest of convenience and simplicity the basic ideas are noted here. Figure 13.B.1 displays the typical phase diagram structure associated with a critical point in a space spanned by the variables (T, h, hs). (The variables are those appropriate for the Ising model, see Eq. (13.1). hs is defined in w 13.2.4.) These are fields (in the thermodynamic sense discussed in Appendix A) which appear in the first law of thermodynamics d E = T d S + h d M + h,dM,
(B.1)
Fig. 13.B.1. Phase diagram for the Ising model of a two-dimensional ferromagnet, h and h~ are the magnetic field and the staggered magnetic field respectively. Scans A, B, and C represent isotherms in which the field is varied through the transition for T < T~, T = T~, and T > Tc respectively.
795
Phase transitions and kinetics of ordering
where S is the entropy, M is the magnetization and Ms is the staggered magnetization. E, the internal energy (i.e. the energy not including the contribution due to the external field h in Eq. (13.1)), is a function of S, M and Ms. The phase diagram imparts the information that for h > 0 we expect the system to display long-range order with positive (spin-up) M, and for h < 0, negative M. The interest centers on the plane at h = 0, in which neither direction is favored. For large values of T or Ihsl the spins are disordered, but within the region bounded by the solid curve in Fig. 13.B. 1, symmetry is broken and the system will choose to display long-range order with either positive or negative magnetization. The boundary of this region is a "seam" where the two phases merge to become one and this is where interesting behavior might be expected. One might imagine investigating this system via scans A, B and C, in which T is held constant while the field h is set initially to a negative value and then increased through 0, stopping on the positive side. If one measured M vs. h along these scans a discontinuity at h = 0 should be seen for scan A and a smooth and gradual increase of M with h for scan C. Precisely what happens for scan B which passes directly through the seam is a more subtle matter and must be settled by calculation or measurement. It is instructive to consider the application of mean field theory. Since, formally at least, the experimentalist has control not of the extensive densities, S, M and M,, but rather of the fields, T, h, hs we must base our considerations on the (magnetic) Gibbs free energy (hereafter simply "free energy") (B.2)
G( T, h, h~) = E - T S - h M - h,M~
(For the relation of G to the partition function see Eq. (13.2).) The equilibrium phase of the system at a particular point in (T, h, hs) space is the one with the lowest value of G at that point, so one can determine the phase diagram from G. The field h, has been included in the treatment in order to characterize the effect of so-called irrelevant fields. That matter will be dealt with below; for the moment we will set h, = 0 for convenience and focus on G(T, h).
B. 1. Mean field theory Mean field theory applied to the nearest neighbor Ising model, see Brout (1965) for example, gives M = tanh
I
4JM + h ]
(B.3)
kBT
an equation that has one, two or three solutions for kBT < 4J, and just one for kBT > 4J. In the former case we need to evaluate G for each of the solutions to determine which represents the equilibrium phase. Using Eq. (B.2) one can express G in terms of M, T and h as
N
--2JM 2
hM + kBT
2
log
2
+
2
log
2
(B.4)
L.D. Roeh?]:~
796
M Curve A B C
T / Tc 0.875 1.000 1.125
! "
9
I
0.25
-0
0.25
-0.5
-
.
4
Fig. 13.B.2. Mean field results for the Ising ferromagnet. The upper panel displays the equilibrium magnetization as a function of h for the isotherms A, B, and C of Fig. 13.B.1. The lower panel display the free energy per spin g = G/N, normalized by kT vs. h for the same three isotherms. See text for further discussion.
Equation (B.3) cannot be solved analytically, but it is straightforward to do so numerically and Fig. 13.B.2 shows the solutions plotted vs. h for three values of T corresponding to the three isotherms in Fig. B. 1. The values of M can be substituted into Eq. (B.4) and the lower panel of Fig. B.2 shows the variation of G for the same three scans as well. One sees that the multiple values of M for T < 4J k---B-'which we shall shortly identify as the Tc for this transition, result in G having a triangular-shaped loop (curve A). At any given ( T , h ) the lowest value of G identifies the equilibrium phase and these values have been indicated with the bold curves. (The continuations of those curves beyond the point at which they represent the equilibrium phase describe metastable phases, leading to the possibility of hysteresis along scan A. The top segment of the loop comprises fully unstable solutions, which are completely unrealizable given their upward curvature which indicates negative susceptibility, see Eq. (13.33a).) The values of M corresponding to the equilibrium phases of G have also been drawn in bold style, and one sees that the slope discontinuity in G at h = 0 corresponds to
Phasetransitionsandkineticsof ordering
797
a magnetization discontinuity, M = M• as expected on the basis of Eq. (13.3a), as the system makes the transition from spin-down order to spin-up order. For T > Tc there is a single solution for all h and no discontinuity in M. Finally, right at Tc we must obviously have singular behavior, in both the h and T directions, as the loop in G shrinks to nothing and disappears. We can display aspects of this symmetry by considering the ~,arious derivatives of G near the critical point. First, one can establish how the M• approach 0 at Tc for h = 0. For this we examine Eq. (B.3) for h = 0 and small M, expanding the hyperbolic tangent to obtain
l (4JM13, M - 4JMkBT--3[k.T
+""
(B.5)
4J whose solutions to lowest order for T < kB are 0 (B.6)
The M - 0 solution is unstable as previously noted and the other two are the equilibrium possibilities previously represented as M• We see that M• go to 0 at Tc 4J = - - , so that our identification of the critical temperature above was correct. kB Moreover, using the standard definition, Eq. (13.16), of t, the reduced temperature we can reexpress the M• as M•
+t)3~43/ll+t-l) (B.7) = + ~f31tl (1 + t + . . )
Thus, the leading singular ~behavior as T---) Tc from below corresponds to a critical exponent 13 of 1/2. See Eq. (13.19) for the definition of 13. One can also readily examine the dependence of M on h at the critical temperature. If we set kBT = 4J in Eq. (B.3) we obtain M - tanh(M + ~)
(B.8a)
where
1 One properly considers this behavior to be singular because although M+do not diverge anywhere, their derivatives with respect to T diverge as Tc is approached from below.
L.D. Roeh?]:~
798
-
h
h =4J
(B.8b)
is a dimensionless field variable. For small M and h, which should apply near the critical point, we can expand the tanh in Eq. (B.8a) to obtain 1
M = (M + T/+ -;- (M
+
~)3 +
9 04
(B.9)
J
so that M = (3 h)I/3
_
_
h + less singular terms
(B.IO)
For this we deduce that mean field theory predicts a value of 3 for the critical exponent 5 (see Eq. (13.21 ). It is worth stopping at this point to emphasize that we these mean field results are being presented because they display the necessity of singular behavior at a phase transition, not because they have any quantitative significance. Note that in this treatment of the Ising model Tc is off (in comparison to Onsager' s exact results) by about 80% (see Eq. (13.4)); and (see Table 13.1 for the exact values) that mean field gives a value for 13that is 4 times too large and for 8 a value that is 5 times too small. Mean field theory also in some cases, including for example the 3-state Potts model, predicts the wrong transition order. Note further that identical predictions for the critical exponents can be derived from Landau theory (see w 13.1.4.2) so that we can identify Landau theory as another sort of mean field theory. The failure of mean field is due to the fact that it ignores correlations and fluctuations. These related concepts can be explained as follows. Correlation refers to the influence of neighboring spins on one another, i.e. if a given spin points upward, its neighbors are more likely than not to do so due the ferromagnetic nearest neighbor interaction. In the mean field treatment each spin interacts with the average of all other spins rather than with its actual neighbors which are more likely than average to be similarly oriented. One can improve the treatment of correlations by basing the mean field calculation on a cluster of sites, rather than a single site. This result in better accuracy for To, but does not change the calculated critical exponents, because of the role played by fluctuations in second order phase transitions. To understand the role played by fluctuations we recall that the correlation length, ~, characterizes the length over which order persists (see w 13.3.1 and Eq. (13.24) for the definition of ~). ~ therefore determines the size of the cluster that one would have to treat accurately in order to correctly predict system behavior. Unfortunately ~ is found to diverge near T~ (see Eq. (13.23)), and this has two important ramifications. First, mean field theory cannot be successful close to Tc, since no finite cluster can encompass the divergence of ~. Secondly, one realizes that the nature of the equilibrium state near Tc must be quite complex. Consider the situation for T close to but slightly less than Tc. We have a small, but nonzero value of M and a nearly divergent correlation length. The fact that the value of M is small means that there are nearly as many spins in the minority direction as in the majority
Phase transitions and kinetics of ordering
799
direction, even though the large value of ~ implies that a given spin influences a vast number of other spins. These apparently contradictory considerations can be synthesized via a conception of the equilibrium state consisting of large, coherent regions of the minority spin direction imbedded within the system which has overall more of the majority oriented spins. Thus if one did a local average ML (i.e. the average value of M within a region of linear size L), the value of M t would be found to fluctuate as one moved about the sample. This observation gives insight into the failure of the Landau version mean field theory. The variation of ML with position suggests that near Tc at least, we should regard the free energy as a function not only of M, but also of VM, the gradient of the order parameter. Treating correlations and fluctuations together presented a theoretical impasse that was not overcome until the development in the '60s and '70s of the idea of scaling and the practical calculation technique of renormalization which grew out of it.
B.2. Scaling The basic idea of scaling arises from the observation that if ~ is the physically important length scale and if it diverges near a critical point, then all other length scales become unimportant. Thus the free energy per site, which formally is a function of several variables G g = N g(t,h,h,)
(B. 1 1)
really somehow depends only on one variable, the correlation length, ~, which itself will be a function of the variables, t, h, etc., which determine how far the system is from its critical point ~. (Note that we express the temperature in terms of the variable t in this analysis to indicate that what matters is the distance from the critical point.) A function which has this property (imposed by nature in this case) is known to mathematicians as a homogeneous function 2. and may only depend on its arguments in the particular fashion flXPx, Xqy, ~,rz.... ) = ~,f(x,y, z .... )
(B.12)
The sense of Eq. (B. 12) applied to the critical free energy is that many combinations of t, h and h, result in the same value of ~ and therefore for those situations the free energy can differ only by the scale factor X. The terminology scaling is meant to imply that a rescaling of the variables implies a particular rescaling of the free energy. The seemingly innocuous Eq. (B.12) has an impressive array of consequences, some of which we detail following.
1 It is conventional to resolve g into regular (analytic) and singular contributions g = gr + gs. The following discussion applies only to the singular part of g. 2 For a good discussion of homogeneous functions and their connection to critical phenomena and a demonstration of Eq. (A. 12) see Chapter 10 of Reichl (1980).
800
L.D. Roeh?]:~
Kadanoff (1966) introduced the notion of b l o c k r e n o r m a l i z a t i o n . Suppose one takes the d-dimensional t lattice and divided it into blocks of linear dimension L, so that each block contains L a sites or spins. A new variable that describes the magnetization of the Ith block can be defined 1
s, = F 2 s,
(B.
iel
St will take on a range of values between +1. Moreover, we could think of determining a new effective Hamiltonian written in terms of the S/'s that would replace Eq. (13.1), but give the same physics, since it describes the same system. This Hamiltonian can be argued to have the same form as Eq. (13.1 ), except that we would have to use rescaled temperatures and field strengths, tL, hL, etc., to get the same degree of order. Given that it arises from a Hamiltonian of the same form, the free energy for a lattice of S/'s, g(tL, he .... ) will have the same functional form as does g(t, h .... ) and since there are fewer blocks than sites by a factor of L a the per block and per site free energies we have g(t/, hL .... ) = L a g(t, h .... )
(B. 14)
Thus we have argued that the free energy does satisfy the defining relationship for a homogeneous function and we can expect variable relationship as given in Eq. (B. 12), which we choose to write using conventional definitions of the exponents tL = T L:'
(B. 15a)
hr. = h Ly'
(B. 15b)
h,.L = h, Ly,
(B. 15c)
The exponents y, and Yh, whose values for several models are given in Table 13.1, thus characterize the rescaling of t and h necessary to preserve the free energy if the length on which the system is investigated is changed by a factor of L. Their values are related to those of the conventional critical exponents defined in Eqs. ( 1 3 . 1 8 24) and that connection will be made shortly. However, let us first deal with the third field h., and its exponent Yi. Note that if the exponent associated with a particular field is negative, as turns out to happen for h, in the case of the Ising model, then as one considers the system on ever larger length scales, the rescaled value, hs.L, grows ever smaller. That particular field must then be irrelevant to the behavior of the system near its critical point where we can go to very large L. Thus the critical behavior of the Ising model
1 For the case of surfaces d = 2, of course, but scaling theory applies to systems of other spatial dimension as well, so we will present this analysis for general d.
Phase transitions and kinetics of ordering
801
is therefore independent of the value of hs, and we will find the same critical exponents whereever we pass through the phase transition line in Fig. 13.B. 1. We term any field whose exponent is negative to be an i r r e l e v a n t field or variable. Changing the value of an irrelevant field experimentally or choosing systems in the same class, but having different values of the various definable irrelevant fields does not change the critical behavior. This is one of the aspects of the idea of u n i v e r s a l i t y defined in w 13.3.2. Irrelevant variables abound in surface phase transitions. The most common example is the coverage in the case of the order-disorder transition of s phase of greater than ( l x l ) periodicity as shown in Fig. 13.6. Having dealt with h, we will no longer include it in our formalism, but before returning to Yt and y;,, note that there is an interesting borderline case to worry about. Suppose the exponent Yl, associated with some v a r i a b l e f i s 0. Then the analogue of Eqs. (B. 15) would give a field that neither grows or diminishes as L increases. Such a field is termed m a r g i n a l and the presence of marginal fields leads to considerable complexity of behavior, including continuous variation of critical exponent values as is seen for example in the XY model (see Table 13.1). Let us return to thermal and magnetic exponents, Yt and Yh. The analysis which lead to Eq. (B. 14) can be repeated for the correlation length, which must rescale by a factor of L under the introduction of blocks CL(tL, hr.) = L-' r
h)
(B. 16)
If h = 0 we expect, according to Eq. (13.23) that ~ ~ Itl-v. Thus Eq. (B. 16) implies itl-~
= (It L:',I)-"
(B. 17)
which can only hold for general L if y , - v -!
(B. 18)
Next let us consider the magnetization, M = - ~)---~gWe rewrite Eq. (B. 14) using Eqs. (B. 15) in the form Oh g(t L",, h L~',)_ = L 't g(t, h)
(B.19)
and differentiate on both sides with respect to h to find M ( t L y,, h L:",)L:", = L a M(t, h)
(B.20)
Then, using the expected form from Eq. ( 1 3 . 1 9 ) , m ~ ( - t ) 13 we have for t < 0 and h =0 ( - t ) ~ L ~:',+:',, = L';(-t) ~
which only holds for arbitrary L if
(B.21 )
802
L.D. Roelofs
d
13= ~
- Yh
(B.22)
Y,
One can similarly derive expressions for other critical exponents in terms of y, and Yh. They are:
d ct = -Y,
(B.23)
2y h - d
7=~
(B.24)
Y, 6=
(B.25)
Yh d-Yh
1"1 = 2 + d - Yh
(B.26)
These equations are consistent with the values for all exactly solved models given in Table 13.1. Finally we conclude by noting that since the 6 measurable critical exponents can all be expressed in terms of just two scaling exponents, Yt and Yh, it should be possible to find relations between the exponents. Thus one can show: ], = 13(8 - 1)
(B.27)
tx + 213 + 7 = 2
(B.28)
v-
2--0~
(B.29)
d
rl=2-d
8+ 1
(B.30)
Equations (B.27-30) are known as scaling laws and they are also consistent with the exponent values given in Table 13.1, of course.
Appendix C" Finite size effects The genuine nonanalyticities associated with phase transitions are possible only in systems of infinite size. Real systems are finite, often distressingly so due to sample defects, and since methods have not been devised in most cases to solve for the equilibrium properties of infinite systems, theoretical results are often also available only for systems of limited size. Therefore the topic of the effect of finite size
Phase transitions and kinetics of ordering
803
considered as a "perturbation" of infinite-system expectations is of importance; this Appendix summarizes that subject briefly and introduces the related topic of finite-size scaling. C. 1. Introduction
Theorists and simulationists are often forced by the inadequacies of their techniques or computational equipment to limit the size of systems considered. One sees, for example, simulations of L• lattice gas models with periodic boundary conditions (pbcs, hereafter) applied, or in the case of transfer matrix studies exact treatment of strips that are infinite in one dimension, but very finite, say 6 or 8 sites wide in the perpendicular direction, again with the application of a periodic condition in that direction. Periodic boundary conditions are a popular choice, not because they are experimentally relevant in most cases, but because they eliminate "special" edge or corner sites, i.e. each site has the same coordination as all the others. Free or fixed boundary conditions and antiperiodic type boundary conditions can also be easily defined and incorporated into calculations, but seldom are because they are in general found to produce larger perturbations of infinite system behavior. These alternatives might, however, be more experimentally relevant, and so they may deserve greater attention that they have received. See, however, Kleban and Flagg (1981) for a simulation study of a lattice gas system with experimentally relevant boundary conditions. The effects of finite size can take quite different forms depending on the details, from rounding the transition and shifting its apparent position, to causing it to be missed entirely ~. This will be discussed in greater detail in w C.3 below. The experimental sources of finite size effects are principally surface defects and impurities if adsorbed layers are under study. These possibilities are probably too numerous and devious to be listed but the principle ones are: surface steps; point lattice defects; surface segregated substrate impurities, and immobilely bound adsorbate impurities or contaminants. Steps obviously interrupt the orderly lattice of sites and interactions, thus producing linear-type defects into the system. Sites adjacent to this defect, because their environment differs from the others contribute differently to the thermodynamic properties of the system. The effect of point defects, either of the substrate or in the form of adsorbed impurities, depends on their mobility and the nature of their interaction with the basic degrees of freedom in terms of which the order parameter is defined as discussed in w C.3. C.2. Length scales
To discuss the effect of these defects on the transitions quantitatively one must identify the associated length scale L. In the case of a surface with steps L would 1 Thisis meant in the sense that an application of a small magnetic field to a ferromagnetic system renders its behavior paramagnetic. The system is magnetized, though in varyingdegrees, in the direction of the applied field at all temperatures and thus cannot display the symmetry breaking associated with a real transition. Also at high temperature the magnetization can diminish, but never vanish entirely.
L.D. Roelo.[i~
804
naturally be the average distance between step edges; in the case of immobile point defects, it would be the typical distance between them, i.e. L --- 1/~t-n--nwhere n is their areal density. Whatever their nature, then the defects can only have an effect when the length scales characterizing the phase transition behavior becomes comparable to L. In the case of a second order transition the important length is ~, the correlation length of the fluctuations (see Eq. (13.24)). The identification of this length scale and its significance leads naturally to the idea of finite-size scaling first proposed by Fisher and Barber (1972). A fuller treatment is given by Barber (1983). The essence of the idea, similar to all scaling ideas, is that if both L and ~ are much larger than the microscopic lattice scale in a phase transition system, that the behavior can be understood in terms of the ratio L/~. Thermodynamic or ordering functions that in general depend on both L and T, can near the transitions be accurately approximated by scaling expressions. For example the free energy could be written G(L,t) = L d g(L/Itl -v)
(C. 1)
where v is the usual correlation length exponent of Eq. (13.23). This important idea can be theoretically exploited to determine critical exponent values from calculations based on finite systems and also provides much insight into the effect of finite size in experimental systems. For first-order transitions, which do not have divergent fluctuations, the above discussion needs some modification. Fisher and Berker (1982) have developed a scaling theory appropriate to this case and their treatment is summarized here. First, it is more conventional, in line with the Ising model paradigm, to think of the field variable, h, rather than temperature as being responsible for taking a system through a first-order transition. Thus the scaling form replacing Eq. (C. 1) will couple L and h, G ( L , h ) = L d g(L/Ihl -v',)
(C.2)
(For surface phase transitions, the variable analogous to h is the one that carries the system across a first-order phase boundary, often the chemical potential.) ,~ Secondly, the significant length scale to which system size is compared is not the correlation length, ~, because that does not become large near a first-order transition. Rather, one looks for the length, ~, at which the free energy cost of a fluctuation to the wrong side of the phase boundary equals some (possibly temperature-dependent) constant. One can argue in most cases that the v h of Eq. (C.2) has the value of 1/d as follows. We suppose that at the temperature of interest the order parameter value on the negative h side of the boundary has value M_. If a small positive field h is being applied, then the free energy cost of a region of dimension fluctuating to the negative order parameter side of the phase boundary is A G = hlM_l~ ~
(C.3)
At fixed temperature the probability of such a fluctuation is proportional to AG and therefore the length scale of such fluctuations is
Phase transitions and kinetics orordering
= Ch -1/'1
805
(C.4)
where C is independent of h (but not t). We can therefore expect rounding of a sharp first-order transition when this ~ is comparable to the size L of a finite system. Comparing Eqs. (C.2) and (C.4) suggests that the value of Vh is 1/d, or 1/2 for the systems of interest in surface science. C.3. Finite size effects
In this section the specific effects of finite size on equilibrium phase transition behavior are discussed in more detail. The effects of finite size limitations depend on what specifically is imposing the limit. We consider surface steps first. The least significant effect size limitations can impose is a rounding of the expected singularity. Surface steps are observed to prevent the passage of adatoms from one terrace to another in many systems. In this case the correlation lengths or ~ cannot exceed the terrace size, L. Thus rounding of the transition will set in when the one appropriate to the transition becomes comparable to L. A stronger effect may be seen when the step edges somehow couple to the order parameter. (An example would be the nucleation of the Si( 111 )(7x7) reconstruction at step edges discussed in w 13.4.2.1.) In this case the step edges act like an applied field whose (average) strength is proportional to the step spacing I/L. This applied field will shift the transition from its infinite system location, the shift being proportional to I/L. Rounding will also be observed in these cases. As noted above, point defects may also constitute a size limitation if they are quenched or immobile and if they couple to the order parameter. One might imagine impurities or defects controlling the development of adlayer order in their vicinity. This amounts to a size limitation, since if the coupling is strong, each defect will control the order in its immediate vicinity. This amounts to the application of randomly directed local fields. One expects then to find the transition shifted in various directions depending on where one is on the sample, and therefore a greater degree of rounding of the transition than would occur as expected on the basis of Eqs. (C. 1) or (C.2). C.4. Finite size scaling
One might assume from the previous subsection that finite size effects are always problematic. This is not the case, as especially Eq. (C.I) can be powerfully exploited in calculations. This primarily theoretical endeavor, known as finite size scaling, is briefly sketched here because it is frequently used in modeling surface phase transitions, where one faces the task of determining transition temperatures and critical exponents of proposed models for surface transitions in order to determine their consistency with experiment. The typical uses are as follows. One popular calculational techniques is based on the transfer matrix approach which allows calculation of all thermodynamic and ordering properties for a system of infinite length, but finite width L. In particular
806
L.D. Roelof~
the correlation length ~ is straightforwardly extracted in the transfer matrix approach. The scaling law embodied in Eq. (C. l) suggested that at the transition point, the ratio of L to ~ will be the same for systems of differing size. Hence looking for the temperature T of intersection L
L' E
(C.5)
for strips of different width, L and L', is an efficient way of estimating the infinite system transition temperature. Another popular approach for solving theoretical models is Monte Carlo simulation which produces distributions of system states with the correct equilibrium probabilities, thus allowing extraction of measurable quantities in the form of averages. Monte Carlo simulation can only be applied to finite lattices, say of size L x L , but one wants to accurately determine infinite system results, including the critical exponents and transition temperatures. Let the scaling ratio of Eq. (C. 1) be defined as the variable x x =- L/Itl -v
(C.6)
Then one can show that the dependence of the order parameter, susceptibility and heat capacity of a system of size L, near its critical point are mr. =L-IVvf,,(x) I
(C.7a)
)~" - T Lv/~fx(x)
(C.7b)
CL = L ' ~ f c ( x )
(C.7c)
where f,,(x), f x ( x ) and f c ( x ) are unknown scaling functions. Simulation calculations of me(T), xL(T) and CL(T) ~ are done for a variety of lattices sizes. One e~an then determine T~, v and 13 from Eq. (C.7a) by plotting on the same graph mLL ~/~ vs. x, adjusting Tc (which changes x since the latter depends on the r e d u c e d temperature) and the exponent values, until the runs at different L superimpose. (For an example of this technique in action see Binder and Landau (1980)). Equations (C.7b,c) can be similarly exploited to determine 7 and a (giving at the same time consistency checks for Tc and v). Note that the forms of Eqs. (C.7) also indicate that the maxima in gL(T) and CL(T) associated with the transition depend on L like L "t/vand L '~/" respectively. This allows determination of the exponent ratios y/v and a/v, which together with the scaling relations (Eqs. (B.27-30)) suffice to determine all the exponents.
I Ct~is the heat capacity per site.
Phase transitions and kinetics q]ordering
807
For first-order transitions finite size scaling can also be useful. For example (see Fisher and Berker (1982) and references therein) the maximum values of the susceptibility and the heat capacity in passing through a first-order transition are both expected to vary with system size like L d, thus allowing determination of transition order from finite size data.
This Page Intentionally Left Blank
Author index Allan, G. 642 Allan, G. s e e Leynaud, M. 646 Allan, G. s e e Lopez, J. 646 Allen, L.R. s e e Conrad, E.H. 358 Allen, S.M. 788 Alloneau, J.M. s e e Thorel, P. 574 Alnot, M. s e e Fargues, D. 569 Als-Nielsen, J. s e e Bohr, J. 568 Als-Nielsen, J. s e e Braslau, A. 358 Als-Nielsen, J. s e e McTague, J.P. 572 Als-Nielsen, J. s e e Pluis, B. 359 Altman, M.S. s e e Robinson, I.K. 135,791 Alverez, M.M. s e e Snyder, E.J. 420 Amer, N.M. s e e Martensson, P. 135,498 Aminpirooz, S. 493 Aminpirooz, S. s e e Becker, L. 494 Aminpirooz, S. s e e Schmalz, A. 500, 648 Ampo, H. s e e Ichninokawa, T. 710 Anazawa, T. s e e Edamoto, K. 225 Ancilotto, F. 133 Andersen, J.N. 493,494 Andersen, J.N. s e e Nielsen, M.M. 499 Andersen, O.K. s e e Nowak, H.J. 135 Anderson, H.L. 418 Anderson, J.A. s e e G6pel, W. 226 Anderson, M.S. s e e Snyder, E.J. 420 Anderson, P.W. 494, 642 Andersson, S. 494 Ando, A. s e e Hirata, A. 226 Andrei, N. 642 Andres, S.R. 358 Andzelm, J. s e e Dunlap, B.I. 181 Angeraud, F. s e e Larher, Y. 572 Angot, T. 567 Angot, T. s e e Sidoumou, M. 574 Anno, K. s e e Kono, S. 134
Aberdam, D. 225 Abraham, F.F. 567 Abraham, F.F. s e e Koch, S.W. 571 Abraham, F.F. s e e Poon, T.W. 98,647 Abraham, K. s e e Dederichs, P.H. 643 Abraham, M. s e e Kroll, C. 226 Abukawa, T. 133,493 Abukawa, T. s e e Kono, S. 134 Achete, C. s e e Niehus, H. 710 Adachi, H. s e e Ellis, D.E. 496 Adachi, H. see Tsukada, M. 183 Adams, D.L. see Aminpirooz, S. 493 Adams, D.L. see Nielsen, M.M. 499 Adams, D.L. s e e Schmalz, A. 648 Adams, D.L. s e e Stampfl, C. 501,792 Adams, J.B. see Liu, C.L. 97 Adams, J.B. see Xu, W. 650 Addato, S.D. s e e Pedio, M. 500 Affleck, I. 642 Agra's't, N. 418 Ahsan, S. s e e Kahn, A. 267 Ai, R. 418 Ai, R. s e e Marks, L.D. 419 Aika, K. s e e Ozaki, A. 500 Aizawa, T. see Itoh, H. 226 Aizawa, T. s e e Souda, R. 227 Akimitsu, J. see Mochrie, S.G.J. 572 Akinci, G. s e e Bak, P. 788 Albano, A.G. s e e Chini, P. 495 Albano, E.V. s e e Miranda, R. 572 Albrecht, T.R. 418 Alder, B.J. s e e Ceperly, D.M. 133 Alerhand, O.L. 96, 265,788 Alerhand, O.L. s e e Vanderbilt, D. 268,710 Alfonso, C. 96 Allan, D.C. s e e Teter, M. 136
809
810
Anton, A.B. s e e Rahman, T.S. 500 Antonik, M.D. 225 Anz, S.J. s e e Snyder, E.J. 420 Aono, M. 225 Aono, M. s e e Katayama, M. 497 Aono, M. s e e Oshima, C. 227 Aono, M. s e e Souda, R. 420 Aono, M. s e e Takami, T. 501 Aono, M. s e e Watanabe, S. 501 Appelbaum, J.A. 642 Applebaum, J.R. 180 Armour, D.G. s e e Verheij, L.K. 501 Arthur, J.R. s e e Cho, A.Y. 266 Artioli, G. s e e Smyth, J.R. 183 Aruga, T. 494 Aspnes, D.E. 265 Astaidi, C. 494 Astaldi, C. s e e Cautero, G. 495 Astaldi, C. see Comicioli, C. 495 Atkins, P.W. 180 Atrei, A. s e e Galeotti, M. 225 Audibcrt, P. 568 Augustyniak, W.M. s e e Martinez, R.E. 97 Aumann, C.E. 265 Aumann, C.E. s e e de Miguel, J.J. 87,710 Aumann, C.E. s e e Saloner, D. 710 Averili, F.W. s e e Ellis, D.E. 496 Avery, N.R. s e e Rahman, T.S. 500 A vouris, P. 265,709 Avouris, P. s e e Hasegawa, Y. 645 A vouris, P. see Lyo, I.-W. 135,498 Avron, J.E. 96 Axiirod, B.M. 568,642 Baberschke, K. s e e D6bler, U. 495 Baca, A.G. s e e Tobin, J.G. 501 Bachelier, V. s e e Courths, R. 643 Bachrach, R.Z. 265 Bachrach, R.Z. s e e Bringans, R.D. 266, 494 Bachrach, R.Z. s e e Olmstead, M.A. 499 Bachrach, R.Z. s e e Uhrberg, R.I.G. 501 Baddorf, A.P. s e e Itchkawitz, B.S. 790 Baddorf, A.P. see Mundenar, J.M. 499 Bader, M. 494 Badcr, S.D. s e e Falicov, L.M. 789 Badt, D. s e e Wilhelmi, G. 501 Badziag, P. 225,265
Author
Baer, D.R. s e e Blanchard, D.L. 180, 225 Baer, D.R. s e e Wang, L.Q. 227 Baerends, E.J. s e e Ellis, D.E. 496 Baetzold, R.C. 180 Bagus, P.S. 494 Bagus, P.S. s e e Bauschlicher, C.W. 494 Bagus, P.S. s e e Hermann, K. 497 Bagus, P.S. s e e Pacchioni, G. 647 Bak, P. 568,788 Bak, W. s e e Corey, E.R. 495 Baker, J.A. s e e Garcia, N. 359 Bakhtizin, R.Z. s e e Park, C. 500 Balibar, S. 96 Bancel, P.A. s e e Keane, D.T. 790 Bancel, P.A. s e e Stephens, P.W. 574 Banerjea, A. s e e Smith, J.R. 98, 136, 648 Bar-Yam, Y. s e e Kaxiras, E. 267 Bfir, M. 494 Baratoff, A. s e e Salvan, F. 500 Barber, M.N. 788 Barber, M.N. s e e Fisher, M.E. 789 Barbieri, A. 225,642 Barbieri, A. s e e Batteas, J.D. 494 Bard, A.J. 418 Bardi, U. 568 Bardi, U. s e e Galeotti, M. 225 Bare, S.R. 494 Bare, S.R. s e e Hofmann, P. 497 Barker, J.A. 642 Barker, R.A. 133,788 Barker, R.A. s e e Felter, T.E. 789 Barnes, C.J. 494 Barnes, C.J. s e e Lindroos, M. 498 Barnes, R.F. 358 Barnett, M.E. s e e Klemperer, O. 359 Bar6, A.M. 642 Baroni, S. 133 Barrett, J.H. 418 Barrett, R.C. s e e Tortonese, M. 420 Barrie, A. 494 Bart, F. 225 Barteit, N.C. 96, 642, 788 Bartelt, N.C. s e e Bak, P. 788 Bartelt, N.C. s e e Hwang, R.Q. 645 Barteit, N.C. s e e .loos, B. 97 Barteit, N.C. s e e Kodiyalam, S. 97 Bartelt, N.C. s e e Ozcomert, J.S. 98
index
Author
index
Bartelt, N.C. s e e Pai, W.W. 98,647 Bartelt, N.C. s e e Phaneuf, R.J. 98 Bartelt, N.C. s e e Roelofs, L.D. 648 Bartelt, N.C. s e e Taylor, D.E. 649 Bartelt, N.C. s e e Wang, X.-S. 98 Bartelt, N.C. s e e Wei, J. 99 Bartelt, N.C. s e e Williams, E.D. 99, 268 Barth, J.V. 133,709 Barth, J.V. s e e Schuster, R. 500, 648,791 Bartos, I. s e e Van Hove, M.A. 136 Barzel, G. s e e Scheffler, M. 500, 648 Baskes, M.I. s e e Daw, M.S. 134, 643 Baskes, M.I. s e e Foiles, S.M. 644 Baskes, M.I. s e e Roelofs, L.D. 648,791 Baski, A.A. s e e Nogami, J. 499 Baski, A.A. s e e Shioda, R. 136, 500 Bassett, D.W. 494, 642 Bassett, W.A. s e e Liu, L.-G. 182 Batchelor, D.R. see Aminpirooz, S. 493 Batchelor, D.R. s e e Schmalz, A. 648 Batra, I.M. s e e Garcia, N. 359 Batra, I.P. 133,494 Batra, I.P. s e e Himpsel, F.J. 267 Batteas, J.D. 494 Baudoing-Savois, R. see Rundgren, J. 227 Bauer, E. 96,418,642, 709, 788 Bauer, E. see Engel, T. 569 Bauer, E. see Kolaczkiewicz, J. 790 Bauer, E. s e e Pinkvos, H. 419 Bauer, E. s e e Telieps, W. 792 Bauer, E. s e e Williams, E.D. 99 Bauer, H.E. s e e Ichimura, S. 226 Bauer, R.S. s e e Bachrach, R.Z. 265 Baumberger, M. 788 Baumberger, M. see Rieder, K.H. 791 B~iumer, M. 225 Bauschlicher, C.W. 494 Bauschlicher, J.C.W. s e e Cox, B.N. 495 Baxter, R.J. 788 Bayer, P. see Muschiol, U. 499 Bayer, P. s e e Wedler, H. 501 Beaume, R. 568 Becher, U. s e e Zeppenfeld, P. 575 Becke, A.D. 642 Becker, L. 494 Becker, L. s e e Aminpirooz, S. 493 Becker, L. s e e Pedio, M. 500
811
Becker, L. s e e Schmalz, A. 500, 648 Becker, R.S. 265,494 Becker, R.S. s e e Kubby, J.A. 267 Beckschulte, M. 418 Bedrossian, P. 133, 494 Bedrossian, P. s e e Zegenhagen, J. 792 BEe, M. 568 Beeby, J. 358 Behm, B.J. 133,494, 642, 788 Behm, R.J. s e e Barth, J.V. 133, 709 Behm, R.J. s e e Brune, H. 495 Behm, R.J. s e e Christmann, K. 643,789 Behm, R.J. s e e Couiman, D.J. 495,789 Behm, R.J. s e e Gritsch, T. 496 Behm, R.J. s e e Hwang, R.Q. 710 Behm, R.J. s e e Imbihl, R. 645 Behm, R.J. s e e Kleinle, G. 498 Behm, R.J. s e e Moritz, W. 646 Behm, R.J. s e e Schuster, R. 500 Behm, R.J. s e e Schuster, R. 648,791 Behm, R.J. s e e Van Hove, M.A. 649 Behm, R.J. s e e Wintterlin, J. 501 Belin, M. s e e Rousset, S. 98 Bellman, A.F. 494 Bellon, R.J. s e e Roelofs, L.D. 648, 791 Benbow, R. s e e Broden, G. 494 Bendt, P. 133 Bennemann, K.H. s e e Dreyss6, H. 643 Bennett, P.A. 788 Bennett, P.A. s e e Robinson, i.K. 43 Bentley, J. s e e Wang, Z.L. 227,421 Berker, A.N. s e e Aierhand, O.L. 265 Berker, A.N. s e e Caflisch, R.G. 568 Berker, A.N. s e e Fisher, M.E. 789 Berlincourt, D.A. s e e Jaffee, H. 419 Bermond, J.M. s e e Alfonso, C. 96 Bermond, J.M. s e e Heyraud, J.C. 97 Bermudez V.M. 180 Bernasek, S.L. s e e Langell, M.A. 226 Berndt, W. s e e Welton-Cook, M.R. 183 Berry, F.J. 180 Berry, S.D. s e e Lind, D.M. 227 Bertei, E. 642 Bertel, E. s e e Bischler, U. 642 Besenbacher, F. 494 Besenbacher, F. s e e Eierdal, L. 495 Besenbacher, F. s e e Feidenhans'l, R. 496
812
Besenbacher, F. s e e Jensen, F. 497,790 Besenbacher, F. s e e Klink, C. 790 Besenbacher, F..Tee Mortensen, K. 499 Besoid, G. 494 Besold, G. s e e Eggeling, von, C. 495 Besold, G..Tee Mtiller, K. 646 Bethge, H. 418 Betts, D.D. 788 Beveridge, D.L..Tee Pople, J.A. 183 Biberian, J.P. s e e Coulomb, J.P. 569 Biberian, J.P. s e e Perdereau, J. 135 Biberian, J.P. s e e Suzanne, J. 574 Biberian, J.P. s e e Van Hove, M.A. 136 Bickcl, N. 225 Bickel, N..Tee Rous, P.J. 360 Biegelsen, D.K. 266 Bienfait, M. 568 Bicnfait, M. s e e Bardi, U. 568 Bienfait, M. s e e Coulomb, J.P. 568,569 B ienfait, M. s e e Gay, J.M. 570 Bicnfait, M. s e e Giachant, A. 570 Bienfait, M. s e e Seguin, J.L. 574 Bicnfait, M. see Suzanne, J. 574 Bienfait, M. s e e Venables, J.A. 575 Bienfait, M. s e e Zeppenfeld, P. 575 Biersack, J.P. 418 Bilalbegovic, G. 96 Bilz, H. s e e Martin, A.J. 182 Binder, K. 642, 788,789 Binder, K. see Kehr, K.W. 790 Binder, K. see Kinzel, W. 646 Binder, K. s e e Selke, W. 648 Binnig, B. 96 Binnig, G. 266,418 Binnig, G. see Ohnesorge, F. 227 Binnig, G.K. 789 Binning, G. see Salvan, F. 500 Binning, G.K. 133 Bird, R.B..Tee Hirschfelder, J.O. 570, 645 Birgeneau, R.J. 568 Birgeneau, R.J. see Dimon, P. 569 Birgeneau, R.J. seeHong, H. 571,790 Birgeneau, R.J. seeHorn, P.M. 571,790 Birgeneau, R.J. s e e Mochrie, S.G.J. 572 Birgeneau, R.J. see Nagler, S.E. 572 Birgeneau, R.J. s e e Robinson, I.K. 135,791 Birgeneau, R.J. s e e Specht, E.D. 574
Author
Birgeneau, R.J. s e e Stephens, P.W. 574 Birkhoff, G. 642 Bischler, U. 642 Bischler, U. s e e Bertel, E. 642 Bisson, C.M. s e e Schwoebel, P.R. 648 Biswas, R. 133 Bjurstrom, M.R. s e e Jin, A.J. 571 Black, J.E. s e e Hall, B. 570 Black, J.E. s e e Rahman, T.S. 500 Blakely, J.M. 96 Blakely, J.M. s e e Keeffe, M.E. 97 B lanchard, D.L. 180, 225 Blanchard, D.L. s e e Conrad, E.H. 358 Blanchin, M.G. s e e Epicier, T. 225 Blank, H. s e e Bienfait, M. 568 Bliznakov, G. s e e Surnev, L. 501 Block, J.D..Tee Robinson, I.K. 135 Bludau, H. 494 Bludau, H. s e e Gierer, M. 496 Bludau, H. s e e Hertel, T. 497 Biudau, H. s e e Over, H. 499, 500 Blugel, S. s e e Kobayashi, K. 134 Blyholder, G. 494 Boato, G. 358 Boato, G. ,Tee Glachant, A. 570 Bobonus, M. s e e Haase, O. 496 Bockel, C. 568 Bockel, C. s e e Dupont-Pavlovsky, N. 569 Bockel, C. s e e Menaucourt, J. 572 Bockel, C. s e e Rdgnier, J. 573 Bf~gh, E. s e e Aminpirooz, S. 493 Br E. s e e Schmalz, A. 648 Bohnen, K.P. 133, 418 Bohnen, K.P. see Chan, C.T. 133 Bohnen, K.P. s e e Ho, K.-M. 134, 790 Bohr, J 266, 568 Bohr, J s e e Braslau, A. 358 Bohr, J s e e D'Amico, K.L. 569 Bohr, J s e e Feidenhans'l, R. 266, 496 Bohr, J s e e Kjaer, K. 571 Bohr, J s e e McTague, J.P. 572 Bohr, J s e e Robinson, I.K. 360 Bol'shov, L.A. 642 Born, M. 358 Bomchil, G. s e e Beaume, R. 568 Bomchil, G. s e e Meehan, P. 572 Bomchii, G. s e e Thorel, P. 574
index
Author
index
Bonevich, J.E. 418 Bonevich, J.E. s e e Marks, L.D. 419 Bonnell, D.A. 225 Bonnell, D.A. s e e Liang, Y. 182, 226 Bonnell, D.A. s e e Rohrer, G.S. 183, 227 Bonnetain, L. s e e Delachaume, J.C. 569 Bonnetain, L. s e e Khatir, Y. 571 Bonzel, H.P. 96, 494, 789 Bonzel, H.P. s e e Breuer, U. 96 Bonzel, H.P. s e e Fetter, S. 496, 789 Bonzel, H.P. s e e Pirug, G. 500 Bonzel, H.P. s e e Wesner, D.A. 501 Bootsma, G.A. 568 Borbonus, M. s e e Koch, R. 498 Bormet, J. 494 Born, M. 180 Bornemann, P. see Engel, T. 569 Bothe, F. see Heinz, K. 497 Bothe, F. s e e Oed, W. 791 Bott, M. 418 Bott, M. s e e Michely, T. 710 B6ttchcr, A. 494 Bouchdoug, M. 568 Bouchet G. see Aberdam, D. 225 Boudart, M. 180 Boudriss, A. 225 Bouldin, C. 568 Bouldin, C.E. s e e Woicik, J.C. 502 Bourdin, J.P. 642 Bouzidi, J. s e e Krim, J. 571 Bowker, M. 642 Bowker, M. s e e Leibsle, F.M. 498 Bowker, M. s e e Murray, P.W. 499 Bozzolo, G. see Rodrigucz, A.M. 135,648 Bozzolo, G. see Smith, J.R. 98,648 Bradshaw, A.M. 494 Bradshaw, A.M. see Bonzel, H.P. 494 Bradshaw, A.M. see Hayden, B.E. 496 Bradshaw. A.M. s e e Hofmann, P. 497 Bradshaw. A.M. s e e Horn, K. 497, 571 Bradshaw. A.M. s e e Persson, B.N.J. 647 Bradshaw, A.M. s e e Pfntir, H. 500 Bradshaw, A.M. s e e Robinson, A.W. 791 Bradshaw A.M. see Rose, K.C. 420 Brand, J.L. 643 Brandes, G.R. see Canter, K.F. 180 Brandes, G.R. see Horsky, T.N. 267
813
Braslau, A. 358 Braun, O.M. 643 Brener, R. 568 Brener, R. s e e Shechter, H. 574 Brener, R. s e e Wang, R. 575 Brenig, W. 643 Brenig, W. s e e Gumhalter, B. 645 Brenig, W. s e e SchOnhammer, K. 648 Brennan, D. 494 Bretz, M. 568, 789 Bretz, M. s e e Campbell, J.H. 568 Bretz, M. s e e Dutta, P. 569 Bretz, M. s e e Quateman, J.H. 573 Breuer, U. 96 Bridge, M.E. 494 Briggs, G.A.D. s e e Burnham, N.A. 418 Bringans, R.D. 266, 494 Bringans, R.D. see Biegelsen, D.K. 266 Bringans, R.D. s e e Olmstead, M.A. 499 Bringans, R.D. see Uhrberg, R.I.G. 136, 501 Brinkman, F. s e e Coppersmith, S.N. 568 Brock, J.D. s e e Robinson, I.K. 791 Brodde, A. s e e B~umer, M. 225 Brodde, A. s e e Wiihelmi, G. 501 Broden, G. 494 Brodskii, A.M. 643 Brodskii, M.I. see Urbakh, A.M. 649 Brommer, K.D. 133 Brook, R.J. 225 Brooks, R.S. s e e Lamble, G.M. 498 Broughton, J.Q. see Brundle, C.R. 494 Brout, R.H. 789 Brown, G.S. s e e Stephens, P.W. 574 Bruch, L.W. 568,643 Bruch, L.W. see Gottlieb, J.M. 645 Bruch, L.W. s e e Klein, J.R. 646 Bruch, L.W. see Unguris, J. 575 Bruckcr, C. s e e Broden, G. 494 Brugger, R.M. s e e Taub, H. 574 Brundle, C.R. 494 Brundle, C.R. s e e Bagus, P.S. 494 Brundle, C.R. s e e Hopster, H. 497 Brune, H. 495 Brune, H. s e e Barth, J.V. 133, 709 Brune, H. s e e Wintterlin, J. 501 Brush, S.G. 789 Buckett, M.I. 225
814 Buckett, M.I. s e e Ai, R. 418 Buckett, M.I. s e e Marks, L.D. 419 Buerger, M.J. 358 Bukshpan, H. s e e Shechter, H. 574 Bullock, E.L. 133 Burandt, B. s e e Claessen, R. 225 Burch, R. 643 Burchhardt, J. s e e Nielsen, M.M. 499 Burchhardt, J. s e e Stampfl, C. 501,792 Burdett, J.K. 643 Burdick, S. s e e EI-Batanouny, M. 134 Burg, B. s e e Koch, R. 498 Btirgler, D. s e e Tarrach, G. 227 Burke, N. 643 Burnham, N.A. 418 Burnham, N.A. s e e Landman, U. 419 Burns, A.R. s e e Jennison, D.R. 645 Burns, G. 43 Burns, M. see Morkoq, H. 227 Bursill, L.A. s e e Smith, D.J. 227 Burt, M.G. 495 Burton, W.K. 709 Busch, H. 789 Busek, P.R. 418 Busing, W.R. 358 Buslaps, T. see Claessen, R. 225 Bustamante, C. 418 Butler, D.M. 568 Bykov, V. see Teplov, S.V. 420 Bylander, D.M. 495 Cabrera, N. see Burton, W.K. 709 Cadoff, 1. s e e Nolder, R. 227 Caflisch, R.G. 568 Cahn, J.W. 96 Cahn, J.W. s e e Allen, S.M. 788 Calisti, S. 568 Calicn, H.B. 96 Calicnfis, A. see LindstrOm, J. 227 Cammarata, R.C. 96 Campargue, R. 358 Campbell, C.T. s e e Over, H. 499 Campbell, D.A. see Eguiluz, A.G. 644 Campbell, I.M. 180 Campbell, J.H. 568 Campuzano, J.C. 495,789 Canter, K.F. 180
Author
index
Canter, K.F. s e e Horsky, T.N. 267 Cantini, P. s e e Boato, G. 358 Cao, R. 495 Cao, Y. 358 Cappus, D. s e e Bfiumer, M. 225 Car, R. 133 Cardenas, R. s e e Gavrilenko, G.M. 644 Cardillo, M.J. s e e Lambert, W.R. 267 Carlsson, A.E. 643 Carneiro, K. 568 Carneiro, K. s e e Taub, H. 574 Carstensen, H. s e e Claessen, R. 225 Carter, E.A. s e e Weakliem, P.C. 268 Casanova, R. 643 Catlow, C.R.A. s e e Lewis, G.V. 182 Caush, M. 180 Cautero, G. 495 Cautero, G. s e e Dhanak, V.R. 495 Celli, V. s e e Hill, N.R. 359 Celli, V. s e e Rieder, K.H. 359 Ceperly, D.M. 133 Ceva, T. 568 Ceva, T. s e e Marti, C. 572 Chadi, D.J. 133, 180, 266, 710 Chadi, D.J. s e e lhm, J. 267 Chadi, D.J. s e e Mailhiot, C. 267 Chadi, D.J. see Qian, G.-X. 268 Chambers, S.A. 180, 225,266 Chambers, S.A. s e e Tran, T.T. 227 Chambliss, D.D. 133, 710 Chan, C.-M. 495 Chan, C.-M. s e e Van Hove, M.A. 183, 360, 501 Chan, C.T. 133 Chan, C.T. see Ding, Y.G. 134, 495,789 Chan, C.T. s e e Ho, K.-M. 134, 790 Chan, C.T. s e e Takeuchi, N. 136 Chan, C.T. see Tomfinek, D. 649 Chan, C.T. s e e Wang, X.W. 136, 792 Chan, C.T. s e e Xu, C.H. 136 Chan, C.T. s e e Zhang, B.L. 136 Chan, M.H.W. 568 Chan, M.H.W. s e e Jin, A.J. 571 Chan, M.H.W. s e e Kim, H.K. 571 Chan, M.H.W. s e e Migone, A.D. 572 Chan, M.H.W. s e e Pestak, M.W. 573 Chan, M.H.W. s e e Zhang, Q.M. 575,792 Chandavarkar, S. 495
Author
index
Chandavarkar, S. s e e Fisher, D. 496 Chandrasekhar, S. 789 Chang, C.C. 225 Chang, C.S. 225 Chang, H.L.M. s e e Guo, J. 226 Chang, S.C. s e e Lubinsky, A.R. 227 Chapon, C. s e e Duriez, C. 225 Chelikowsky, J.R. 133 Chelikowsky, J.R. s e e Cohen, M.L. 180 Chen, C.-L. 643 Chen, C.-L. s e e Tsong, T.T. 792 Chen, C.J. 418, 710 Chen, D.M. s e e Bedrossian, P. 133,494 Chen, D.M. s e e Zegenhagen, J. 792 Chen, J. s e e Weitering, H.H. 136 Chen, J.G. 495 Chen, S.P. s e e Voter, A.F. 649 Chen, W. 266 Chen, W. s e e Kahn, A. 267 Chen, Y.C. see Flynn, C.P. 570 Chern, G. s e e Lind, D.M. 227 Chcrnov, A.A. 96 Chernov, A.A. s e e Haneman, D. 267 Chester, G.V. s e e Tobochnik, J. 792 Chester, M. 133,495 Chestcrs, M.A. 568 Chctty, N. 133 Chetty, N. s e e Stokbro, K. 136 Chetwynd, D.G. s e e Smith, S.T. 420 Chevary, J.A. s e e Perdew, J.P. 647 Chiang, S. see Chambliss, D.D. 133,710 Chiang, S. s e e Johnson, K.E. 710 Chiang, S. s e e Wilson, R.J. 136 Chiang, S. s e e W611, Ch. 136 Chiang, T.C. see Hirschorn, E.S. 267 Chiang, T.C. see Samsavar, A. 710 Chiaradia, P. s e e Bachrach, R.Z. 265 Chiarrello, R.P. s e e You, H. 711 Ching, W.Y. 643 Chini, P. 495 Chinn, M.D. 358,568 Chinn, M.D. s e e Fain, S.C. 569 Chinn, M.D. s e e Shaw, C.G. 574 Cho, A.Y. 266 Chou, Y.C. s e e Hong, I.H. 497 Christensen, A.N. 225 Christensen, A.N. s e e Johansson, L.I. 226
815
Christensen, A.N. s e e LindstrOm, J. 227 Christman, K. s e e Imbihl, R. 645 Christmann, K. 495,643,789 Christmann, K. s e e Behm, R.J. 494, 642, 788 Christmann, K. s e e Gierer, M. 496 Christmann, K. s e e Koch, R. 498 Christmann, K. s e e Over, H. 499 Christmann, K. s e e Schwarz, E. 500 Christmann, K. s e e Van Hove, M.A. 649 Chua, F.M. s e e Kuk, Y. 646 Chubb, S.B. 495 Chung, J.W. 789 Chung, J.W. s e e Roelofs, L.D. 791 Chung, S. 568 Chung, T.T. 568 Chung, Y.W. 225 Chung, Y.W. s e e Zschack, P. 228 Citrin, P.H. s e e Riffe, D.M. 500 Clabes, J. s e e Hahn, P. 710 Claessen, R. 225 Clark, A. 97 Clark, D.E. 789 Clark, R. s e e Nagler, S.E. 572 Clarke, D.R. 225 Cochran, W. s e e Lipson, H. 359 Cocke, D.L. 225 Coddens, G. s e e Zeppenfeld, P. 575 Coenen, F.P. s e e Wesner, D.A. 501 Cohen, J.B. s e e Zschack, P. 228 Cohen, J.M. s e e Liu, C.L. 97 Cohen, M.L. 180 Cohen, M.L. s e e lhm, J. 134, 267 Cohen, M.L. s e e Louie, S.G. 135 Cohen, M.L. s e e Northrup, J.E. 267 Cohen, P.I. 568 Cohen, P.I. s e e Lent, C.S. 359 Cohen, P.I. s e e Pukite, P.R. 359 Cohen, R.E. 643 Colbourn, E.A. 180 Cole, M.W. s e e Jung, D.R. 571 Cole, M.W. s e e Klein, J.R. 646 Cole, M.W. s e e Vidali, G. 575,649 Colella, N.J. 568 Colenbrander, B.G. s e e Turkenburg, W.C. 420 Collart, E. 225 Collazo-Davila, C. 418 Collins, I.R. s e e Fisher, D. 496
816
Collins, J.B. 643 Collins, J.B. s e e Rikvold, P.A. 648 Colton, R.J. see Burnham, N.A. 418 Colton, R.J. s e e Hues, S.M. 419 Colton, R.J. s e e Landman, U. 419 Comelli, G. 495 Comelli, G. s e e Comicioli, C. 495 Comelli, G. s e e Dhanak, V.R. 495 Comicioli, C. 495 Comrie, C.M. 495 Comrie, C.M. s e e Bridge, M.E. 494 Comsa, G s e e B o t t , M . 418 Comsa, G s e e David, R. 569 Comsa, G s e e Kern, K. 571, 710 Comsa, G s e e Kunkel, R. 710 Comsa, G s e e Michely, T. 710 Comsa, G see Poelsema, B. 573,710 Comsa, G see Zeppenfeld, P. 575 Condon, N.G. s e e Murray, P.W. 227 Cong, S. 643 Conrad, E.H. 358,710 Conrad, E.H. see Cao, Y. 358 Conrad, E.H. s e e Robinson, I.K. 360 Considine, D.M. 180 Convert, P. s e e Razafitianamaharavo, A. 573 Conway, K.M. 495 Cook, M.R. s e e Himpsel, F.J. 267 Cooper, B.R. s e e Fernando, G.W. 134 Copel, M. 133,495 Coppersmith, S.N. 568 Cord, B. 180, 225 Cord, B. s e e Courths, R. 643 Corey, E.R. 495 Cornsa, G. s e e Poelsema, B. 359 Corson, D. s e e Lorrain, P. 182 Cotton, F.A. 43 Coulman, D. s e e Gritsch, T. 496 Couiman, D.J. 495,789 Coulomb, J P. 568,569 Coulomb, J P. s e e Beaume, R. 568 Coulomb, J P. s e e Bienfait, M. 568 Coulomb, J P. see Gay, J.M. 570 Coulomb, J P. s e e Glachant, A. 570 Coulomb, J.P. s e e Krim, J. 571 Coulomb, J.P. see Madih, K. 572 Coulomb, J.P. s e e Razafitianamaharavo, A. 573
Author
index
Coulomb, J.P. s e e Suzanne, J. 574 Coulomb, J.P. s e e Trabelsi, M. 575 Coulon, M. s e e Delachaume, J.C. 569 Coulon, M. s e e Khatir, Y. 571 Coulon, M. s e e Tabony, T. 574 Courths, R. 643 Courths, R. s e e Cord, B. 180, 225 Courths, R. s e e Pollak, P. 500 Cowan, P.L. 643 Cowan, P.L. s e e Woicik, J.C. 502 Cowie, B. s e e Kerkar, M. 497 Cowley, J.M. 225, 358 Cowley, J.M. s e e Busek, P.R. 418 Cowley, J.M. s e e Gajdardziska-Josifovska, M. 225 Cowley, J.M. s e e Liu, J. 419 Cowley, J.M. s e e Yao, N. 184, 228 Cowley, R.A. s e e Andres, S.R. 358 Cox, B.N. 495 Cox, D.F. 180, 225 Cox, D.F. s e e Henrich, V.E. 181 Cox, D.F. s e e Semancik, S. 183 Cox, P.A. s e e Henrich, V.E. 226 Cremaschi, P.L. 643 Crommie, M.F. 643 Croset, B. see Madih, K. 572 Croset, B. 569 Croset, B. s e e Coulomb, J.P. 569 Croset, B. s e e Marti, C. 572 Croset, B. s e e Razafitianamaharavo, A. 573 Croweil, J.E. s e e Chen, J.G. 495 Crozier, P.A. 418 Crozier, P.A. s e e Gajdardziska-Josifovska, M. 225 Cui, J. 495,569 Cui, J. s e e Jung, D.R. 571 Cunningham, S.L. 643 Cunningham, S.L. s e e Williams, E.D. 650 Curtiss, C.F. s e e Hirschfelder, J.O. 570, 645 Cvetko, D. s e e Bellman, A.F. 494 Czanderna, A.W. s e e Lu, C. 572 D'Amico, K.L. 569 D'Amico, K.L. s e e Liang, K.S. 359 D'Amico, K.L. s e e Specht, E.D. 574 Daboul, D. s e e Seehofer, L. 500 Dabrowski, J. 266
Author
index
Dahl, L.F. s e e Corey, E.R. 495 Dai, X.Q. 643 Daimon, H. 133 Daiser, S. s e e Miranda, R. 572 Daley, R.S. 133,789 Daley, R.S. see Hildner, M.L. 790 Danner, H.R. s e e Coulomb, J.P. 569 Danner, H.R. s e e Taub, H. 574 Danner, H.R. s e e Trott, G.J. 575 Darville, J. see de Fr6sart, E. 180 Das Sarma, S. s e e Kodiyalam, S. 97 Dash, J.G. 97, 569 Dash, J.G. see Bienfait, M. 568 Dash, J.G. s e e Bretz, M. 568 Dash, J.G s e e Chung, T.T. 568 Dash, J.G see Ecke, R.E. 569 Dash, J.G s e e Gay, J.M. 570 Dash, J.G s e e Glachant, A. 570 Dash, J.G s e e Goodstein, D.L. 570 Dash. J.G s e e Huff, G.B. 571 Dash. J.G s e e Hulburt, S.B. 571 Dash. J.G s e e Kjems, J.K. 571 Dash. J.G s e e Krim, J. 571 Dash. J.G s e e Migone, A.D. 572 Dash J.G s e e Motteler, F.C. 572 Dash J.G s e e Muirhead, R.J. 572 Dash J.G s e e Pengra, D.B. 573 Dash J.G see Seguin, J.L. 574 Dash, J.G s e e Shechter, H. 574 Dash, J.G s e e Zhu, D.M. 575 David, R. 569 David, R. s e e Kern, K. 571 David, R. s e e Zeppenfeld, P. 575 Davidov, A.S. 418 Davidson, E.R. 133 Davidson, E.R. s e e Feller, D. 181 Davies, J.A. s e e Jackman, T.E. 419, 497 Davies, J.A. s e e L'Ecuyer, J. 419 Davies, J.A. s e e Norton, P.R. 647 Davis, H.L. s e e Gruzalski, G.R. 226 Davis, H.L. s e e Itchkawitz, B.S. 790 Davis, R.F. s e e Chang, C.S. 225 Davis, R.F. s e e Kevan, S.D. 498 Davis, R.F. s e e Liaw, H.P. 226 Davis, R.F. s e e Tobin, J.G. 501 Davison, S.G. 643 Davison, S.G. s e e Liu, W.-K. 646
817
Davison, S.G. s e e Sulston, K.W. 649 Davisson, C.J. 359 Davydov, S.Yu. 643 Daw, M.S. 133, 134, 643,789 Daw, M.S. s e e Einstein, T.L. 644 Daw, M.S. s e e Fallis, M.C. 644 Daw, M.S. s e e Felter, T.E. 644 Daw, M.S. s e e Foiles, S.M. 644 Daw, M.S. s e e Gumbsch, P. 97 Daw, M.S. s e e Roelofs, L.D. 648, 791 Daw, M.S. s e e Wright, A.F. 650 Dayan, M. 225 De Andres, P. s e e Oed, W. 499 De Beauvais, C. s e e Marti, C. 572 De'Bell, K. s e e Piercy, P. 647 De Cheveigne, S. s e e Rousset, S. 98 De Fr6sart, E. 180 De Miguel, J.J. 87, 710 De Miguei, J.J. s e e Aumann, C.E. 265 De Paola, R.A. s e e Heskett, D. 497 De Souza, E.P. s e e Rapp, R.E. 573 De Tacconi, N.R. s e e Liu, S.H. 646 De Vita, A. s e e Manassidis, I. 227 De Wette, F.W. 180 Debe, M.K. 134, 789 Deckert, A.A. s e e Brand, J.L. 643 Dederichs, P.H. 643 Degenhardt, D. 569 Delachaume, J.C. 569 Delachaume, J.C. s e e Tabony, T. 574 Delbouille, A. s e e Boudart, M. 180 Delchar, T.A. s e e Woodruff, D.P. 183 Delley, B. s e e Spiess, L. 136 Delley, B. s e e Ye, L. 136 Delly, B. 180 Demmin, R.A. s e e Song, K.-J. 98 Demuth, J.E. 495,643,789 Demuth, J.E. s e e Alerhand, O.L. 265 Demuth, J.E. s e e Hamers, R.J. 496 Demuth, J.E. s e e Marcus, P.M. 498 Demuth, J.E. s e e van Loenen, E.J. 136 Denier, A.W. s e e Viieg, E. 136 Denier van der Gon, A.W. 266 Denier van der Gon, A.W. s e e Tromp, R.M. 420 Dennison, J.R. s e e Larese, J.Z. 572 DePristo, A.E. s e e Raeker, T.J. 647
818
DePristo, A.E. s e e Stave, M.S. 136 Derouane, E.G. s e e Boudart, M. 180 Derry, T.E. 180 Desjonqu~res, M.C. 643 Desjonqu~res, M.C. s e e Bourdin, J.P. 642 Desjonqu~res, M.C. s e e Oils, A.M. 647 Deutsch, M. s e e Braslau, A. 358 Dev, B.N. 789 DeWette, F.W. s e e Reiger, R. 183 Dhanak, V.R. 495 Dhanak, V.R. s e e Comelli, G. 495 Dhanak, V.R. s e e Comicioli, C. 495 Dhanak, V.R. s e e Murray, P.W. 499 Dick, G.B. 180 DiDio, R.A. s e e Gruzalski, G.R. 226 DiDio, R.A. s e e Mundenar, J.M. 499 Diederich, F.N. s e e Snyder, E.J. 420 Diehl, R.D. 495,569 Diehl, R.D. s e e Barnes, C.J. 494 Diehl, R.D. s e e Chandavarkar, S. 495 Diehl, R.D. s e e Cui, J. 495 Diehl, R.D. s e e Fain, S.C. 569 Diehl, R.D. s e e Fisher, D. 496 Diehl, R.D. s e e Kerkar, M. 497 Dieleman, J. s e e Dijkkamp, D. 97 Dietrich, S. 569 Dijkkamp, D. 97 Dijkkamp, D. s e e Elswijk, H.B. 496 Dillon, J.A. s e e Farnsworth, H.E. 789 Dimon, P. 569 Dimon, P. s e e Mochrie, S.G.J. 572 DiNardo, N.J. 418 DiNardo, N.J. s e e Weitering, H.H. 136 Ding, M.Q. s e e Barnes, C.J. 494 Ding, Y.G. 134, 495,789 Dtibler, U. 495 Dobrzynski, L. 643 Dobrzynski, L. s e e Cunningham, S.L. 643 Dobson, P.J. s e e Joyce, B.A. 267 Doering, D.L. 495 Dolle, P. 569 Dolle, P. s e e Fargues, D. 569 Domany, E. 569 Dong, C.-Z. s e e Madey, T.E. 97 Dong, C.-Z. s e e Song, K.-J. 98 Dornisch, D. 134, 495 Dose, V. s e e Scheid, H. 500
Author
Dtitsch, B. s e e Oed, W. 499 Dovesi, R. s e e Cause, M. 180 Dovesi, R. s e e Pisani, C. 183 Dow, J.D. s e e Vogl, P. 183 Doyen, G. 789 Drabold, D. s e e Ordej6n, P. 135 Draper, C.F. s e e Hues, S.M. 419 Drechsler, M. 97 Dresselhaus, G. s e e Henrich, V.E. 181 Dreyss6, H. 643 Dreyss6, H. s e e Stauffer, L. 648 Drir, M. 569 Drir, M. s e e Nham, H.S. 573 Drittler, B. s e e Dederichs, P.H. 643 Droste, Ch. s e e Scheffler, M. 500, 648 Dubois, L.H. 495 Ducros, P. s e e Aberdam, D. 225 Duffy, D.M. s e e Tasker, P.W. 227 Dufour, L.-C. 180, 225 Dufour, L.-C. s e e Boudriss, A. 225 Dufour, L.-C. s e e Nowotny, J. 182 Duke, C.B. 180, 181,266, 418 Duke, C.B. s e e Godin, T.J. 266 Duke, C.B. s e e Horsky, T.N. 267 Duke, C.B. s e e Kahn, A. 267 Duke, C.B. s e e LaFemina, J.P. 182, 267 Duke, C.B. s e e Lessor, D.L. 267 Duke, C.B. s e e Lubinsky, A.R. 227, 267 Duke, C.B. s e e Mailhiot, C. 267 Dumas, M. s e e Chen, W. 266 Dumas, M. s e e Kahn, A. 267 Dumas, Ph. s e e Thibaudau, F. 501 Dunham, D. s e e Stoehr, J. 420 Dunlap, B.I. 181 Dunn, D. s e e Ai, R. 418 Dunn, D. s e e Marks, L.D. 419 Dunn, D.N. 418 Dunphy, J. s e e Barbieri, A. 642 Dupont-Pavlovsky, N. 569 Dupont-Pavlovsky, N. s e e Razafitianamaharavo, A. 573 Dupont-Pavlovsky, N. s e e R6gnier, J. 573 Duraud, J.P. s e e Bart, F. 225 Duriez, C. 225 Dutheil, A. s e e Gay, J.M. 570 Dutta, P. 569 Duval, X. s e e Matecki, M. 572
index
Author
index
Duval, X. s e e Menaucourt, J. 572 Duval, X. s e e R6gnier, J. 573 Duvai, X. s e e Thomy, A. 574 Dzioba, S. 225 Eades, J.A. s e e Samsavar, A. 710 Ealet, B. 225 Ealet, B. s e e Gillet, E. 225 Eastman, D.E. s e e Himpsel, F.J. 181 Eastman, D.E. s e e Lang, N.D. 571 Eberhardt, W. 495 Eberhardt, W. s e e Heskett, D. 497 Ebina, K. 643 Ecke, R.E. 569 Eckert, J. 569 Eckert, J. s e e Grier, B.H. 570 Eckert, J. s e e Satija, S.K. 573 Economou, E.N. 644 Edamoto, K. 225 Eden, V.L. 569 Edmonds, T. 569 Edwards,J.C. see Antonik, M.D. 225 Egdell, R.G. 181 Eggeling, von, C. 495 Eggleston, C.M. see Johnsson, P.A. 226 Eguiluz, A.G. 644 Ehrenreich, H. 644 Ehrhardt, J.J. see Fargues, D. 569 Ehrlich, G. 644, 789 Ehrlich, G. s e e Fink, H.-W. 644 Ehrlich, G. s e e Wang, S.C. 501 Ehrlich, G. see Watanabe, F. 649 Eierdal, L. 495 Eigler, D.M. s e e Crommie, M.F. 643 Einstein, T.L. 97, 134, 644, 789 Einstein, T.L. s e e Bak, P. 788 Einstein, T.L. s e e Bartelt, N.C. 96,642, 788 Einstein, T.L. s e e Eisner, D.R. 97 Einstein, T.L. s e e Joos, B. 97 Einstein, T.L. see Khare, S.V. 97, 645 Einstein, T.L. s e e Nelson, R.C. 98 Einstein, T.L. see Pai, W.W. 98,647 Einstein, T.L. s e e Roelofs, L.D. 648,791 Einstein, T.L. s e e Taylor, D.E. 649 Einstein, T.L. s e e Wang, X.-S. 98 Einstein, T.L. s e e Williams, E.D. 99 Eisner, D.R. 97
819
Eiswirth, M. 496 Eiswirth, M. s e e B~ir, M. 494 Eiswirth, M. s e e Krischer, K. 498 EI-Batanouny, M. 134 El-Batanouny, M. s e e Hsu, C.-H. 790 Elgin, R.L. 569 Ellenson, W. s e e Nielsen, M. 573 Ellenson, W. s e e Satija, S.K. 573 Ellenson, W.D. s e e Eckert, J. 569 Ellis, D.E. 181,496 Ellis, D.E. s e e Guo, J. 181,226 Ell[is, T.H. 569 Elsaesser, C. s e e Ho, K.M. 134 Elswijk, H.B. 496 Elyakhloufi, M.H. s e e Ealet, B. 225 Emmett, P.H. 496 Eng, P. s e e Meyerheim, H.L. 498 Eng, P.J. s e e Smilgies, D.-M. 792 Engel, T. 359, 496, 569 Engel, T. s e e Conrad, E.H. 358,710 Engel, T. s e e James, R.W. 359 Engel, T. s e e Kaufman, R. 359 Engel, T. s e e Sander, M. 227 Engel, T. s e e Szabo, A. 227 Engel, W. 418,496 Engel, W. s e e Griffith, O.H. 419 Engel, W. see Jakubith, S. 497 Engel, W. s e e Rose, K.C. 420 Engel, W. s e e Rotermund, H.H. 500 Engelhard, M.H. s e e Wang, L.Q. 227 Engelhardt, H.A. s e e Madey, T.E. 498 Engelhardt, H.A. s e e Pfn0r, H. 500 English, C.A. s e e Venables, J.A. 575 Enta, Y. s e e Kinoshita, T. 498 Epicier, T. 225 Ercolessi, F. 134, 644 Ercolessi, F. s e e Bilalbegovic, G. 96 Ercolessi, F. s e e Garofalo, M. 790 Ercolessi, F. s e e Tosatti, E. 136, 649 Eriksen, S. s e e Egdell, R.G. 181 Erley, W. s e e Bar6, A.M. 642 Ernst, H.J. 710 Ernst, K.H. s e e Over, H. 499 Ernst, K.H. s e e Schwarz, E. 500 Ertl, G. 43, 359, 418, 496,789 Erti, G. s e e B~ir, M. 494 Ertl, G. s e e Barth, J.V. 133,709
820 Erti, G. s e e Behm, B.J. 133,494, 642, 788 Ertl, G. s e e Bludau, H. 494 Ertl, G. s e e Bonzel, H.P. 494 Ertl, G. s e e B/Sttcher, A. 494 Ertl, G. s e e Brune, H. 495 Ertl, G. s e e Christmann, K. 495,643,789 Ertl, G. s e e Coulman, D.J. 495,789 Ertl, G. s e e Doyen, G. 789 Ertl, G. s e e Eiswirth, M. 496 Erti, G. s e e Engel, T. 496 Ertl, G. s e e Gierer, M. 496 Ertl, G. s e e Gritsch, T. 496 Ertl, G. s e e Hertel, T. 497 Ertl, G. s e e Imbihl, R. 497,645 Ertl, G. s e e Jacobi, K. 497 Ertl, G. s e e Jakubith, S. 497 Ertl, G s e e Kieinle, G. 498 Ertl, G s e e Krischer, K. 498 Ertl, G s e e Miranda, R. 572 Ertl, G s e e Moritz, W. 646 Ertl, G s e e Over, H. 499, 500 Ertl, G s e e Rotermund, H.H. 420, 500 Ertl, G see Schuster, R. 500, 648,791 Ertl, G s e e Shi, H. 500 Ertl, G. s e e Trost, J. 501 Ertl, G. see Van Hove, M.A. 649 Ertl, G. s e e Woratschek, B. 502 Erwin, S.C. 134 Estrup, P.J. 496, 789 Estrup, P.J. s e e Barker, R.A. 133,788 Estrup, P.J. s e e Chung, J.W. 789 Estrup, P.J. s e e Daley, R.S. 133,789 Estrup, P.J. see Felter, T.E. 789 Estrup, P.J. s e e Hildner, M.L. 790 Estrup, P.J. see Robinson, I.K. 135,791 Estrup, P.J. see Roelofs, L.D. 791 Evans-Lutterodt, K. s e e Robinson, I.K. 135, 791 Everts, H.-U. see Sandhoff, M. 648 Ewald, P.P. 181 Eyring, L. s e e Busek, P.R. 418 Fabre, F. s e e Ernst, H.J. 710 Fadley, C.S. s e e Bullock, E.L. 133 Fahnle, M. s e e Ho, K.M. 134 Fain, S.C. 569 Fain, S.C. s e e Chinn, M.D. 358,568
Author
Fain, S.C. s e e Cui, J. 569 Fain, S.C. s e e Diehl, R.D. 569 Fain, S.C. s e e Eden, V.L. 569 Fain, S.C. s e e Osen, J.W. 573 Fain, S.C. s e e Shaw, C.G. 574 Fain, S.C. s e e Taub, H. 792 Fain, S.C. s e e Toney, M.F. 575 Fain, S.C. s e e You, H. 575 Faisal, A.Q.D. 569 Faisal, A.Q.D. s e e Hamichi, M. 570 Faisal, A.Q.D. s e e Venables, J.A. 575 Falicov, L.M. 789 Falkenberg, G. s e e Seehofer, L. 500 Fallis, M.C. 644 Family, F. 710 Fan, F.R. s e e Bard, A.J. 418 Fan, W.C. s e e Over, H. 499 Fargues, D. 569 Farnsworth, H.E. 789 Farnsworth, H.E. s e e Schlier, R.E. 135,268 Farrell, H.H. 266 Fasolino, A. s e e Wang, C.Z. 136, 792 F~issler, T.F. s e e Burdett, J.K. 643 Faul, J.W.O. 569 Fedak, D.G. 134 Fedorus, A.G. s e e Bol'shov, L.A. 642 Fedyanin, V.K. s e e Gavrilenko, G.M. 644 Feenstra, R.M. 266, 418 Feenstra, R.M. s e e Pashley, M.D. 268 Fehlner, F.P. 181 Feibelman, P.J. 134, 644, 789 Feibelman, P.J. s e e Knotek, M.L. 226 Feibelman, P.J. s e e Williams, A.R. 649 Feidenhans'l, R. 266, 359, 496, 789 Feidenhans'l, R. s e e Bohr, J. 266 Feidenhans'l, R. s e e Dornisch, D. 134, 495 Feidenhans'l, R. s e e Grey, F. 266 Feile, R. 570 Fein, A.P. s e e Feenstra, R.M. 418 Feldman, L.C. 418 Feldman, L.C. s e e Headrick, R.L. 496 Feldman, L.C. s e e Stensgaard, I. 136 Feller, D. 181 Felter, T.E. 644, 789 Felter, T.E. s e e Daley, R.S. 133, 789 Felter, T.E. s e e Hildner, M.L. 790 Felton, R.C. s e e Prutton, M. 183,227
index
Author
index
Feng, Y.P. s e e Kim, H.K. 571 Fenter, P. s e e Hfiberle, P. 790 Fernando, G.W. 134 Ferrante, J. s e e Rodriguez, A.M. 135,648 Ferrante, J. s e e Rose, J.H. 648 Ferrante, J. s e e Smith, J.R. 98, 136, 648 Ferreira, O. s e e Tejwani, M.J. 574, 792 Ferrer, S. 496, 789 Ferrer, S. s e e Bonzel, H.P. 494 Ferret, P. s e e Epicier, T. 225 Feulner, P. s e e Pfntir, H. 500 Feynman, R. 181 Feynman, R.P. 134 Fink, H.-W. 418,644 Fink, J. see Claessen, R. 225 Finney, M.S. see Howes, P.B. 134 Finnis, M.W. 134, 644 Finzel, H.-U. 359 Fiolhais, C. s e e Perdew, J.P. 647 Fiorentini, V. 134 Firment, L.E. 225 Fisher, A.J. 181 Fisher, D. 496 Fisher, D. s e e Kerkar, M. 497 Fisher, D.S. s e e Coppersmith, S.N. 568 Fisher, D.S. s e e Fisher, M.E. 97 Fisher, G.B. s e e Root, T.W. 500 Fisher, H.J. s e e Murray, P.W. 227 Fisher, M.E. 97,789 Fisher, M.E. s e e Huse, D.A. 571 Flagg, R. s e e Kieban, P. 790 Flavell, W.R. s e e Egdeil, R.G. 181 Fleming, R.M. 359 Fleszar, A. s e e Scheffler, M. 500, 648 Flipse, C.F.J. see Murray, P.W. 227 Flodstr6m, S.A. s e e Hammar, M. 226 Flores, F. 644 Flores, F. s e e Garcia-Moliner, F. 134 Flores, F. see Joyce, K. 645 Flynn, C.P. 570 Flynn, C.P. s e e Y adavalli, S. 228 Flynn, D.K. s e e Behm, B.J. 133 Flytzani-Stephanopoulos, M. 97 Fock, V.A. 181 Fogedby, H.C. 789 Foiles, S.M. 644 Foiles, S.M. s e e Daw, M.S. 134, 643
821 Foiles, S.M. s e e Einstein, T.L. 644 Foiles, S.M. s e e Felter, T.E. 644 Foiles, S.M. s e e Roelofs, L.D. 648,791 Foiles, S.M. s e e Schwoebel, P.R. 648 Folkets, R. s e e Ernst, H.J. 710 Folman, M. s e e Shechter, H. 574 Folman, M. s e e Uram, K.J. 501 Ftilsch, S. s e e Schimmelpfenning, J. 574 Fong, C.Y. s e e Fallis, M.C. 644 Fong, C.Y. s e e Wright, A.F. 650 Fontes, E. s e e Guryan, C.A. 570 Ford, W.K. s e e Blanchard, D.L. 180, 225 Ford, W.K. s e e Lessor, D.L. 267 Ford, W.K. s e e Wan, K.J. 501 Frahm, R. s e e Greiser, N. 570 Francis, S.M. s e e Leibsle, F.M. 498 Frank, F.C. 570 Frank, H. s e e Finzel, H.-U. 359 Frank, H.H. 496 Frank, K.H. s e e Ferrer, S. 496 Frankel, D. s e e G6pel, W. 226 Frankl, D.R. s e e Chung, S. 568 Frankl, D.R. s e e Jung, D.R. 571 Franz, R.U. s e e Rotermund, H.H. 420 Frauenfelder, H. 570 Frederiske, H.P.R. 97 Freeland, P.E. s e e Zegenhagen, J. 502, 792 Freeman, A.J. s e e Fu, C.L. 134, 710, 789 Freeman, A.J. s e e Spiess, L. 136 Freeman, A.J. s e e Wimmer, E. 136, 501,650 Freeman, A.J. s e e Ye, L. 136 Freeman, D.L. 644 Freimuth, H. 570 Freimuth, H. s e e Cui, J. 569 French, T.M. 225 Frenken, F.W.M. s e e Kuipers, L. 97 Frenken, J.W.M. 496 Frenken, J.W.M. s e e Denier van der Gon, A.W. 266 Frenken, J.W.M. s e e Pluis, B. 359 Frenken, J.W.M. s e e Smeenk, R.G. 500 Frenken, J.W.M. s e e Van Pinxteren, H.M. 98 Freund, H.J. s e e Btiumer, M. 225 Fricke, A. s e e Mendez, M.A. 498 Friedel, J. 644 Friedel, J. s e e Bourdin, J.P. 642 Friedel, P. s e e Lanoo, M. 182
822 Friedman, D.J. s e e Bullock, E.L. 133 Fritsche, L. s e e Noffke, J. 499 Fritzsche, V. s e e Wedler, H. 501 Frohn, J. 97,644 Frohn, J. s e e Poensgen, M. 98 Fryberger, T.B. s e e Cox, D.F. 180, 225 Fu, C.L. 134, 710, 789, 790 Fu, C.L. s e e Wimmer, E. 501 Fuchs, G. s e e Epicier, T. 225 Fuchs, H. s e e Salvan, F. 500 Fuggle, J.C. 496 Fuggle, J.C. s e e Steinkilberg, M. 501 Fujiwara, T. see Nowak, H.J. 135 Fukushi, D. see Takami, T. 501 Fuoss, P.H. 359 Fuoss, P.H. s e e Robinson, I.K. 43 Furuya, K. s e e Andrei, N. 642 Fuselier, C.R. 570 Gadzuk, J.W. 496 Gajdardziska-Josifovska, M. 225 Gajdardziska-Josifovska, M. see Crozier, P.A. 418 Galatry, L. 570 Galeotti, M. 225 Gallagher, J. 644 Galli, G. s e e larlori, S. 181 Gameson, I. 570 Ganachaud, J.P. see Bourdin, J.P. 642 Ganz, E. 496, 790 Gao, G.B. s e e Morkor H. 227 Gao, Y. s e e Chambers, S.A. 225 Garcia, A. 418 Garcia-Moliner, F. 134 Garcia, N. 359 Garcia, N. see Rieder, K.H. 359 Garfunkei, E. s e e Novak, D. 227 Garfunkel, E. s e e Song, K.-J. 98 Garibaldi, V. 359 Garofalini, S.H. 181 Garofalo, M. 790 Garrett, B.C. s e e Truong, T.N. 649 Gauthier, S. s e e Rousset, S. 98 Gauthier, Y. s e e Hammar, M. 226 Gauthier, Y. s e e Rundgren, J. 227 Gautier, M. s e e Bart, F. 225 Gavrilenko, G.M. 644
Author
index
Gawlinski, E.T. s e e Saxena, A. 791 Gawlinski, G.T. 710 Gay, J.G. s e e Ricter, R. 135 Gay, J.M. 570 Gay, J.M. s e e Bienfait, M. 568 Gay, J.M. s e e Denier van der Gon, A.W. 266 Gay, J.M. s e e Krim, J. 571 Gay, J.M. s e e Meichel, T. 572 Gay, J.M. s e e Pluis, B. 359 Gay, R.R. 225 Geisinger, K.L. 181 George, J. s e e Kern, K. 710 George, S.M. s e e Brand, J.L. 643 Gerber, C. s e e Binnig, G. 266, 418 Gerber, Ch. s e e Binnig, G.K. 789 Gerber, Ch. s e e Binning, G.K. 133 Gerlach, R.L. 496 Germer, L.H. 496 Germer, L.H. see Davisson, C.J. 359 Giannozzi, P. s e e Baroni, S. 133 Gibbs, D. s e e D'Amico, K.L. 569 Gibbs, D. s e e Zehner, D.M. 792 Gibbs, G.V. s e e Geisinger, K.L. 181 Gibbs, G.V. s e e Hill, R.J. 181 Gibbs, J.W. 97 Gibson, A. 181 Gibson, J.M. see Pohland, O. 98 Gibson, K.D. 570 Gibson, W.M. see Narusawa, T. 647 Gierer, M. 496 Gierer, M. see Bludau, H. 494 Gierer, M. s e e Hertel, T. 497 Gierer, M. see Over, H. 499, 500 Gierlotka, S. s e e Pluis, B. 359 Giesen, M. s e e Frohn, J. 97,644 Giesen, M. see Poensgen, M. 98 Giesen-Seibert, M. 644 Gijzeman, O.L.J. see Bootsma, G.A. 568 Gillan, M.J. s e e Manassidis, I. 227 Gilles, J.M. s e e de Frdsart, E. 180 Gilles, N.S. see Fuselier, C.R. 570 Gillet, E. 225 Gillet, E. s e e Ealet, B. 225 Giimer, G.H. s e e Leamy, H.J. 43 Gilmer, G.H. s e e Weeks, J.D. 711 Giiquin, B. s e e Larher, Y. 572 Gilquin, B. s e e Ser, F. 57
Author
823
index
Girard, C. 570, 644 Girard, C. s e e Galatry, L. 570 Girard, C. s e e Girardet, C. 570 Girard, C. s e e Meichel, T. 572 Girard, J.C. s e e Rousset, S. 98 Girardet, C. 570 Girardet, C. s e e Girard, C. 570, 644 Girardet, C. s e e Lakhlifi, A. 571 Girardet, C. s e e Meichei, T. 572 Gittes, F.T. 570 Gjostein, N.A. s e e Fedak, D.G. 134 Glachant, A. 570 Glachant, A. s e e Bardi, U. 568 Glachant, A. s e e Beaume, R. 568 Glander, G. see Tong, S.Y. 268 Glander, G.S. s e e Wei, C.M. 136 Glanz, G. s e e Hofmann, P. 497 Glasser, M.L. s e e Gumbs, G. 645 Gl6bl, M. s e e Scheid, H. 500 Glueckstein, J.C. see Nogami, J. 499 Gobcli, G.W. s e e Lander, J.J. 135 Godfrey, M.J. s e e Needs, R.J. 97 Godin, T.J. 181,266 Goldberg, J.L. see Wang, X.-S. 98 Goldenfeld, N. 790 Goldman, A.I. see Guryan, C.A. 570 Goldman, A.I. see Stephens, P.W. 574 Goldmann, M. s e e Ceva, T. 568 Gollisch, H. 645 Golovchenko, J. s e e Ganz, E. 496, 790 Golovchenko, J.A. s e e Bedrossian, P. 133,494 Golovchenko, J.A. s e e Hwang, I.-S. 497 Golovchenko, J.A. see Martinez, R.E. 97 Goluvchenko, J.A. s e e Zegenhagen, J. 792 Golze, M. 570 Gomer, R. 418,645 Gomer, R. s e e Tringides, M. 649 Gomer, R. s e e Uebing, C. 649 Gomer, R. s e e Wang, C. 575 Gonser-Buntrock, C. s e e Schwarz, E. 500 Goodstein, D.L. 570 Goodstein, D.L. s e e Elgin, R.L. 569 Goodstein, D.L. s e e Hamilton, J.J. 570 Goodwin, L. 134 G6pel, W. 226 Gi3pel, W. see Kroll, C. 226 Gordon, R.G. 645
Gossler, J. s e e Hofmann, P. 497 Gotoh, T. 181 Gotoh, Y. 710 Gotter, U. s e e Horn, M. 710 Gottfried, K. 645 Gottlieb, J.M. 645 Graham, W.R. s e e Copel, M. 495 Grant, M. s e e Gawlinski, G.T. 710 Gray, H.B. 266 Greber, T. s e e Btittcher, A. 494 Greene, R.L. s e e Greiser, N. 570 Greenler, R.G. s e e Campuzano, J.C. 495 Greg, F. s e e Feidenhans'l, R. 266 Grehk, T.M. 496 Greiser, N. 570 Gremaud, G. s e e Burnham, N.A. 418 Grempel, D.R. s e e Villain, J. 360 Grest, G.S. 790 Grey, F. 266 Grey, F. s e e Dev, B.N. 789 Grey, F. s e e Dornisch, D. 134, 495 Grey, F. s e e Feidenhans'l, R. 496, 789 Grier, B.H. 570 Griffith J.E. 267 Griffith, J.E. s e e Kubby, J.A. 267 Griffith, O.H. 419 Griffith, O.H. s e e Rempfer, G.F. 419 Griffith, O.H. s e e Skoczylas, W.P. 420 Griffiths, K. 790 Griffiths, K. s e e Bare, S.R. 494 Griffiths, K. s e e Jackman, T.E. 419 Griffiths, R.B. 97,790 Griffiths, R.B. s e e Butler, D.M. 568 Griffiths, R.B. s e e Domany, E. 569 Griffiths, R.B. s e e Niskanen, K.J. 573 Grimley, T.B. 645 Grimsby, D. 496 Gritsch, T. 496 Grobecker, R. s e e B6ttcher, A. 494 Gronsky, R. s e e Falicov, L.M. 789 Gronwald, K.D. 359 Grout, P.J. s e e Joyce, K. 645 Groves, G.W. s e e Kelly, A. 226 Grozea, D. s e e Collazo-Davila, C. 418 Gruber, E.E. 97 Grumbach, M.P. s e e Ordej6n, P. 135 Grunze, M. s e e Golze, M. 570
824 Gruyters, M. s e e Jacobi, K. 497 Gruzalski, G.R. 226 Guan, J. s e e Madey, T.E. 97 Guillermo, B. s e e Smith, J.R. 136 Guinea, F. s e e Rose, J.H. 648 Guinier, A. 359 Gumbs, G. 645 Gumbsch, P. 97 Gumhalter, B. 645 Gunnarsson, O. s e e Jones, R.O. 134, 181 Gunther, S. s e e Hwang, R.Q. 710 Gtintherodt, H.-J. 645 Gtintherodt, H.-J. s e e Tarrach, G. 227 Gunton, J.D. s e e Collins, J.B. 643 Gunton, J.D. s e e Gawlinski, G.T. 710 Gunton, J.D. s e e Rikvold, P.A. 648 Gunton, J.D. s e e Saxena, A. 791 Guo, J. 181,226 Guo, J. s e e Ellis, D.E. 181 Guo, Q. s e e Murray, P.W. 499 Guo, T. s e e Blanchard, D.L. 180, 225 Guo, T. s e e Wan, K.J. 501 Gurney, R.W. 496 Guryan, C.A. 570 Gustafsson, T. s e e Chester, M. 133,495 Gustafsson, T. see Copel, M. 133,495 Gustafsson, T. s e e Haberle, P. 134, 790 Gustafsson, T. see Novak, D. 227 Gustafsson, T. see Zhou, J.B. 228 Guthmann, C. s e e Balibar, S. 96 Gwanmesia, G.D. s e e Susman, S. 183 Gygi, F. see lariori, S. 181 Haas, G. s e e Rotermund, H.H. 420 Haase, J. see Aminpirooz, S. 493 Haase, J. s e e Bader, M. 494 Haase, J. see Becker, L. 494 Haase, J. s e e Pangher, N. 500 Haase, J. see Pedio, M. 500 Haase, J. s e e Schmalz, A. 500, 648 Haase, O. 496 Haase, O. s e e Koch, R. 498 Haberen, K.W. s e e Pashley, M.D. 267, 268 Haberland, H. s e e Woratschek, B. 502 Haberle, P. 134, 790 H~iberle, P. s e e Zhou, J.B. 228 Haensel, R. s e e Degenhardt, D. 569
Author
Haftel, M.I. 134, 645 Haga, Y. 419 Hfiglund, J. s e e Hammar, M. 226 Hagstrum, H.D. 419 Hahn, P. 710 Hfikansson, K.L. s e e Hammar, M. 226 Haler, J.F. 226 Hall, B. 570 Halperin, B.I. 570, 790 Halperin, B.I. s e e Coppersmith, S.N. 568 Halperin, B.I. s e e Nelson, D.R. 573 Halpin-Healy, T. 570 Hamann, D. s e e Feibelman, P.J. 644 Hamann, D.R. 134 Hamann, D.R. s e e Appelbaum, J.A. 642 Hamann, D.R. s e e Applebaum, J.R. 180 Hamann, D.R. s e e Biswas, R. 133 Hamann, D.R. s e e Lambert, W.R. 267 Hamann, D.R. s e e Tersoff, J. 136, 420 Hamers, R.J. 496 Hamers, R.J. s e e Alerhand, O.L. 265 Hamers, R.J. s e e van Loenen, E.J. 136 Hamichi, M. 570 Hamichi, M. s e e Faisal, A.Q.D. 569 Hamichi, M. s e e Venables, J.A. 575 Hamilton, J.J. 570 Hammar, M. 226 Hammer, L. see Eggeling, von, C. 495 Hammer, L. s e e Mendez, M.A. 498 Hammer, L. s e e Oed, W. 499 Hammonds, E.M. s e e Horn, P.M. 571,790 Han, W.K. 134 Hanada, T. s e e Hikita, T. 181,226 Hanayama, M. s e e Morishige, K. 572 Hanekamp, L.J. s e e Bootsma, G.A. 568 Haneman, D. 134, 181,267, 359, 790 Hanke, G. s e e Lang, E. 359 Hannaman, D.J. 496 Hansen, F.Y. 570 Hansen, F.Y. s e e Coulomb, J.P. 569 Hansen, F.Y. s e e Trott, G.J. 575 Hansen, F.Y. s e e Wang, R. 575 Hansen, G.D. s e e Rikvold, P.A. 648 Hansson, G.V. s e e Bachrach, R.Z. 265 Hansson, G.V. s e e Nichoils, J.M. 135,499 Hansson, G.V. s e e Uhrberg, R.I.G. 268 Hara, S. 226
index
Author
825
index
Harbison, J.P. s e e Farrell, H.H. 266 Hardiman, M. s e e Venables, J.A. 575 Harp, G.R. s e e Stoehr, J. 420 Harris, J. s e e Liebsch, A. 359 Harrison, W.A. 134, 181,267 Harten, U. 134 Harten, V. 710 Hartman, J.K. s e e Gadzuk, J.W. 496 Hartung, V. s e e Schtinhammer, K. 648 Hasegawa, F. s e e Kumagai, Y. 498 Hasegawa, T. 496 Hasegawa, Y. 645 Htiser, W. s e e Heidberg, J. 570 Hashizume, T. s e e Jeon, D. 134, 790 Hashizume, T. s e e Park, C. 500 Hashizume, T. s e e Taniguchi, M. 501 Hastings, J.B. see Eckert, J. 569 Hastings, J.M. s e e Larese, J.Z. 572 Hathaway, K.B. s e e Falicov, L.M. 789 Hau, U. see Courths, R. 643 Hawkes, P.W. 419 Hayami, W. s e e Souda, R. 227 Hayden, B.E. 496, 790 Haydock, R. s e e Gallagher, J. 644 Haydock, R. s e e Gibson, A. 181 Haydock, R. s e e Haneman, D.R. 359 Hayes, F.H. s e e Brennan, D. 494 Hayes, W. 226 Hazma, A.V. see Schildbach, M.A. 183 He, Y.L. see Zuo, J.K. 792 Headrick, R.L. 496 Hehre, W.J. 181 Heidberg, J. 570 Heidemann, A.D. s e e Larese, J.Z. 571 Heilman, P. s e e Lang, E. 359 Heimann, P. see Himpsel, F.J. 181 Heine, V. s e e Burt, M.G. 495 Heine, V. s e e Finnis, M.W. 134 Heiney, P. s e e Horn, P.M. 571,790 Heiney, P.A. s e e Birgeneau, R.J. 568 Heiney, P.A. s e e Guryan, C.A. 570 Hciney, P.A. see Stephens, P.W. 574 Heinz, K. 497 Heinz, K. see Besold, G. 494 Heinz, K. s e e Bickel, N. 225 Heinz, K. s e e Chubb, S.B. 495 Heinz, K. s e e Eggeling, von, C. 495
Heinz, K. s e e Lang, E. 359 Heinz, K. s e e Mendez, M.A. 498 Heinz, K. s e e MUller, K. 646 Heinz, K. s e e Muschiol, U. 499 Heinz, K. s e e Oed, W. 499, 791 Heinz, K. s e e Pendry, J.B. 791 Heinz, K. s e e Rous, P.J. 360 Heinz, K. s e e Wedler, H. 501 Held, G. s e e Barbieri, A. 642 Held, G. s e e Feenstra, R.M. 266 Held, G. s e e Greiser, N. 570 H61d, G. s e e JiJrgens, D. 645 Held, G. s e e Keane, D.T. 790 Held, G. s e e Lindroos, M. 498 Held, G. s e e Pfntir, H. 500 Held, G. s e e Schwennicke, C. 648 Heigesen, G. s e e Ocko, B.M. 135 Hellmann, H. 134 Henderson, B. 181 Henderson, M.A. 226 Hennig, D. s e e Methfessel, M. 97 Henrich, V E. 181,226 Henrich, V E. s e e Gay, R.R. 225 Henrich, V E. s e e Lad, R.J. 226 Henrich, V E. s e e Rohrer, G.S. 183,227 Henrich, V E. s e e Zhang, Z. 184 Henriot, M s e e Bart, F. 225 Henry, C.R. s e e Duriez, C. 225 Henzler, M. 359, 710 Henzler, M. s e e Busch, H. 789 Henzler, M. s e e Hahn, P. 710 Henzler, M. s e e Horn, M. 710 Henzler, M. s e e Schimmelpfenning, J. 574 Hering, S.V. 570 Herman, F. 645 Herman, G.S. s e e Bullock, E.L. 133 Hermann, K. 497, 570 Hermann, K. s e e Watson, P.R. 649 Hermanson, J.C. s e e Wan, K.J. 501 Hermsmeier, B.D. s e e Stoehr, J. 420 Herrera-Gomez, A. s e e Woicik, J.C. 502 Herring, C. 97 Hertel, T. 497 Hertel, T. s e e Bludau, H. 494 Hertel, T. s e e Gierer, M. 496 Hertel, T. s e e Over, H. 499 Hertz, J.A. s e e Einstein, T.L. 644
826
Heskett, D. 497 Heskett, D. s e e Frank, H.H. 496 Heslinga, D.R. 497 Hess, G.B. 570 Hess, G.B. s e e Drir, M. 569 Hess, G.B. s e e Nham, H.S. 573 Hess, G.B. s e e Youn, H.S. 575 Heydenreich J. s e e Bethge, H. 418 Heyraud, J.C. 97, 134 Heyraud, J.C. s e e Alfonso, C. 96 Heyraud, J.C. s e e Metois, J.E. 97 Hibma, T. s e e Heslinga, D.R. 497 Hibma, T. s e e Peacor, S.D. 227 Hickernell, D.C. see Bretz, M. 568 Higashiyama, K. 497 Higashiyama, K. s e e Kono, S. 135 Hikita, T. 181,226 Hildner, M.L. 790 Hildner, M.L. s e e Daley, R.S. 133,789 Hill, N.R. 359 Hill, R.J. 181 Hill, T.L. 97 Hillert, B. s e e Becker, L. 494 Hillert, B. see Pedio, M. 500 Himpsel, F. see Eberhardt, W. 495 Himpsel, F.J. 181,267 Himpscl, F.J. s e e Lang, N.D. 571 Himpsel, F.J. s e e McLean, A.B. 498 Hinncn, C. see Liu, S.H. 646 Hirabayashi, K. 181 Hirata, A. 226 Hirschfelder, J.O. 570, 645 Hirschom, E.S. s e e Samsavar, A. 710 Hirschorn, E.S. 267 Hirth, J.P. s e e Srolovitz, D.J. 648 Hjalmarson, H.P. s e e Vogl, P. 183 Hjclmberg, H. 497,645 Hjeimberg, H. s e e Johansson, P.K. 645 Hnace, B.K. see Poirer, G.E. 227 Ho, K.M 134, 790 Ho, K.M s e e Bohnen, K.P. 133,418 Ho, K.M s e e Chan, C.T. 133 Ho, K.M s e e Ding, Y.G. 134, 495,789 Ho, K.M s e e Fu, C.L. 134, 790 Ho, K.M s e e Louie, S.G. 135 Ho, K.M s e e Liu, S.H. 646 Ho, K.M. s e e Takeuchi, N. 136
A uthor
Ho, K.M. s e e Wang, X.W. 136, 792 Ho, K.M. s e e Xu, C.H. 136 Ho, K.M. s e e Zhang, B.L. 136 Ho, P.S. s e e Poon, T.W. 98,647 Ho, W. s e e Richter, L.J. 647 Hobson, J.P. s e e Edmonds, T. 569 H~che, H. s e e Keller, K.W. 710 Hochella, Jr., M.F. 181 Hochella, Jr., M.F. s e e Johnsson, P.A. 226 Hoegen, V. s e e Henzler, M. 710 Hoeven, A.J. s e e Dijkkamp, D. 97 HOfer, H. s e e Wintterlin, J. 501 Hoffmann, F.M. 497 Hoffmann, F.M. s e e Bradshaw, A.M. 494 Hoffmann, F.M. s e e Heskett, D. 497 Hoffmann, F.M. s e e Pfntir, H. 500 Hoffmann, R. 181,497,645 Hoffmann, R. s e e Halet, J.F. 226 Hoffmann, R. s e e Jansen, S.A. 226 Hoffmann, R. s e e Wong, Y.-T. 650 Hofmann, P. 497 Hofmann, P. s e e Bare, S.R. 494 Hofmann, P. s e e Bradshaw, A.M. 494 Hofmann, S. s e e Ichimura, S. 226 Hohage, M. s e e Michely, T. 710 Hohenberg, P. 134, 181,497 Hohlfeld, A. s e e Horn, K. 497 Hoinkes, H. 359 Hoinkes, H. s e e Finzel, H.-U. 359 Holland, B.W. s e e Onuferko, J.H. 791 Holland, B.W. s e e Zimmer, R.B. 360 Hollins, P. s e e Horn, K. 497 Holloway, P.H. 497, 645 Holloway, S. s e e N~rskov, J.K. 499, 647 Holmstr(Sm, S. s e e Nordlander, P. 647 Holub-Krappe, E. s e e Frenken, J.W.M. 496 Hong, H. 571,790 Hong, I.H. 497 Honjo, G. s e e Hopster, H. 497 Hopster, H.J. s e e Falicov, L.M. 789 Horn, K. 497,571 Horn, K. s e e Frenken, J.W.M. 496 Horn, K. s e e Hermann, K. 570 Horn, M. 710 Horn, M. s e e Henzler, M. 710 Horn, P.M. 571,790
index
Author
827
index
Horn, P.M. s e e Dimon, P. 569 Horn, P.M. s e e Greiser, N. 570 Horn, P.M. s e e Hong, H. 571,790 Horn, P.M. s e e Mochrie, S.G.J. 572 Horn, P.M. s e e Nagler, S.E. 572 Horn, P.M. s e e Specht, E.D. 574 Horn, P.M. s e e Stephens, P.W. 574 Horng, S.F. s e e Horsky, T.N. 267 Horsky, T.N. 267 Horsky, T.N. s e e Canter, K.F. 180 Horton, L.L. s e e Wang, Z.L. 227 Hosaka, S. s e e Hasegawa, T. 496 Hoshino, T. s e e Dederichs, P.H. 643 Hosokawa, Y. s e e Kirschner, J. 419 Hosoki, S. s e e Hasegawa, T. 496 Hou, Y. s e e Aono, M. 225 Houm~ller, A. s e e NCrskov, J.K. 499 Houston, J.E. s e e Park. R.L. 359 Howes, P.B. 134 Hrbek, J. 497 Hsu, C.-H. 790 Hsu, T. 226 Hsu, T. s e e Kim, Y. 226 Hu, G.Y. 790 Hu, P. see Barnes, C.J. 494 Hu, P. s e e Lindroos, M. 498 Hu, W.Y. see Tong, S.Y. 268 Huang, H. 134, 497 Huang, H. see Over, H. 499 Huang, H. s e e Tong, S.Y. 268,501 Huang, H. s e e Wei, C.M. 136 Huang, H. see Wu, H. 502 Huang, Y. s e e Tobin, J.G. 501 Huang, Z. 497 Hubbard, A.T. 419 Huber, D.L. s e e Ching, W.Y. 643 Hudson, J.B. see Holloway, P.H. 497, 645 Huerta-Garnica, M. s e e Rousset, S. 98 Hues, S.M. 419 Huff, G.B. 571 Huff, W.T. s e e Huang, Z. 497 Htifner, S. s e e Courths, R. 643 Hui, K.C. 497 Hui, K.C. s e e Wong, P.C. 502 Hulburt, S.B. 571 Humbert, A. s e e Thibaudau, F. 501 Hurst, J. s e e Pinkvos, H. 419
Hurych, Z. s e e Broden, G. 494 Hurych, Z. s e e Soukiassian, P. 136 Huse, D.A. 571,790 Hussain, M. s e e Horn, K. 497 Hussain, Z. s e e Huang, Z. 497 Hwang, I.-S. 497 Hwang, I.-S. s e e Ganz, E. 496, 790 Hwang, J. s e e Pate, B.B. 182 Hwang, R.Q. 645,710 Hwang, R.Q. s e e Ogletree, D.F. 647 Hwang, R.Q. s e e Williams, E.D. 360 Hybertsen, M.S. 134 Hybertsen, M.S. s e e Becker, R.S. 494 Hybertsen, M.S. s e e Zegenhagen, J. 502 Hyde, B.G. s e e O'Keefe, M. 182 lannotta, S. s e e Ellis, T.H. 569 larlori, S. 181 Ibach, H s e e Frohn, J. 97,644 Ibach, H s e e Giesen-Seibert, M. 644 lbach, H s e e Lehwald, S. 498,791 Ibach, H s e e Poensgen, M. 98 Ibach, H s e e Rahman, T.S. 500 Ibach, H see Sander, D. 98 lbach, H s e e Voigtltinder, B. 649 lchimura, S. 226 lchinokawa, T. s e e Itoh, H. 226 Ichinokawa, T. s e e Kirschner, J. 419 Ichinose, T. s e e Itoh, H. 226 lchninokawa, T. 710 Ignatiev, A. 571 Ignatiev, A. s e e Over, H. 499 lhm, G. s e e Jung, D.R. 571 lhm, G. s e e Vidali, G. 575, 649 lhm, J. 134, 267 lhm, J. s e e Aspnes, D.E. 265 Ikeda, N. s e e Kirschner, J. 419 ll'chenko, L.G. s e e Braun, O.M. 643 lllas, F. s e e Bagus, P.S. 494 Imbeck, R. s e e B6ttcher, A. 494 lmbihl, R. 497,645 lmbihl, R. s e e Eiswirth, M. 496 Imbihi, R. s e e Moritz, W. 646 Imbihl, R. s e e Rose, K.C. 420 Indovina, V. s e e Boudart, M. 180 Inglesfield, J.E. 790 Ino, S. s e e Daimon, H. 133
828 Ino, S. s e e Gotoh, Y. 710 Ipatova, I.P. s e e Maradudin, A.A. 359 Ishida, H. 497 lshikawa, T. s e e Takahashi, T. 136, 501 Ishikawa, Y. s e e Kirschner, J. 419 Ishimoto, K. s e e Kumagai, Y. 498 lshizawa, Y. s e e Aono, M. 225 lshizawa, Y. s e e Otani, S. 227 lshizawa, Y. s e e Souda, R. 227,420 Israelachvili, J.N. 419 Itchkawitz, B.S. 790 Itoh, H. 226 Jackman, T.E. 419, 497 Jackman, T.E. s e e Norton, P.R. 647 Jackson, A.G. 43,226, 359 Jackson, D.P. s e e Jackman, T.E. 497 Jackson, K. 134 Jackson, K.A. s e e Leamy, H.J. 43 Jackson, K.A. s e e Perdew, J.P. 647 Jacobi, K. 497 Jacobi, K. s e e Ranke, W. 268 Jacobi, K. s e e Shi, H. 500 Jacobsen, K.W. 497, 790 Jacobsen, K.W. s e e Feidenhans'i, R. 496 Jacobsen, K.W. s e e Stokbro, K. 136 Jacoby, M. s e e Marks, L.D. 419 Jaehnig, M. s e e Gt~pel, W. 226 Jaffee, H. 419 Jtiger, R. s e e St6hr, J. 501 Jahns, V. s e e Meyerheim, H.L. 498 Jakubith, S. 497 Jakubith, S. s e e Rotermund, H.H. 500 Jaloviar, S.G. s e e De Miguel, J.J. 87 James, R. s e e Kaufman, R. 359 James, R.W. 359 Jamison, K.D. s e e Behm, B.J. 133 Janak, J.L. s e e Moruzzi, V.L. 646 Janata, J. 181 Jansen, S.A. 226 Jardin, J.P. s e e Desjonqu/~res, M.C. 643 Jasnow, D. s e e Ohta, T. 791 Jasperson, S.N. 571 Jaszczak, J.A. s e e Wolf, D. 99, 650 Jaubert, M. s e e Glachant, A. 570 Jayaprakash, C. 97,645 Jayaram, G. s e e Collazo-Davila, C. 418
Author
index
Jefferson, D.A. s e e Smith, D.J. 227 Jefferson, D.A. s e e Zhou, W. 228 Jeng, S.-P. s e e Zhang, Z. 184 Jennings, G. s e e Campuzano, J.C. 789 Jennison, D.R. 645 Jensen, F. 497, 790 Jensen, F. s e e Feidenhans'l, R. 496 Jensen, F. s e e Mortensen, K. 499 Jensen, L.H. s e e Stout, G.H. 360 Jentjens, R. s e e Giesen-Seibert, M. 644 Jentz, D. s e e Barbieri, A. 642 Jeon, D. 134, 790 Jepsen, D.W. s e e Demuth, J.E. 495 Jepsen, D.W. s e e Marcus, P.M. 498 Jin, A.J. 571 Joannopoulos, J.D. s e e Alerhand, O.L. 96, 265,788 Joannopoulos, J.D. s e e Brommer, K.D. 133 Joannopoulos, J.D. s e e Kaxiras, E. 267 Joannopoulos, J.D. s e e Needels, M. 267 Joannopoulos, J.D. s e e Rappe, A.M. 135 Joannopoulos, J.D. s e e Vanderbilt, D. 268, 710 Johansson, L.I. 226 Johansson, L.I. s e e Hammar, M. 226 Johansson, L.I. s e e Lindberg, P.A.P. 227 Johansson, L.I. s e e Lindstrtim, J. 227 Johansson, L.I. s e e Rundgren, J. 227 Johansson, P.K. 645 Johansson, P.K. s e e NCrskov, J.K. 499 Johnson, D.D. 645 Johnson, E.D. s e e Cocke, D.L. 225 Johnson, K.E. 710 Johnson, K.L. 419 Johnson, R.J. s e e Feidenhans'l, R. 496 Johnson, R.L. s e e Bohr, J. 266 Johnson, R.L. s e e Dev, B.N. 789 Johnson, R.L. s e e Dornisch, D. 134, 495 Johnson, R.L. s e e Feidenhans'l, R. 266, 496, 789 Johnson, R.L. s e e Grey, F. 266 Johnson, R.L. s e e Seehofer, L. 500 Johnsson, P.A. 226 Joly, Y. s e e Rundgren, J. 227 Jona, F. s e e Himpsel, F.J. 267 Jona, F. s e e Huang, H. 497 Jona, F. s e e Kleinle, G. 498 Jona, F. s e e Over, H. 499, 500
Author
index
Jona, F. s e e Quinn, J. 135,500 Jona, F. s e e Sokolov, J. 500 Jona, F. s e e Yang, W.S. 502 Jona, F. s e e Zanazzi, E. 360 Jones, A.V. s e e Ignatiev, A. 571 Jones, B.A. 645 Jones, E.R. s e e McKinney, J.T. 359 Jones, R.G. s e e Kerkar, M. 497 Jones, R.O. 134, 181 Joos, B. 97 Jordan-Sweet, J.L. s e e Keane, D.T. 790 Jos6, J.V. 790 Josell, D. 97 Joyce, B.A. 267 Joyce, K. 645 Joyner, R.W. s e e MacLaren, J.M. 646 Jung, D.R. 571 Jupille, J. 790 JiJrgens, D. 645 J~irgens, D. see Schwennicke, C. 648 J~irgens, D. s e e Sklarek, W. 648 Kaburagi, M. 645 Kaburagi, M. s e e Ebina, K. 643 Kaburagi, M. s e e Urano, T. 183 Kadanoff, L.P. 790 Kadanoff, L.P. s e e Jos6, J.V. 790 Kahn, A. 267 Kahn, A. s e e Chen, W. 266 Kahn, A. s e e Duke, C.B. 266 Kahn, A. s e e Horsky, T.N. 267 Kahn, A. s e e Lessor, D.L. 267 Kalkstein, D. 645 Kampshoff, E. s e e Heidberg, J. 570 Kanaji, T. see Urano, T. 183 Kanamori, J. s e e Kaburagi, M. 645 Kang, H.C. 645 Kang, J.M. s e e Gay, J.M. 570 Kang, M.C. s e e Zhang, T. 650 Kang, W.M. s e e Tobin, J.G. 501 Kang, W.M. s e e Tong, S.Y. 501 Kao, C.-T. s e e Ohtani, H. 647 Kaplan, R. 226 Kappus, W. 645 Kara, A. s e e Chung, S. 568 Kardar, M. s e e Caflisch, R.G. 568 Kardar, M. s e e Halpin-Healy, T. 570
829 Kariotis, R. 97 Kariotis, R. s e e Aumann, C.E. 265 Kariotis, R. s e e de Miguel, J.J. 87, 710 Kariotis, R. s e e Hamichi, M. 570 Kariotis, R. s e e Lagally, M.G. 267 Kariotis, R. s e e Swartzentruber, B.S. 98,268, 710 Kariotis, R. s e e Venables, J.A. 575 Karlin, B.A. s e e Woicik, J.C. 502 Karlsson, C.J. s e e Northrup, J.E. 135 Kaski, K. s e e Gawlinski, G.T. 710 Kaski, K. s e e Rikvold, P.A. 648 Kasper, E. s e e Hawkes, P.W. 419 Katayama, M. 497 Kato, H. s e e Edamoto, K. 225 Kato, M. s e e Katayama, M. 497 Kaufman, R. 359 Kaufman, R. s e e James, R.W. 359 Kaukasoina, P. s e e Fisher, D. 496 Kawai, M. s e e Hikita, T. 181,226 Kawai, S. s e e Matsumoto, T. 182, 227 Kawai, S. s e e Nakamatasu, H. 182 Kawai, S. s e e Oshima, C. 227 Kawai, S. s e e Tanaka, H. 227 Kawai, T. s e e Matsumoto, T. 182, 227 Kawai, T. s e e Tanaka, H. 227 Kawasaki, K. s e e Ohta, T. 791 Kawazu, A. 497 Kawazu, A. s e e Sakama, H. 268 Kawazu, A. s e e Shioda, R. 136, 500 Kaxiras, E. 267 Kaxiras, E. s e e Lyo, I.-W. 135,498 Kaxiras, E. s e e Martensson, P. 135,498 Kaxiras, E. s e e Rappe, A.M. 135 Keane, D.T. 790 Keat, P.P. s e e Shropshire, J. 183 Keating, P.N. 134 Keeffe, M.E. 97 Kehr, K.W. 790 Keller, D. s e e Bustamante, C. 418 Keller, K.W. 710 Kellogg, G. 790 Kellogg, G. s e e Schwoebel, P.R. 648 Kelly, A. 226 Kemmochi, M. s e e Kirschner, J. 419 Kendall, K. s e e Johnson, K.L. 419 Kendelewicz, T. s e e Soukiassian, P. 136
830 Kendelewicz, T. s e e St6hr, J. 501 Kendelewicz, T. s e e Woicik, J.C. 502 Kenik, E.A. s e e Wang, Z.L. 227 Kerkar, M. 497 Kern, K. 571,710 Kern, K. s e e David, R. 569 Kern, K. s e e Zeppenfeld, P. 575 Kern, R. 97 Kern, R. s e e Quentel, G. 573 Kersten, H.H. s e e Turkenburg, W.C. 420 Kesmodel, L.L. s e e Van Hove, M.A. 136 Kevan, S.D. 267, 498 Kevan, S.D. s e e Skelton, D.C. 648 Kevan, S.D. s e e Wei, D.H. 649 Khare, S.V. 97,645 Khare, S.V. s e e Einstein, T.L. 134 Khare, S.V. s e e Nelson, R.C. 98 Khatir, Y. 571 Khokonov, K.B. s e e Kumikov, V.K. 97 Khor, K.E. s e e Kodiyalam, S. 97 Kikuta, S. s e e Takahashi, T. 136, 501 Kilcoyne, A.L.D. s e e Robinson, A.W. 791 Kim, H.-Y. s e e Vidali, G. 649 Kim, H.K. 571 Kim, H.K. s e e You, H. 711 Kim, H.K. see Zhang, Q.M. 575,792 Kim, H.Y. s e e Jung, D.R. 571 Kim, H.Y. s e e Vidali, G. 575 Kim, Y. 226 Kim, Y. s e e Hsu, T. 226 Kim, Y.S. s e e Gordon, R.G. 645 Kimura, Y. s e e Shibata, A. 791 King, D.A. s e e Bare, S.R. 494 King, D.A. s e e Barnes, C.J. 494 King, D.A. s e e Bowker, M. 642 King, D.A. s e e Debe, M.K. 134, 789 King, D.A. s e e Griffiths, K. 790 King, D.A. s e e Hofmann, P. 497 King, D.A. s e e Jupille, J. 790 King, D.A. s e e Lamble, G.M. 498 King, D.A. s e e Lindroos, M. 498 King, D.A. s e e Stensgaard, I. 792 King-Smith, R.D. s e e Stich, I. 136 King-Smith, R.D. s e e Ramamoorthy, M. 227 Kingdon, K.H. s e e Langmuir, I. 498 Kingsbury, D.L. s e e Ma, J. 572 Kinniburgh, C.G. 181,359
Author
Kinoshita, T. 498 Kinosita, K. s e e Gotoh, T. 181 Kinzel, W. 645,646 Kinzel, W. s e e Selke, W. 648 Kirkpatrick, S. s e e Jos6, J.V. 790 Kirschner, J. 419 Kiskinova, M. 498 Kiskinova, M. s e e Cautero, G. 495 Kiskinova, M. s e e Comeili, G. 495 Kiskinova, M. s e e Dhanak, V.R. 495 Kittaka, S. s e e Morishige, K. 572 Kittel, C. 134, 182, 498,571 Kittel, C. s e e Ruderman, M.A. 648 Kjaer, K. 571 Kjaer, K. s e e Bohr, J. 568 Kjems, J.K. 571 Kjems, J.K. s e e Taub, H. 574 Klapwijk, T.M. s e e Heslinga, D.R. 497 Kleban, P. 710, 790 Kleban, P. s e e Bak, P. 788 Kleban, P.H. s e e Clark, D.E. 789 Klebanoff, L.E. s e e Tobin, J.G. 501 Klein, J. s e e Rousset, S. 98 Klein, J.R. 646 Klein, M.L. s e e Peters, C. 573 Klein, M.L. s e e Ruiz-Suarez, J.C. 573 Kleiner, J. s e e Mo, Y.W. 710 Kleinle, G. 498 Kleinle, G. s e e Over, H. 499 Kleinman, L. s e e Batra, I.P. 494 Kleinman, L. s e e Bylander, D.M. 495 Klemperer, O. 359 Klier, K. 498 Klimov, A s e e Galeotti, M. 225 Kiink, C. 790 Klink, C. s e e Mortensen, K. 499 Klitsner, T. 267 Klitsner, T. s e e Becker, R.S. 265 Knorr, K. 571 Knorr, K. s e e Faul, J.W.O. 569 Knorr, K. s e e Koort, H.J. 571 Knorr, K. s e e Shirazi, A.R.B. 574 Knorr, K. s e e Volkmann, U.G. 575 Knorr, K. s e e Weimer, W. 575 Knotek, M.L. 226 Kn~Szinger, H. 182 Kobayashi, K. 134
index
Author
831
index
Koch, J. s e e Ertl, G. 496 Koch, R. 498 Koch, R. s e e Haase, O. 496 Koch, S.W. 571 Koch, S.W. s e e Abraham, F.F. 567 Kochanski, G.P. s e e Griffith J.E. 267 Kodiyalam, S. 97 Koel, B.E. s e e Parrott, L. 500 Koestner, R.J. 498 Koestner, R.J. s e e Van Hove, M.A. 136, 501 Kogut, J.B. 646 Kohler, U. s e e Henzler, M. 710 Kohn, W. 134, 182, 498,646 Kohn, W. s e e Hohenberg, P. 134, 181,497 Kohn, W. s e e Lang, N.D. 498 Kohn, W. s e e Lau, K.H. 646 Kolaczkiewicz, J. 790 Kolaczkiewicz, J. s e e Rogowska, J.M. 648 Kolasinski, K.W. s e e Kubiak, G.D. 182 Koma, A. s e e Hirata, A. 226 Komolov, S.A s e e Mr P.J. 227 Kono, S. 134, 135,498 Kono, S. s e e Abukawa, T. 133,493 Kono, S. s e e Higashiyama, K. 497 Kono, S. s e e Kinoshita, T. 498 Koort, H.J. 571 Koranda, S. s e e Stoehr, J. 420 Kordesch, M. s e e Engel, W. 496 Kordesch, M.E. 419 Kordesch, M.E. s e e Garcia, A. 418 Kordesch, M.E. s e e Rotermund, H.H. 500 Korringa, J. 646 Kortan, A.R. s e e Roelofs, L.D. 791 Korte, U. s e e Schwegmann, S. 500 Kose, R. s e e Over, H. 500 Koster, G.F. 646 Koster, G.F. s e e Slater, J.C. 183 Koster, G.F. s e e Slater, J.R. 136 Kosterlitz, J.M. 359 Kosterlitz, J.M. 790 Kosterlitz, M. 571 Kotliar, B.G. s e e Jones, B.A. 645 Kouteck, J. 646 Krakauer, H. s e e Fernando, G.W. 134 Krakauer, H. s e e Roelofs, L.D. 135 Krakauer, H. s e e Roelofs, L.D. 791 Krakauer, H. s e e Singh, D. 136
Krakauer, H. s e e Singh, D. 791 Krakauer, H. s e e Wimmer, E. 136 Krakauer, H. s e e Wimmer, E. 650 Krakauer, H. s e e Yu, R. 136 Kramer, H.M. 571 Kramer, H.M. s e e Venables, J.A. 575 Krans, R.L. s e e Frenken, J.W.M. 496 Kress, W. s e e de Wette, F.W. 180 Kress, W. s e e Reiger, R. 183 Kreuzer, H.J. s e e Payne, S.H. 98, 647 Kriebel, D.L. s e e Roelofs, L.D. 791 Krim, J. 571 Krim, J. s e e Bienfait, M. 568 Krim, J. s e e Gay, J.M. 570 Krim, J. s e e Migone, A.D. 572 Krim, J. s e e Muirhead, R.J. 572 Krischer, K. 498 Krischer, K. s e e Eiswirth, M. 496 Kroll, C. 226 Kronberg, M.L. 226 Kruger, P. see Landemark, E. 267 Kubala, S. s e e Engel, W. 496 Kubby, J.A. 267 Kubby, J.A. s e e Soukiassian, P. 136 Kubiak, G.D. 182 Kubiak, G.D. s e e Sowa, E.C. 183 Kudo, M. s e e Hikita, T. 181,226 Kuhlenbeck, H. s e e B~iumer, M. 225 Ktihnemuth, R. s e e Heidberg, J. 570 Kuipers, L. 97 Kuk, Y. 646 Kulik, A.J. s e e Burnham, N.A. 418 Kumagai, Y. 498 Kumar, S. s e e Gawlinski, G.T. 710 Kumikov, V.K. 97 Kunkel, R. 710 Kunz, A.B. 182 Kuppers, J. s e e Ertl, G. 43, 359, 496 Ktippers, J. s e e Woratschek, B. 502 Kurtz, R.L. 226 Kvick, ~. s e e Smyth, J.R. 183 L'Ecuyer, J. 419 Lackey, D. s e e Hayden, B.E. 790 Lad, R.J. 226 Lad, R.J. s e e Antonik, M.D. 225 Ladas, S. s e e Imbihl, R. 497
832 Laegsgaard, E. s e e Klink, C. 790 LaFemina, J.P. 182, 267 LaFemina, J.P. s e e Blanchard, D.L. 180, 225 LaFemina, J.P. s e e Gibson, A. 181 LaFemina, J.P. s e e Godin, T.J. 181,266 Lagally, M.G. 267, 359, 571, 710, 790 Lagally, M.G. s e e Aumann, C.E. 265 Lagally, M.G. s e e Barnes, R.F. 358 Lagally, M.G. s e e Ching, W.Y. 643 Lagally, M.G. s e e de Miguel, J.J. 87, 710 Lagally, M.G. s e e Haneman, D. 267 Lagally, M.G. s e e Martin, J.A. 359 Lagally, M.G. s e e Mo, Y.W. 710 Lagally, M.G. s e e Saloner, D. 710 Lagally, M.G. s e e Swartzentruber, B.S. 98, 268,710 Lagally, M.G. s e e Tringides, M.C. 710 Lagaily, M.G. s e e Wang, G.-C. 98,649 Lagally, M.G. s e e Webb, M.B. 268,360, 792 Lagally, M.G. s e e Welkie, D.G. 711 Lagally, M.G. see Wu, P.K. 711 Lagerof, K.P.D. s e e Lee, W.E. 226 Lahee, A.M. 498 Lahee, A.M. s e e Campuzano, J.C. 789 Lahee, A.M. see Harten, U. 134, 710 Lakhlifi, A. 571 Lam, D.J. s e e Ellis, D.E. 181 Lain, D.J. s e e Guo, J. 181,226 Lambert, R.M. 43 Lambert, R.M. s e e Bridge, M.E. 494 Lambert, R.M. s e e Comrie, C.M. 495 Lambert, W.R. 267 Lambeth, D.N. s e e Falicov, L.M. 789 Lamble, G.M. 498 Landau, D.P. s e e Binder, K. 642, 788,789 Landau, L.D. 571,790 Landemark, E. 267 Lander, J.J. 135 Landman, U. 419 Landree, E. s e e Collazo-Davila, C. 418 Lanclskron, H. s e e Pendry, J.B. 791 Lang, E. 359 Lang, N.D. 419, 498, 571,646 Lang, N.D. s e e Klink, C. 790 Lang, N.D. see N~rskov, J.K. 499, 647 Lang, N.D. s e e Williams, A.R. 649 Lang, P. s e e Dederichs, P.H. 643
Author
Langell, M.A. 226 Langmuir, I. 498 Langreth, D.C. 135 Lanoo, M. 182 Lapeyre, G.J. s e e Wu, H. 502 Lapujoulade, J. 790 Lapujoulade, J. s e e Ernst, H.J. 710 Lapujoulade, J. s e e Villain, J. 360 Larese, J.Z. 571,572 Larese, J.Z. s e e Chung, S. 568 Larese, J.Z. s e e Guryan, C.A. 570 Larher, Y. 572 Larher, Y. s e e Nardon, Y. 572 Larher, Y. s e e Ser, F. 57 Larher, Y. s e e Teissier, C. 574 Larher, Y. s e e Terlain, A. 574 Larsen, P.K. s e e Joyce, B.A. 267 Larson, B.E. s e e Brommer, K.D. 133 Larson, B.E. s e e Hsu, C.-H. 790 Lau, K.H. 646 Lau, K.H. s e e Kohn, W. 646 Lauter, H.J. s e e Croset, B. 569 Lauter, H.J. s e e Cui, J. 569 Lauter, H.J. s e e Degenhardt, D. 569 Lauter, H.J. s e e Feile, R. 570 Lauter, H.J. s e e Freimuth, H. 570 Lauter, H.J. s e e Gay, J.M. 570 Lauter, H.J. s e e Kjaer, K. 571 Lauter, H.J. s e e Madih, K. 572 Lauter, H.J. s e e Taub, H. 792 Lauter, H.J. s e e Tiby, C. 574, 575 Lauter, H.J. s e e Wiechert, H. 575 Law, D.S.L. s e e Lindberg, P.A.P. 227 Law, D.S.L. s e e Lindstr6m, J. 227 Lazneva, E.F. s e e MOiler, P.J. 227 La~gsgaard, E. s e e Eierdal, L. 495 L~egsgaard, E. s e e Feidenhans'i, R. 496 Ltegsgaard, E. s e e Jensen, F. 497, 790 Le Boss6, J.C. 646 Leadbetter, A.J. 182 Leamy, H.J. 43 Lebehot, A. s e e Campargue, R. 358 Lee, B.W. s e e Lubinsky, A.R. 227,267 Lee, K.B. s e e Guryan, C.A. 570 Lee, P.A. s e e Coppersmith, S.N. 568 Lee, T.D. 791 Lee, W.E. 226
index
Author
833
index
LeGoues, F.K. s e e Tromp, R.M. 420 Lehmann, M.S. s e e Wright, A.F. 184 Lehmpfuhl, G. 419 Lehmpfuhl, G. s e e Wang, N. 421 Lehwald, S. 498, 791 Lehwald, S. s e e Rahman, T.S. 500 Lehwald, S. s e e Voigtl~inder, B. 649 Leibsle, F.M. 498 Leibsle, F.M. s e e Hirschorn, E.S. 267 Leibsle, F.M. s e e Murray, P.W. 227, 499 Leibsle, F.M. s e e Samsavar, A. 710 Leidheiser, Jr. H. s e e Klier, K. 498 Lelay, G. 135 Lemonnier, J.C. s e e Campargue, R. 358 Lenc, M. s e e Muellerova, I. 419 Lenglart, P. s e e Allan, G. 642 Lenssinck, J.M. s e e Dijkkamp, D. 97 Lent, C.S. 359 Lent, C.S. s e e Pukite, P.R. 359 Lenz, J. see Schwarz, E. 500 Lcrner, E. s e e Bienfait, M. 568 Lerner, E. see Krim, J. 571 Lerner, E. see Rapp, R.E. 573 Lessor, D.L. 267 Lessor, D.L. s e e Blanchard, D.L. 180, 225 Lessor, D.L. see Horsky, T.N. 267 Leung, W.Y. s e e Chung, S. 568 Levi, A.C. 359 Levi, A.C. s e e Garibaldi, V. 359 Levy, H.A. see Busing, W.R. 358 Lewis, G.V. 182 Leynaud, M. 646 Li, C.H. see Kevan, S.D. 498 Li, X.P. 135 Li, Y. s e e Murray, P.W. 499 Li, Y.S. s e e Jennison, D.R. 645 Li, Z.G. s e e Smith, D.J. 420 Li, Z.R. s e e Chan, M.H.W. 568 Li, Z.R. see Migone, A.D. 572 Liang, K.S. 359 Liang, W.Y. s e e Zhou, W. 228 Liang, Y. 182, 226 Liang, Y. s e e Chambers, S.A. 225 Liao, D.K. s e e Hong, I.H. 497 Liaw, H.P. 226 Liebau, F. 182 Liebermann, R.C. s e e Susman, S. 183
Liebsch, A. 359, 646 Lieske, N.P. 182 Liew, Y.F. 791 Lifshitz, I.M. 791 Lifshitz, I.M. s e e Landau, L.D. 790 Lighthill, M.J. 646, 791 Lin, D.L. s e e Zheng, H. 650 Lin, D.S. s e e Hirschorn, E.S. 267 Lin, M.E. s e e Morkoq, H. 227 Lin, X.F. s e e Wan, K.J. 136, 501 Lind, D.M. 227 Lindberg, P.A.P. 227 Linder, B. s e e MacRury, T.B. 646 Lindgren, S.A. 498 Lindhard, J. 419 Lindner, H. s e e Oed, W. 499 Lindner, Th. s e e Horn, K. 497 Lindroos, M. 498 Lindroos, M. s e e Barnes, C.J. 494 Lindroos, M. s e e Fisher, D. 496 Lindroos, M. s e e PfniJr, H. 500 Lindsley, D.H. 227 Lindstri3m, J. 227 LindstrOm, J.B. s e e Lindberg, P.A.P. 227 Linke, U. s e e Sander, D. 98 Lippel, P.H. s e e Canter, K.F. 180 Lippel, P.H. s e e W~311,Ch. 136 Lipson, H. 359 Littleton, M.J. s e e Mott, N.F. 182 Litzinger, J.A. 572 Litzinger, J.A. s e e Butler, D.M. 568 Liu, C.L. 97 Liu, F.C. s e e Ma, J. 572 Liu, F.C. s e e Zeppenfeld, P. 575 Liu, H. s e e Himpsel, F.J. 267 Liu, J. 419 Liu, L.-G. 182 Liu, S.H. 646 Liu, W.-K. 646 Liu, W.-K. s e e Dai, X.Q. 643 Liu, W.-K. s e e Sulston, K.W. 649 Lo, W.J. s e e Chung, Y.W. 225 Lo, W.K. 419 Ltiffler, U. s e e Wedler, H. 501 Lohmeir, M. s e e van der Vegt, H.A. 710 Longoni, V. s e e Chini, P. 495 Lopez, J. 646
834 Lopez, J. s e e Le Boss6, J.C. 646 Lopez Vazquez-de-Parga, A. s e e Ogletree, D.F. 647 Lorrain, P. 182 Louie, S.G. 135 Louie, S.G. s e e Becker, R.S. 494 Louie, S.G. s e e Chelikowsky, J.R. 133 Louie, S.G. s e e Hybertsen, M.S. 134 Louie, S.G. s e e Tomfinek, D. 649 Louie, S.G. s e e Vanderbilt, D. 183 Louie, S.G. s e e Zhu, X. 268 Lovesey, S.W. s e e Marshall, W. 572 L6wdin, P.O. 646 Lowe, J.P. 182 Lowenstein, J. s e e Andrei, N. 642 Lu, C. 572 Lu, H.C. s e e Zhou, J.B. 228 Lu, Ping s e e Smith, D.J. 420 Lu, T.-M. s e e Presicci, M. 359 Lu, T.-M. s e e Wang, G.-C. 649, 792 Lu, T.M. see Lagally, M.G. 710 Lu, T.M. see Wang, G.C. 710 Lu, T.M. s e e Yang, H.N. 711 Lu, T.M. see Zuo, J.K. 711 Lu, W. see Smith, R.L. 227 Lu, W.C. s e e Dai, X.Q. 643 Lu, W.C. see Zhang, T. 650 Lu, Y.-T. s e e Zhang, Z. 99 Lubinsky, A.R. 227, 267 Lucovsky, G. s e e Pantelides, S.T. 182 Ludwig, A.W.W. s e e Affleck, I. 642 Luedtke, W.D. s e e Landman, U. 419 Lundgren, E. s e e Andersen, J.N. 493,494 Lundgren, E. s e e Nielsen, M.M. 499 Lundqvist, B. 646 Lundqvist, B.I. s e e NCrskov, J.K. 499 Lundqvist, S. 646 Lundqvist, S. s e e Lundqvist, B. 646 Lundqvist, S. s e e March, N.H. 135 Lupis, C.H.P. 572 Luschka, M. s e e Finzel, H.-U. 359 Luscombe, J.G. s e e Tringides, M.C. 710 Lutz, C.P. s e e Crommie, M.F. 643 Lutz, M.A. s e e Feenstra, R.M. 266 Lyo, I.W. 135,498 Lyo, I.W. s e e Avouris, Ph. 709 Lyo, S.-W. s e e Avouris, P. 265
Author
index
Ma, J. 572 Maca, F. s e e Scheffler, M. 500 Mfica, F. s e e Scheffler, M. 648 Macdonald, J.E. s e e Conway, K.M. 495 Macdonald, J.E. s e e Vlieg, E. 360 MacDonald, J.E. s e e Pluis, B. 359 MacDonald, J.E. s e e Van Silfhout, R.G. 268 MacDowell, A.A. s e e Robinson, I.K. 135, 791 MacGillavry, C.H. 43 MacLachlan, A.D. 572 MacLane, S. s e e Birkhoff, G. 642 MacLaren, J.M. 267,498,646 MacRae, A.U. s e e Germer, L.H. 496 MacRury, T.B. 646 Madden, H.H. s e e Park, R.L. 43, 573 Madelung, E. 182 Madey, T.E. 97, 498 Madey, T.E. s e e Maschhoff, B.L. 227 Madey, T.E. s e e Song, K.-J. 98 Madey, T.E. s e e Thiel, P.A. 183 Madhavan, P. 646 Madih, K. 572 Maglietta, M. 498 Magnanelli, S. s e e Bardi, U. 568 Mahanty, J. 646 Mailhiot, C. 267 Maiihiot, C. s e e LaFemina, J.P. 267 Mak, A. s e e Hong, H. 571,790 Mak, A. s e e Specht, E.D. 574 Maksym, P.A. 182 Malic, R.A. s e e McRae, E.G. 267 Manassidis, I. 227 Mangat, P. s e e Soukiassian, P. 136 Mansfield, M. s e e Needs, R.J. 97 Manzke, R. s e e Claessen, R. 225 Mao, D. s e e Chen, W. 266 Maradudin, A.A. 359, 646 Maradudin, A.A. s e e Cunningham, S.L. 643 Maradudin, A.A. s e e Dobrzynski, L. 643 Maradudin, A.A. s e e Eguiluz, A.G. 644 Maradudin, A.A. s e e Portz, K. 647 March, N.H. 135,646 March, N.H. s e e Flores, F. 644 March, N.H. s e e Joyce, K. 645 March, N.H. s e e Lundqvist, S. 646 March, N.H. s e e Mahanty, J. 646 Marchenko, V.I. 97,646
Author
index
Marcus, P.M. 498 Marcus, P.M. s e e Chubb, S.B. 495 Marcus, P.M. s e e Demuth, J.E. 495 Marcus, P.M. s e e Himpsel, F.J. 267 Marcus, P.M. s e e Quinn, J. 135,500 Marcus, P.M. s e e Sokolov, J. 500 Marcus, P.M. s e e Yang, W.S. 502 Maree, P.M.J. 267 Mark, P. s e e Duke, C.B. 266 Mark, P. see Lubinsky, A.R. 227, 267 Marks, L.D. 419 Marks, L.D. s e e At, R. 418 Marks, L.D. see Bonevich, J.E. 418 Marks, L.D. s e e Buckett, M.I. 225 Marks, L.D. see Collazo-Davila, C. 418 Marks, L.D. s e e Dunn, D.N. 418 Marshall, W. 572 Mfirtensson, P. 135,498 Mfirtensson, P. see Grehk, T.M. 496 Mfirtensson, P. see Nicholls, J.M. 135,499 Marti, C. 572 Marti, C. s e e Ceva, T. 568 Marti, C. s e e Coulomb, J.P. 569 Marti, C. see Croset, B. 569 Marti, C. s e e Thorel, P. 574 Marti, O. s e e Spatz, J.P. 420 Martin,, R.M. s e e Ordej6n, P. 135 Martin, A.J. 182 Martin, J.A. 359 Martin, J.A. s e e Lagally, M.G. 359 Martin, J.A. s e e Saloner, D. 710 Martin, J.l. s e e Roelofs, L.D. 791 Martin, R.M. s e e Chetty, N. 133 Martin-Rodero, A. s e e Joyce, K. 645 Martinez, R.E. 97 Martinez, V. s e e Soria, F. 500 Martini, K.M. s e e EI-Batanouny, M. 134 Martins, J.L. s e e Troullier, N. 136 Martir, E.I. s e e Roelofs, L.D. 648 Maschhoff, B.L. 227 Mashkova, E.S. 419 Mason, B.F. 572 Mason, M.G. s e e Kevan, S.D. 498 Mason, R. 498 Masri, P. s e e Coulomb, J.P. 568 Matecki, M. 572 Matecki, M. s e e Coulomb, J.P. 569
835 Matecki, M. s e e Dolle, P. 569 Materer, N. s e e Barbieri, A. 642 Materlik, G. s e e Dev, B.N. 789 Mathias, H. s e e Lind, D.M. 227 Mathiez, Ph. s e e Thibaudau, F. 501 Matsumoto, T. 182, 227 Matsumoto, T. s e e Tanaka, H. 227 Matsunami, N. s e e L'Ecuyer, J. 419 Matsushima, T. s e e Imbihl, R. 645 Matsushima, T. s e e Moritz, W. 646 Mattera, L. s e e Boato, G. 358 Mauck, M.S. s e e Rempfer, G.F. 419 Mayer, J.E. s e e Born, M. 180 McAlpine, N. s e e Haneman, D. 267 McCartney, M.R. s e e Gajdardziska-Josifovska, M. 225 McCartney, M.R. s e e Smith, D.J. 420 McColm, l.J. 227 McCormick, W.D. s e e Goodstein, D.L. 570 McCoy, B.M. s e e Tracy, C.A. 792 McGrath, R. s e e Diehl, R.D. 495 McKay, S.R. s e e Unertl, W.N. 360 McKee, R.A. s e e Wang, Z.L. 227 McKinney, J.T. 359 McLachlan, A.D. 646 McLean, A.B. 498 McLean, E.O. s e e Bretz, M. 568 McMillan, W.L. s e e Anderson, P.W. 642 McMurry, H.L. s e e Taub, H. 574 McRae, E.G. 267,791 McTague, J.P. 572 McTague, J.P. s e e Kjaer, K. 571 McTague, J.P. s e e Nielsen, M. 573 McTague, J.P. s e e Novaco, A.D. 499, 573 McTague, J.P. s e e Taub, H. 574 Meade, R.D. 97,498 Meade, R.D. s e e Alerhand, O.L. 96, 788 Meade, R.D. s e e Bedrossian, P. 133, 494 Meade, R.D. s e e V anderbilt, D. 268,710 Mednick, K. s e e Bylander, D.M. 495 Medvedev, V.K. s e e Braun, O.M. 643 Meehan, P. 572 Mehl, M.J. s e e Cohen, R.E. 643 Mehl, M.J. s e e Langreth, D.C. 135 Met, M.N. s e e Tong, S.Y. 268 Meichel, T. 572 Meichel, T. s e e Gay, J.M. 570
836 Meier, F. s e e Pierce, D.T. 500 Melle, H. 135 Melmed, A. 419 Men, F.K. 791 Men, F.K. s e e Tong, S.Y. 268 Men, F.K. s e e Webb, M.B. 268 Menaucourt, J. 572 Menaucourt, J. s e e Bockel, C. 568 Menaucourt, J. s e e Bouchdoug, M. 568 Menaucourt, J. s e e R6gnier, J. 573 Mendez, M.A. 498 Mendez, M.A. s e e Wedler, H. 501 Menzel, and E. s e e Melle, H. 135 Menzel, D. 498 Menzel, D. s e e Fuggle, J.C. 496 Menzel, D. s e e Hofmann, P. 497 Menzel, D. s e e Lindroos, M. 498 Menzel, D. see Madey, T.E. 498 Menzel, D. s e e Michalk, G. 498 Menzel, D. see Pfntir, H. 500 Menzel, D. s e e Steinkilberg, M. 501 Mcrrill, R.P. s e e Cocke, D.L. 225 Mcrzbacher, E. 419 Methfessel, M. 97, 135 Methfessel, M. s e e Fiorentini, V. 134 Metiu, H. see Zhang, Z. 99 Mctois, J.E. 97 Metois, J.J. see Alfonso, C. 96 Mctois, J.J. see Heyraud, J.C. 97, 134 Meyer, G. see Mfirtensson, P. 135,498 Meyer, J.A. 646 Meyer, J.A. s e e Kuk, Y. 646 Meyer, R.J. s e e Duke, C.B. 266 Meyerheim, H.L. 498 Michalk, G. 498 Michely, T. 710 Michely, T. s e e Bott, M. 418 Miedema, A.R. 498 Migone, A.D. 572 Migone, A.D. s e e Chan, M.H.W. 568 Migone, A.D. s e e Zhang, S. 575 Miller, J.S. s e e Plummer, E.W. 500 Miller, M. s e e Vlieg, E. 360 Miller, T . s e e Samsavar, A. 710 Miller, W.A. s e e Tyson, W.R. 98 Millis, A.J. s e e Jones, B.A. 645 Mills, D.L. s e e Hall, B. 570
Author
index
Mills, D.L. s e e Rahman, T.S. 500 Mills, Jr., A.P. s e e Canter, K.F. 180 Mills, Jr., A.P. s e e Horsky, T.N. 267 Milne, R.H. s e e Hui, K.C. 497 Mimata, K. s e e Morishige, K. 572 Miner, K.D. s e e Chan, M.H.W. 568 Mintmire, J.W. s e e Dunlap, B.I. 181 Miranda, R. 499, 572 Misawa, S. s e e Hara, S. 226 Mitchell, K.A.R. s e e Grimsby, D. 496 Mitchell, K.A.R. s e e Hui, K.C. 497 Mitchell, K.A.R. s e e Parkin, S.R. 500 Mitchell, K.A.R. s e e Vu Grimsby, D.T. 649 Mitchell, K.A.R. s e e Wong, P.C. 502 Miura, S. s e e Ichninokawa, T. 710 Miyano, K.E. s e e Woicik, J.C. 502 Miyazaki, E. s e e Edamoto, K. 225 Mo, Y.W. 710 Mo, Y.W. see Lagally, M.G. 267 Mo, Y.W. s e e Swartzentruber, B.S. 98,268, 710 Mochida, A. s e e Edamoto, K. 225 Mochrie, S.G.J. 359, 572 Mochrie, S.G.J. s e e Feidenhans'l, R. 496 Mochrie, S.G.J. s e e Nagler, S.E. 572 Mochrie, S.G.J. s e e Song, S. 98 Mochrie, S.G.J. s e e Yoon, M. 99 Mochrie, S.G.J. s e e Zehner, D.M. 792 Modesti, S. see Astaldi, C. 494 Molchanov, V.A. s e e Mashkova, E.S. 419 Moler, E.J. s e e Huang, Z. 497 Moliere, G. 419 Moiler, M. s e e Spatz, J.P. 420 Moiler, M.A. s e e Ruiz-Suarez, J.C. 573 M611er, P. s e e Eiswirth, M. 496 Mr P.J. 227 M6nch, W. 499 Moncton, D.E. s e e D'Amico, K.L. 569 Moncton, D.E. s e e Dimon, P. 569 Moncton, D.E. s e e Mochrie, S.G.J. 572 Moncton, D.E. s e e Nagler, S.E. 572 Moncton, D.E. s e e Stephens, P.W. 574 Moncton, D.E. s e e Specht, E.D. 574 Monkenbusch, M. 572 Montrol, E.W. s e e Maradudin, A.A. 359 Monty, C. s e e Dufour, L.C. 225 Moog, E.R. 572
Author
837
index
Moog, E.R. s e e Unguris, J. 575 Moore, A.J.W. 97 Moore, I.D. 646 Moore, I.D. s e e Flores, F. 644 Moore, W.T. s e e Hui, K.C. 497 Morgante, A. s e e Bellman, A.F. 494 Morgante, A. s e e Btittcher, A. 494 Mori, R. s e e Kumagai, Y. 498 Morikawa, Y. 135 Morikawa, Y. s e e Kobayashi, K. 134 Morishige, K. 572 Morita, S. s e e Ueyama, H. 420 Moritz, W. 135,646 Moritz, W. s e e Dornisch, D. 134, 495 Moritz, W. s e e Gierer, M. 496 Moritz, W. s e e Kleinle, G. 498 Moritz, W. s e e Meyerheim, H.L. 498 Moritz, W. s e e Michalk, G. 498 Moritz, W. s e e Over, H. 499 Moritz, W. s e e Stampfl, C. 501,792 Moritz, W. see Zuschke, R. 502 Morkoq, H. 227 Morkot/, H. see Strite, S. 227 Morrison, J. s e e Estrup, P.J. 496 Morrison, J. s e e Lander, J.J. 135 Morrison, J.A. s e e Peters, C. 573 Morse, P.M. 791 Mortensen, K. 499 Mortensen, K. s e e Bedrossian, P. 133,494 Mortensen, K. s e e Zegenhagen, J. 792 Moruzzi, V.L. 646 Moser, H.R. s e e Heskett, D. 497 Mott, N.F. 182 Motteler, F.C. 572 Mouritsen, O.G. 710 Mouritsen, O.G. s e e Fogedby, H.C. 789 Mowforth, C. s e e Morishige, K. 572 Mowforth, C.W. 572 Mross, W. 499 Mueilerova, I. 419 Muirhead, R.J. 572 Muller, K. s e e Bickel, N. 225 Muller, K. s e e Lang, E. 359 Muller, K. s e e Rous, P.J. 360 MOiler, E.W. 419 Mtiller, J. 499 Mtiller, J.E. 646
Mtiller, K. 646 Mtiller, K. s e e Besold, G. 494 Miiller, K. s e e Chubb, S.B. 495 Mtiller, K. s e e Eggeling, von, C. 495 MOiler, K. s e e Gruzalski, G.R. 226 Miiller, K. s e e Mendez, M.A. 498 Mtiller, K. s e e Oed, W. 499 MiJller, K. s e e Pendry, J.B. 791 Mullins, W.W. 97 Mullins, W.W. s e e Gruber, E.E. 97 Mundenar, J.M. 499 Munoz, M.C. s e e Soria, F. 500 Murakami, S. s e e Gotoh, T. 181 Murata, Y. s e e Aruga, T. 494 Murata, Y. s e e Gotoh, T. 181 Murray, P.W. 227,499 Murray, P.W. s e e Leibsle, F.M. 498 Muryn, C.A. s e e Murray, P.W. 227 Muscat, J.-P. 647 Muscat, J.P. 499 Muschiol, U. 499 Muto, Y. 572, 647 Mykura, H. s e e Blakely, J.M. 96 Myshlyavtsev, A.V. 647 Nagayoshi, H. 499 Nagayoshi, H. s e e Kono, S. 135 Nagler, S.E. 572 Nahm, H.S. s e e Drir, M. 569 Nahr, H. s e e Finzel, H.-U. 359 Najafabadi, R. 647 Nakagawa, K. s e e Maree, P.M.J. 267 Nakamatasu, H. 182 Nakamura, N. s e e Kono, S. 134 Nakatani, S. s e e Takahashi, T. 136, 501 Nakayama, T. s e e Takami, T. 501 Napartovich, A.P. s e e Bol'shov, L.A. 642 Narasimhan, S. 135 Nardon, Y. 572 Narusawa, T. 647 Nastasi, M. s e e Tesmer, J.R. 420 Naumovets, A.G. 499, 647 Naumovets, A.G. s e e Bol'shov, L.A. 642 Neddermeyer, H. s e e Btiumer, M. 225 Neddermeyer, H. s e e Wilhelmi, G. 501 Needels, M. 267 Needels, M. s e e Brommer, K.D. 133
838 Needs, R.,I. 97 Neilsen, M. s e e Dornisch, D. 495 Nelson, D.R. 573 Nelson, D.R. s e e Halperin, B.I. 570, 790 Nelson, D.R. s e e Jos6, J.V. 790 Nelson, J.S. s e e Klitsner, T. 267 Nelson, R.C. 98 Nenow, D. 573 Neubert, M. s e e Schwarz, E. 500 Neugebauer, J. 499 Neugebauer, J. s e e Bormet, J. 494 Neugebauer, ,i. see Schmalz, A. 500, 648 Neve, J. s e e Lindgren, S.A. 498 Newns, D.M. s e e Muscat, J.P. 499 Ncwns, D.M. see Nc~rskov, ,I.K. 499 Newton, J.C. s e e Wang, R. 575 Newton, M.D. s e e Hill, R.J. 181 Ng, Lily s e e Uram, K.J. 501 Nham, H.S. 573 Nicholas, J.F. 98, 135,499 Nicholls, J.M. s e e Grehk, T.M. 496 Niehus, H. 710 Niehus, H. s e e Kern, K. 710 Nielsen, M. 573 Nielsen, M. s e e Bohr, J. 266, 568 Nielsen, M. s e e Dornisch, D. 134 Nielsen, M. see Dutta, P. 569 Nielsen, M. s e e Feidenhans'l, R. 496, 789 Nielsen, M. s e e Grey, F. 266 Nielsen, M. s e e Kjaer, K. 571 Nielsen, M. s e e McTague, J.P. 572 Nielsen, M. s e e Stampfl, C. 792 Nielsen, M.M. 499 Nielsen, M.M. s e e Aminpirooz, S. 493 Nielsen, M.M. s e e Stampfl, C. 501 Nielson, M. s e e Feidenhans'l, R. 266 Nienhuis, B. 791 Niessen, A.K. s e e Miedema, A.R. 498 Nightingale, M.P. 573,647 Niskanen, K.,I. 573 Nodine, M.H. s e e Gay, R.R. 225 Noffke, J. 499 Nofke, J. s e e Hermann, K. 570 Nogami, J. 499 Nogami, J. s e e Kono, S. 135 Nogami, J. s e e Shioda, R. 136, 500 Nogami, ,i. s e e Wan, K.J. 136, 501
Author
Nogler, S.E. s e e Specht, E.D. 574 Nolden, I. s e e Van Beijeren, H. 98 Nolder, R. 227 Nomura, E. s e e Katayama, M. 497 Noonan, J.R. s e e Gruzalski, G.R. 226 Nordlander, P. 647 Norman, D. s e e Lamble, G.M. 498 Norris, C. s e e Conway, K.M. 495 Norris, C. s e e Howes, P.B. 134 Norris, C. s e e Van Silfhout, R.G. 268 Nc~rskov, J. s e e Stoltze, P. 648 N~rskov, J.K. 135,499, 647 Nr J.K. s e e Besenbacher, F. 494 N~rskov, J.K. s e e Feidenhans'l, R. 496 Nc~rskov, J.K. s e e Jacobsen, K.W. 497, 790 NCrskov, .I.K. s e e Stokbro, K. 136 Northrup, J.E. 135,267, 499 Northrup, J.E. s e e Biegelsen, D.K. 266 Northrup, J.E. s e e Bringans, R.D. 494 Northrup, J.E. s e e Nicholls, J.M. 135,499 Northrup, J.E. s e e Uhrberg, R.I.G. 501 Northrup, J.E. s e e Zegenhagen, ,i. 792 Northrup, Jr., C.J.M. 182 Norton, P.R. 647 Norton, P.R. s e e Jackman, T.E. 419, 497 Novaco, A.D. 499, 573 Novaco, A.D. s e e Kjems, J.K. 571 Novaco, A.D. s e e McTague, J.P. 572 Novak, D. 227 Nowak, H.,I. 135 Nowotny, J. 182 Nowotny, J. s e e Dufour, L.-C. 180 Nozieres, P. 98 Nunes, R.W. s e e Li, X.P. 135 Nussbaum, R.H. 573 Nyberg, G.L. s e e Bare, S.R. 494 Nyholm, R. s e e Andersen, J.N. 493, 494 O'Keefe, M. 182 Ocai, C. s e e Bader, M. 494 Ochab, J. s e e Bak, P. 788 Ocko, B.M. 135 Ocko, B.M. s e e Zehner, D.M. 792 Oed, W. 499, 791 Oed, W. s e e Mendez, M.A. 498 Oed, W. s e e Muschiol, U. 499 Oed, W. s e e Pendry, J.B. 791
index
839
A uthor index
Oen, O.S. 419 Ogletree, D.F. 499, 647 Ogletree, D.F. s e e Barbieri, A. 642 Ogletree, D.F. s e e Van Hove, M.A. 268, 649 Ohdomari, I. s e e Hara, S. 226 Ohmura, Y. s e e Flores, F. 644 Ohnesorge, F. 227 Ohta, H. s e e Kinoshita, T. 498 Ohta, M. s e e Ueyama, H. 420 Ohta, T. 791 Ohtani, H. 499, 647 Okamoto, N. s e e Takahashi, T. 136, 501 Oils, A.M. 647 Olesen, L. s e e Klink, C. 790 Olmstead, M.A. 267,499 Oimstead, M.A. s e e Uhrberg, R.I.G. 136, 501 Ong, P.J. s e e Smilgies, D.M. 136 Onuferko, J.H. 791 Oppenheimer, J.R. s e e Born, M. 180 Ordej6n, P. 135 Ortega, A. s e e Pfntir, H. 500 Osakabe, N. 98,791 Osen, J.W. 573 Oshima, C. 227 Oshima, C. see Aono, M. 225 Oshima, C. s e e ltoh, H. 226 Oshima, C. s e e Souda, R. 227,420 Osuch, K. 227 Otani, S. 227 Otani, S. s e e Aono, M. 225 Otani, S . s e e Souda, R. 227,420 Outka, D.A. s e e D6bler, U. 495 Over, H 499, 500 Over, H s e e Bludau, H. 494 Over, H s e e Gierer, M. 496 Over, H s e e Hertel, T. 497 Over, H s e e Huang, H. 134, 497 Over, H s e e Stampfl, C. 501,792 Overhauser, A.W. s e e Dick, G.B. 180 Overney, R.M. 419 Ozaki, A. 500 Ozcomert, J.S. 98 Ozcomert, J.S. s e e Pal, W.W. 98,647 Pacchioni, G. 647 Packard, W.E. s e e Men, F.K. 791 Packard, W.E. s e e Tong, S.Y. 268
Pai, W.W. 98,647 Pai, W.W. s e e Ozcomert, J.S. 98 Palmari, J.P. s e e Bienfait, M. 568 Palmberg, P.W. 573 Palmer, R.L. s e e Kern, K. 571 Pan, J.M. s e e Maschhoff, B.L. 227 Pandey, K.C. 135 Pandey, K.C. s e e Kaxiras, E. 267 Pandey, K.C. s e e Mfirtensson, P. 135,498 Pandit, R. 573 Pandy, K.C. 182, 267 Pangher, N. 500 Pangher, N. s e e Aminpirooz, S. 493 Pangher, N. s e e Comelli, G. 495 Pantelides, S.T. 182 Paolucci, G. s e e Comelli, G. 495 Paolucci, G. s e e Dhanak, V.R. 495 Papaconstantopoulos, D.A. 135 Papaconstantopoulos, D.A. s e e Cohen, R.E. 643 Papanikolaou, N. s e e Dederichs, P.H. 643 Park, C. 500 Park, C.Y. s e e Abukawa, T. 493 Park, R.L. 43,359, 573,791 Park, R.L. s e e Hwang, R.Q. 645 Park, R.L. s e e Roelofs, L.D. 791 Park, R.L. s e e Taylor, D.E. 649 Park, R.L. s e e Williams, E.D. 360 Park, S.I. s e e Nogami, J. 499 Parkin, S.R. 500 Parkin, S.S.P. s e e Falicov, L.M. 789 Parmigiani, F. s e e Bagus, P.S. 494 Parmigiani, F. s e e Pacchioni, G. 647 Parr, R.G. 182 Parrinello, M. s e e Car, R. 133 Parrinello, M. s e e Ercolessi, F. 134, 644 Parrinello, M. s e e larlori, S. 181 Parrott, L. 500 Parry, D.E. 182 Parshin, A.Y. s e e Marchenko, V.I. 97,646 Pashitskii, E.A. s e e Braun, O.M. 643 Pashley, M.D. 267, 268,419 Passell, L. 573 Passell, L. s e e Carneiro, K. 568 Passeli, L. s e e Dutta, P. 569 Passell, L. s e e Eckert, J. 569 Passell, L. s e e Grier, B.H. 570
840 Passell, L. s e e Kjems, J.K. 571 Passell, L. s e e Larese, J.Z. 571,572 Passell, L. s e e Satija, S.K. 573 Passell, L. s e e Taub, H. 574 Passell, L. s e e You, H. 575 Passler, M.A. s e e Hannaman, D.J. 496 Pate, B.B. 182 Patel, J.R. s e e Zegenhagen, J. 502, 792 Paton, A. s e e Duke, C.B. 266 Paton, A. s e e Horsky, T.N. 267 Paton, A. s e e Kahn, A. 267 Patterson, H. s e e Grier, B.H. 570 Patterson, H. s e e Satija, S.K. 573 Paul, A. 182 Pauling, L. 227, 268, 573,647 Payne, M.C. s e e Needels, M. 267 Payne, M.C. s e e Stich, I. 136 Payne, M.C. s e e Teter, M. 136 Payne, S.H. 98,647 Peacor, S.D. 227 Pcdersen, J.S. 359 Pcderson, J.S. see Feidenhans'l, R. 266 Pederson, J.S. s e e Grey, F. 266 Pcderson, M.R. see Jackson, K. 134 Pcderson, M.R. s e e Perdew, J.P. 647 Pedio, M. 500 Pedio, M. s e e Becker, L. 494 Pchike, E. s e e Tersoff, J. 792 Peierls, R. 182, 573 Pelz, J.P. s e e B irgeneau, R.J. 568 Pendry, J.B. 359, 791 Pendry, J B s e e Andersson, S. 494 Pendry, J B s e e MacLaren, J.M. 267,498,646 Pendry, J.B s e e Muschiol, U. 499 Pendry, J B s e e Oed, W. 499 Pendry, J B see Rous, P.J. 360 Pendry, J.B s e e Wedler, H. 501 Pengra, D.B. 573 Pbpe, G. s e e Gay, J.M. 570 Perdereau, J. 135 Perdew, J.P. 135,647 Perez-Sandoz, R. s e e Weitering, H.H. 136 Perry, T. s e e Smith, J.R. 98, 136, 648 Pershan, P.S. s e e Braslau, A. 358 Persson, B.N.J. 647, 791 Pestak, M.W. 573 Peter, J. s e e Harten, U. 134
Author
Peters, C. 573 Peters, C. s e e Specht, E.D. 574 Peters, C.J. s e e Hong, H. 571,790 Peterson, L.D. s e e Farrell, H.H. 266 Petot-Ervas, G. s e e Dufour, L.C. 225 Pettifor, D.G. s e e Goodwin, L. 134 Pflanz, S. s e e Over, H. 499 Pfntir, H. 500, 791 Pfntir, H. s e e Jtirgens, D. 645 Pfntir, H. s e e Lindroos, M. 498 Pfn~ir, H. s e e Michalk, G. 498 Pfntir, H. s e e Piercy, P. 647 Pfniar, H. s e e Sandhoff, M. 648 Pfn~ir, H. s e e Schmidtke, E. 648 Pfntir, H. s e e Schwennicke, C. 648 Pfntir, H. s e e Sklarek, W. 648 Pfntir, H. s e e Sokolowski, M. 648 Phaneuf, R.J. 98,268, 791 Phaneuf, R.J. s e e Williams, E.D. 99 Phillips, J.C. 268 Phillips, J.M. 227 Phillips, J.M. s e e Bruch, L.W. 643 Phillips, K. s e e G6pel, W. 226 Pianetta, P. s e e Cao, R. 495 Pianetta, P. s e e Nogami, J. 499 Pianetta, P. s e e Woicik, J.C. 502 Pickett, W.E. 135 Pierce, D.T. 500 Pierce, D.T. s e e Falicov, L.M. 789 Piercy, P. 647 Piercy, P. s e e Pfntir, H. 500, 791 Piggins, N. s e e Pluis, B. 359 Pignet, T. see Christmann, K. 643 Pinkvos, H. 419 Pirug, G. 500 Pisani, C. 183 Pisani, C. s e e Causer, M. 180 Pisani, C. s e e Grimley, T.B. 645 Pitzer, K.S. s e e Sinanoglu, O. 574, 648 Plancher, M. s e e Doyen, G. 789 Plass, R. s e e Collazo-Davila, C. 418 Plass, R. s e e Marks, L.D. 419 Pluis, B. 359 Plummer, E.W. 500 Plummer, E.W. s e e Heskett, D. 497 Plummer, E.W. s e e Itchkawitz, B.S. 790 Plummer, E.W. s e e Mundenar, J.M. 499
index
Author
841
index
Poeisema, B. 359, 573,710 Poelsema, B. s e e Kunkel, R. 710 Poensgen, M. 98 Poensgen, M. s e e Frohn, J. 97, 644 Poensgen, M. s e e Giesen-Seibert, M. 644 Pohland, O. 98 Poirer, G.E. 227 Polanyi, M. 500 Polatoglou, H.M. s e e Tserbak, C. 649 Pollak, P. 500 Polli, M. s e e Bellman, A.F. 494 Pollmann, J. s e e Landemark, E. 267 Polzonetti, G. s e e Bagus, P.S. 494 Poon, T.W. 98,647 Pople, J.A. 183 Pople, J.A. s e e Hehre, W.J. 181 Popova, S.V. s e e Stishov, S.M. 183 Poppa, H. s e e Pinkvos, H. 419 Portz, K. 647 Prade, J. see Reiger, R. 183 Praline, G. s e e Parrott, L. 500 Preikszas, D. s e e Rose, H. 420 Presicci, M. 359 Press, W. 573 Preuss, E. s e e Bonzel, H.P. 96 Price, G.L. 573 Price, G.L. s e e Venables, J.A. 575 Prigge, D. s e e Ertl, G. 496 Prince, K.C. s e e Bellman, A.F. 494 Prince, K.C. s e e Comelli, G. 495 Prince, K.C. s e e Comicioli, C. 495 Prince, K.C. s e e Dhanak, V.R. 495 Prince, K.C. s e e Murray, P.W. 499 Prinz, G. s e e Falicov, L.M. 789 Pritchard, J. see Chesters, M.A. 568 Pritchard, J. s e e Horn, K. 497 Pritchard, J. s e e Roberts, R.H. 573 Prokrovskii, V.L. 573 Prutton, M. 183,227 Prutton, M. see Welton-Cook, M.R. 183 Pu Hu, Zi s e e Zuschke, R. 502 Puga, M.W. s e e Tong, S.Y. 268 Pukite, P.R. 359 Pulay, P. 135 Purcell, K.G. s e e Jupille, J. 790 Purcell, K.G. s e e Stensgaard, I. 792 Puschmann, A. s e e Bader, M. 494
Qian, G.-X. 268 Quate, C.F. 419 Quate, C.F. s e e Albrecht, T.R. 418 Quate, C.F. s e e Binnig, G. 418 Quate, C.F. s e e Nogami, J. 499 Quate, C.F. s e e Shioda, R. 136, 500 Quate, C.F. s e e Tortonese, M. 420 Quateman, J.H. 573 Quentel, G. 573 Quinn, J. 135,359, 500 Quinn, J. s e e Huang, H. 497 Quinn, J. s e e Over, H. 500 Qvarford, M. s e e Andersen, J.N. 493,494 Rabe, K.M. s e e Rappe, A.M. 135 Rablais, J.W. s e e Teplov, S.V. 420 Radom, L. s e e Hehre, W.J. 181 Rae, A.I.M. s e e Mason, R. 498 Raeker, T.J. 647 Raeker, T.J. s e e Stave, M.S. 136 Rahman, T.S. 500 Rahman, T.S. s e e Lehwald, S. 791 Raich, J.C. s e e Fuselier, C.R. 570 Rakova, E.V. 227 Ramamoorthy, M. 227 Ramana, M.V. s e e Fernando, G.W. 134 Ramsey, J.A. s e e Prutton, M. 183, 227 Ramseyer, T. s e e Roelofs, L.D. 135,791 Rangelov, G. 500 Rangelov, G. s e e Kiskinova, M. 498 Rangelov, G. s e e Surnev, L. 501 Ranke, W. 268 Rapp, R.E. 573 Rappe, A.M. 135 Rastomjee, C.S. s e e Rose, K.C. 420 Rausenberger, B. 419 Rausenberger, B. s e e Rose, K.C. 420 Ravikumar, V., D. Wolf, V.P. Dravid, Rayment, T. s e e Gameson, I. 570 Rayment, T. s e e Meehan, P. 572 Rayment, T. s e e Mowforth, C.W. 572 Raynerd, G. s e e Venables, J.A. 575 Razafitianamaharavo, A. 573 Redfield, A.C. 98,647 Reed, D.S. s e e Robinson, I.K. 360 Reeder, R.J. 183 R6gnier, J. 573
842 R6gnier, J. s e e Thomy, A. 574 Reichl, L.E. 791 Reider, K.H. 183 Reif, F. 791 Reiger, R. 183 Reihl, B. s e e Ferrer, S. 496 Reihl, B. s e e Lang, N.D. 571 Reihl, B. see Nicholls, J.M. 135 Reinecke, T.L. s e e Tiersten, S.C. 649 Reineker, P. s e e Spatz, J.P. 420 Rempfer, G.F. 419 Rempfer, G.F. see Skoczylas, W.P. 420 Rettner, C.T. s e e Barker, J.A. 642 Reuter, M.C. s e e Tromp, R.M. 420, 792 Reutt-Robey, J.E. s e e Ozcomert, J.S. 98 Reutt-Robey, J.E. s e e Pai, W.W. 98,647 Rhead, G.E. s e e Perdereau, J. 135 Rhodin, T.N. s e e Broden, G. 494 Rhodin, T.N. s e e Demuth, J.E. 643 Rhodin, T.N. see Gadzuk, J.W. 496 Rhodin, T.N. s e e Gerlach, R.L. 496 Rhodin, T.N. s e e lgnatiev, A. 571 Ricci, M. s e e Ealet, B. 225 Richardson, N.V. s e e Bare, S.R. 494 Richter, D. see Grier, B.H. 570 Richter, D. s e e Larese, J.Z. 571 Richter, L.J. 647 Rickard, J.M. see Duriez, C. 225 Rickard, J.M. s e e Quentel, G. 573 Ricken, D.E. s e e Robinson, A.W. 791 Rickman, J.M. 647 Rioter, R. 135 Ricdel, E.K. s e e Nienhuis, B. 791 Rieder, K.H 359, 360, 791 Rieder, K.H s e e Baumberger, M. 788 Ricder, K.H s e e Engel, T. 359, 496 Rieder, K.H s e e Haase, O. 496 Ricder, K.H s e e Koch, R. 498 Rieder, K.H s e e Swendsen, R.H. 360 Riedinger, R. s e e Dreyss6, H. 643 Riedinger, R. s e e Stauffer, L. 648 Rifle, D.M. 500 Rikvoid, P.A. 647, 648 Rikvold, P.A. s e e Collins, J.B. 643 Riley, F.L. 227 Roberts, A.D. s e e Johnson, K.L. 419 Roberts, R.H. 573
Author
index
Robinson, A.W. 791 Robinson, I.K. 43, 135, 360, 791 Robinson, I.K. s e e Bohr, J. 266 Robinson, I.K. s e e Fuoss, P.H. 359 Robinson, I.K. s e e Headrick, R.L. 496 Robinson, I.K. s e e Meyerheim, H.L. 498 Robinson, I.K. s e e Smilgies, D.M. 136, 792 Robinson, M.T. 420 Rocca, M. s e e Lehwald, S. 791 Rocker, G. s e e G6pel, W. 226 Rodge, W.E. s e e Abraham, F.F. 567 Rodriguez, A.M. 135 Rodriguez, A.M. 648 Roelofs, L.D. 135,648, 791 Roelofs, L.D. s e e Bartelt, N.C. 642, 788 Roelofs, L.D. s e e Payne, S.H. 98,647 Roetti, C. s e e Caus,~, M. 180 Roetti, C. s e e Pisani, C. 183 Rogowska, J.M. 648 Rohrer, G. 420 Rohrer, G.S. 183, 227 Rohrer, G.S. s e e Smith, R.L. 227 Rohrer, H. s e e Binnig, B. 96, 266, 418 Rohrer, H. s e e Binnig, G.K. 789 Rohrer, H. s e e Binning, G.K. 133 Rojo, J.M. s e e Miranda, R. 499 Rollefson, R.J. s e e Grier, B.H. 570 Rolley, E. s e e Balibar, S. 96 Root, T.W. 500 Rose, H. 420 Rose, J.H. 648 Rose, K.C. 420 Rosei, R. s e e Bellman, A.F. 494 Rosei, R. s e e Cautero, G. 495 Rosei, R. s e e Comelli, G. 495 Rosei, R. s e e Comicioli, C. 495 Rosei, R. s e e Dhanak, V.R. 495 Rosei, R. s e e Murray, P.W. 499 Rosenbaum, T.F. s e e Nagler, S.E. 572 Rosenblatt, D.H. s e e Kevan, S.D. 498 Rosenblatt, D.H. s e e Tobin, J.G. 501 Rosenblatt, D.H. s e e Tong, S.Y. 501 Rotermund, H.H. 420, 500 Rotermund, H.H. s e e B~ir, M. 494 Rotermund, H.H. s e e Engel, W. 496 Rotermund, H.H. s e e Jakubith, S. 497 Rottman, C. 98
Author
843
index
Rottman, C. s e e Bartelt, N.C. 96 Rottman, C. s e e Jayaprakash, C. 97,645 Rouquerol, J. 573 Rouquerol, J. s e e R6gnier, J. 573 Rous, P.J. 98, 360 Rous, P.J. s e e MacLaren, J.M. 267,498 Rous, P.J. s e e Nelson, R.C. 98 Rousina, R. s e e Dzioba, S. 225 Rousseau-Violet, J. s e e Le Boss6, J.C. 646 Rousset, S. 98 Rovida, G. s e e Bardi, U. 568 Rovida, G. s e e Galeotti, M. 225 Rownd, J.J. s e e Haneman, D. 267 Rowntree, P. 573 Rowntree, P.A. s e e Ruiz-Suarez, J.C. 573 Rubin, Y. s e e Snyder, E.J. 420 Rubio, G. s e e Agra'ft, N. 418 Rudberg, E. 360 Ruderman, M.A. 648 Rudge, W.E. s e e Koch, S.W. 571 Rudnick, J. 648 Rudolf, P. see Astaldi, C. 494 Rudolf, P. s e e Cautero, G. 495 Ruiz-Suarez, J.C. 573 Ruland, W. 573 Rundgren, J. 227 Rundgren, J. s e e Hammar, M. 226 Rundgren, J. s e e Lindgren, S.A. 498 Ruska, E. 420 Russo, J. s e e Liang, K.S. 359 Ryberg, R. s e e Persson, B.N.J. 647 Saam, W.F. s e e Jayaprakash, C. 97, 645 Saam, W.F. see Nightingale, M.P. 573 Sacedon, J.L. s e e Soria, F. 500 Sacks, W. s e e Rousset, S. 98 Sacramento, P. s e e Collins, J.B. 643 Sagawa, S. s e e Higashiyama, K. 497 Sagawa, T. s e e Kono, S. 135 Sagncr, H.-J. s e e Frank, H.H. 496 Saiki, K. see Hirata, A. 226 Saito, Y. 573 Sakai, A. s e e Lambert, W.R. 267 Sakama, H. 268 Sakawa, H. s e e Kawazu, A. 497 Sakuma, E. s e e Hara, S. 226 Sakurai, T. s e e Jeon, D. 134, 790
Sakurai, T. s e e Park, C. 500 Sakurai, T. s e e Taniguchi, M. 501 Salahub, D.R. 183 Salaneck, W.R. s e e Plummer, E.W. 500 Salanon, B. s e e Lapujoulade, J. 790 Salanon, M. s e e Falicov, L.M. 789 Saldin, D.K. s e e MacLaren, J.M. 267, 498 Saldin, D.K. s e e Oed, W. 499 Saldin, D.K. s e e Rous, P.J. 360 Salmeron, M. s e e Barbieri, A. 642 Salmeron, M. s e e Miranda, R. 499 Salmeron, M. s e e Ogletree, D.F. 647 Saloner, D. 710 Salvan, F. 500 Salvan, F. s e e Nicholls, J.M. 135 Salvan, F. s e e Thibaudau, F. 501 Samant, M.G. s e e Stoehr, J. 420 Samsavar, A. 710 Samsavar, A. s e e Hirschorn, E.S. 267 Sander, D. 98 Sander, M. 227 Sanders, D.E. s e e Stave, M.S. 136 Sandhoff, M. 648 Santucci, A. s e e Galeotti, M. 225 Sarid, D. 420 Sarikaya, M. 420 Saris, F.W. see Smeenk, R.G. 500 Saris, F.W. s e e Turkenburg, W.C. 420 Saris, F.W. see Van der Veen, J.F. 501 Satija, S. s e e You, H. 575 Satija, S.K. 573 Satija, S.K. s e e Passell, L. 573 Sato, H. 98 Satoko, C. s e e Tsukada, M. 183 Satti, D. s e e Thibaudau, F. 501 Sautet, P. s e e Barbieri, A. 642 Savage, D.E. s e e Saloner, D. 710 Savage, T.S. s e e Ai, R. 418 Saxena, A. 791 Schabel, M.C. s e e Northrup, J.E. 135 Schabes-Retchkiman, P.S. 573 Schabes-Retchkiman, P.S. s e e Venables, J.A. 575 Schtifer, J.A. s e e G6pel, W. 226 Schaffroth, Th. s e e Besold, G. 494 Schardt, B. s e e Ocko, B.M. 135 Schart, A. s e e Kern, K. 710
844 Schaub, T. s e e Tarrach, G. 227 Scheffler, M. 500, 648 Scheffler, M. s e e Bormet, J. 494 Scheffler, M. s e e Dabrowski, J. 266 Scheffler, M s e e Fiorentini, V. 134 Scheffler, M s e e Horn, K. 571 Scheffler, M s e e Methfessel, M. 97 Scheffler, M s e e Neugebauer, J. 499 Scheffler, M s e e Schmalz, A. 500, 648 Scheffler, M s e e Stampfl, C. 501,792 Scheffler, M. s e e Stumpf, R. 649 Scheid, H. 500 Schick, M. 573, 791 Schick, M. s e e Dietrich, S. 569 Schick, M. see Domany, E. 569 Schick, M s e e Gittes, F.T. 570 Schick, M s e e Kinzel, W. 645 Schick, M s e e Nienhuis, B. 791 Schick, M s e e Nightingale, M.P. 573 Schick, M s e e Pandit, R. 573 Schiff, L.I 360 Schildbach, M.A. 183 Schildberg, H.P. s e e Cui, J. 569 Schildberg, H.P. s e e Freimuth, H. 570 Schinlmelpfenning, J. 574 Schleyer, P.V.R. s e e Hehre, W.J. 181 Schlier, R.E. 135,268 Schlicr, R.E. s e e Farnsworth, H.E. 789 Schl6gi, R. see Erti, G. 496 Schluter, M. 268 Schmalz, A. 500, 648 Schmalz, A. s e e Aminpirooz, S. 493 Schmalz, A. s e e Becker, L. 494 Schmicker, D. 574 Schmidt, G. see Besold, G. 494 Schmidt, G. s e e Bickel, N. 225 Schmidt, G. see Eggeling, von, C. 495 Schmidt, K. s e e Koch, R. 498 Schmidt, L.D. see Flytzani-Stephanopoulos, M. 97 Schmidt, L.D. s e e Root, T.W. 500 Schmidtke, E. 648 Schmidtlein, G. s e e Pendry, J.B. 791 Schnatterly, S.E. s e e Jasperson, S.N. 571 Schnell-Sorokin, A.J. 98 Schober, O. s e e Christmann, K. 495 SchOnektis, O. s e e Heidberg, J. 570
Author
Schtinhammer, K. 648 Schott, J. s e e Hayden, B.E. 790 Schreiner, D.G. s e e Park. R.L. 359 Schrieffer, J.R. 648 Schrieffer, J.R. s e e Einstein, T.L. 644 Schrieffer, R. s e e Herman, F. 645 Schrtider, J. s e e Hwang, R.Q. 710 Schr6der, U. s e e de Wette, F.W. 180 Schr6der, U. s e e Reiger, R. 183 Schr6dinger, E. 183 Schuller, I.K. s e e Falicov, L.M. 789 Schulman, L.S. s e e Avron, J.E. 96 Schultz, H. s e e Dornisch, D. 495 Schultz, P.A. s e e Jennison, D.R. 645 Schulz, H. s e e Dornisch, D. 134 Schuster, R. 500, 648, 791 Schwartz, C. 574 Schwartz, L.M. s e e Ehrenreich, H. 644 Schwarz, E. 500 Schwarz, E. s e e Gierer, M. 496 Schwarz, E. s e e Koch, R. 498 Schwarz, E. s e e Over, H. 499 Schwegmann, S. 500 Schwennicke, C. 648 Schwennicke, C. s e e Schmidtke, E. 648 Schwennicke, C. s e e Sklarek, W. 648 Schwoebel, P.R. 648 Schwoebei, R.L. s e e Blakely, J.M. 96 Scire, R. s e e Young, R. 421 Scoles, G. s e e Ellis, T.H. 569 Scoles, G. s e e Rowntree, P. 573 Scoles, G. s e e Ruiz-Suarez, J.C. 573 Sears, M.P. s e e Jennison, D.R. 645 Seehofer, L. 500 Seguin, J.L. 574 Seguin, J.L. see Bienfait, M. 568 Seguin, J.L. s e e Suzanne, J. 574 Seguin, J.L. s e e Venables, J.A. 575 Seifert, R.L. s e e Yang, Y.-N. 99 Seiler, H. s e e Ichimura, S. 226 Seitz, F. 183 Selke, W. 648 Selke, W. s e e Kinzel, W. 646 Selloni, A. s e e Ancilotto, F. 133 Selloni, A. s e e Takeuchi, N. 268 Semancik, S. 183 Semancik, S. s e e Cox, D.F. 180, 225
index
Author
845
index
Semancik, S. s e e Doering, D.L. 495 Septier, A. 420 Ser, F. 57 Sesselmann, W. s e e Woratschek, B. 502 Shah, P.J. s e e Mouritsen, O.G. 710 Sham, L.J. s e e Kohn, W. 134, 182, 498,646 Sharma, Y.P. s e e Taub, H. 574 Shaw, C.G. 574 Shechter, H. 574 Shechter, H. s e e Brener, R. 568 Shechter, H. s e e Krim, J. 571 Shechter, H. s e e Wang, R. 575 Sheiko, S. s e e Spatz, J.P. 420 Shek, M.L. s e e Johansson, L.I. 226 Shelton, J.C. s e e Blakely, J.M. 96 Sheth, R. s e e Roelofs, L.D. 791 Shi, H. 500 Shi, H. s e e Jacobi, K. 497 Shiba, H. 574 Shibata, A. 791 Shibata, Y. s e e Oshima, C. 227 Shih, H.D. s e e Huang, H. 497 Shioda, R. 136, 500 Shirazi, A.R.B. 574 Shirley, D.A. s e e Huang, Z. 497 Shirley, D.A. s e e Kevan, S.D. 498 Shirley, D.A. s e e Tobin, J.G. 501 Shirley, D.A. s e e Tong, S.Y. 501 Shivaprasad, S.M. s e e Madey, T.E. 97 Shkrcbtii, A.I. s e e Takeuchi, N. 268 Shore, J.D. s e e Roelofs, L.D. 648 Shropshirc, J. 183 Shu, Q.S. s e e Ecke, R.E. 569 Shuttleworth, R. 98 Sibcncr, S.J. s e e Gibson, K.D. 570 Siboulet, O. s e e Rousset, S. 98 Sidoumou, M. 574 Sidoumou, M. s e e Audibert, P. 568 Sigmund, P. 420 Silverman,, P.J. s e e Stensgaard, 1. 136 Siiverman, P.J. see Kuk, Y. 646 Siivi, B. 183 Simon, A. 500 Sinanoglu, O. 574, 648 Sinclair, J.E. s e e Finnis, M.W. 134, 644 Singh, D. 136,791 Singh, D. s e e Roelofs, L.D. 135,791
Singh, D. s e e Yu, R. 136 Singh, D.J. s e e Perdew, J.P. 647 Sinha, S.K. s e e Dutta, P. 569 Sinha, Sunil K. 791 Skelton, D.C. 648 Skelton, D.C. s e e Wei, D.H. 649 Skibowski, M. s e e Claessen, R. 225 Skinner, A.J. s e e Goodwin, L. 134 Sklarek, W. 648 Skoczylas, W.P. 420 Skottke-Klein, M. s e e Over, H. 499 Slater, J.C. 183 Slater, J.R. 136 Siavin, A.J. s e e Feenstra, R.M. 266 Slijkerman, W.F.J. s e e Hara, S. 226 Slyozov, V.V. s e e Lifshitz, I.M. 791 Smeenk, R.G. 500 Smeenk, R.G. s e e Van der Veen, J.F. 501 Smilgies, D.M. 136, 792 Smit, L. s e e Derry, T.E. 180 Smith, D.J. 227,420 Smith, G.W. s e e Weakliem, P.C. 268 Smith, J.R. 98, 136, 648 Smith, J.R. s e e Ricter, R. 135 Smith, J.R. s e e Rose, J.H. 648 Smith. J.V. s e e Smyth, J.R. 183 Smith, P.V. s e e Zheng, X.M. 184 Smith, R.L. 227 Smith. S.T. 420 Smouluchowski, R. 136 Smyth, J.R. 183 Sneddon, L.G. s e e Mundenar, J.M. 499 Snyder, E.J. 420 Sokolov, J. 500 Sokolowski, M. 648 Solomon, E.I. s e e Gay, R.R. 225 Somers, J. s e e Horn, K. 497 Somers, J.S. s e e Robinson, A.W. 791 Somorjai, G.A. 43, 98, 183 Somorjai, G.A. s e e Barbieri, A. 225,642 Somorjai, G.A. s e e Batteas, J.D. 494 Somorjai, G.A. s e e Chung, Y.W. 225 Somorjai, G.A. s e e Dubois, L.H. 495 Somorjai, G.A. s e e French, T.M. 225 Somorjai, G.A. s e e Koestner, R.J. 498 Somorjai, G.A. s e e MacLaren, J.M. 267,498 Somorjai, G.A. s e e Ogletree, D.F. 499, 647
846 Somorjai, G.A. s e e Ohtani, H. 499,647 Somorjai, G.A. s e e Toyoshima, I. 649 Somorjai, G.A. s e e Van Hove, M.A. 98, 136, 268, 501,649 Song, K.-J. 98 Song, K.-J. s e e Madey, T.E. 97 Song, S. 98 SCrensen, O.T. 227 Soria, F. 500 Soszka, W. s e e Turkenburg, W.C. 420 Souda, R. 227,420 Souda, R. s e e Aono, M. 225 Soukiassian, P. 136 Soven, P. 648 Soven, P. s e e Kalkstein, D. 645 Sowa, E.C. 183 Spackman, M.A. s e e Geisinger, K.L. 181 Spadicini, R. s e e Garibaldi, V. 359 Spaepen, F. s e e Josell, D. 97 Spanjaard, D. see Bourdin, J.P. 642 Spanjaard, D. see Desjonqu6res, M.C. 643 Spanj,'lard, D. see Oils, A.M. 647 Spatz, J.P. 420 Spccht, E.D. 574 Spcnce, J.C.tt. 420 Spcnce, J.C.H. see Lo, W.K. 419 Spcnce, J.C.H. see Zuo, J.M. 421 Spicer, W.E. 501 Spicer, W.E. s e e Woicik, J.C. 502 Spiess, L. 136 Srolovitz, D.J. 648 Srolovitz, D.J. s e e Najafabadi, R. 647 Srolovitz, D.J. s e e Rickman, J.M. 647 Stair, P.C. 360 Stair, P.C. see Collazo-Davila, C. 418 St:lit, P.C. see Marks, L.D. 419 Stair, P.C. s e e Van Hove, M.A. 136 Stampfl, C. 501,792 Stancioff, P. s e e EI-Batanouny, M. 134 Stanlcy, H.E. 574 Stark, J.B. s e e Robinson, I.K. 43 Starke, U. see Heinz, K. 497 Starke, U. see Oed, W. 499,791 Starkey, E.K. s e e Batteas, J.D. 494 Stauffer, L. 648 Stave, M.S. 136 Stechel, E.B. s e e Jennison, D.R. 645
Author
Steele, W.A. 574 Stefan, P.M. s e e Johansson, L.I. 226 Stefanou, N. s e e Dederichs, P.H. 643 Steffen, B. s e e Bonzel, H.P. 96 Steinkilberg, M. 501 Steinkilberg, M. s e e Fuggle, J.C. 496 Stensgaard, I. 136, 792 Stensgaard, I. s e e Eierdal, L. 495 Stensgaard, I. s e e Feidenhans'l, R. 496 Stensgaard, I. s e e Jensen, F. 497, 790 Stensgaard, I. s e e Klink, C. 790 Stensgaard, I. see Mortensen, K. 499 Stephens, P.W. 574 Stephens, P.W. s e e Guryan, C.A. 570 Stern, E.A. s e e Bouldin, C. 568 Stern, E.A. s e e Rudnick, J. 648 Steslicka, M. s e e Davison, S.G. 643 Stevens, K. s e e Horsky, T.N. 267 Stewart, G.A. s e e Butler, D.M. 568 Stewart, G.A. s e e Litzinger, J.A. 572 Stich, I. 136 Stiles, K. see Duke, C.B. 266 Stiles, K. see Horsky, T.N. 267 Stiles, M.D. 648 Stillingcr, F. 136, 648 Stishov, S.M. 183 Stocker, W. see Baumberger, M. 788 Stocker, W. s e e Rieder, K.H. 791 Stockmeyer, R. see Monkenbusch, M. 572 St6hr, J. 420, 501 Stt~hr, J. s e e Di3bler, U. 495 Stokbro, K. 136 Stoll, E. s e e Binnig, G.K. 133, 789 Stoltenberg, J. 574 Stoltze, P. 648 Stoneham, A.M. 648 Stoneham, A.M. s e e Flores, F. 644 Stoneham, A.M. see Hayes, W. 226 Stoner, N. s e e Van Hove, M.A. 501 Stott, M. 648 Stout, G.H. 360 Strite, S. 227 Strite, S. s e e Morkog:, H. 227 Stroscio, J.A. s e e Feenstra, R.M. 418 Sttihn, B. see Wiechert, H. 575 Stulen, R.H. s e e Felter, T.E. 644 Stulen, R.H. s e e Sowa, E.C. 183
index
A uthor
847
index
Stumm, W. 183 Stumpf, R. 648,649 Sudo, A. s e e Nakamatasu, H. 182 Suematsu, H. s e e Hong, H. 571,790 Sugawara, Y. s e e Ueyama, H. 420 Suhl, H. s e e Levi, A.C. 359 Suhren, M. s e e Heidberg, J. 570 Sullivan, T.S. s e e Coulomb, J.P. 569 Sullivan, T.S. s e e Ecke, R.E. 569 Sulston, K.W. 649 Sun, Q. 649 Surnev, L. 501 Surnev, L. see Kiskinova, M. 498 Surnev, L. s e e Rangelov, G. 500 Susman, S. 183 Suter, R.M. s e e Greiser, N. 570 Sutter, R.M. s e e Colella, N.J. 568 Sutton, L.E. 501 Sutton, M. see Dimon, P. 569 Sutton, M. s e e Hong, H. 571 Sutton, M. s e e Mochrie, S.G.J. 572 Sutton, M. see Nagler, S.E. 572 Sutton, M. see Passell, L. 573 Sutton, M. see Specht, E.D. 574 Suzanne, J 574 Suzanne, J see Angot, T. 567 Suzanne, J see Audibert, P. 568 Suzanne, J s e e Beaume, R. 568 Suzanne, J s e e Bienfait, M. 568 Suzanne, J s e e Brener, R. 568 Suzannc, J. s e e Calisti, S. 568 Suzanne, J. see Coulomb, J.P. 568,569 Suzannc, J. s e e Gay, J.M. 570 Suzanne, J s e e Kramer, H.M. 571 Suzanne, J see Krim, J. 571 Suzanne, J see Meichei, T. 572 Suzanne, J s e e Migone, A.D. 572 Suzanne, J s e e Passell, L. 573 Suzanne, J s e e Seguin, J.L. 574 Suzanne, J s e e Shechter, H. 574 Suzanne, J s e e Sidoumou, M. 574 Suzanne, J s e e Venables, J.A. 575 Suzanne, J see Wang, R. 575 Suzuki, S. see Kinoshita, T. 498 Sverdlov, B. see Morkoq, H. 227 Swartz, L.E. see Biegelsen, D.K. 266 Swartzentruber, B.S. 98, 268, 7 l0
Swartzentruber, B.S. s e e Becker, R.S. 265,494 Swartzentruber, B.S. s e e Lagally, M.G. 267 Swartzentruber, B.S. s e e Webb, M.B. 268 Swendsen, R.H. 360, 574 Swiech, W. s e e Williams, E.D. 99 Szabo, A. 227 Szasz, L. 183 Szeftel, J.M. s e e Rahman, T.S. 500 Tabony, T. 574 Taglauer, E. s e e Beckschulte, M. 418 Takahashi, M. s e e Takayanagi, K. 136, 268, 710, 792 Takahashi, T. 136, 501 Takami, T. 501 Takata, K. s e e Hasegawa, T. 496 Takayanagi, K. 136, 268, 7 I0, 792 Takayanagi, K. s e e Haga, Y. 419 Takayanagi, K. s e e Shibata, A. 791 Takayanagi, K. s e e Yamazaki, K. 792 Takeuchi, N. 136, 268 Talapov, A.L. see Prokrovskii, V.L. 573 Tan, Y.T. see Baetzold, R.C. 180 Tanaka, H. 227 Tanaka, H. see Matsumoto, T. 182, 227 Tanaka, K. s e e Taniguchi, M. 501 Tanaka, T. s e e Otani, S. 227 Tang, S.P. s e e Spiess, L. 136 Taniguchi, M. 501 Tanishiro, Y. s e e Osakabe, N. 98 Tanishiro, Y. s e e Sato, H. 98 Tanishiro, Y. s e e Takayanagi, K. 136, 268, 710, 792 Tanishiro, Y. s e e Yamazaki, K. 792 Tarrach, G. 227 Tasker, P.W. 183,227 Tasker, P.W. s e e Baetzold, R.C. 180 Taub, H 574, 792 Taub, H s e e Carneiro, K. 568 Taub, H s e e Coulomb, J.P. 569 Taub, H s e e Hansen, F.Y. 570 Taub, H see Kjems, J.K. 571 Taub, H s e e Krim, J. 571 Taub, H s e e Larese, J.Z. 572 Taub, H s e e Suzanne, J. 574 Taub, H s e e Trott, G.J. 575 Taub, H. s e e Wang, R. 575
848 Tayler, L.I. s e e Roelofs, L.D. 135 Taylor, D.E. 649 Taylor, L.L. s e e Roelofs, L.D. 791 Taylor, T.N. s e e Parrott, L. 500 Teissier, C. 574 Teitel, S. s e e Jayaprakash, C. 97 Tejwani, M.J. 574 Tejwani, M.J. 792 Telieps, W. 792 Telieps, W. s e e Bauer, E. 418,709 Teller, E. s e e Axilrod, B.M. 568, 642 Teplov, S.V. 420 Terakura, K. s e e Ishida, H. 497 Terakura, K. s e e Kobayashi, K. 134 Terlain, A. 574 Terminello, L.J. s e e McLean, A.B. 498 Tersoff, J. 136, 420, 792 Tersoff, J. s e e Feenstra, R.M. 418 Tesmer, J.R. 420 Testardi, L.R. s e e Lind, D.M. 227 Teter, M. 136 Thciss, S.K. see Ganz, E. 496, 790 Thcodorou, G. 649 Thcodorou, G. s e e Tserbak, C. 649 Thibaudau, F. 501 Thicl, P.A. 183 Thici, P.A. see Behm, B.J. 133 Thiel, P.A. s e e Wang, W.D. 711 Thomas, G. see Griffiths, K. 790 Thomas, G.E. 501 Thomas, R.K. s e e Meehan, P. 572 Thomas, R.K. s e e Morishige, K. 572 Thomas, R.K. s e e Mowforth, C.W. 572 Thomlinson, W. s e e Carneiro, K. 568 Thomson, R.M. see Wert, C.A. 43 Thorny, A 574 Thorny, A see Bockel, C. 568 Thorny, A see Bouchdoug, M. 568 Thorny, A see Coulomb, J.P. 569 Thorny, A s e e Dolle, P. 569 Thorny, A s e e Dupont-Pavlovsky, N. 569 Thorny, A s e e Marti, C. 572 Thorny, A s e e Matecki, M. 572 Thorny, A s e e Menaucourt, J. 572 Thorny, A s e e Razafitianamaharavo, A. 573 Thomy, A s e e R6gnier, J. 573 Thorel, P. 574
Author
index
Thorel, P. s e e Coulomb, J.P. 568,569 Thorel, P. s e e Croset, B. 569 Thornton, G. s e e Leibsle, F.M. 498 Thornton, G. s e e Murray, P.W. 227,499 Thornton, J.M.C. s e e van der Vegt, H.A. 710 Thouless, D.J. s e e Kosterlitz, J.M. 359, 571, 790 Tiby, C. 574, 575 Tiersten, S.C. 649 Tobin, J.G. 501 Tobin, J.G. s e e Kevan, S.D. 498 Tobin, J.G. s e e Tong, S.Y. 501 Tobochnik, J. 792 Toennies, J.P. 360, 575 Toennies, J.P. s e e Harten, V. 710 Toennies, J.P. s e e Lahee, A.M. 498 Toennis, J.P. s e e Schmicker, D. 574 Tom~inek, D. 649 Tom~inek, D. s e e Dreyss6, H. 643 Tommasini, F. s e e Bellman, A.F. 494 Tommei, G.E. s e e Garibaldi, V. 359 Tompa, H. s e e Ruland, W. 573 Toney, M. see Bohr, J. 266 Toney, M.F. 575 Toney, M.F. s e e Shaw, C.G. 574 Tong, S.Y. 268,501 Tong, S.Y. see Hong, I.H. 497 Tong, S.Y. s e e Huang, H. 134, 497 Tong, S.Y. s e e Kevan, S.D. 498 Tong, S.Y. s e e Over, H. 499, 500 Tong, S.Y. s e e Tobin, J.G. 501 Tong, S.Y. see Van Hove, M.A. 360, 501 Tong, S.Y. s e e Wei, C.M. 136 Tong, S.Y. s e e Wu, H. 502 Tong, W.M. s e e Snyder, E.J. 420 Tong, X. s e e Pohland, O. 98 Tonner, B.D. s e e Stoehr, J. 420 Tonner, B.P. 420 Toppe, W. s e e Schwegmann, S. 500 T/3rnevik, C. s e e Hammar, M. 226 TCJrnqvist, E. s e e Narusawa, T. 647 Torrini, M. s e e Galeotti, M. 225 Torrini, M. s e e Grimley, T.B. 645 Tortonese, M. 420 Torzo, G. s e e Taub, H. 792 Tosatti, E. 136, 649 Tosatti, E. s e e Ancilotto, F. 133
Author
849
index
Tosatti, E. s e e Bilalbegovic, G. 96 Tosatti, E. s e e Ercolessi, F. 134, 644 Tosatti, E. s e e Garofalo, M. 790 Tosatti, E. s e e larlori, S. 181 Tosatti, E. s e e Takeuchi, N. 268 Tosatti, E. s e e Trayanov, A. 575 Tosatti, E. s e e Wang, C.Z. 136, 792 Toth, L.E. 227 Toyoshima, I. 649 Trabelsi, M. 575 Tracy, C.A. 792 Tracy, J.C. 501 Trafas, B.M. s e e Yang, Y.-N. 99 Tran, T.T. 227 Trayanov, A. 575 Trayanov, A. s e e Nenow, D. 573 Tr6glia, G. s e e Ol/~s, A.M. 647 Trevor, P.R. see Lambert, W.R. 267 Tringides, M.C. 649, 710 Tringides, M.C. s e e Saloner, D. 710 Tringides, M.C. see Wang, W.D. 711 Tringides, M.C. see Wu, P.K. 711 Tromp, R. s e e Himpsel, F.J. 267 Tromp, R.M 420, 792 Tromp, R.M s e e Maree, P.M.J. 267 Tromp, R.M s e e Rotermund, H.H. 420 Tromp, R.M see Schnell-Sorokin, A.J. 98 Tromp, R.M s e e Smeenk, R.G. 500 Tromp, R.M s e e van Loenen, E.J. 136 Tromp, R.M. s e e van der Veen, J.F. 501 Trost, J. 501 Trost, J. s e e Brune, H. 495 Trott, G.J. 575 Trott, G.J. s e e Coulomb, J.P. 569 Troullier, N. 136 Truhlar, D.G. s e e Truong, T.N. 649 Truong, T.N. 649 Tsen, S.C.Y. s e e Smith, D.J. 420 Tserbak, C. 649 Tsong, I.S.T. s e e Chang, C.S. 225 Tsong, T.T. 420, 649, 792 Tsong, T.T. see Casanova, R. 643 Tsong, T.T. s e e Chen, C.-L. 643 Tsong, T.T. s e e Cowan, P.L. 643 Tsong, T.T. s e e Mtiller, E.W. 419 Tsukada, M. 183 Tsukada, M. s e e Watanabe, S. 501
Tsuno, K. 420 Tucker, Jr., C.W. 501 Tung, R.T. s e e Robinson, I.K. 360 Tung, R.T. s e e Wei, J. 99 Turkenburg, W.C. 420 Ttishaus, M. s e e Persson, B.N.J. 647 Tyson, W.R. 98 Uchida, Y. s e e Lehmpfuhl, G. 419 Uchida, Y. s e e Wang, N. 421 Ude, M. s e e Takami, T. 501 Ueba, H. 649 Uebing, C. 649 Ueda, K. s e e Sakama, H. 268 Ueyama, H. 420 Uhrberg, R.I.G. 136, 268,501 Uhrberg, R.I.G. s e e Bringans, R.D. 494 Uhrberg, R.I.G. s e e Landemark, E. 267 Uhrberg, R.I.G. s e e Northrup, J.E. 135 Uhrberg, R.I.G. s e e Olmstead, M.A. 499 Umbach, C.C. s e e Keeffe, M.E. 97 Umbach, E. see Tobin, J.G. 501 Unertl, W. s e e Golze, M. 570 Unertl, W.N. 360, 420 Unertl, W.N. s e e Bak, P. 788 Unertl, W.N. s e e Clark, D.E. 789 Unertl, W.N. s e e Jackman, T.E. 497 Unguris, J. 575 Unguris, J. s e e Cohen, P.I. 568 Uram, K.J. 501 Urano, T. 183 Urbakh, A.M. 649 Urbakh, M.I. s e e Brodskii, A.M. 643 Uzunov, D.I. 792 Vainshtein, B.K. 43 Valbusa, U. s e e Ellis, T.H. 569 Van Acker, J.F. s e e Andersen, J.N. 493 Van Beijeren, H. 98 Van Beijeren, H. s e e A vron, J.E. 96 Van den Berg, J.A. s e e Verheij, L.K. 501 Van der Merwe, J.H. s e e Frank, F.C. 570 Van der Veen, J.F. 183,420, 501,792 Van der Veen, J.F. s e e Conway, K.M. 495 Van der Veen, J.F. s e e Denier van der Gon, A.W. 266 Van der Veen, J.F. s e e Derry, T.E. 180
850
Van der Veen, J.F. s e e Frenken, J.W.M. 496 Van der Veen, J.F. s e e Hara, S. 226 Van der Veen, J.F. s e e Himpsel, F.J. 181 Van der Veen, J.F. s e e Maree, P.M.J. 267 Van der Veen, J.F. s e e Pluis, B. 359 Van der Veen, J.F. s e e Smeenk, R.G. 500 Van der Veen, J.F. s e e Tromp, R.M. 420 Van der Veen, J.F. s e e Van Silfhout, R.G. 268 Van der Veen, J.F. s e e Vlieg, E. 360 Van der Vegt, H.A. 710 Van der Werf, D.P. s e e Heslinga, D.R. 497 Van Hove, M.A. 98, 136, 183, 268,360, 501, 649 Van Hove, M.A. s e e Barbieri, A. 225,642 Van Hove, M.A. s e e Batteas, J.D. 494 Van Hove, M.A. s e e Christmann, K. 643, 789 Van Hove, M.A. s e e Koestner, R.J. 498 Van Hove, M.A. s e e MacLaren, J.M. 267,498 Van Hove, M.A. s e e Ogletree, D.F. 499 Van Hove, M.A. see Ohtani, H. 499, 647 Van Hove, M.A. s e e Somorjai, G.A. 98 Van Hove, M.A. see Sowa, E.C. 183 Van Hove, M.A. see Van der Veen, J.F. 792 Van Hove, M.A. see Watson, P.R. 649 Van Huong, C.N. s e e Liu, S.H. 646 Van Loenen, E.J. 136 Van Loenen, E.J. s e e Dijkkamp, D. 97 Van Loenen, E.J. s e e Elswijk, H.B. 496 Van Pinxteren, H.M. 98 Van Pinxtesen, H.M. s e e van der Vegt, H.A. 710 Van Schilfgaarde, M. s e e Methfessel, M. 135 Van Sciver, S.W. 575 Van Sciver, S.W. s e e Hering, S.V. 570 Van Silfhout, R.G. 268 Vanderbilt, D. 98, 136, 183,268,710 Vanderbilt, D. s e e Alerhand, O.L. 96, 265,788 Vanderbilt, D. s e e Bedrossian, P. 133,494 Vanderbilt, D. s e e Li, X.P. 135 V anderbiit, D. s e e Meade, R.D. 97,498 Vanderbilt, D. s e e Narasimhan, S. 135 V anderbilt, D. see Ramamoorthy, M. 227 Vandervoort, K.G. s e e You, H. 711 Vannerberg, N.-G. 501 V aughn, P.A. s e e Shropshire, J. 183 V edula, Yu.S. s e e Naumovets, A.G. 499, 647 Venables, J.A. 575
Author
index
Venables, J.A. s e e Calisti, S. 568 Venables, J.A. s e e Faisal, A.Q.D. 569 Venables, J.A. s e e Hamichi, M. 570 Venables, J.A. s e e Price, G.L. 573 Venables, J.A. s e e Schabes-Retchkiman, P.S. 573 Venables, J.A. s e e Seguin, J.L. 574 Veneklasen, L. 420 Verhei ~, L.K. 501 Verheij, L.K. s e e Kunkel, R. 710 Verheij, L.K. s e e Poeisema, B. 573 Vernitron Corp., 421 Verway, E.J.W. 183 V erwoerd, W.S. s e e Badziag, P. 265 Verwoerd, W.S. s e e Osuch, K. 227 V ickers, J.S. s e e Becker, R.S. 265,494 Vickers, J.S. s e e Kubby, J.A. 267 Victora, R.H. s e e Falicov, L.M. 789 Vidali, G. 575,649 Vieira, S. s e e Agra'it, N. 418 Vilches O.E. s e e Bretz, M. 568 Vilches O.E. s e e Coulomb, J.P. 568, 569 Vilches O.E. s e e Ecke, R.E. 569 Vilches O.E. s e e Hering, S.V. 570 Vilches O.E. see Ma, J. 572 Viiches O.E. s e e Stoltenberg, J. 574 Vilches O.E. s e e Tejwani, M.J. 574, 792 Vilches O.E. s e e Van Sciver, S.W. 575 Vilches. O.E. s e e Zeppenfeid, P. 575 Vilfan, . s e e Villain, J. 792 Villain, J. 360, 792 Villain, J. s e e Wolf, D.E. 650 V issar, R.J. s e e Collart, E. 225 Vlieg, E. 136, 360 Vlieg, E. see Conway, K.M. 495 Viieg, E. s e e Headrick, R.L. 496 Vlieg, E. s e e van der Vegt, H.A. 710 Vogl, P. 183 Voigtlfinder, B. 649 Volin, K.J. see Susman, S. 183 Volkmann, U.G. 575 Volkmann, U.G. s e e Faul, J.W.O. 569 Volimer, R. s e e Schmicker, D. 574 Vollmer, R. see Toennies, J.P. 575 Volokitin, A.I. 649 V on Oertzen, A. s e e Engel, W. 496 Von Oertzen, A. s e e Jakubith, S. 497
Author
index
Von Oertzen, A. s e e Rotermund, H.H. 500 V ora, P. s e e Dutta, P. 569 Voronkov, V.V. 98,649 Vosko, S.H. s e e Perdew, J.P. 647 Voter, A.F. 649 Voter, A.F. s e e Liu, C.L. 97 Vu Grimsby, D.T. 496, 649 V vedensky, D.D. s e e MacLaren, J.M. 267, 498,646 Wachutka, G. s e e Scheffler, M. 500, 648 Wagner, R. 421 Walker, J.A. s e e Prutton, M. 183,227 Walker, J.S. s e e Domany, E. 569 Walker, S.M. s e e Grimley, T.B. 645 Wallden, L. s e e Lindgren, S.A. 498 Wailis, R.F. s e e Eguiluz, A.G. 644 Wailis, R.F. s e e Maradudin, A.A. 646 Waiters, A.B. s e e Boudart, M. 180 Wan, K.J. 136, 501 Wandclt, K. s e e Miranda, R. 572 Wang, C. 575 Wang, C.P. s e e Over, H. 499 Wang, C.Z. 136, 792 Wang, C.Z. s e e Xu, C.H. 136 Wang, G.C. 98,649, 710, 792 Wang, G.C. s e e Ching, W.Y. 643 Wang, G.C. see Lagally, M.G. 710 Wang, G.C. s e e Liew, Y.F. 791 Wang, G.C. s e e Wendelken, J.F. 792 Wang, G.C. see Yang, H.N. 711 Wang, G.C. s e e Zuo, J.K. 711,792 Wang, J. s e e Ocko, B.M. 135 Wang, L.Q. 227 Wang, M. s e e Dai, X.Q. 643 Wang, N. 421 Wang, R. 575 Wang, R. s e e Gay, J.M. 570 Wang, R. s e e Krim, J. 571 Wang, S.C. 501 Wang, S.K. s e e Wang, R. 575 Wang, S.W. s e e Van Hove, M.A. 268,649 Wang, W.D. 711 Wang, X.-S. 98 Wang, X.-S. see Wei, J. 99 Wang, X.W. 136, 792 Wang, Y. s e e Perdew, J.P. 135
851 Wang, Y. s e e Susman, S. 183 Wang, Y.C. s e e Chang, C.S. 225 Wang, Y.R. s e e Duke, C.B. 266 Wang, Y.R. s e e Kahn, A. 267 Wang, Z.L. 227, 421 Wang, Z.L. s e e Yao, N. 184 Ward, J. s e e Young, R. 421 Warmack, R.J. s e e Zuo, J.K. 228, 711 Warren, B.E. 360, 575 Washkiewicz, W.K. s e e Robinson, I.K. 43, 360 Wasterbeck, E. s e e Simon, A. 500 Watanabe, F. 649 Watanabe, F. s e e Ehrlich, G. 644 Watanabe, S. 501 Watson, P.R. 649 Watt, I.M. 420 Weakliem, P.C. 268 Weaver, J.H. s e e Yang, Y.-N. 99 Webb, M.B. 268, 360, 711,792 Webb, M.B. s e e Barnes, R.F. 358 Webb, M.B s e e Bennett, P.A. 788 Webb, M.B s e e Cohen, P.I. 568 Webb, M.B s e e Lagally, M.G. 267 Webb, M.B s e e McKinney, J.T. 359 Webb, M.B s e e Men, F.K. 791 Webb, M.B s e e Mo, Y.W. 710 Webb, M.B s e e Moog, E.R. 572 Webb, M.B s e e Phaneuf, R.J. 268, 791 Webb, M.B s e e Swartzentruber, B.S. 98,268, 710 Webb, M.B. s e e Tong, S.Y. 268 Webb, M.B. s e e Unguris, J. 575 Webber, P.R. s e e Bassett, D.W. 494 Weber, T. s e e Stillinger, F. 136, 648 Weber, W. s e e Wang, X.W. 136, 792 Wedler, H. 501 Weeks, J.D. 98, 711 Wegner, F.J. 792 Wei, C.M. 136 Wei, C.M. s e e Hong, I.H. 497 Wei, C.M. s e e Tong, S.Y. 268,501 Wei, D.H. 649 Wei, D.H. s e e Skelton, D.C. 648 Wei, J. 99 Wei, J. s e e Williams, E.D. 99 Wei, S.H. s e e Singh, D. 136 Weibel, E. s e e Binnig, G. 266, 789
852
Weimer, W. 575 Weinberg, W H. s e e Chan, C.-M. 495 Weinberg, W H. s e e Christmann, K. 643, 789 Weinberg, W H. s e e Comrie, C.M. 495 Weinberg, W H. s e e Kang, H.C. 645 Weinberg, W H. s e e Rahman, T.S. 500 Weinberg, W H. s e e Thomas, G.E. 501 Weinberg, W.H. s e e Van Hove, M.A. 183, 501,360, 649 Weinberg, W.H. s e e Williams, E.D. 501,650 Weinert, B. s e e Noffke, J. 499 Weinert, M. s e e Fu, C.L. 134, 710 Weinert, M. s e e Wimmer, E. 136, 650 Weiss, A.H. s e e Braslau, A. 358 Weiss, G.H. s e e Maradudin, A.A. 359 Weiss, H. see Ertl, G. 496 Weiss, H. see Schmicker, D. 574 Weiss, W. see Barbieri, A. 225 Weitcring, H.H. 136 Wcitcrling, H.H. s e e Heslinga, D.R. 497 Welkie, D.G. 711 Wclton-Cook, M.R. 183 Wclton-Cook, M.R. s e e Prutton, M. 183,227 Wcndeken, J.F. s e e Zuo, J.K. 711 Wendclken, J.F. 792 Wcndclken, J.F. s e e Zuo, J.K. 228 Wengelnik, H. see Wilhelmi, G. 501 Wern, H. see Courths, R. 643 Wcrt, C.A. 43 Wcrtheim, G.K. see Rifle, D.M. 500 Wesner, D.A. 501 W estrin, P. s e e Lindgren, S.A. 498 Wetzl, K. see Eiswirth, M. 496 Wheeler, J.C. see GriMths, R.B. 790 Whetten, R.L. see Snyder, E.J. 420 White, A.F. s e e Hochella, Jr., M.F. 181 White, G.K. 575 White, J.D. s e e Cui, J. 495 White, J.M. s e e Parrott, L. 500 White, J.M. s e e Poirer, G.E. 227 White, J.W. 575 White, J.W. see Meehan, P. 572 White, J.W. s e e Tabony, T. 574 Whitten, J.L. 649 Whitten, J.L. s e e Cremaschi, P.L. 643 Whitten, J.L. s e e Madhavan, P. 646 Wicksted, J.P. see Larese, J.Z. 571
Author
index
Widom, M. s e e Grest, G.S. 790 Wiechers, J. s e e Brune, H. 495 Wiechert, H. 575 Wiechert, H. s e e Cui, J. 569 Wiechert, H. s e e Feile, R. 570 Wiechert, H. s e e Freimuth, H. 570 Wiechert, H. s e e Knorr, K. 571 Wiechert, H. s e e Koort, H.J. 571 Wiechert, H. s e e Tiby, C. 575 Wiechert, H. s e e Weimer, W. 575 Wieckowski, A. s e e Rikvold, P.A. 647 Wiesendanger, R. s e e Gtintherodt, H.-J. 645 Wiesendanger, R. s e e Tarrach, G. 227 Wilhelmi, G. 501 Wilhelmi, G. s e e Btiumer, M. 225 Willenborg, K. s e e Dederichs, P.H. 643 Williams, A.A. s e e Pluis, B. 359 Williams, A.R. 649 Williams, A.R. s e e Lang, N.D. 4 9 8 , 5 7 1 , 6 4 6 Williams, A.R. s e e Moruzzi, V.L. 646 Williams, B.R. s e e Mason, B.F. 572 Williams, E.D. 99, 268,360, 501,650 Williams, E.D. s e e Barteit, N.C. 96 Williams, E.D. s e e Hwang, R.Q. 645 Williams, E.D. s e e Kodiyalam, S. 97 Williams, E.D. s e e Phaneuf, R.J. 98 Williams, E.D. s e e Taylor, D.E. 649 Williams, E.D. s e e Wang, X.-S. 98 Williams, E.D. s e e Wei, J. 99 Williams, R.S. s e e Katayama, M. 497 Williams, R.S. s e e Snyder, E.J. 420 Williams, W.S. 228 Willis, C.R. s e e Hsu, C.-H. 790 Willis, R.F. s e e Campuzano, J.C. 789 Willis, R.F. s e e Jeon, D. 134, 790 Wilsch, H. s e e Finzel, H.-U. 359 Wilson, E.B. s e e Pauling, L. 573 Wilson, K.G. 650, 792 Wilson, R.J. 136 Wilson, R.J. s e e Chambliss, D.D. 133, 710 Wilson, R.J. s e e Johnson, K.E. 710 Wilson, R.J. s e e WOlI, Ch. 136 Wimmer, E. 136, 501,650 Wimmer, E. ,see Fu, C.L. 134, 710 Winkelmann, K. s e e Toennies, J.P. 360 Winkler, R.G. s e e Spatz, J.P. 420 Wintterlin, J. 501
Author
853
index
Wintterlin, J. s e e Behm, R.J. 494 Wintterlin, J. s e e Brune, H. 495 Wintterlin, J. s e e Coulman, D.J. 495,789 Wintterlin, J. s e e Kleinle, G. 498 Wintterlin, J. s e e Trost, J. 501 Witzel, St. s e e Pollak, P. 500 Wohlgemuth, H. 502 Wohlgemuth, H. s e e Gierer, M. 496 Wohigemuth, H. s e e Over, H. 499 Wohlgemuth, H. s e e Schwarz, E. 500 Woicik, J. s e e Pate, B.B. 182 Woicik, J.C. 502 Wojciechowski, K.F. s e e Rogowska, J.M. 648 Wold, A. 183 Wolf, D. 99, 650 Wolf, D. s e e Moritz, W. 135 Wolf, D. s e e Zuschke, R. 502 Wolf, D.E. 650 Wolf, E. s e e Bom, M. 358 Wolf, J.F. s e e Frohn, J. 97, 644 Wolf, J.F. s e e Poensgen, M. 98 Wolff, P.A. s e e Schrieffer, J.R. 648 Wolken, G., Jr. 360 Wolkow, R. 711,792 W611, Ch. 136 W611, Ch. s e e Harten, V. 134, 710 W611, Ch. s e e Lahee, A.M. 498 Wong, C.W. 792 Wong, P.C. 502 Wong, Y.-T. 650 Wonka, U. s e e Finzel, H.-U. 359 Wood, E.A. 43 Woodruff, D.P. 183,421 Woodruff, D.P. s e e Kerkar, M. 497 Woodruff, D.P. s e e Onuferko, J.H. 791 Woodruff, D.P. s e e Robinson, A.W. 791 Woratschek, B. 502 Wortis, M. 99 Wortis, M. s e e Pandit, R. 573 Wortis, M. s e e Rottman, C. 98 Wright, A.F. 184, 650 Wright, A.F. s e e Leadbetter, A.J. 182 Wright, C.J. s e e Flores, F. 644 Wu, F.Y. 792 Wu, H. 502 Wu, J. s e e Pate, B.B. 182 Wu, N.J. s e e Wang, W.D. 711
Wu, P.K. 711 Wu, Y. s e e Stoehr, J. 420 Wu, Y.K. s e e Vu Grimsby, D.T. 496, 649 Wyckoff, R.G.W. 228 Wyckoff, W.G. 268 Wycoff, R.W.G. 184 Wyrobisch, W. s e e Bradshaw, A.M. 494 Wyrobisch, W. s e e Hofmann, P. 497 Xie, J. s e e Sun, Q. 649 Xiong, F. s e e Ganz, E. 790 Xdng, F. s e e Ganz, E. 496 Xu, C.H. 136 Xu, G. s e e Tong, S.Y. 268 Xu, J. s e e Rowntree, P. 573 Xu, J. s e e Ruiz-Suarez, J.C. 573 Xu, P. s e e Dunn, D.N. 418 Xu, W. 650 Yadavalli, S. 228 Y aegashi, Y. s e e Kinoshita, T. 498 Yagi, K. 421 Y agi, K. s e e Osakabe, N. 98,791 Y agi, K. s e e Sato, H. 98 Yagi, K. s e e Takayanagi, K. 136 Y agi, K. s e e Y amazaki, K. 792 Y alabik, M.C. s e e Rikvoid, P.A. 648 Yalisove, S. s e e Copel, M. 495 Yamada, M. s e e Bullock, E.L. 133 Yamazaki, K. 792 Y ang, C.N. s e e Lee, T.D. 791 Yang, H.N. 711 Yang, H.N. s e e Zuo, J.K. 228 Y ang, M.H. s e e Y adavalli, S. 228 Yang, W. s e e Parr, R.G. 182 Yang, W.S. 502 Y ang, W.S. s e e Huang, H. 497 Y ang, X. s e e Cao, R. 495 Yang, Y.-N. 99 Yaniv, A. 650 Yao, N. 184, 228 Yates, Jr., J.T. s e e Chen, J.G. 495 Y ates, Jr., J.T. s e e Uram, K.J. 501 Ye, L. 136 Ying, S.C. 792 Ying, S.C. s e e Chung, J.W. 789 Ying, S.C. s e e Han, W.K. 134
854 Ying, S.C. s e e Hu, G.Y. 790 Ying, S.C. s e e Roelofs, L.D. 791 Ying, S.C. s e e Tiersten, S.C. 649 Yip, S. s e e Poon, T.W. 98, 647 Yoon, M. 99 Yoshida, S. s e e Hara, S. 226 Yoshikawa, S. s e e Nogami, J. 499 You, H. 575,711 Youn, H.S. 575 Young, R. 421 Yu, R. 136
Zaima, S. s e e Oshima, C. 227 Zanazzi, E. 360 Zanazzi, E. s e e Galeotti, M. 225 Zangwili, A. 184, 650, 792 Zangwill, A. s e e Redfield, A.C. 98,647 Zaremba, E. s e e Stott, M. 648 Zartner, A. s e e Hofmann, P. 497 Zastavnjuk, V.V. s e e Teplov, S.V. 420 Zegenhagen, J. 502,792 Zeglinski, D M. s e e Ogletree, D.F. 647 Zehner, D.M 792 Zchner, D.M s e e Gruzalski, G.R. 226 Zehner, D.M s e e Mundenar, J.M. 499 Zehner, D.M s e e Yoon, M. 99 Zehner, D.M s e e Zuo, J.K. 228, 711 Zeiger, H.J. s e e Gay, R.R. 225 Zcller, R. s e e Dederichs, P.H. 643 Zemansky, M.W. 99 Zeng, H.C. s e e Parkin, S.R. 500 Zcppenfeld, P. 575 Zeppcnfeld, P. s e e Bienfait, M. 568 Zeppenfeld, P. s e e David, R. 569 Zeppenfeld, P. s e e Kern, K. 571, 710 Zerner, M.C. s e e Salahub, D.R. 183
Author
Zettlemeyer, A.C. s e e Klier, K. 498 Zhang, B.L. 136 Zhang, J.P. s e e Marks, L.D. 419 Zhang, Q.M. 575,792 Zhang, Q.M. s e e Kim, H.K. 571 Zhang, Q.M. s e e Larese, J.Z. 572 Zhang, S. 575 Zhang, T. 650 Zhang, T. s e e Dai, X.Q. 643 Zhang, T. s e e Sun, Q. 649 Zhang, Z. 99, 184 Zhang, Z.P. s e e Ai, R. 418 Zhdanov, V.P. s e e Myshlyavtsev, A.V. 647 Zheng, H. 650 Zheng, X.M. 184 Zhou, J.B. 228 Zhou, M.Y. s e e Hui, K.C. 497 Zhou, M.Y. s e e Parkin, S.R. 500 Zhou, M.Y. s e e Wong, P.C. 502 Zhou, W. 228 Zhu, D.M. 575 Zhu, D.M. s e e Pengra, D.B. 573 Zhu, X. 268 Zia, R.K.P. s e e Avron, J.E. 96 Zieger, H.J. s e e Henrich, V.E. 181 Ziegler, J.F. s e e Biersack, J.P. 418 Ziman, J.M. 136 Zimmer, R.B. 360 Zlati, V. s e e Gumhalter, B. 645 Zschack, P. 228 Zunger, A. 184 Zunger, A. see Bendt, P. 133 Zunger, A. s e e ihm, J. 134 Zuo, J.-K. s e e Wang, G.-C. 792 Zuo, J.K. 228,711,792 Zuo, J.M. 421 Zuschke, R. 502
index
Subject index adsorbed surfaces 159 adsorption 64, 68, 142, 322, 453,465,476, 53, 579, 592, 665,773 adsorption/desorption rate 633 adsorption effects on reconstruction 767 adsorption energies 594, 641 adsorption, heat and entropy of 533 adsorption, isosteric heat of 533, 534, 535 adsorption isotherms 509, 511,533,549, 554, 555, 561-563,566 adsorption sites 317,434, 460, 472, 474, 615 adsorption-induced dipoles 584 advanced Green's function 589 AES (Auger electron spectroscopy) 166, 217, 219, 456, 507,554, 661 affinity energy 587 AFM 195, 197, 223, 363, 373 Ag 67, 91, 105, 121, 127,676 Ag adsorption 124 Ag/Ag(l 11) 708 Ag(100) 83 Ag(110) 67, 116, 122, 449, 449, 637, 760 A g ( l l l ) 83,558 Ag( 11 l)-Cs 469, 470 ",/-3-x'~-f3Ag/Si( 111 ) and Au/Si( 111 ) 125 AIP 249 AI 111, 116, 128 AI adsorption 124 A! and Sn on GaAs(llO) 124 AI(OOI)-Na 464 AI(II0) 116 AI(I 11) 460, 474, 476, 476, 763 AI(II I)-K 469, 473,475,476 AI( 11 l)-Na 476 AI(I 11)-O 448,456 AI(I 1 l)-Rb 469
ab initio calculations 153 ab initio computations 156 ab initio methods 141, 144 ab-initio molecular dynamics simulations 110 ab initio psuedopotentials 148 ab initio SCF-LCAO method 147
aberrations in emission microscopes 417 accommodation 552 accommodation coefficient 689 activated process 438,450, 453 activation barrier 456, 582 activation energy 247, 450, 689, 772 adaptive grids 111 adatom interactions 659, 787 adatom model 487 adatom-substrate hopping 590 adatom-substrate coupling 601,604 adatoms 239, 289, 451,471,579, 665 adatom-adatom interactions 515 added-row 451 adhesion images 376 adiabatic calorimetry 535 adiabatic cleavage 58 adiabatic demagnetization 531 adiabatic process 57 adparticles 471 adsorbate-defect complex 177, adsorbate ordering 718 adsorbate-induced reconstructions 132, 718 adsorbate-adsorbate interactions 616, 697, 473,518, 517,564, 567 adsorbates 176, 325 adsorbate-substrate interactions 515, 517, 518, 564, 567 adsorbed layers 427 adsorbed overlayers 248 855
856 o~-A1203171, 194, 198, 207, 209 ct-A1203 (0012) 208 ot-Al203 (0001) 199, 200, 208,224 a-alumina 171, 172 y-alumina 207 AlAs 249 alkali adatoms 582 alkali adsorption 760, 761,769 alkali halides 175 alkali metal/Si(111) 127 alkali metals 116, 120 alkali metals on semiconductors 130 alkali-metal atoms 428,459 allotropes 174 alloy formation 476 AIN 219,220 ammonia 459 ammonia synthesis 477 ammonia synthesis reaction 459 amplitude ratios 746 amplitudes 737 Anderson model 586, 591,609, 638 angle-resolved photoemission 129 angle resolved photoemission spectroscopy (ARPES) 233 angular dependence of the surface free energy 77 angular-resolved UPS 440 anharmonic effects 666 anion terminated 258 anionic vacancies 175 anions 248,249, 253,261,264 an~sotropic temperature effects 437 antsotropy in the dimerization 671 anJsotropy of the surface tension 68 annealing 244 anti-NiAs structure 218 anti-Bragg condition 295 antibonding 430, 452, 484, 638,639, 697 antibonding electronic states 246 antibonding molecular orbital 479, 582 antiferromagnetic field 731 antiferromagnetic Ising model 730 antiferromagnetic Ising model phase diagrams 732 antiphase walls 684 Ar 511,518,540, 564
Subject index
Arrhenius plot 621 As 129 As capping 262 As dimers 262 As trimer 259 asymptotic interaction 607, 608 asymptotic regime 638 atom-atom correlation function 273 atom basis 194 atom diffraction 37 atom probe 387 atom scattering 273,555 atomic beam scattering 659 atomic connectivity 234 atomic coordinates 113 atomic force microscope s e e AFM atomic forces 144 atomic form factor 274, 327 atomic height steps 199 atomic packing 189 atomic polarizabilities 505 atomic relaxations 246 atomic resolution 380 atomic scale resolution 376 atomic scattering factor 289, 290, 658 atomic sizes 233 atomic steps 289 atomic vacancies and substitutions 174 atomic vibrations 79 atoms, rows of 389 atoms, strings of 389 atop sites 21,580, 587,639, 640, 768 attenuation coefficient 279, 284, 325 attenuation length 281 Au 91, 105, 127, 755 Au adsorption 124 Au(100) 120, 122, 755 Au(110) 120, 449, 619, 760, 771 Au(110)(Ix2) 116 A u ( l l l ) 120, 122,405,700 Au(111)(22xq-33) reconstruction 699 Auger de-excitation 478 Auger electron spectroscopy see AES Auger electrons 307 auto-correlation function 37,299 autocatalytic process 441 autocatalytic reaction process 483
Subject index
autocompensation 142, 164, 167, 17 I, 172, 174, 177. 247,250, 256, 259, 265 autocompensation principle 248 autocorrelation function 508,686, 687 average T-matrix approximation (ATA) 599 Axilrod and Teller 507 azimuthal ordering 468, 482 B. AI, Ga, In on Si(100) and Si(l I 1) 128 B 5 site 488 back focal plane 400 back-bonding 431,432, 483 back-donation 430, 431,468,485 background intensity 658 backscattering spectra 393 ballistic surface erosion 199 band gap 115 band structure 106, 154, 626 band structure effects 105 bandwidth 609 bare interaction 634 basal planes 532 basis functions 109 basisscts 110, 148 basis vectors 22 basis vectors of the reciprocal lattice 22 BaTiO 3 (100) 211 bcc 8.604, 611. 615. 620. 621. 623-625 bcc substrates 624 Bc 111 Beeby approximation 349 Beeby matrix inversion 323 bending-vibrational modes 435 bent bonding 435 benzoate 365 BeO 202 Bi 233 Bi atoms on Si 403 Bi/Si(l I 1) 129 Bi2Sr2CaCu208_ x 211 binary collisions 387,388 binding energy 6 5 , 5 4 7 , 5 4 8 , 5 8 0 binding sites 684, 695, 768 binding states 613 block renormalization 800 blocking 195 blocking dips 392
857 Blyholder model 429, 431 BN 219, 532 body-centered-cubic see bcc Boltzmann factor 35 o-bond 241,246 bond angles 104, 105, 142, 590 bond distortions 490 bond lengths 104, 107, 113, 142, 162, 169, 171, 172, 175, 178, 253, 447, 460, 469-471,474, 481,590, 606, 626, 630 bond-length-conserving rotations 255, 263,265 bond orientational order 521 bond-saturation model (BSM) 618 bond strength 456 bonded atoms 668 bonding 245,430, 587,639, 690, 697 bonding in ceramics 189 bonding orbital 582 Born approximation 274, 344 Born-Mayer repulsive term 155 Born-Oppenheimer approximation 145 bouncing mode 512 bound state 584 bound state resonances 348 Bragg angle 338 Bragg peak 659 Bragg points 293. 295 Bravais lattices 5. 19, 193, 233 points 5 unit cell vectors 5 bridge bonds 604 bridge sites 434, 587,593,640, 769 bright and dark field imaging 401,405 Brillouin zone 23, 288,667, 676, 686 Brillouin zone boundary 23 broken symmetry 793 brownian mobility 542 brownian motion 513 Brush 719 Buckingham potential 155 buckled dimers 662 buckled surface 475 buckling 178,485,606 buffer layers 675 bulk crystallography 5 bulk density of states 132 bulk elastic constants 155
858 bulk lattice constant 254 bulk modulus 672 bulk phonons 285 bulk states 115 Burger vector 41,663,672, 673,700 buried interfaces 344, 406 buried oxygen 459 butadiene-iron-tricarbonyl (BIT) 546 butane 512, 512, 512 c-BN 220 c-BN (001) 220 C ( I I I ) 161 C(I 1 l)-(2xl) 162 c(2x2) 581,598,627, 631 C2H 4 513 C2H 6 513,515,521,542 CaCO 3 196 cadmium chloride 511 calorimetry 533,535,566, 747 adiabatic 535 isothermal 535 canonical partition functions 726 canonical ensemble 65 cantilevers 378,379 CaO 202 capillary condensation 531,532, 566 carbides 143,212,222 carbon monoxide, adsorption of 428 carbonyl chemistry 427 carrier transport 655 Car-Parrinello method 109, 110 catalysis 139, 166, 176, 443,716 catalysts 171 catalytic CO oxidation reaction 440 catalytic poisoning 632 catalytic processes 117 catalytic reactions 174, 427 cation terminated surface 258 cations 248,249, 253, 261,264 CD 4 539 CdS 231,249, 263 CdSe 231,249 CdTe 249,403 centered meshes 13 centered sites 21,587,626 ceramic materials 187
Subject index ceramic synthesis 139 ceramics, bonding in 188 cesium peroxide (Cs20 2) 478 cesium suboxide (Cs! iO3) 478 cesium superoxide (CsO2) 478 Cf3CI 517 CF3CI 517 CH 4 509,513,514,559 chains 130, 237 chalcogens 446, 630 channeling 195,392, 393, 397 channeling/blocking 392 charge corrugation 349 charge density 107, 108, 116 charge density distribution 104 charge gradient 619 charge neutral surface 167 charge neutrality 178,200 charge state 171 charge transfer 247,449, 457,459, 464, 483, 486, 582, 600, 630, 633 charge transfer effects 105 charge-induced reconstruction 465 charged neutral surface 142 chemical adsorption 65 chemical bonding 159, 235,427,507 chemical forces 245 chemical potential 53, 57, 64, 65,580, 588, 633,723, 726, 729, 737, 773,804 chemical properties 273 chemical sensor applications 166 chemical sensors 176 chemical turbulence 443 chemical vapor deposition (CVD) 326 chemical waves 442 chemisorption 115, 174, 176, 427,443,448, 459, 505,580, 582, 584, 585,587,600, 636, 729, 787,788 chemisorption bond 582 chemisorption energy 431,450 chemisorption state 445 chemisorption, weak 596 classical force fields 174 classical force models 122 classical models 104 classical potential models 140, 154 classical potentials 104, 117, 145, 155, 175, 176
Subject index classical turning point 345,348 classification 747 classification of overlayer meshes 30 Clausius-Clapeyron line 533, 551,552 clean solid surfaces 55 cleavage 56, 58, 160, 172, 188, 233,235, 243, 247-249, 679 cleavage faces 236, 258,265 cleavage process 142 cleavage surfaces 142, 256 cleaved surfaces 234 cluster binding energy 627 cluster calculations 144, 448,582 cluster density 688 cluster methods 112 cluster models 156, 158, 159 cluster variation 631 clusters 112, 473,600, 613 Co on transition metals 427 CO covered Pt(ll0) 364 CO on Ni(100) 431 CO on Pt(100) 413,417 CO on Pt(ll 1) 659, 660 CO/Ru(0001) 430 Co(1010) 453 coadsorbate phase 480 coadsorption 428, 461,483,485,586 coesite ! 74 coexistence 509, 510, 563,567, 615 coexistence regions 533,730, 780 coherence length 532, 553, 686 coherence limitations 733 coherent domains 277 coherent lattices 30 coherent potential approximation 583 cohesive energy 104, 189, 507, 618,629, 725, 771 coincidence lattices 30, 686 collective excitations 541 collective motion 513 Coloumb and exchange interactions 150 combined space method 323 commensurability 483, 517 commensurate adlayer 515 commensurate lattices 30 commensurate phase 555,561 commensurate solid phase 559
859 commensurate solids 510, 513,514, 522, 545, 567 commensurate structure 515, 560 commensurate-incommensurate transition 516, 519, 554, 567 complete wetting 562 completely ionic material 163 complex meshes 15 composition 53 compound formation 456, 477 compound semiconductors 247, 248 cohapressibility 701 concentration 53 concerted movements 771 condensation 534, 550, 551,564, 688 configurational entropy 65, 81,633 conjugate honeycomb-chained-trimer 126 constituents 508 contamination 549 continuous phase transitions 567 continuous transition 525,561,720, 766 continuum approximation 604 continuum elasticity 605 contractions 448 contractive reconstructions 119, 132 contrast transfer function 402 convergence 108, ! 12 convergent beam-REM 405 convexity 69, 72 convolution theorem 304 CoO 201,202 coordination 618,639 coordination number 470, 474, 617, 624 coordination polyhedra 189, 190 core-electrons 108 core-level-shift spectroscopy 758 corner-cube- and face-centered-anisotropy 754 corrected effective medium theory 616 correlated collisions 392 correlated roughness 709 correlation effects 112, 600 correlation energy 148 correlation function 580, 632, 633, 665,737, 738,741,751 correlation length 701,740, 749, 753, 798, 799, 804, 806 correlation length exponent 804
860 correlations 38,545, 581,600, 625,678,747, 798,799 corrosion 139, 176, 443 corrugation 348, 349, 468, 469, 471--473, 481, 483 corrugation function 345 corundum 171, 179, 193, 198,207 Coulomb forces 163,374 Coulomb potential 154, 155 Coulomb repulsion 120, 609, 616, 623 Coulombic interactions 144, 148, 154, 155 coupled channel calculation 348 coupled channel techniques 275,344, 348 covalent bonding 106, 132, 188, 231,764 covalent radius 471 coverage 64, 510, 550, 561,665,678,680, 726, 774, 785 CPA 599 Cr20 3 207, 209 Cr3C 2 213 or- and 13-cristobalite 174 critical angle 335 for X-rays 330 critical behavior 448 critical exponents 508,522, 535, 718, 736, 737,742, 743,746-748,766, 788,798, 800, 801,804-806 critical hopping 601 critical line 751 critical mismatch 516 critical phenomena 508 critical point 511,531,567, 716, 718,720, 728,736, 797,794, 799 critical scattering 739, 741,749, 766 critical temperature 509, 533, 797 critical wetting temperature 563 crossover point 445 crystal growth 691,704, 705 crystal lattices 5 crystal shape 53 crystal structure factor 276, 289 crystal symmetry 189 crystal truncation rods 280, 299, 305 crystalline metal oxides 142 crystallographic definition of the step 71 crystallographic orientation 68 crystallography of a plane 10, 11
Subject index Cu 91,105 Cu on Cu(100) 684 Cu(100) 684 Cu(110) 116, 365,364, 449, 464, 761 Cu(110)(2• reconstruction 364, 697,698 Cu(110)-K 761 Cu(110)p(2• 450 Cu(110)-O 451 Cu(111) 673, 674 Cu(l I l)-Cs 469 Cu(113) 667 Cu(115) 667 Cu3Au(001) 679 cubic anisotropy 751 cubic sphalerite structure 215 cubium 600, 601 CuCI 249, 251 d-bands 591 d-shells 105 dangling bond electrons 141 dangling bonds 127, 129, 132, 141, 142, 163, 167, 170, 177, 217, 231,234, 236, 242, 246-248,250, 488,668,669, 758 rehybridization of 234, 246 saturation of 234 dangling orbitals 770 dark field imaging 401 Darwin width 335,339 Davidson scheme 109 deBroglie relation 309, 356 deBroglie wavelength 304, 382 Debye temperature 286 Debye-Waller 352 Debye-Waller effect 666 Debye-Waller exponent 286 Debye-Waller factor 326, 511,548,743,744 decomensuration 231 defect interactions 223,641 defect microstructure 197 defect sites 174 defect states 131 defect structures 157 defects 35, 142, 159, 194, 195, 221,224, 231, 245,277, 289, 290, 298,343, 456, 458, 655,656, 674, 699, 704, 708, 725,784, 803, 805
Subject index
of the first kind 38, 39, 655,658 of the second kind 38, 39, 655 of the third kind 656 defocus 401 degree of ionicity 189 delocalized rr bonding 243 dendritic islands 691 densities 793 density function theory 242 density functional calculations 474 density functional formalism 105, 107, 464 density functional method 144, 145, 147, 149, 151 density functional pseudopotential 173 density functional theories 108, 115 density matrix methods 107, 111 density of states 581,588,616, 655 depolarization 467, 468,470 of dipoles 462 depth of field 412 desorption 461,467, 549,630, 633,773 desorption temperature 479 devil's staircase 517 diagonalization procedures 109 diamond 179,231 diamond cleavage surface 160 diamond lattice 236 diamond surfaces 142, 160 dielectric 139 dielectric constant 139 dielectric permittivity 139, 165 difference map 340 diffraction 29, 273,537,538,655-666, 674, 678,683,688,697, 706, 708 Laue condition for 279 diffraction intensity oscillations 707,708 diffraction lineshape 288, 319 diffraction methods 33 diffraction oscillations 680 diffraction pattern 400 diffraction peaks 327 diffraction profiles 325,538 diffraction rod 305 diffuse intensity 739 diffusion 53, 124, 513,515,634, 658,686, 691,707,716, 718,730, 760, 761, 771-773,776
861 diffusion anisotropy 689, 690, 691 diffusion barriers 690, 693,771 diffusion coefficients 358,530, 547,635,688 diffusion constant 787 diffusion mechanisms 122 diffusion of defects 709 diffusion time 787 diffusivity 93 dimensionality 511 dimer pair energy 624 dimer rows 130, 668,688, 689, 697 dimerization 178 dimers 237, 239, 246, 247,249, 260, 261,582, 630, 661,662, 677 dimer-adatom-stacking fault model 123, 237 diperiodic groups 18 dipolar repulsions 85 dipole moment 459, 464, 466, 467, 470, 472, 507,582 dipole-dipole coupling 436, 583 dipole-dipole interactions 88,469, 582, 583, 635,636 dipole-dipole repulsion 467,471,473,603 dipole-quadrupole force 583 dipoles 505 dipoles, screening between the 471 direct imaging 398 direct interactions 579, 581,580, 586, 587, 626 direct lattice 21 direct methods 363 direct space 273 directional bonding 105 directional d-bonding 119 directions 23 directions of a form 6 disclinations 523,567 discrete Gaussian model 86 disk of confusion 402 dislocation, edge orientation 41 dislocation line 39 dislocation, mixed 41 dislocation networks 709 dislocation, screw orientation 41 dislocations 199, 219, 221,223, 523,523,545, 554, 567, 655,672, 673,705,707 disorder 35,242, 680, 701,703, 788 disorder effects 703
862 disordered structures 289 disordering transitions 760 dispersion 609 dispersion curves 155, 513, 558 dispersion forces 506 dispersion relations 513 displacement sensors 371 displacive reconstructions 118, 119, 757, 759 dissociation 431,445,549, 725 dissociation energy 445,621 dissociative adsorption 580 dissociative chemisorption 448 distorted rhenium oxide 193 dividing surface 60, 61 domain boundary pinning 704 domain boundary 42 domain boundary pinning 704 domain degeneracy 695 domain growth kinetics 693 domain size 671,675,698,701,703,704 domain size distribution 687,703,704, 775 domain walls 42, 520, 523, 545,554, 567,656, 672,688,693,695,696, 701,709 domains 281,317,655,656 domains 664, 669, 674, 684-687,693, 694, 696, 697, 702, 703 domains on stepped surfaces 696 o donation 485 donor-acceptor mechanism 429 double alignment technique 391,392 double-layer missing-row reconstruction 454 dynamic LEED 322 dynamic linear dielectric function 507 dynamic polarizability 507 dynamical coupling 513 dynamical matrix 605 dynamical scattering 326 EAM (embedded atom method) 105, 122, 614, 616, 617,619, 625,629, 636, 771 edge dislocation 39 EDIM (embedded diatomics-in-molecules) 618 EELS (electron energy loss spectroscopy) 143, 211,615,634 effective medium 599 effective medium theory s e e EMT effective radius 469
Subject index
Eikonal approximation 345,346 Einstein mode 513 elastic constants 104, 604 elastic deformation 375 elastic distortion 602 elastic effects 601,604, 635 elastic incoherent structure factor (EISF) 541 elastic interactions 88, 90, 606, 698 elastic mean free path 279 elastic medium 636 elastic moduli 635 elastic repulsions 85 elastic strain, release of 490 elastic stress in the surface region 486 elastic yield 307 elasticity theory 93 electric dipole moments 582 electric multipole 508 electrical insulator 139 electrochemical cells 641 electrochemical methods 117, 369 electrochemistry 176 electron bombardment 175 electron charge density 351 electron correlation 141, 149, 151, 162 electron correlation effects 149 electron correlation energy 148 electron counting 142 electron density 151 electron density gradient 342 electron diffraction 733 electron double counting terms 154 electron-electron interactions 146, 148, 151, 152, 153 electron-energy loss spectroscopy see EELS electron exchange coupling 610 electron gas 109 electron-gas model 156 electron-ion energy 151 electron microscopy 91,363, 398, 766 electron spin 146 electron-stimulated desorption (ESD) 194, 205,310, 555 electron-stimulated reactions 199 electron tunneling 364 electronegative adsorbates 582 electroneutrality 430
863
Subject index
electronic charge density 108 electronic correlations 623 electronic friction 448 electronic hopping 579 electronic properties 21,273 electronic structure 106, 235, 367,655 electronic wavefunctions 104, 108 electronic work function 461 electropositive adsorbates 582 electrostatic dipole layer 461 electrostatic interaction 146, 486, 582 elemental semiconductors 236 elementary excitations 112 ellipsometry 560, 566 embedded atom method s e e EAM embedded cluster model 613-616 embedded cluster technique 600 embedded diatomics-in-molecules s e e EDIM embedding functions 617-619 emission electron microscopy 408 empirical classical potential models 144, 159 empirical quantum-mechanical methods 141 empirical schemes 619 empirical techniques 107 cmpiricai tight-binding methods 153, 159 EMT 105,613,614, 615,629, 770 corrected 616 energetics 234 energetics of surface defects 122 energy barrier 684 energy barriers 771 energy expansion 721 energy potential relief 471 enthalpy 534, 535 entropic repulsion 91 entropic step repulsion 81 entropy 60, 79, 81, 82, 85,534, 535,657,658, 665 epitaxial growth 210, 224, 242, 684, 699, 702, 773 equilibrium 53,767 equilibrium crystal shape 53 equilibrium faceted surfaces 68 equilibrium surface atomic geometry 144 equivalent-crystal-theory 105 ESDIAD 436 ethane 510, 514, 521,524, 525,525,542, 543,
550, 553 evaporation 451 Ewald construction 278, 279, 281,309 Ewald sphere 279, 315, 320 Ewald term 108 exchange constants 719 exchange-correlation energy 107, 109, 152 exchange-correlation functionals 107 exchange-correlation potentials 153 exchange energies 151 exchange interaction 150 exchange mechanisms 629 exchange-overlap forces 505 exfoliated graphite 532 extended defects 175, 363 extended Htickel model 594, 640 extended systems 112 extensive variables 70 extrinsic stacking fault 676 face-centered cubic s e e fcc facet planes 197 faceted TaC( 11O) 216 faceting 55, 68, 69, 74, 122, 199, 224, 242, 681,682 faceting transitions 73,581 facets 30, 67, 197, 201,221,655 family of bulk lattice planes 35 far asymptotic region 609 Faraday Cup 313 fcc 8,606, 619 fcc(100) 446 f c c ( l l l ) 116,446 Fe(211) 449 Fej_xO 190, 201,202 Fel_xO(l 11) 206 ot-Fe203(O001) (10]-2) 194 ~-Fe203(O001) 194 ot-Fe203 171, 196, 207 Fe304 206 Fe304(001) 207 Fe304(111) 206 FeAI20 4 206 feedback systems 371 Fermi energy 382, 440, 588,595,601,607, 638 Fermi level 115, 132, 464, 488 Fermi surface 106, 580, 592, 610, 636, 640
864 Fermi surface domination 591 Fermi surface effects 105 ferrimagnetic 206 ferroelectrics 210 ferromagnetic phase 720 ferromagnetic system 719 FeTiO 3 209 field desorption 386 field emission 315 field evaporation 386 field ion microscopy s e e FIM film growth 406 FIM 363, 380, 381,385, 611,620, 623,626, 627,632, 663, 771 FIM, resolution of 384 FIM tip 621,623 finite-size effects 343, 701,702, 743,749, 767, 783 finite-size limitations 702 finite-size rounding 743 finite-size scaling 803,804, 805 finite-size scaling analysis 702 Finnis-Sinclair models 105 first ionization potential 462 first-order condensation 563 first-order discontinuity 727 first-order melting 525 first-order phase transition 243, 718 first-order transition 550, 561,672, 720, 723, 724, 749, 750, 765,766 first-principles calculations 618 first-principles methods 104, 107, 141, 144 Fischer-Tropsch reaction 428 five-dimensional transition and noble metal surfaces 120 FLAPW (full potential linear augmented plane wave) 108 flood electron gun 307 fluctuation-dissipation theorem 740 fluctuations 681,722, 723,738,742, 748, 778, 784, 798, 799, 804 fluid phase 726, 727 fluorite 193, 202 focal length 402 Fock operator 150 force sensor 372 force sensor, cantilever beam 373
Subject index
forces 110 forces on each atom 158 foreshortening 405 form factor 274 four-circle geometry 331 fourfold hollow site 684 fourfold sites 455 fourfold-coordinated hollow sites 448 Fourier transform 738-740 fractal 691,708 fractal dimensionality 691 fracture 178 Frank and Van der Merwe theory 516, 519 Fraunhofer diffraction 400 Fredholm determinant 589 free-electron models 116 free energy 53, 76, 658,675,723, 736, 740, 746, 776, 777,799, 800, 804 free energy barrier 723 free energy expansion 747 free-electron gas 591,614 free-fermion approximation 92 freezing 565 Fresnel reflectivity 342 Fresnel theory 561 Friedel (1958) sum rule 599 Friedel oscillations 608, 617 Friedel sum rule 638 frontier orbitals 581,596 full potential linear augmented plane wave s e e FLAPW fullerene 375 Ga 128 Ga adsorption 124 Ga trimer 259 Ga vacancy 258 GaAs 124, 130, 231,249, 254, 367 G a A s ( - l - l - l ) 259 GaAs(100) 260 GaAs(100)-c(2• 235 GaAs(110) 123 GaAs(l I 1)-(2• 258 G a A s ( l l l ) and (-1-1-1) 258 GaN 218-220 GaP 249 GaP(I 11)-(2x2) 258
Subject index
gas phase 726 GaSb 249 Gaussian approximation 277, 278 Gaussian functions 147 Ge 111, 160, 231,233, 236, 246 Ge(100) 244 Ge( 111 )-(2xl) 247 Ge(l 11) 487, 765 Ge( 111 )-c(2x8) 247 Ge(l 11)(x/3-• 127 Ge(lll)-(2• 162, 163 germanium 243 Gibbs 55, 60, 68 Gibbs adsorption equation 64, 66 Gibbs dividing surface 60, 64 Gibbs free energy 719, 726, 795 glass formation 139 glassy phases 224 glide lines 16, 18 glide plane 455 glide-plane symmetry 481 global minimum ! 14 glue model 105, 122,617 gold 338 gradient corrections 107 gradient corrections 619 grain boundaries 35, 159, 174, 219, 291,363, 387 Gram-Schmidt 587 grand canonical partition functions 726 grand partition function 726 grand potentials 63, 69, 726 grand thermodynamic potential 57 graphite 377, 509-514, 516-518, 520, 521, 524, 525,532, 540, 542, 543,545,546, 550, 560, 563,564, 566 graphitic (2x2) 615 graphitic (2x2) or (2x2)-2H 618 Green's function 111, 112, 156, 157, 158, 588, 589, 592, 596, 604, 605,609, 610, 611, 612, 624, 628,635,639, 640 advanced 589 matching techniques 112 techniques 144 Grimley 584 ground state 114, 581 ground state energy 109
865 groundwater transport of contaminants 139 Group III elements AI, Ga, In on Si(100) 128 groups 16, 724 growth exponent 703 growth kinetics 703 growth mode 708 growth rate 705 GW approximation 115 H 3 site 487 halides 143 Halperin-Nelson model 525 Hamaker constant 374 Hamiltonian 108, 146 Hamiltonian matrix 106, 107, 146, 153 hard hexagon lattice gas model 750 hard wall approximation 345 hard wall model 345 hard-sphere radii 469, 471,473 hard-square model 631 harmonic approximation 518 harmonic-oscillator approximation 471 harpooning 462 Hartree (or Coulomb) interaction 150 Hartree and exchange-correlation potential 108 Hartree and exchange interactions 153 Hartree energy 152, 153 Hartree-Fock 162, 587, 600 Hartree-Fock approximation 505 Hartree-Fock bonding and antibonding resonances 599 Hartree-Fock calculations 107 Hartree-Fock computations 173 Hartree-Fock level 149 Hartree interaction 148 Hartree-type calculation 107 hcp 8, 606 hcp crystals, lattice planes 8 hcp sites 447,448,480 hcp(0001) 446 He 116 3He 511 4He 511 He diffraction 165,456, 659, 707 He scattering 116, 566, 657, 660, 666, 667,684 heat and entropy of adsorption 533 heat capacity 511, 59, 522, 535, 806, 807
866 heats of adsorption 431,549, 551,552 heats of cleavage and adsorption 64 heavy domain walls 42 height-height distribution functions 37 Heisenberg model 749, 753,754 Heisenberg uncertainty principle 384 helium atom scattering 275 helium atoms, wavelength of the 356 Hellmann-Feynman theorem 110 Heimholtz coils 308 Helmholtz free energy 56, 63, 69 Hermann-Mauguin notation 18 Hermitian matrix eigenvalue equation 109 Herring 55, 68 herringbone packing 521 herringbone pattern 122, 521 herringbone reconstructions 700 herringbone solid 521,525 herringbone structures 699 heteroepitaxial growth 124, 248 heteroepitaxial systems 687 hetcrogeneous catalysis 428,459 heterolytic dissociation 177 heterostructures 605 hexagonal aligned incommensurate phase 555 hexagonal-close-packed s e e hcp hexagonal rotated incommensurate phase 555 hexagonal rotated phase 557 hexagonal wurtzite structure 215 hexatic liquid crystal 523 hexatic orientational order 525 hexatic phases 559 hexto interaction 619 HfC 213,214 HfN 218 high-density surfaces 446 high energy electron diffraction (HEED) 273 high-order commensurate adlayer 515 high resolution electron energy loss spectroscopy s e e HREELS high resolution electron microscopy s e e HREM high resolution LEED 313 high-temperature annealing 175 high-symmetry sites 448,465,471 higher order commensurabilities 517 highest occupied molecular orbitals s e e HOMO hill- and valley structure 68
Subject index
[hkl] zone 33 holding potential 507, 641 hollow sites 21,434, 446, 465,471,699 HOMO (highest occupied molecular orbital) 429, 596 homoepitaxial growth 684, 688,690, 692, 697 homoepitaxy 689 homogeneous function 799 homologous classes 235 honeycomb-chained-trimers 125, 126, 492 honeycomb structure 447 honeycomb symmetry 83 hopping 384 hopping translations 542 hot adatoms 448 HREELS (high resolution electron energy loss spectroscopy) 440, 447, 455,464, 471, 480, 484 HREM (high resolution electron microscopy) 197, 198 Hi.ickel model 594 m extended 594 hybridization 429,464, 465,486, 490, 612, 641 hydrogen 160 hyperoxide ions 0 2 478 hysteresis 370, 723,724, 796
ICISS 165 ideal crystal 8 ideal gas 64 ~deal substrate lattice 25 deal surface 8 dentity transformation 16 iluminated area 332 iimenite 193,209 mage charge 465 image formation 400 image intensifiers 386 image plane 584 image shift 462 imperfections 35 improper rotations 16 impurities 42, 202, 586, 655,704, 708,725, 784, 803, 805 impurity atoms 39 In 128 In adsorption 124
Subject index
in-phase condition 302, 667,679, 692, 693 in-phase scattering 295 lnAs 249 incoherent lattices 30 incoherent scattering 540 incommensurability 516, 517, 557 incommensurate adlayer 515 ncommensurate chains 517 ncommensurate close-packed surface atomic layer 120 incommensurate-commensurate transition 547 incommensurate lattices 30 incommensurate layer 490 incommensurate overlayers 358 incommensurate phase 490, 521,545,559, 560, 561,724 with a hexagonal network of walls 520 incommensurate reconstruction 759 ncommensurate solids 510, 513, 523,545,567 ncommensurate structures 515, 518,686 ncommensurate superstructure 479 ncomplete wetting 562 ndependent particle partition function 84 ndex of refraction 330 ndirect interactions 579, 585, 607,611,612, 637 inelastic mean free path 279 inelastic mean free path length 306 inelastic neutron scattering 540 inelastic scattering 306, 659 InN 219,220 inner potential 319, 320, potential 322 inP 249 lnSb 249 InSb(l 11)-(2• 258 instantaneous atomic positions 37 instrument resolution 303 instrument response function 304, 743 insulators 115 integrated intensities 335 intensive variables 70 interaction energies 82, 640 interaction potentials 505,507 lnteratomic interactions 104 interface thickness 680 Interface width 683 interfaces 55, 159
867 interference fringes 405 ~nterference function 276, 290 ~nterferometry 380 nterlayer diffusion 707 nternal energy 55, 56, 63, 658 nternal polarization 464 international notation 18 International Union of Pure and Applied Physics 5 interplanar distances 113 interplanar spacings 62 interstitial atoms 189 interstitials 39, 655 ~ntrinsic stacking fault 676 ~ntrinsic step anisotropy 665 nverse photoemission 115, 129, 440, 465,638 nverse spinel structure 191 nversions 16 ion backscattering spectra 390 ion backscattering techniques 363,387 Ion beam induced damage 197 ion bombardment 175,239, 241,244, 249, 258 ion neutralization spectroscopy 601 ion scattering 113, 125, 143,717,758 ion scattering spectrometry see ISS 213, 233, 396 ionic bonding 188, 231,464, 470 Ionic bonding framework 163 ionic insulator surface 154 ionic insulators 144 ~onlc interaction 428 ionic materials 169, 178 ionic potential 108 iomc radius 189 ionically bonded insulating systems 153 ionlcity 233,254, 471 iomzation 587 ~omzation cross section 551 ionization gauge 551 ionization level 383 ionization probability 381 IPES 457 Ir 122, 755 Ir(001) 403 Ir(l 11) 447 iron oxide 206 irrelevant fields 795,801
868
Subject index
Ising antiferromagnet 730 Ising behavior 522 Ising model 83, 84, 632, 704, 705,716, 718-722, 726, 747,750, 760, 766, 794, 798,800 lsing transition 524 sland formation 448,476 island growth 486 sland nucleation 706 sland phase 730 sland separation distribution 687 island shape 690 islands 440, 458,627,687,688,689, 705 :sosteric heat of adsorption 507, 533,534, 535, 633 isothermal calorimetry 535 isothermal process 57 isotherms 509, 550, 553,555,564, 566 ISS (ion scattering spectrometry) 203,213, 233,396
kink density 73, 79, 85, 88, 89 kink energy 74, 82, 83, 84, 90, 91,670 kink formation energy 242, 665,669 kink-kink interactions 83, 670 kink-kink separation 93 kink sites 223 kinks 33, 71, 79, 87, 122, 174, 176, 239, 565, 663,665,666, 669, 670 kinks, thermal excitation of 85 KMnF 3 169, 170 knife edge singularity 85 Kohn-Sham self-consistent approach 609 Kohn-Sham equation 108, 109 Kohn-Sham method 151, 152 Kondo problem 638 Kosterlitz-Thouless point 751 Kosterlitz-Thouless transition 747 krypton 516, 520, 545,558,563 krypton/graphite 522 KZnF 3 170
Jahn-Teller transition 697 jcllium 119, 507,582, 583,585,586, 590, 601, 607,608,609,611,612,613,616, 631 jeilium model 461 jump to contact 375
La203 198, 199 lamellar halides 532 Landau classification 746 Landau rules 723,766 Landau theory 747, 798 Landau's first rule 724 Landau's second rule 723, 724 Landau's third rule 724, 765,766 Langdau expansion 722 Langmuir adsorption model 65 Langmuir-Gurney picture 476 Langmuir-Gurney model 461,464, 466, 483 laser-induced diffusion 635 late-time growth 777 latent heat 534 lateral disorder 674 lateral force mode 376 lateral interaction energy 555 lateral interactions 580, 584, 617, 626, 627, 631,634, 635,638, 641 lateral length distribution 680 lateral resolution of the STM 367 lattice 655 lattice constant 257 lattice, direction in a 6 lattice dynamics 557,634
K 111, 120 K(110) 759 keatite 174 kinematic analysis 273 kinematic approximation 287, 322, 659, 678, 687 kinematic approximation in 3-d 274 kinematic diffraction amplitude 733 kinematic diffraction intensity 733, 738 kinematic model 326 kinematic scattering 336, 345 kinematic scattering intensity 742 kinematic scattering theory 557 kinetic accessibility 261 kinetic energy 151 kinetic limitations 187 kinetic oscillations 428,435,441 kinetic phenomena 772 kinetics 141, 162, 163,656, 704, 788 kink corner energies 83
Subject index
lattice gas 581,585,620, 631,632, 695 lattice gas analogy 716 lattice gas Hamiltonian 725 lattice gas models 615,703,716, 725,773 lattice gas phase 728 lattice gas system 803 lattice gas transformation 730 lattice line 16 lattice liquid 620 lattice liquid phase 728 lattice mismatch 41,290, 516, 567,697 lattice planes 6 family of 6 lattice sites 79, 581,678 lattice-gas 580, 633 lattice-gas order parameters 733 Laue conditions 537 for diffraction 279 layer groups 18 layer spacings 326 layer-by-layer growth 680, 705-708 layered perovskite 211 LCAO (linear combination of local orbital) 108,591,599 LDA (local density approximation) 107, 109, 112, 115, 116, 117, 122, 628,640 lead zirconate titinate ceramic 369 ledges 33 LEED (low energy electron diffraction) 37, 54, 113, 121, 125-127, 143, 160, 162, 165, 166, 172, 194, 203-205,207,208, 210, 211,214, 215,217-219, 223,233, 234, 241,254, 258,273, 305,358,436-438, 447,448,450, 453,455-458,468, 471-476,480, 482, 485,507, 511,514, 518,548,566, 601,606,615,631,661, 694, 697,732, 738,742,755,758,760, 762, 763, 765 m resolution 320 LEEM (low energy electron microscopy) 408, 766 m contrast mechanisms in 408,409 resolution in 409 samples for 411 left-handed coordinate system 28 Legendre transformations 57 LEIS (low energy ion scattering) 194, 396, 399
869 Lennard-Jones 505 Lennard-Jones one-dimensional diagram 445 Lennard-Jones potentials 104, 105,506, 548, 640 Lennard-Jones repulsion 583 Lennard-Jones interaction 93 Lenz 719 LEPD (low energy positron diffraction) 143 level mixing 462 Li 111 Li and Na on Ru(0001) 469 LiF(001) 352 lifetime-broadening 462 Lifshitz criterion 724 light domain walls 42 limitations to resolution 363 LiNbO 3 171,209 line tension 84 linear bridge sites 21 linear chains 482 linear-combination of atomic orbitals s e e SCF-LCAO linear combination of local orbital s e e LCAO linear sites 580 linear transformation 27 liquid phase 726 liquid-gas critical point 727 liquid-like structures 469 LiTaO 3 171 LMTO (linear muffin-tin orbitais) 108 local coordination 189 local density approximation s e e LDA local density functional method 431 local density of states 132, 623 local distortions 615 local equilibration 53 local functional 109 local minimum 113 local orbital based methods 111, 113 local relaxations 590 local roughness 295 localization of a state 115 London dispersion forces 505 long-bridge sites 450, 452, 615 long-range correlations 699 long-range forces 565 long-range interactions 637, 697
870 long-range order 289, 325,343,469, 479, 508, 632,655,697, 701,718,719, 732 long-range translational order 523 longitudinal mode 513 Lorentz factor 303, 338 Lorentzian function 542, 739 low coverage 112 low energy electron diffraction see LEED low energy electron microscopy ,tee LEEM low energy ion scattering see LEIS low energy positron diffraction see LEPD low index directions 31 low index surfaces 8 lowest unoccupied molecular orbitals s e e LUMO LUMO (lowest unoccupied molecular orbital) 429, 596 M203 207 Madclung sum 155 magnctlc fields 325 magnetlc forces 374 magnetic ordering 587 magnetxc sandwiches 641 magnetic shielding 308 magnettte 206 magnetization 719, 720, 726, 795,797, 800, 801 magnification 381 marginal fields 801 mass flow 53 mass transport 53,450 matrix method 317 matrix notation 25, 27 Maxwell relationship 58, 64 MBE (molecular beam epitaxy) 235,249, 258, 259, 260, 261,262, 706 template 239 McLachlan modification 583 mean field 107,631 mean field theory 587,747, 795,796 mean square displacement 511 mean square displacements of steps 91 mean square fluctuation 663 mean squaref height fluctuations 299, 300, 678 mean squared vibrational amplitude 286 medium energy ion scattering s e e MEIS
Subject index
MEIS (medium energy ion scattering) 126, 162, 194, 392, 457, 770 melting 231,514, 523,524, 525,546 melting point 565 melting transition 342, 469 meso-scale structures 376 mesoscopic ordered domains 122 metal carbonyls 428,429,430, 434, 435 metal dimers 128 metal overlayers on semiconductors 123 metal-semiconductor transition 242 metal surfaces 104, 116, 427, 660 metal trimer overlayer 489 metal-carbonyl complexes, vibrational spectra of 430 metal-semiconductor bonds 489 metallic bonding 188,464, 469 metallic contacts 124 metallization 473 metals 115 metal-metal bonds 127, 489, 491 metal-semiconductor bonds 127,491 metal-semiconductor interfaces 104, 123,486 metastable (1• structure 455 metastable phases 113, 124, 476, 796 methane 511,542 Mg 111 MgO 163, 177, 201,406, 507,513, 533, 539, 559, 566 MgO(001) 164, 165, 176 MgO(100) 187, 1 9 5 , 2 0 1 , 2 2 3 , 5 1 3 , 5 1 4 , 5 4 2 MgO(l 11) 224 mica 676 microchannel plate 386 microelectronics 174, 239 microfacets 205 microscopy 656 Miller indices 6, 22, 33, 35 mirror electron microscopy 408 mirror plane 17 mirror-reflection planes 17 miscut angle 242 mismatch 516 critical 516 lattice 516 missing row 441 missing row model 454
871
Subject index
missing row phase 759 missing row reconstructions 119, 132, 449, 456, 465,759, 761,768,769, 771 missing row structure 451 mixed-basis pseudopotential approach 110 mixed-basis technique 111 mixed dislocation 674 mixed phases 42 mixed representation 587,605 MnO 201,202 Mo 132 Mo(001) 106, 118 Mo(100) 116 Mo(310) 416, 416 Mo2C 213 ]'- MoC 192 mobility 468,546 molecular adsorbate systems 427 molecular adsorption 431 molecular beam epitaxy see MBE molecular coordination chemistry 255 molecular dynamics 105, 119, 159, 162, 508 molecular dynamics simulations 110 molecular orbitals of CO 429 molecular orientational order 521 molecular vibrations 512 Moliere potential 394 momentum transfer 300, 347 momentum transfer vector 274 Monte Carlo 631 Monte Carlo calculations 745 Monte Carlo simulations 79, 119, 6 ! 3, 614, 615, 618, 619,627,632, 633,634, 639, 703,704, 705,706, 806 Monte Carlo techniques 394 mosaic 41,289, 290, 291,674, 676 mosaic structure 675 MOssbauer spectroscopy 511,545 Mott-Littleton approach 156, 159, 175, 176 mound structures 684 muffin tin 631 muffin-tin orbitals 106 muffin-tin spheres 613, 614 Mulliken (1934) electronegativity 587 Mullins 81 multicomponent systems 62 multigrids 111
multiple height steps 679 multiple scattering 284, 322, 326, 346, 347, 678,733,738 multisite interactions 604 Na 111 NaCI 507, 708 NaF(001) 354 NaxWO 3 (100) 211 NbC 213 NbN 218 Ne 518 nearest neighbors 189 nearest-neighbor repulsion 581 negative surface excesses 62 neon 511 neutralization 397 neutron scattering 535,566, 717 neutron scattering experiments 533 next-nearest-neighbor attraction 581 Ni 105 Ni island nucleation 699 Ni(001) 316, 317 Ni(00 I)-C 764 Ni(001)c(2• 457 Ni(100) 457,686 Ni( 100)-C 763 Ni( 100)-K 470 Ni(100)-O 448,456 Ni(l 10) 449, 762 Ni(110)(2• 440 Ni(110)(2• 453,455 Ni(l 10)-CO 440 Ni(110)p(2• 450, 453 Ni(110)p(3• 453 Ni(l 11)-O 742 Ni(l 11)-K 468,469 Ni(l 13) 667 Ni(115) 666, 667 Ni(771) 453 NiO 201,202 NiO(100) 202, 222, 457,686 N i O ( l l l ) 457 niobium pentoxide 193 nitrides 143, 218,222 nitrogen 511,521 Ni-CO 430
872 no load point 375 noble-metal adsorption on silicon and germanium surfaces 125 noble metals 116, 117,637, 640 noble metals on Si or Ge 127 non-rotated hexagonal phase 520 non-stoichiometric surfaces 142 non-bonded atoms 668 non-bonding electronic states 246 non-channeling 393 non-contact imaging 376 non-directional bonding 581 non-equilibrium structure factors 7903 non-interacting steps 302 non-registered binding 729 non-uniform strain 675 non-universal behavior 747,753 non-wetting 562 normal mode 286 normal vibrational modes 285 notation 35 Novaco-McTague rotation 518,523, 524 Novaco-McTague epitaxial rotation 549 nucleated islands 688 nucleation 473,478,688, 766, 775,776,778, 788 nucleation and growth 775,785 nucleation processes 689 number of the space group 19 O/Ru(001) 736 O/W(100) 693,694 O/W(110) 685,694 O/W(211) 704 02-anion sublattice 199 object wavefunction 400 occupied density of states 364 octahedral coordination 188 octahedral interstices 207 octahedral interstitial sites 192 octahedrai sites 191 on-top position 484, 485 on-top sites 20, 434, 469, 471,475,476, 580 one-electron density 151 one-electron eigenfunctions 152 one-electron eigenvalues 151 one-electron electronic energy 148
Subject index one-electron functions 151 one-electron kinetic energy and electron-ion (nuclear) attraction 149 one-electron wavefunctions 147, 149, 150 one-dimensional defects 655 one-electron energies 613, 616 one-phonon scattering 287 Onsager 719 optical interferometry 378 optical lever 378, 380 optical transfer 400 order parameters 719, 721,723, 732, 733, 746, 747, 804-806 order-disorder transition 119, 448,521-523, 535,567, 671,697 order(N) methods 111 ordered phase 42, 613 ordering kinetics 661 orientational order 521 orientationai order-disorder transition 525 orientational phase diagram 68 orientationai phase separation 55, 68 orientational phase separation of vicinal Au(l 11) 79 orientational phase separation of vicinai Pb(l 11) 79 orientational phase separation of vicinal Si(l 11) 79 out-of-phase condition 297, 302, 328,663, 666, 667,669, 679, 692, 693 out-of-phase scattering 295 overhang 81 overlap matrix 106, 147 overlayer lattice 42 overlayer mesh 25 overlayer meshes, classification of 30 overlayer structures 457 overlayer unit mesh vectors 25 overlayers 655 oxidation 427,457 oxide structures 191,456 oxides 143,222, 427 oxygen adsorption 443 oxygen deficient surface 166 oxygen-induced relaxations 447 p(2x2) oxygen on Ni(l 11) 754 oxygen penetration 457
Subject index
oxygen vacancies 175 p 3 conformation 264 pa~r correlation function 300, 523 pa~r distribution function 37, 39, 298 pair energy 586, 598 pa~r interaction energy 593 pa~r interactions 579, 581,586, 594, 638,639 pa~r potential 154, 625 pair-wise interactions 79 parallel imaging 398 paramagnetism 720 partial dislocations 674, 676 particle-hole symmetry 594 particle-vacancy interchange 729 particle-vacancy transcription 728 partition function 719 Patterson function 37, 299 Pauli exclusion principle 149, 505 Pauling law of electronegativity 189 Pauling radius 469, 471-474 Pb(110) 342, 343 Pd 105, 122 Pd(100)-H EAM 626 Pd(110) 449 Pd(110)-CO 440 Peierls distortion 162 perfect absorption 280 periodic boundary conditions 803 Periodic Table 431 perovskite (100) surfaces 169 perovskite 175, 178, 193,210 peroxide ions 02- 478 phase boundaries 221,511,533,535,628, 631,640 phase coexistence 784 phase contrast 401 phase diagrams 128, 468,509, 510, 521,523, 533,535,552, 582, 613,615,617,618, 626, 631,639, 686, 693, 702, 718, 720, 726, 728, 729, 735, 751,788, 793,794 phase equilibria 74 phase separation 69, 77, 78 phase shifts 275,590, 613,614 phase transitions 53,242, 473,508,516, 521, 522,535,549, 551,552, 567, 604, 681, 693,701,704, 788
873 phase transition (7x7) ~ ( l x l ) 242 phases 581 phases of a form 7 phenacite structure 193, 220 phonon dispersion 557, 634 phonon dynamics 709 phonon energies 155 phonon excitation 448 phonons 330, 511, 513, 601 photocatalysis 210 photocathodes 459, 478 photoemission 114, 115, 162, 448,601 photoemission data 159 photoemission electron microscope (PEEM) 442 photoemission spectroscopy 476 photon stimulated desorption (PSD) 194 phthalocyanine 241 physisorbed films 565 physisorption 505,536, 548,580, 583,583, 729, 742, 773,778 physisorption state 445 physisorption studies 531 7t and 7t* orbitals 246 rt and n:* states 247 rt-bonded chain model 123, 131, 132 7t-bonded chain S i ( l l l ) ( 2 x l ) system 114 rt-bonded chains 160, 161, 162, 163, 236, 240, 243,247 ~-bonds 160, 161, 241,246 piezoelectric materials 369 piezoelectric scanning device 369, 370 pinwheel reconstruction 763 planar model 750 planar sp 2 bonded chains 240 Planck's constant 146 plane wave based methods 108 plane wave pseudopotential 130 plane waves 147 plastic deformation 375 point and space group symmetry 16 point defects 174, 175,202, 221,223,289, 290, 358,363, 386, 655,657, 658, 805 point groups 16, 19 point lattice defects 803 point of general position 18 point of special position 18
Subject index
874 point operation 16 point-group symmetry 579 point-to-point resolution 368,402 poisoning 616 Poisson's equation 599 Poisson's ratio 94, 635,672 polar surfaces 207, 210, 236, 248,249, 258 polarizability 154, 464, 583 polarization 155,610 polarization factor 326 polarization potential 155 polycrystalline surfaces 416 polymorphism 521,567 polytypes 215 polytypes of SiC 217 porosity 224 positional correlation 525 positional disordering 522 positional order 521 positive surface excesses 62 potential barrier 445 Ports models 750 3-state Potts model 522, 724, 747, 748, 754, 798 4-state clock model 750, 751 4-state Potts model 448, 724 clock model 750 planar Potts 750 q-state Ports models 750 power law 680, 703 power-law decay exponents 611 power-law line shape 301 preconditioned conjugate gradient scheme 109 premelting 547 premeiting transition 243 pressure 70 pressure for ideal gases 729 pressure measurement 551 primitive cells 579 primitive lattice vectors 317 primitive meshes 13 primitive overlayer unit mesh 29 principles of semiconductor surface reconstruction 246 projected band structure 115 projected electron density 340 Prokrovskii and Talapov 519
promoters in catalytic reactions 477 promoting 616 pseudo-potential plane wave methods 110 pseudo-potential plane waves 108 pseudo-hexagonal layer 755 pseudo-hexagonal structure 755,757 pseudo-potential methods 108, 113 pseudo-potentials 148 Pt 53, 54, 105, 122, 755 Pt(ll0) (1• 397 Pt Co 382 P t o n P t ( l l l ) 689 Pt/Pt(l 11) 680 Pt(001) 396 Pt( 100)c(4• 413 Pt(110) 414, 415,449, 760 Pt(110) (1• 398,440, 441 Pt(110) (2• 440 Pt(110)-CO 435,438 P t ( l l l ) 520, 557 Pt(I 1 I)-CO 639 Pt(l 1 I)-K 468,469 Pulay forces 110 pull-off force 376 Q-resolution 312 quantum chemistry 592 quantum chemistry codes 112 quantum gases 531 quantum mechanical calculations 233 quantum mechanical methods 144 quantum mechanical models 179 quantum mechanical potentials 144 quantum size effect 478 quarto interaction energy 586 quartos 619 quartz 194 or- and I]-quartz 174, 193 quartz microbalance 566 quasi-atom approach 616 quasi-chemical methods 631,634 quasi-one-dimensional states 610, 637, 641 quasielastic incoherent neutron scattering (QENS) 541 quench 780, 805 radial scan mode 339
Subject index
rainbow angles 351 rainbow scattering 351 RAIRS 440 random direction 393 random phase approximation 687,693 random steps 300 random field Ising model (RFIM) 704 rare gas adsorption 583 rare gases 509 Rayleigh ansatz 347 reaction kinetics 773 real and reciprocal space lattices, transformations between 29 real space images 363 real space lattice 21,555 rebonding 127, 132,591 reciprocal lattice 21 reciprocal lattice, basis vectors of the 22 reciprocal lattice rods 537, 553,669, 682, 683, 692 reciprocal lattice vectors 22, 278, 315,347 reciprocal mesh vectors 22 reciprocal space 273,291 Recknagel formula 409 reconstructed (110) fcc metals 637 reconstructed Au( I 11) 405,406 reconstructed phases 209 reconstructed Si(100) 364 reconstructed surfaces 141, 281 reconstruction 68, 112, 122, 131, 132, 194, 208, 214, 220, 233,241,243, 247,248, 253, 2 8 1 , 4 4 7 , 4 5 3 , 4 5 5 , 6 1 9 , 661,669, 718,725,730, 741,787 (7• 764, 765, 770 contractive type 120 effects on adsorption on 768 recursion method 107 reduced correlation function 737 reduced free energy 71 rcduced surface free energy 72 reduced surface tension 74, 76, 78, 81 reduced temperature 736, 737,776, 806 reentrant 563 reentrant aligned incommensurate solid phase 559 reentrant fluid 523 reflection anisotropy microscopy 416
875 reflection electron microscopy see REM reflection high-energy electron diffraction s e e RHEED reflections 16 refractory metals 675 registry 730 regular point system 18 rehybridization 235,243,248, 251, 616, 630, 639 rehybridization of the dangling bonds 234 relativistic effects 118 relaxation 62, 112, 116, 122, 131, 132, 200, 206, 208, 223,233, 246, 281,327,447, 569, 476, 606, 629, 641,664 relevant operators 750 reliability (R) factors 323 REM (reflection electron microscopy) 93, 198, 200, 209, 404, 636 u holography 405 m resolution 405 renormalization 738, 799 renormalization eigenvalues 747 renormalization fixed point 749 renormalization group 638 renormalized forward scattering (RFS) 323 repulsive dipole--dipole interaction 472 resistive anode 312 resolution 544 FIM 384 function 305 LEEM 409 REM 405 SFM 366, 376 TEM 402 resonance 353 resonance energies 610 resonant bond 488 resonant bonding 487 resonant scattering 352, 557 rest atoms 239, 243 retarded Green's function 589 reverse scattering perturbation (RSP) 323 Rh 122 Rh(100)-Cs 469 RHEED (reflection high energy electron diffraction) 125, 165, 170, 194, 209, 234, 456, 553, 566
876 ridge-and-valley morphology 681 ring structure 693 rippling 215 RKKY interaction energy 591 rocking surface mode 512 rocksalt 163, 175, 178, 179, 193,214 rocksalt structure 190, 192, 201, 213, 218 rocksalt structure alkali halides 142 rod scans 331 root-mean-square displacement 39 rotated hexagonal phase 520 rotated incommensurate solid phase 559 rotated structures 469 rotation 16, 511,514, 542 rotational axes 17 rotational diffusion 522, 542, 543 rotational diffusion coefficient 515 rotational epitaxy 469 rotational motions 514 rotational scattering 541 rough stcps 668 roughened surfaces 705 roughening 231 roughening temperature 81,301,565,706 roughening transition 665--678,705 roughness 219,669, 674, 679 roughness parameter 301 row-pairing reconstruction 761 Ru relaxation 606 Ru(0001) 447,460, 461,476, 477 Ru(0001)('~3• 469, 479 Ru(0001 )(~/3• 470 Ru(0001 )(2x2)-Cs 472 Ru(0001)-CO 435,436, 437 Ru(0001)p(2x2)-Cs system 472 Ru(0001)-Cs 468,470, 470, 472 Ru(0001)-K 470, 472, 473 Ru(0001 )-Li 473 Ru(0001)-Na 470, 473 Ru(0001 )-O 447 Ru(0001 )-Rb 473 Ru(001) 691,743 Ru3(CO)!2 435 rumpling 195,201, 219, 486 Rutherford backscattering spectroscopy (RBS) 392 Rutherford scattering cross section 392
Subject index
rutile 179, 193, 203 rutile (110) surface 166 rutile surface 175 sapphire 194, 207, 209 saturation of the dangling bonds 241 Sb 129, 233,708 Sb adsorption 124 scaled structure factors 744 scaling 235,716, 738,744, 780, 782, 799 scaling exponents 802 scaling function 779, 806 scaling laws 257,265, 802, 806 scanning electron microscopy s e e SEM scanning force microscopy s e e SFM scanning probe microscopies 224, 363 scanning tunneling microscopy s e e STM scanning tunneling spectroscopy (STS) 233 scattered-wave theory 628 scattering angle 275 scattenng cross section 344, 555 scattenng length 274, 630 scattenng matrix 322 scattenng plane 303,305 scattenng theory 589, 630 scattenng theory methods 628,639 SCF-LCAO (linear-combination of atomic orbitals) 144, 145, 147, 149, 151, 152 Scherzer contrast transfer function 401 Scherzer defocus 404 Sch6nflies notation 17, 18 Schottky barrier 124, 130 Schrtidinger equation 108, 146, 344 one-electron 147, 157 screened Coulomb potential 387 screening 392, 471,472, 476 screening between the dipoles 471 screening charge 465 screening length 388 screening potential 108 screw and edge dislocations 35,672 screw dislocation 39, 705,706 second moment 625 second-order phase transition 718,736, 798 second-order transition 701,702, 720 second-phase precipitates 224 secondary electrons 307
Subject index
secondary emission ratio 307 secondary ion mass spectroscopy see SIMS segregation 224 selection rules 540 selective adsorption 352 selenides 142 self-adsorption 473 self-affine scaling 681 self-consistency 599 self-consistent field 144 self-consistent Hartree-Fock 587 self-consistent matrix Green's-function (MGF) scattering theory 629 self-consistent pseudopotential calculations 129 self-similarity 704 SEM (scanning electron microscopy) 273 semiconductor surface reconstruction 234, 261,265 principles of 246 semiconductor surfaces 91, 104-106, 108, 111, 114, 123,231,427,660 semiconductor systems 131 semiconductors 115, 601,611,635,675,676 metal overlayers on 123 semiempirical methods 640 semiempiricai quantum-mechanical methods 141 semiempirical potentials 507 semiempirical techniques 107 SEXAFS 449, 474 SFM (scanning force microscopy) 363,372, 373 resolution of the 376 shadow cone 396, 388-390, 397 shape transition 691 shear 523 shell models 155, 165 short-bridge site 615 short-range correlation 632 short interaction range 104 short-range potential 155 Si !11, 123, 160, 231,233,236, 246, 338 Si dimer chains 128 Si(001) 660, 668-671,688,689 Si(0010) 663 Si(lO0) (2• 241
877 Si(100) 90, 123, 124, 130, 662, 689, 693 Si(110) 660 S i ( l l l ) (2• 123, 132, 162, 163,235,240, 247 Si(l I 1) (7• 132, 241,247, 328, 329, 513, 662, 677, 696, 725, 805 S i ( l l l ) 91, 93, 123, 124, 127-129, 161,412, 487,488, 717 Si(l I 1) surface reconstructions with ('~-•162 124 Si(l I 1) ('43-•162 129 Si(l I 1) ('43-• 696 Si(111) (5• 404 SiC, polytypes of 217 2H-SiC 215 3C-SiC 215,217 4H-SiC 215,217 6H-SiC 215,217 6H-SiC (0001) 218 or-SiC 215 [3-SIC 215,217 13-SiC(l 11) 217,218 [3-Si3N4 220 oc-SiO2 194 ot-Si3N 220 I3-Si3N4 221 silica 174,212 silicates 212 silicide layer 124 silicon 236 silicon carbide 215 silicon nitride 220 silicon wafers 123 silicon-on-sapphire 209 silver 524, 525 simple cubium 592 simple metals 108, 132 s~mple lattices 30 SIMS (secondary ion mass spectroscopy) 456 simulation techniques 105 s~ngle alignment technique 390 s~ngle-bond scission 161, 162, 236 s~ngle crystal substrates 427 s~ngle height steps 665 s~ngle scattering limit 274 s~ngularity 793 site binding energy 726
878 site switching 485 Si-Si dimers 217 skimmer 355 slab 113 slab calculations 144, 156, 158,601 slab thickness 158 Slater determinant 149, 151 Slater exchange-correlation potential 152 Siater-Koster 2-center approximation 106 Slater-type orbitals 147 Smoluchowski 117 Smoluchowski smoothing effect 132 SnO 2 166, 203 SnO 2(110) 166, 168,203 soft metals 675 softening of a surface phonon 118 solid l-solid 2 transitions 567 solid-on-solid (SOS) model 79 solid phase 726 solid state physics 5 solid-on-solid model 80, 299,665 solid-ovcrlayer interface potentials 325 solid-solid interface 342 solid-vacuum interface 112, 342 solid-vapor interface 61 solitary wave 443 solubility 474 source extension 320 s,p electrons 116 sp 2 bonding 237,490 sp 2 chains 237, 241,246, 258,263 sp 2 conformation 264 sp 2 coordinated Si atoms 237 sp 2 coordination 256 sp 2 hybridization 173 sp 2 surface chain 237 sp2-bonded surface layer 475 sp 3 bonding 237,488,489 sp 3 orbitals 487 sp3-type dangling bonds 486 SPA-LEED (spatially analyzed LEED) 202, 315 space charge 315 space groups 16 notation 19 number of 19 short form symbol for 19
Subject index
spatial coherence 314 spatial self-organization 441 spatially analyzed LEED see SPALEED specific heat 535, 633,749 specific heat exponent 741 spectromicroscopy 416 spectroscopic ionicity 257 speed ratio 355 sphalerite structure 193, 219, 220 spherical aberration 401,402 spin interactions 631 spin-polarized electrons 459 spin-polarized low energy electron microscopy 416 spinel lattice 206 spinel structure 193, 206 spiral dislocation 707 spiral reaction fronts 414 split positions concept 471 split-off states 597 spot profile analysis 669 spreading pressure 66 sputtering 397 SrO 210 SrO (Type I1) terminations 170 SrTiO 3 169,210 SrTiO 3 (100) 210 SrTiO 3 (2• 171 stability of surfaces 55 stacking faults 15, 199, 239, 386, 6 5 5 , 6 7 5 , 6 9 6 staggered chemical potential 734 staggered coverage 733 staggered field 731 staggered magnetic field 734 standing waves 638 static lattice 223 statistical distributions 90 statistical mechanics 55, 79 step bunching 78, 79, 681 step collisions 82 step crossing 81 step density 72, 73, 76, 79, 80 step diffusivity 85, 87, 88, 91 step distributions 94 step edge barrier 702, 707, 708 step edge diffusion 690 step edge roughness 663
879
Subject index
step edge stiffness 88 step edges 33, 71, 83, 223,239, 663,681 step energy 93,665,670 step fluctuations 665 step formation 82 step formation energy 82, 242 step free energy 73, 81, 83, 85 step height 663,664 step height distribution 678 step height multiplicity 665 step interaction coefficient 94, 95 step interaction energy 82 step interactions 77 step lengths 302 step meandering 665,669, 672, 697 step pressure 73, 77 step roughness 669 step-step dipole interactions 672 step-step interactions 81, 83, 91,93,636, 664, 665 step-step interaction energies 242 step-step repulsions 83 step-step separation 93 step-step repulsion 292,635 step stiffness 82, 88, 91 step-terrace array 197 step wandering 55, 82, 85, 90 stepped surfaces 80, 159, 231,236, 242, 302, 453,662, 666 steps 30, 35, 67, 79, 96, 122, 174, 176, 197-199, 221,223, 224, 291,363,565, 606,607,635,636, 655,663,665,668, 679,688,689,696, 705,725 stereographic projection 30, 31,663 stcreographic triangle 33 sticking coefficient 479, 552 sticking probability 441,457 stimulated desorption 358 stishovite 174 STM (scanning tunneling microscopy) 54, 67, 90, 91, 93, 116, 122, 125, 126, 130, 195, 196, 202-205,207, 211,214-216, 218, 223,234, 239, 241,242, 259, 273,363, 364, 371,372,438,449, 451,453,454, 456-458,476, 490, 492, 601,606, 623, 633,636, 655, 661-664, 669, 673,674, 688,689, 690, 693,697,699, 700, 709,
755,760, 761,764, 766, 771 lateral resolution 367 resolution 366 tip 369 stoichiometry 235 strain 93, 175, 178, 235, 481,626, 665, 671, 672, 674, 675,686 strain energy 699 strain gauge 380 stress 93, 104, 177,242, 245,290, 428,488, 491,635,636, 663, 664, 665,671,675, 698,699 stress anisotropy 668,672 stress relaxation 697 stress-mediated interactions 93 striped incommensurate phase 519, phase 555 striped phase 520 structure 775 structure factor 738,740, 744, 779 subgroup 724 sublimation 534 substitutional adsorption 763 substitutional atoms 289 substitutional model 490, 491 substitutional site 474, 476 substrate lattices 686 substrate relaxation 618,627, 628 substrate-mediated interaction 483 subsurface oxygen 456 subsurface sites 617 sulfides 142 superlattice 692 superlattice reflections 327 superlattice rods 284, 293, 297,339 superlattices 199, 208 superperiodicity 698 superstructures 661,685 surface, definition of 55 surface atomic geometry 235 surface band structure 114 surface barrier shift 462 surface Brillouin zone 114, 246, 598,605, 724, 741,743 surface charge 194 surface charging 140, 179 surface chemical bonding 233,235,249, 253, 256 m
Subject index
880 surface crystallography 3, 9, 339 surface dangling bonds 261 surface decomposition 179 surface defects 171, 174, 176, 281,414, 657 surface diffraction rods 280 surface diffusion 56, 438,442, 580, 681 surface dynamics 179 surface electron charge distribution 344 surface electron density 119 surface electronic properties 113 surface electronic structure 104, 112 surface energy 114, 142, 671, 681 surface excess 60, 63 surface excess quantities 60 surface faceting 55 surface force constant 512 surface free energy 242, 635 angular dependence of the 77 surface imperfections 363 surface magnetism 717 surface mechanical properties 375 surface melting 77, 122,565-567 surface mesh 194 surface i~lolccule 591 surfacc morphology 96 surface orientation 53 surface peak 393,394 surface phonon spectrums 358 surface phonons 285,286 surfacc plasmons 307 surface poisoning 630 surface rate processes 634 surface reactions 716 surface reactive site 176 surface reconstruction 110, 117, 122, 131, 14(), 141, 164, 169, 172, 177, 178, 197-200, 223,255,389, 390, 395,427, 438,441,450, 451,453,665,681,717, 753,755,771 surface relaxation 110, 117, 122, 140, 14 !, 164, 169, 170, 172, 173, 175-178, 212, 219, 249, 254, 389, 390, 395,660, 755 surface resonance scattering 406 surface resonances 114 surface roughening 55, 122,563,565,666, 717 surface roughness 195,328 surface segregated substrate impurities 803
surface segregation 175 surface stability 69 surface-state bands 248 surface states 112, 114, 115, 162, 211,234, 246, 250, 259, 261,450, 585,599, 601, 610, 611,635,637, 638,641 surface steps 803,805 surface stoichiometries 140, 188 surface strain tensor 671 surface strains 142 surface stress 77, 94, 95,767 changes in 94 surface structural properties 113 surface structure 140 properties 113 surface symmetry 325 surface tensile stress 122 surface tension 55, 56, 57, 58, 63, 65, 69, 81, 114, 301 anisotropy of the 68 as a function of orientation 83 changes in 56 values of 56 surface terminations 200 surface thermodynamics 55, 60 surface topography 366 surface topology 172, 173, 178, 179 surface V I center 176 surface X-ray diffraction 125-128,273, 326 surface X-ray scattering 113, 126 surface X-ray scattering system 331 surface-force constants 457 surfaces states 112 surfactants 705,708 survival of the largest 785 susceptibility 739, 796, 806, 807 symmetry 516 symmetry frustration 515 symmetry operations 724 synchrotron radiation 326 T 4 site 487 TaC 192, 195,213,214 TaC(I 10) 682 TaC(110) transition 683 TaN 218 ~5-TaN 192
Subject index
tantalates 212 tapping mode 376 TED (transmission electron diffraction) 172, 241,765 TEM (transmission electron microscopy) 214, 273,401,755 resolution in 402 temperature 53, 68 tensor LEED 324 terrace-ledge-kink model 33 terrace size 663,678,702 terrace size distribution 665,678 terrace-step-kink 86 terrace-step-kink model 637 terrace-width distribution 91, 93, 636 terraces 33, 79, 81, 198,239, 297,679, 689, 703 tetrahedral bond 487 tetrahedral coordination 188 tetrahedral sites 191 tetrahedral structure 264 tetrahedrally coordinated compound semiconductors 235,236, 248 tetrahedrally coordinated semiconductors 231 THEED (transmission high energy electron diffraction) 554 thermal accommodation 381 thermal average 285 thermal desorption 580 thermal desorption spectra 467 thermal diffuse scan 319 thermal diffuse scattering 330 thermal disorder 675 thermal equilibrium 53 thermal excitation of kinks 85 thermal expansion coefficients 326 thermal vibration 341, 351 thermal vibrational amplitudes 326, 393 thermal vibrations 39, 322 thermodynamic densities 793 thermodynamic faceting 53 thermodynamic fields 793 thermodynamic phase separation 68 thermodynamics 53, 55 thin film processing 567 Thomson scattering formula 326 three-body terms 105
881 three-dimensional metals 111 three-phase coexistence 563 three-way automotive exhaust catalyst 454 threefold adatom site 487 threefold hollow 488 threefold site 487 threefold-coordinated hcp site 485 threefold-coordinated site 455 threefold-fcc hollow sites 449 threefold-hollow position 476 threefold-hollow site 481 through bond 579 Ti20 3 171,207, 209 TiC 192, 195,213 TIC(100) 195,213 T i C ( I l l ) 214 tie bar 72, 76, 78 tight-binding 144, 611, 612, 618, 619, 620, 624, 634 tight-binding calculations 110, 118,623 tight-binding chain 595 tight-binding methods 104, 106, 141, 144, 145, 148, 149, 153, 154, 162, 599 tight-binding models 179, 586,609, 617,636, 640, 670 t~ght-binding substrate 610 t~ght-binding total energy calculations 258 t~lt angle 253 t~lted dimer model 244 tilted dimers 239, 240, 241,242, 246 t~me-of-flight methods 397 TiN 218,219 TiO 2 166, 194,203,210 TiO2(001) 205 TiO2(100) 197, 198 TiO2(110) 166, 168,203,204, 205 TiO2(l 11) 224 TiO 2 (Type I) terminations 170 tip 378, 379, 380, 386 tip-substrate interactions 374 topological reconstruction 178 topology 142, 171, 175, 176 torsions 512 total energy 104, 108, 111, 113, 114, 144 total energy calculations 125 total energy functional 144 transfer function 304
882 transfer matrices 633,634, 803, 805 transfer-matrix techniques 632, 635 transfer width 304, 339 transformations 29 transition metal carbides 213 transition metal carbonyl complexes 430 transition metal nitrides 218 transition metal substrate 630 transition metal surfaces 111,474 transition metals 105, 106, 111, 116, 117, 579, 592, 616, 6 1 8 , 6 2 3 , 6 2 5 , 6 4 0 transition temperature 759, 805,806 translation 511 translation operation 11 translation vector 5, 42 translational and orientational order 525 translational and rotational motions 513 translational diffusion 542 translational diffusion coefficient 514, 515, 542 translational motions 514 transmission electron diffraction s e e TED transmission electron microscopy s e e TEM transmission high energy electron diffraction see TtlEED transversc modc 513 transverse scan 339 trial wavcfunctions 109 triangulation methods 387 tricritical point 733 a- and [5-tridymite 174 trigonai prism 192 trimcrs 247,489, 491,492, 618,621,623,624, 685 trio energies 598,624, 632 trio interaction energy 586 trio interactions 597, 607,618,620, 621,624, 632,728 trtos 619 triple-bond scission 161 triple dipole 507,583 triple point 509, 531,533 triple point wetting 562, 567 triple point wetting transition 562 tripod scanner 369 true atomic resolution 372, 376 tube scanner 369
S u b j e c t index
tungsten bronzes 211 tungsten carbide 193 tungsten carbide structure 192 tunneling 380 tunneling current 363, 364 tunneling probability 366, 382 tunneling tip 378 tunneling transmission probability 366 twin boundaries 41,386, 677 twinning axis 41 twins 674 two-dimensional Bravais lattice 11 two-dimensional Cs oxides 482 two-dimensional defects 655 two-dimensional melting 751 two-dimensional melting transitions 567 two-dimensional phase transitions 358, 508 two-dimensional point groups 18 two-dimensional space groups 18, 43 two-electron exchange interaction 149 two-electron Hartree screening (or Coulomb) interaction 149 Type I termination 169 Type !1 termination 170 ultrahigh vacuum (UHV) 123,241 ultraviolet photoelectron spectroscopy see UPS uncertainty principle 478 uniaxial rotation 515 uniform stepped surface model 295 uniform strain 676 uniformly stepped surface 78 unit cells 5, 8 basis of 8 positions 8 unit matrix 28 unit mesh transformation 25 unit mesh vectors 11 unit meshes 11, 13,684, 685,696 centered rectangular 13 hexagonal 13 oblique 13 rectangular 13 square 13 unit vectors 684 universal binding curve 616 universality 716, 744, 746, 801
883
Subject index
universality classes 448, 508,522, 716, 736, 746, 747 unoccupied density of states 364 unpaired bonds 486, 489 unpaired electrons 428, 491 unpaired orbitals 487 untiited dimers 242 up-quenching 718,780 UPS (ultraviolet photoemission spectroscopy) 143,430, 436, 456, 457,464, 465 V203 171,207, 209 V205 195, 196 vacancies 38, 42, 79, 208, 213, 214, 217, 218, 219, 221,222, 247, 289, 290, 474, 475, 629, 655,661,662, 665,705 vacancy defects 211 vacancy form factor 290 vacancy-vacancy interactions 630 vacuum 139 vacuum level 115 vacuum-solid interface potentials 325 valence band transitions 307 valence electrons 106, 116 van der Waals 344, 377,507, 579 van der Waals attractive term 155 van der Waals diameters 516 van der Waals dispersion force 374 van der Waals interaction 445,505,506, 583 van der Waais potential 275,583,584 van der Waals radius 516,548 van der Waals systems 563 vanadates 193, 212 vaporization 534 variance 508 VC 195,213 VC x 214 very low energy electron diffraction see VLEED vibration 511 vibration isolation 371,378 vibration properties 21 vibrational amplitudes 286, 325 vibrational force constants 257 vibrational frequencies 583 vibrational mode 511,540 vibrational motion 39, 709
vibrational partition function 65 vibrational spectra of metal-carbonyl complexes 430 vibrational spectroscopy 436 vibrational states 634 vibrations 547 vicinal 231 vicinal Ag(110) 681 vicinal plane 663 vicinal Si(l I 1) 681 vicinal surfaces 30, 33, 74, 79, 205,239, 242, 606, 635,636, 665, 681,696, 717 vicinal ZnO surfaces 223 vicinality 669, 705 Vidicon 310 virial coefficient 507 VLEED (very low energy electron diffraction) 409 VN 192,218 void channels 220 volumetry 533 vortex 751 W 53, 132 W/W(100) 692, 693 W(001) 106, 328,329, 768 W(001), c(2• reconstruction of 118 W(001) reconstruction 697,757,758 W(100) reconstructions 717 W(110) 693 W(110)-H 762 W ( I I I ) 54 Warren approximation 277 wave vectors 274 wavelength of the helium atoms 356 wavelets 111 WC 192, 213 weak chemisorption 596 weak coupling 591 wetting 562, 564-567 complete 562 u incomplete 562 triple point 567 white graphite 219 Wigner-Seitz cells 23, 117 WKB approximation 366 WO 3 211
884 Wood notation 25 work function 115,262, 365,382, 442, 449, 462, 466, 470, 478, 633 work function change 114 work-function lowering 461 Wulff plots 69, 224 wurtzite 193,202, 231,239, 241,246 wurtzite cleavage faces 262 wurtzite structure 220 X-ray diffraction 37 X-ray photoelectron diffraction see XPD X-ray photoemission spectroscopy see XPS X-ray diffraction 204, 453,557,729, 733,742, 757,758, 760 X-ray reflectivity 330, 342 X-ray resolution function 339 X-ray scattering 321,358, 511,543, 708,755 X-rays, critical angle for 330 Xc 511, 516, 518,520, 524, 525,557,566 XPD (X-ray photoelectron diffraction) 143 XPS (X-ray photoemission spectroscopy) 143, 448,456, 457
Subject index
XY model 750, 751,753,801 XY model with cubic anisotropy 747, 751,758 YBa2Cu3OT_x 211 Young's modulus 94, 635 zero and one-phonon intensities 287 zero creep 56 zero force 113 zero-point energy 584 zigzag chains 440, 455, 481,482 zi,,zao,:, ,:, type displacement 118 zincblende 215,220, 231,239, 248,256, 260 zincblende (110) cleavage faces 250 zincite 231 zirconia 212 ZnO 202, 203, 231,249, 263 ZnS 249 ZnSe 249 ZnTe 249 ZrC 195,213 ZrN 192,218, 219 ZrO 2 212
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