Low-Energy Electron Diffraction : Experiment, Theory and Surface Structure Determination

Surface crystallography plays the same fundamental role in surface science which bulk crystallography has played so successfully in solid-state physics and chemistry. The atomic-scale structure is one of the most important aspects in the understanding of the behavior of surfaces in such widely diverse fields as heterogeneous catalysis, microelectronics, adhesion, lubrication, cor­ rosion, coatings, and solid-solid and solid-liquid interfaces. Low-Energy Electron Diffraction or LEED has become the prime tech­ nique used to determine atomic locations at surfaces. On one hand, LEED has yielded the most numerous and complete structural results to date (almost 200 structures), while on the other, LEED has been regarded as the "technique to beat" by a variety of other surface crystallographic methods, such as photoemission, SEXAFS, ion scattering and atomic diffraction. Although these other approaches have had impressive successes, LEED has remained the most productive technique and has shown the most versatility of application: from adsorbed rare gases, to reconstructed surfaces of sem­ iconductors and metals, to molecules adsorbed on metals. However, these statements should not be viewed as excessively dogmatic since all surface­ sensitive techniques retain untapped potentials that will undoubtedly be explored and exploited. Moreover, surface science remains a multi-technique endeavor. In particular, LEED never has been and never will be self­ sufficient. LEED has evolved considerably and, in fact, has reached a watershed.

1. The Relevance and Historical Development of LEED.- 1.1 The Relevance of Surface Crystallography.- 1.2 The Historical Development of LEED.- 1.2.1 The Period Before Wave Mechanics.- 1.2.2 The Discovery of Electron Diffraction.- 1.2.3 The Aftermath of the Discovery of Electron Diffraction.- 1.2.4 The Period 1930 – 1965.- 1.2.5 The Renaissance of LEED: Experimental Advances in the Mid-1960s.- 1.2.6 The Theoretical Solution: The Late 1960s and Early 1970s.- 1.2.7 The Era of Structural Determination: The 1970s and 1980s.- 2. The LEED Experiment.- 2.1 General Features of LEED Experiments.- 2.2 Sample Mounting.- 2.3 Electron Gun and Display System.- 2.3.1 Electron Gun.- 2.3.2 Display System.- 2.4 Methods of Data Acquisition.- 2.4.1 Faraday-Cup Collector and Spot Photometer.- 2.4.2 Photographic Technique.- 2.4.3 Vidicon Camera Method.- 2.4.4 Position-Sensitive Detector.- 2.5 Instrumental Response Function.- 2.5.1 Basic Concepts.- 2.5.2 Contributions to the Response Width.- 2.6 Determination of Angle of Incidence.- 2.6.1 Different Methods.- 2.6.2 Theory.- 2.6.3 An Example.- 2.7 Determination of the Debye Temperature.- 2.7.1 The Debye Temperature Normal to the Crystal Surface.- 2.7.2 The Debye Temperature Parallel to the Crystal Surface.- 3. Ordered Surfaces: Structure and Diffraction Pattern.- 3.1 Two-Dimensional Periodicity and the LEED Pattern.- 3.1.1 Miller and Miller-Bravais Indices.- 3.1.2 Lattice and Basis.- 3.1.3 Direct and Reciprocal Lattices.- 3.2 Superlattices at Surfaces.- 3.3 Stepped and Kinked Surfaces.- 3.3.1 The Step Notation.- 3.3.2 The Microfacet Notation for Cubic Materials.- 3.3.3 Unit Cells of Stepped and Kinked Surfaces.- 3.4 Symmetries and Domains at Surfaces.- 3.4.1 Symmetries in Two Dimensions.- 3.4.2 Domains.- 3.5 Interpretation of LEED Patterns.- 3.5.1 Patterns with a Bravais Array of Spots.- 3.5.2 Patterns with Multiple Bravais Arrays of Spots ― Domains.- 3.5.3 Patterns Exhibiting Extinctions Due to Glide-Plane Symmetry.- 3.5.4 Rationally Related Lattices and Coincidence Lattices.- 3.5.5 An Instructive Example of Pattern Interpretation.- 3.5.6 Incommensurate Lattices.- 3.5.7 Split Spots.- 3.5.8 An Example: Compact Structures vs. Antiphase Domain Structures of Adsorbed Carbon Monoxide Overlayers.- 3.5.9 Patterns with Multiple Specular Spots.- 3.5.10 Laser Simulation of LEED Patterns.- 4. Kinematic LEED Theory and Its Limitations.- 4.1 Definition of Kinematic Theory.- 4.1.1 Atomic Scattering Factor.- 4.1.2 Elastic Scattering.- 4.1.3 Amplitude of Diffraction.- 4.1.4 Surface Sensitivity.- 4.1.5 From Amplitudes to Intensities of Diffraction.- 4.2 The Kinematic Structure Factor for Ordered Surfaces.- 4.2.1 Two-Dimensional Bragg Conditions.- 4.2.2 General Derivation of Two-Dimensional Bragg Conditions in LEED from the Schrödinger Equation.- 4.2.3 Plane Waves, Beams, and the LEED Pattern.- 4.2.4 I-V, I-?, I-?, and Other Collections of Data.- 4.2.5 Kinematic Diffraction by Bravais Lattices of Atoms.- 4.2.6 The Case of Non-Bravais Lattices.- 4.2.7 Surface Structures Deviating from the Bulk Structure.- 4.2.8 Surfaces with Superlattices.- 4.2.9 Modulated Structures.- 4.2.10 The Simple Effect of Multiple Scattering on LEED Patterns.- 4.2.11 The Ewald Sphere.- 4.2.12 Further Applications of the Kinematic Theory of LEED.- 4.3 The Scattering Processes in LEED.- 4.3.1 Inelastic Scattering Processes.- 4.3.2 Modeling the Effect of the Mean Free Path.- 4.3.3 Spin Effects.- 4.4 The Elastic Scattering Potential.- 4.4.1 Atomic Potentials.- 4.4.2 The Muffin-Tin Constant.- 4.4.3 Potential Steps.- 4.5 Atomic Scattering.- 4.5.1 Spherical-Wave Scattering.- 4.5.2 Plane Wave Scattering.- 4.5.3 Phase Shifts.- 4.5.4 Atoms as Point Scatterers.- 4.6 The Inner Potential and the Muffin-Tin Constant.- 4.7 Temperature Effects.- 4.7.1 The Debye-Waller Factor.- 4.8 From Kinematic to Dynamical LEED.- 4.8.1 Clean Crystals and Bragg Reflections in One Dimension.- 4.8.2 Three-Dimensional Effects.- 4.8.3 Overlayer Effects.- 5. Dynamical LEED Theory.- 5.1 Multiple Scattering.- 5.2 Diffraction in Crystalline Lattices.- 5.2.1 Expansion in Spherical Waves.- 5.2.2 Expansion in Plane Waves.- 5.2.3 Expansion in Bloch Waves.- 5.2.4 Forward vs. Backward Scattering.- 5.3 Multiple Scattering in the Spherical-Wave Representation ― Self-Consistent Formalism.- 5.3.1 Scattering by Two Atoms.- 5.3.2 Scattering by N Atoms.- 5.3.3 One Periodic Plane of Atoms.- 5.3.4 Several Periodic Planes of Atoms.- 5.3.5 Change to Plane-Wave Amplitudes.- 5.3.6 Layer Diffraction Matrices for Plane Waves.- 5.3.7 One-Center Expansion.- 5.4 Perturbation Expansion of Multiple Scattering in the Spherical-Wave Representation: Reverse-Scattering Perturbation (RSP) Method.- 5.4.1 The Principle of RSP.- 5.4.2 The Formalism of RSP.- 5.4.3 The Use of RSP.- 5.5 Diffraction by a Stack of Layers: Transfer-Matrix and Bloch-Wave Method.- 5.5.1 The Bloch Condition.- 5.5.2 The Bloch Functions.- 5.5.3 The Transfer Matrix.- 5.5.4 Wave Matching at the Surface.- 5.5.5 Small Layer Spacings.- 5.5.6 Relation to Band Structure.- 5.6 Diffraction by a Stack of Layers: Layer-Stacking and Layer-Doubling Method.- 5.6.1 The Case of Two Layers.- 5.6.2 The Case of Many Layers.- 5.7 Diffraction by a Stack of Layers: Renormalized-Forward-Scattering (RFS) Perturbation Method.- 5.7.1 The Principle of RFS.- 5.7.2 The Formalism of RFS.- 5.8 Efficiency of Computation and the Combined-Space Method.- 5.9 Superlattices and Domains.- 5.9.1 Diffraction and Superlattices.- 5.9.2 Domains.- 5.10 Symmetries.- 5.10.1 Types of Symmetry.- 5.10.2 The Formalism of Symmetrization.- 5.10.3 Glide-Plane Symmetry.- 5.11 Thermal Effects.- 5.11.1 Temperature-Dependent Phase Shifts.- 5.11.2 Illustrations of Multiple-Scattering Effects in Temperature-Dependent LEED.- 5.12 Potential Steps, Surface States, Surface Resonances and LEED Fine Structure.- 5.12.1 Potential Steps.- 5.12.2 Surface States, Surface Resonances and LEED Fine Structure.- 5.13 Relativistic and Spin-Dependent Effects in LEED.- 5.14 Some Other Theoretical Techniques.- 5.14.1 Bootstrapping.- 5.14.2 The Chain Method.- 5.14.3 Multiple Scattering in Disordered Systems.- 5.14.4 Pseudopotentials.- 5.14.5 A Semiclassical Theory of LEED.- 5.15 Outstanding Theoretical Problems in LEED.- 5.16 Application of LEED Theory to Other Electron Spectroscopies.- 5.17 Computer Programs.- 6. Methods of Surface Crystallography by LEED.- 6.1 The Kinematic Approach to Surface Crystallography.- 6.1.1 Kinematic Simulation of Intensity Data.- 6.1.2 Layer Spacings from Sequences of Bragg Peaks.- 6.2 Averaging Methods.- 6.2.1 Constant-Momentum-Transfer Averaging (CMTA).- 6.2.2 CMTA with Azimuthal Averaging at Constant Energy.- 6.3 Fourier-Transform Methods.- 6.3.1 The Patterson Function.- 6.3.2 The Convolution-Transform Method.- 6.3.3 The Transform-Deconvolution Method.- 6.3.4 Fourier Transform of Intensity Beats from Overlayer and Substrate.- 6.4 The Dynamical Approach to Surface Crystallography.- 6.4.1 Dynamical Effects on Intensity Data.- 6.4.2 Information Content of Measured Data.- 6.4.3 Extraction of Structural Information from Dynamical LEED Intensities.- 6.5 Reliability Factors (R-Factors).- 6.5.1 Various R-Factors.- 6.5.2 Reliability of Reliability Factors.- 6.5.3 Dealing with Different Experiments and Different Beams.- 6.5.4 Noise and Smoothing.- 6.5.5 The Use of R-Factors.- 6.6 Accuracy and Precision of Structural Determination.- 7. Results of Structural Analyses by LEED.- 7.1 Clean Unreconstructed Surfaces.- 7.1.1 The Rh(111) Surface.- 7.1.2 Multilayer Relaxations.- 7.2 Reconstructed Surfaces.- 7.2.1 The Ir(110)-(1 × 2) Reconstructed Surface.- 7.2.2 The Si(100)-(2 × 1) Reconstructed Surface.- 7.2.3 The GaAs(110)-(1 × 1) Reconstructed Surface.- 7.3 Adsorbed Atomic Layers.- 7.3.1 The Ir(110)-(2 × 2)-2S Atomic Overlayer.- 7.3.2 The Ir(110)-c(2 × 2)-O and Ir(111)-(2 × 2)-O Atomic Overlayers.- 7.3.3 The Ti(0001)-(1 × 1)-N Atomic Underlayer.- 7.4 Adsorbed Molecular Layers.- 7.4.1 The Ni(100)-c(2 × 2)-CO Molecular Overlayer.- 7.4.2 The Pd(100)-($$2\sqrt 2 \times \sqrt 2$$)R45°-2CO Molecular Overlayer.- 7.4.3 Molecular Overlayers of C2H2 and C2H4 on Pt(111) and Rh(111).- 8. Two Dimensional Order-Disorder Phase Transitions.- 8.1 Introduction to Order-Disorder Phase Transitions at Surfaces...- 8.1.1 Chemisorption and Ordering Principles.- 8.1.2 Universality, Nonuniversality, Critical Exponents and Scaling.- 8.1.3 Applicability to Actual Surfaces.- 8.2 The Interaction of Hydrogen with the (111) Surface of Nickel.- 8.2.1 An Optimum Case.- 8.2.2 Experimental Results for Hydrogen Chemisorption on Ni(111).- 8.2.3 Parameters for LEED Analysis.- 8.2.4 The Geometry of Chemisorbed Hydrogen on Ni(111).- 8.2.5 Thermal Motion and Disorder in the Hydrogen Overlayer.- 8.2.6 The Order-Disorder Phase Transition and Adatom-Adatom Interaction Energies.- 8.2.7 A Renormalization-Group Theory Description of the Order-Disorder Transition of Hydrogen on Ni(111).- 8.2.8 A Cluster-Variational Description of the Order-Disorder Transition of Hydrogen on Ni(111).- 8.2.9 An Atomic Band Structure Description of Hydrogen on Ni(111).- 8.3 The Interaction of Hydrogen with the (100) Surface of Palladium.- 8.3.1 Significance of the H/Pd(100) System.- 8.3.2 An Experimental Characterization of Hydrogen on Pd (100).- 8.3.3 The Order-Disorder Phase Transition.- 8.3.4 The Connection Between the Ising Model and the Lattice-Gas Model.- 8.3.5 The Lattice-Gas Model with First- and Second-Neighbor Interactions.- 8.3.6 Effects of Three-Body Interactions.- 8.3.7 Effects of Third-Neighbor Interactions.- 8.3.8 Comparison Between Experiment and Theory for Hydrogen on Pd (100).- 8.4 The Interaction of Hydrogen with the (110) Surface of Iron.- 8.4.1 Significance of the H/Fe (110) System.- 8.4.2 An Experimental Characterization of Hydrogen on Fe(110).- 8.4.3 LEED Observations and Order-Disorder Phase Transitions of Hydrogen on Fe(110).- 8.4.4 Theoretical Predictions: A Lattice Gas on a Centered-Rectangular Lattice.- 8.4.5 Comparison Between Experiment and Theory for Hydrogen on Fe(110).- 9. Chemical Reactions at Surfaces and LEED.- 9.1 Monitoring Surface Reactions by LEED.- 9.2 The Adsorption of Oxygen on Rh(111) at 335 K.- 9.2.1 First-Order Langmuir Adsorption.- 9.2.2 The Structure of Oxygen on Rh(111).- 9.2.3 LEED Intensity Proportional to Oxygen Coverage.- 9.3 The Reaction Between Hydrogen and Ordered Oxygen on Rh(111).- 9.3.1 Reaction Threshold Temperature.- 9.3.2 First-Order Catalytic Reaction.- 9.3.3 Model for the Catalytic Reaction.- 9.3.4 Activation Energies and Preexponential Factors.- 9.3.5 Experimental Determination.- 9.4 The Reaction Between Hydrogen and Both Ordered and Disordered Oxygen on Rh(111).- 9.4.1 Order-Dependent Kinetics.- 9.4.2 Relative Amounts of Ordered and Disordered Oxygen.- 10. Island Formation of Adspecies and LEED.- 10.1 The Nature of Islands on Surfaces.- 10.2 LEED Beam Profiles for Arrays of Ordered Islands.- 10.2.1 Distributions of Islands.- 10.2.2 One-Dimensional Overlayers.- 10.2.3 Two-Dimensional Overlayers.- 10.2.4 Dependence on Surface Coverage.- 10.2.5 Summary of Theoretical Results for Beam Profiles.- 10.3 Island Formation in a Real System: CO on Ru(0001).- 10.3.1 Conditions of Island Formation.- 10.3.2 Experimental Results.- 10.3.3 Analysis and Discussion of Results.- 10.3.3a The Step-Limited Model of Island Formation.- 10.3.3b Dissolution of Islands.- 10.3.4 Summary of Island Formation Properties for CO/Ru (0001).- 11. The Future of LEED.- 11.1 Experimental Outlook.- 11.1.1 Improvements in Experimental Techniques.- 11.1.2 New Experimental Directions.- 11.2 Theoretical Outlook.- 11.2.1 Survival of the Kinematic Theory.- 11.2.2 Partial Multiple Scattering.- 11.2.3 Developments in the Dynamical Theory.- 11.2.3a Coherent Kinematic Summation of Amplitudes over Different Local Configurations.- 11.2.3b Reduced Unit Cell.- 11.2.3c Asymptotic Regime.- 11.2.4 New Directions.- 11.3 Progress in Structural Determination.- 11.3.1 Degree of Completeness of Structural Determinations.- 11.3.2 R-Factors and Structural Search Techniques.- 11.3.2a Projection Improvement.- 11.3.2b Functional Fitting of R-Factors.- 11.3.2c Steepest Descent.- 11.3.2d Least Squares.- 11.4 LEED vs. Other Surface-Sensitive Techniques.- 11.4.1 Individual Techniques.- 11.4.2 Comparisons Between Surface-Sensitive Techniques.- 11.4.3 Complementary and Competitive Techniques.- 12. Reference List and Table for Surface Structures.- Appendix A: Acronyms of Techniques Related to Surface Science.- Appendix B: A Computer Program to Determine the Angle of Incidence in LEED.- List of Major Symbols.- References.