Mechanical Stress on the Nanoscale

Edited by Margrit Hanbu¨cken, Pierre Mu¨ller, and Ralf B. Wehrspohn Mechanical Stress on the Nanoscale

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Edited by Margrit Hanbücken, Pierre Müller, and Ralf B. Wehrspohn

Mechanical Stress on the Nanoscale Simulation, Material Systems and Characterization Techniques

The Editors Dr. Margrit Hanbücken CINaM-CNRS Campus Luminy Marseille, Frankreich Dr. Pierre Müller Université Paul Cézanne Campus Saint-Jérôme Marseille, Frankreich Prof. Dr. Ralf B. Wehrspohn Fraunhofer Inst. für Werkstoffmechanik Halle Halle, Germany

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V

Contents Preface XV List of Contributors

XVII

Part One Fundamentals of Stress and Strain on the Nanoscale 1 1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1 1.3.2 1.4 1.4.1 1.4.2 1.4.3 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.6 1.6.1

Elastic Strain Relaxation: Thermodynamics and Kinetics 3 Frank Glas Basics of Elastic Strain Relaxation 3 Introduction 3 Principles of Calculation 4 Methods of Calculation: A Brief Overview 6 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys 7 Spinodal Decomposition with No Elastic Effects 8 Elastic Strain Relaxation in an Alloy with Modulated Composition 9 Strain Stabilization and the Effect of Elastic Anisotropy 11 Elastic Relaxation in the Presence of a Free Surface 11 Diffusion 12 Diffusion without Elastic Effects 12 Diffusion under Stress in an Alloy 13 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers 14 Introduction 14 Elastic Strain Relaxation 15 Critical Thickness 16 Morphological Relaxation of a Solid under Nonhydrostatic Stress 17 Introduction 17 Calculation of the Elastic Relaxation Fields 18 ATG Instability 19 Kinetics of the ATG Instability 21 Coupling between the Morphological and Compositional Instabilities 21 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures 22 Quantum Dots 23

VI

Contents

1.6.2

Nanowires 24 References 24

2

Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity 27 Pierre Müller Introduction 27 Theoretical Background 28 Bulk Elasticity: A Recall 28 Stress and Strain Definition 29 Equilibrium State 29 Elastic Energy 30 Elastic Constants 30 How to Describe Surfaces or Interfaces? 31 Surfaces and Interfaces Described from Excess Quantities 34 The Surface Elastic Energy as an Excess of the Bulk Elastic Energy 34 The Surface Stress and Surface Strain Concepts 35 Surface Elastic Constants 37 Connecting Surface and Bulk Stresses 39 Surface Stress and Surface Tension 40 Surface Stress and Adsorption 41 The Case of Glissile Interfaces 42 Surfaces and Interfaces Described as a Foreign Material 42 The Surface as a Thin Bulk-Like Film 43 The Surface as an Elastic Membrane 43 Applications: Size Effects Due to the Surfaces 44 Lattice Contraction of Nanoparticles 44 Effective Modulus of Thin Freestanding Plane Films 46 Bending, Buckling, and Free Vibrations of Thin Films 48 General Equations 48 Discussion 50 Static Bending of Nanowires: An Analysis of the Recent Literature 52 Young Modulus versus Size: Two-Phase Model 52 Young Modulus versus Size: Surface Stress Model 53 Prestress Bulk Due to Surface Stresses 53 A Short Overview of Experimental Difficulties 54 Conclusion 55 References 56

2.1 2.2 2.2.1 2.2.1.1 2.2.1.2 2.2.1.3 2.2.1.4 2.2.2 2.2.3 2.2.3.1 2.2.3.2 2.2.3.3 2.2.3.4 2.2.3.5 2.2.3.6 2.2.3.7 2.2.4 2.2.4.1 2.2.4.2 2.3 2.3.1 2.3.2 2.3.3 2.3.3.1 2.3.3.2 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.3.5 2.4

3 3.1 3.2

Onset of Plasticity in Crystalline Nanomaterials 61 Laurent Pizzagalli, Sandrine Brochard, and Julien Godet Introduction 61 The Role of Dislocations 63

Contents

3.3 3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.7 3.8

4

4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.5

Driving Forces for Dislocations 63 Stress 64 Thermal Activation 64 Combination of Stress and Thermal Activation 64 Dislocation and Surfaces: Basic Concepts 65 Forces Related to Surface 65 Balance of Forces for Nucleation 66 Forces Due to Lattice Friction 66 Surface Modifications Due to Dislocations 68 Elastic Modeling 68 Elastic Model 68 Predicted Activation Parameters 70 What is Missing? 70 Peierls–Nabarro Approaches 72 Atomistic Modeling 72 Examples of Simulations 73 Determination of Activation Parameters 74 Comparison with Experiments 75 Influence of Surface Structure, Orientation, and Chemistry 76 Extension to Different Geometries 78 Discussion 79 References 80 Relaxations on the Nanoscale: An Atomistic View by Numerical Simulations 83 Christine Mottet Introduction 84 Theoretical Models and Numerical Simulations Energetic Models 85 Numerical Simulations 87 Definitions of Physical Quantities 89 Relaxations in Surfaces and Interfaces 91 Surface Reconstructions 92 Surface Alloys: a Simple Case of Heteroatomic Adsorption 94 Heteroepitaxial Thin Films 96 Relaxations in Nanoclusters 98 Free Nanoclusters 99 Supported Nanoclusters 100 Nanoalloys 101 Conclusions 103 References 104

85

VII

VIII

Contents

Part Two Model Systems with Stress-Engineered Properties 5

5.1 5.2 5.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5

6

6.1 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.3.1 6.3.3.2 6.4 6.5

7 7.1 7.2 7.2.1 7.2.2 7.2.2.1 7.2.2.2 7.2.3

107

Accommodation of Lattice Misfit in Semiconductor Heterostructure Nanowires 109 Volker Schmidt and Joerg V. Wittemann Introduction 109 Dislocations in Axial Heterostructure Nanowires 111 Dislocations in Core–Shell Heterostructure Nanowires 113 Roughening of Core–Shell Heterostructure Nanowires 115 Zeroth-Order Stress and Strain 117 First-Order Contribution to Stress and Strain 120 Linear Stability Analysis 122 Results and Discussion 124 Conclusion 127 References 127 Strained Silicon Nanodevices 131 Manfred Reiche, Oussama Moutanabbir, Jan Hoentschel, Angelika Hähnel, Stefan Flachowsky, Ulrich Gösele, and Manfred Horstmann Introduction 131 Impact of Strain on the Electronic Properties of Silicon 132 Methods to Generate Strain in Silicon Devices 135 Substrates for Nanoscale CMOS Technologies 135 Local Strain 136 Global Strain 139 Biaxially Strained Layers 139 Uniaxially Strained Layers 142 Strain Engineering for 22 nm CMOS Technologies and Below 142 Conclusions 146 References 146 Stress-Driven Nanopatterning in Metallic Systems 151 Vincent Repain, Sylvie Rousset, and Shobhana Narasimhan Introduction 151 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales 152 Surface Stress 152 Surface Reconstruction and Misfit Dislocations 153 Homoepitaxial Surfaces 153 Heteroepitaxial Systems 155 Stress Domains 156

Contents

7.2.4 7.3 7.3.1 7.3.1.1 7.3.1.2 7.3.1.3 7.4 7.4.1 7.4.2 7.5

8

8.1 8.2 8.2.1 8.2.1.1 8.2.1.2

8.2.1.3 8.2.2 8.2.2.1 8.2.2.2 8.2.2.3 8.3 8.3.1 8.3.2 8.3.2.1 8.3.2.2 8.3.2.3 8.3.3 8.3.3.1 8.3.3.2 8.4

Vicinal Surfaces 157 Nanopatterned Surfaces as Templates for the Ordered Growth of Functionalized Nanostructures 158 Metallic Ordered Growth on Nanopatterned Surface 158 Introduction 158 Nucleation and Growth Concepts 159 Heterogeneous Growth 160 Stress Relaxation by the Formation of Surface-Confined Alloys 162 Two-Component Systems 162 Three-Component Systems 162 Conclusion 164 References 165 Semiconductor Templates for the Fabrication of Nano-Objects 169 Joël Eymery, Laurence Masson, Houda Sahaf, and Margrit Hanbücken Introduction 169 Semiconductor Template Fabrication 170 Artificially Prepatterned Substrates 170 Morphological Patterning 170 Silicon Etched Stripes: Example of the Use of Strain to Control Nanostructure Formation and Physical Properties 171 Use of Buried Stressors 171 Patterning through Vicinal Surfaces 173 Generalities 173 Vicinal Si(111) 173 Vicinal Si(100) 173 Ordered Growth of Nano-Objects 175 Growth Modes and Self-Organization 175 Quantum Dots and Nanoparticles Self-Organization with Control in Size and Position 176 Stranski–Krastanov Growth Mode 176 Au/Si(111) System 177 Ge/Si(001) System 179 Wires: Catalytic and Catalyst-Free Growths with Control in Size and Position 179 Strain in Bottom-Up Wire Heterostructures: Longitudinal and Radial Heterostructures 181 Wires as a Position Controlled Template 183 Conclusions 184 References 184

IX

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Contents

Part Three Characterization Techniques of Measuring Stresses on the Nanoscale 189 9

9.1 9.1.1 9.1.2 9.1.3 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.4 9.4.1 9.4.2 9.4.2.1 9.4.2.2 9.4.3 9.4.3.1 9.4.3.2 9.4.3.3 9.4.4 9.5

Strain Analysis in Transmission Electron Microscopy: How Far Can We Go? 191 Anne Ponchet, Christophe Gatel, Christian Roucau, and Marie-José Casanove Introduction: How to Get Quantitative Information on Strain from TEM 192 Displacement, Strain, and Stress in Elasticity Theory 192 Principles of TEM and Application to Strained Nanosystems 192 A Major Issue for Strained Nanostructure Analysis: The Thin Foil Effect 193 Bending Effects in Nanometric Strained Layers: A Tool for Probing Stress 194 Bending: A Relaxation Mechanism 194 Relation between Curvature and Internal Stress 195 Using the Bending as a Probe of the Epitaxial Stress: The TEM Curvature Method 196 Occurrence of Large Displacements in TEM Thinned Samples 197 Advantages and Limits of Bending as a Probe of Stress in TEM 199 Strain Analysis and Surface Relaxation in Electron Diffraction 199 CBED: Principle and Application to Determination of Lattice Parameters 199 Strain Determination in CBED 201 Use and Limitations of CBED in Strain Determination 202 Nanobeam Electron Diffraction 203 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils 203 Principle 203 What Do We Really Measure in an HREM Image? 205 Image Formation 205 Reconstruction of the 3D Strain Field from a 2D Projection 205 Modeling the Surface Relaxation in an HREM Experiment 206 Full Relaxation (Uniaxial Stress) 206 Intermediate Situations: Usefulness of Finite Element Modeling 207 Thin Foil Effect: A Source of Incertitude in HREM 207 Conclusion: HREM is a Powerful but Delicate Method of Strain Analysis 208 Conclusions 209 References 210

Contents

10

10.1 10.2 10.3 10.4 10.4.1 10.4.2 10.5 10.6 10.7 10.8

11

11.1 11.2 11.3 11.4 11.4.1 11.4.2 11.4.3 11.5 11.5.1 11.5.1.1 11.5.1.2 11.5.1.3 11.5.2 11.6 11.7

12

12.1 12.2 12.2.1 12.2.2

Determination of Elastic Strains Using Electron Backscatter Diffraction in the Scanning Electron Microscope 213 Michael Krause, Matthias Petzold, and Ralf B. Wehrspohn Introduction 213 Generation of Electron Backscatter Diffraction Patterns 214 Strain Determination Through Lattice Parameter Measurement 215 Strain Determination Through Pattern Shift Measurement 216 Linking Pattern Shifts to Strain 216 Measurement of Pattern Shifts 219 Sampling Strategies: Sources of Errors 221 Resolution Considerations 222 Illustrative Application 225 Conclusions 229 References 230 X-Ray Diffraction Analysis of Elastic Strains at the Nanoscale 233 Olivier Thomas, Odile Robach, Stéphanie Escoubas, Jean-Sébastien Micha, Nicolas Vaxelaire, and Olivier Perroud Introduction 233 Strain Field from Intensity Maps around Bragg Peaks 234 Average Strains from Diffraction Peak Shift 236 Local Strains Using Submicrometer Beams and Scanning XRD 240 Introduction 240 High-Energy Monochromatic Beam: 3DXRD 241 White Beam: Laue Microdiffraction 243 Local Strains Derived from the Intensity Distribution in Reciprocal Space 248 Periodic Assemblies of Identical Objects with Coherence Length > Few Periods 248 Introduction 248 Reciprocal Space Mapping 249 Applications 251 Single-Object Coherent Diffraction 252 Phase Retrieval from Strained Crystals 254 Conclusions and Perspectives 255 References 256 Diffuse X-Ray Scattering at Low-Dimensional Structures in the System SiGe/Si 259 Michael Hanke Introduction 259 Self-Organized Growth of Mesoscopic Structures 259 The Stranski–Krastanow Process 260 LPE-Grown Si1xGex/Si(001) Islands 261

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XII

Contents

12.3 12.3.1 12.3.2 12.3.3 12.4 12.5 12.5.1 12.5.2 12.5.3 12.5.4 12.6

X-Ray Scattering Techniques 262 High-Resolution X-Ray Diffraction 262 Grazing Incidence Diffraction 263 Grazing Incidence Small-Angle X-Ray Scattering 264 Data Evaluation 265 Results 266 The Influence of Shape and Size on the GISAXS Signal 266 HRXRD Measurement of Strain and Composition 269 Positional Correlation Effects in HRXRD 270 Iso-Strain Scattering 271 Summary 273 References 274

13

Direct Measurement of Elastic Displacement Modes by Grazing Incidence X-Ray Diffraction 275 Geoffroy Prévot Introduction 275 Elastic Displacement Modes: Analysis and GIXD Observation 276 Fundamentals of Linear Elasticity in Direct Space 276 Basic Equations 276 Atomic Displacements and Elastic Interactions 277 Green’s Tensor in Reciprocal Space 279 Grazing Incidence X-Ray Diffraction of Elastic Modes 280 Diffraction by a Surface 280 Contribution of the Elastic Modes 280 Procedure for Analyzing the Systems 281 Self-Organized Surfaces 282 Force Distribution and Interaction Energy for Self-Organized Surfaces 282 A 1D Case: OCu(110) 283 A 2D Case: NCu(001) 286 Vicinal Surfaces 289 Force Distribution and Interaction Energy for Steps 289 Experimental Results for Vicinal Surfaces of Transition Metals 292 Conclusion 294 References 295

13.1 13.2 13.2.1 13.2.1.1 13.2.1.2 13.2.2 13.2.3 13.2.3.1 13.2.3.2 13.2.3.3 13.3 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2 13.5

14

14.1 14.2 14.2.1 14.2.2

Submicrometer-Scale Characterization of Solar Silicon by Raman Spectroscopy 299 Michael Becker, George Sarau, and Silke Christiansen Introduction 299 Crystal Orientation 300 Qualitative Maps 300 Quantitative Analysis 302

Contents

14.2.3 14.3 14.3.1 14.3.2 14.3.2.1 14.3.2.2 14.3.3 14.3.4 14.3.4.1 14.3.4.2 14.3.5 14.3.5.1 14.3.5.2 14.3.5.3

14.3.6 14.4 14.4.1 14.4.2 14.4.2.1 14.4.2.2 14.4.2.3 14.4.3 14.4.4 14.5

15 15.1 15.2 15.3 15.3.1 15.3.2 15.3.3 15.4

Comparison with Other Orientation Measurement Methods 306 Analysis of Stress and Strain States 307 General Theoretical Description 307 Quantitative Strain/Stress Analysis in Polycrystalline Silicon Wafers 309 Assumptions 309 Numerical Determination of Stress Components 310 Experimental Procedure to Determine Phonon Frequency Shifts 311 Additional Influences on the Phonon Frequency Shifts 311 Temperature 311 Drift of the Spectrometer Grating 313 Applications 313 Mechanical Stresses at the Backside of Silicon Solar Cells 313 Stress Fields at Microcracks in Polycrystalline Silicon Wafers 315 Stress States at Grain Boundaries in Polycrystalline Silicon Solar Cell Material and the Relation to the Grain Boundary Microstructure and Electrical Activity 316 Comparison with other Stress/Strain Measurement Methods 318 Measurement of Free Carrier Concentrations 318 Theoretical Description 319 Experimental Details 321 Small-Angle Beveling and Nomarski Differential Interference Contrast Micrographs 321 Evaluation of the Raman Data 322 Calibration Measurements 324 Experimental Results 324 Comparison with other Dopant Measurement Methods 328 Concluding Remarks 328 References 329 Strain-Induced Nonlinear Optics in Silicon 333 Clemens Schriever, Christian Bohley, and Ralf B. Wehrspohn Introduction 333 Fundamentals of Second Harmonic Generation in Nonlinear Optical Materials 334 Second Harmonic Generation and Its Relation to Structural Symmetry 336 Sources of Second Harmonic Signals 337 Bulk Contribution to Second Harmonic Generation 338 Surface Contribution to Second Harmonic Generation 341 Strain-Induced Modification of Second-Order Nonlinear Susceptibility in Silicon 343

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Contents

15.5 15.5.1 15.5.2 15.6

Strained Silicon in Integrated Optics 348 Strain-Induced Electro-Optical Effect 348 Strain-Induced Photoelastic Effect 350 Conclusions 352 References 353 Index

357

XV

Preface The development of future integrated (‘‘smart’’) micro- and nanosystems is generally focusing on further improvements of functionality and performance, enhancement of miniaturization and integration density, and extension into new application fields. In addition to any of these technological developments, reliability, quality, and manufacturing yield are key prerequisites for the development of any complex innovative (‘‘smart’’) micro-/nanosystem application. Consequently, new methods, instruments, and tools adjusted to the specific boundary conditions of the miniaturization level down to the nanoscale have to be provided allowing the investigation and understanding of the microstructure, possible failure processes, and reliability risks. In addition, methods and tools allowing the addressing and measurement of locally affected material properties, such as residual stresses, in combination with the microstructure are required. Such instruments and techniques are required to support a focused and rapid technological development and the time-efficient design of components and smart systems. The particular results of microstructure and stress characterization do not only provide the basis for technological process step improvement but are also required for advanced simulation approaches and models that can be used to consider reliability properties already during the product development stage (‘‘design for reliability’’ concept). Such concepts gain increasing importance since they allow to reduce time-to-market and development cost. Present local stress and strain measurements on the nanoscale are based on special transmission electron microscopy techniques such as CBED, HRTEM-GPA, or holographic dark field technology, special scanning electron microscopy techniques such as EBSD or adapted X-ray diffraction techniques such as coherent X-ray diffraction. This book brings together leading groups in these different disciplines to apply these techniques for local strain and stress measurement and its theoretical background. The book consists of three parts. Part One addresses the fundamentals of stress and strain on the nanoscale including an introduction to thermodynamics, kinetics, and models of elasticity, plasticity, and relaxation. Part Two addresses applications where stress and strain on the nanoscale are relevant such as SiGe devices or nanowires. In Part Three, techniques for measuring stress and strain on the

XVI

Preface

nanoscale are presented such as CBED-TEM, EBSD-REM, different ways to use X-rays, Raman, and nonlinear optical methods. To our knowledge, it is for the first time that this compendium combines theory, measurement techniques, and applications for stress and strain on the nanoscale. We believe that with increasing complexity of nanoscale devices, the increasing amount of the integration of various technologies, and various aspect ratios, it will be crucial to understand in detail processes and phenomena of nanostress. This work was stimulated by the cooperation of the Fraunhofer Society, the MaxPlanck-Society, the Carnot Association, and the CNRS via the C’Nano-PACA. This book is dedicated to Prof. Ulrich Gösele, who coinitiated this project. February 28, 2011 Halle and Marseille

Ralf Wehrspohn Margrit Hanbücken Pierre Müller

XVII

List of Contributors Michael Becker Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

Marie-José Casanove CNRS-UPS Centre d’Elaboration de Matériaux et d’Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Sandrine Brochard Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France

Silke Christiansen Max Planck Institute for the Science of Light Guenther-Scharowsky – Str. 1 91058 Erlangen Germany

Christian Bohley Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany and Martin-Luther-University Centre for Innovation Competence SiLi-nano Karl-Freiherr-von-Fritsch-Str. 3 06120 Halle (Saale), Germany

Stéphanie Escoubas Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France and CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

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List of Contributors

Joël Eymery CEA/CNRS/Université Joseph Fourier CEA, INAC, SP2M 17 rue des Martyrs 38054 Grenoble Cedex 9 France

Angelika Hähnel Max Planck Institute of Microstructure Physics Weinberg 2 06120 Halle Germany

Stefan Flachowsky GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany

Margrit Hanbücken CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France

Christophe Gatel CNRS-UPS Centre d’Elaboration de Matériaux et d’Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Michael Hanke Paul-Drude-Institute for Solid State Electronics Hausvogteiplatz 5-7 10117 Berlin Germany

Frank Glas CNRS Laboratoire de Photonique et de Nanostructures Route de Nozay 91460 Marcoussis France Julien Godet Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France Ulrich Göseley Max Planck Institute of Microstructure Physics Weinberg 2 06120 Halle Germany

Jan Hoentschel GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany Manfred Horstmann GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany Michael Krause Fraunhofer IWM Walter-Hülse – Str. 1 06120 Halle Germany Laurence Masson CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France

List of Contributors

Jean-Sébastien Micha INAC/SPrAM UMR 5819 (CEA-CNRS-UJF) CEA-Grenoble 17 rue des Martyrs 38054 Grenoble Cedex 9 France Christine Mottet CINaM – CNRS Campus de Luminy, Case 913 13288 Marseille Cedex 9 France Oussama Moutanabbir Max Planck Institute of Microstructure Physics Weinberg 2 06120 Halle Germany Pierre Müller Aix Marseille Université Center Interdisciplinaire de Nanoscience de Marseille UPR CNRS 3118 Campus de Luminy, Case 913 13288 Marseille Cedex 9 France Shobhana Narasimhan JNCASR Theoretical Sciences Unit Jakkur 560 064 Bangalore India

Olivier Perroud Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France and CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France Laurent Pizzagalli Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France Anne Ponchet CNRS-UPS Centre d’Elaboration de Matériaux et d’Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France Matthias Petzold Fraunhofer Institute for Mechanics of Materials Halle Walter-Hülse-Str.1 06120 Halle

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List of Contributors

Geoffroy Prévot Université Pierre et Marie Curie-Paris 6 UMR CNRS 7588, Institut des NanoSciences de Paris Campus Boucicaut, 140 rue de Lourmel 75015 Paris France Manfred Reiche Max Planck Institute of Microstructure Physics Weinberg 2 06120 Halle Germany

Houda Sahaf CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France George Sarau Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany and

Vincent Repain CNRS et Université Paris Diderot Matériaux et Phénomènes Quantiques Bâtiment Condorcet – Case 7021 75205 Paris France Odile Robach CEA-Grenoble INAC/SP2M/NRS 17 rue des Martyrs 38054 Grenoble Cedex 9 France Christian Roucau CNRS-UPS Centre d’Elaboration de Matériaux et d’Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France Sylvie Rousset CNRS et Université Paris Diderot Matériaux et Phénomènes Quantiques Bâtiment Condorcet – Case 7021 75205 Paris France

Max Planck Institute for the Science of Light Guenther-Scharowsky – Str. 1 91058 Erlangen Germany Volker Schmidt Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany Clemens Schriever Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany and Martin-Luther-University Centre for Innovation Competence SiLi-nano Karl-Freiherr-von-Fritsch-Str. 3 06120 Halle (Saale), Germany

List of Contributors

Olivier Thomas Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France and CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France Nicolas Vaxelaire Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France and CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

Ralf B. Wehrspohn Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany and Fraunhofer Institute for Mechanics of Materials Halle Walter-Hülse-Str. 1 06120 Halle Germany Joerg V. Wittemann Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

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Index a adsorption 41 Asara-Tiller-Grinfeld-method 20 atomistic simulation 72, 87 atomistic valence force field simulations 181

b Beltz and Freund formula 71 biaxially strained layers 139 Bragg-Williams adsorption 41

free surfaces 11 Frenkel-Kontorova-model 92, 152, 153, 154

g geometrical phase analysis 203 Gibbs surface 32 Glissile interfaces 42 Green’s function 6 growth modes 175, 176

h c cantilever 53 convergent beam electron diffraction (CBED) 199 cord construction 8 crystal truncation rods 266 curvature method 196

heterogeneous growth 160 Hookean solid 35 Hough transformation 216

i inclusion 5 inhomogeneous substitutional alloys 7

d

k

defect energy 89 density functional theory 83 diffusion 10 dislocations 63 – critical thickness 16, 113 – nucleation 72 distorted wave Born approximation 269

Kikuchi lines 214 Kikuchi pattern 220 Kirchhoff theory 50 Kohn-Sham equation 86

e eigenstrain 4 elastic energy 30 electron backscatter diffraction 215 Eshelby’s method 5

f Fick’s law 12 finite element methods 52, 158, 271 Fourier method 7

l Langmuir adsorption 41 lattice friction 66 linear stability analysis 9, 122

m mechanical equilibrium 12 Misfit dislocations 15, 153 mixing free enthalpy 8 mobility enhancement 132, 138 modulated composition 9 Monte Carlo simulations 85 morphocompositional instability 22

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanbücken, Pierre Müller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j Index

358

morphological relaxation 17 – strain 17 MOSFET 133

n nanobeam electron diffraction 203 nanoelasticity 27 nanoparticle 44 nanowires 24, 52, 98, 111, 181 nonlinear optics 334 nucleation 66, 159 nudged elastic band 75

o orientation map 214

p Papkovich-Neuber potential 120 Peach-Koehler formula 64 Peierls-Nabarro-theory 72 phase retrieval 254 phonon frequency shifts 311 photoelastic effect 350 piezoresistance coefficients 133 Pikus-Bir hamiltonian 133 prestress 53

strained silicon devices 136, 253, 348 stress domains 156 stress stabilization 11 stress-free strain 4 surface energy 34, 38 surface modifications 68 surface reconstruction 92, 119, 153, 173, 206, 289 surface stress 35, 40, 41, 53, 91, 152, 156

t template fabrication 170 tetragonal strain 16 thin films 46, 96, 140 tight binding second momentum approximation 87 transmission electron microscopy 193 – high-resolution 203 transport flux 12 true fracture strain 62

v Vegard’s law 10 vicinal surfaces 157 Voronoi area 161

w q

Wafer bonding 172

quantum dots 23, 176, 261, 267

x r Raman-spectroscopy 299 reciprocal space maps 248 roughening 115

s second harmonic generation 334 solar cell 299 spinodal decomposition 8 strain engineering 142, 171 strain relaxation – elastic 3, 9, 11, 15, 18, 195 – plastic 63, 65, 162

x-ray diffraction 240 – 3DXRD 241 – GI-SAXS 266 – grazing incidence 264, 275 – HRXRD 269 – Laue microdiffraction 243 – reciprocal space mapping 249

y yield strength 62

z zero-order Laue zone 200

j1

Part One Fundamentals of Stress and Strain on the Nanoscale

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn.  2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Elastic Strain Relaxation: Thermodynamics and Kinetics Frank Glas 1.1 Basics of Elastic Strain Relaxation 1.1.1 Introduction

Although frequently used, the phrase elastic strain relaxation is difficult to define. It usually designates the modification of the strain fields induced in a solid by a transformation of part or whole of this solid. At variance with plastic relaxation, in crystals, elastic relaxation proceeds without the formation of extended defects, thereby preserving lattice coherency in the solid. Elastic strain relaxation is intimately linked with the notion of instability. Indeed, the transformation considered is often induced by the change of a control parameter (temperature, forces applied, flux of matter, etc.). It may imply atomic rearrangements. Usually the realization of the instability is conditioned by kinetic processes (in particular, diffusion), which themselves depend on the stress state of the system. Elastic relaxation may also occur during the formation of part of a system, for instance, by epitaxial growth. The state with respect to which the relaxation is assessed may then exist not actually, but only virtually, as a term of comparison (e.g., the intrinsic state of a mismatched epitaxial layer grown on a substrate). Moreover, it is often only during growth that the kinetic processes are sufficiently active for the system to reach its optimal configuration. In the present introductory section, we give a general principle for the calculation of strain relaxation and briefly discuss some analytical and numerical methods. In the next sections, we examine important cases where elastic strain relaxation plays a crucial part. Section 1.2 deals with strain relaxation in substitutional alloys with spatially varying compositions and with the thermodynamics and kinetics of the instability of such alloys against composition modulations. Section 1.3 introduces a kinetic process of major importance, namely, diffusion, and summarizes how it is affected by elastic effects. Section 1.4 treats the case of a homogeneous mismatched layer of uniform thickness grown on a substrate. Section 1.5 shows how a system with a planar free surface submitted to a nonhydrostatic stress is unstable with respect to Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn.  2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j 1 Elastic Strain Relaxation: Thermodynamics and Kinetics

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the development of surface corrugations. Finally, Section 1.6 briefly recalls how the presence of free surfaces in objects of nanometric lateral dimensions, such as quantum dots or nanowires (NWs), permits a much more efficient elastic strain relaxation than in the case of uniformly thick layers. 1.1.2 Principles of Calculation

At given temperature and pressure, any single crystal possesses a reference intrinsic mechanical state E 0 in which the strains and stresses are zero, namely, the state defined by the crystal lattice (and the unit cell) of this solid under bulk form. If the crystal experiences a transformation (change of temperature, phase transformation, change of composition, etc.), this intrinsic mechanical state changes to E 1 , where again strains and stresses are zero (Figure 1.1a). The corresponding deformation is the stress-free strain (or eigenstrain) e*ij with respect to state E 0 ; for instance, for a change of temperature dT, e*ij ¼ dij adT, where a is the thermal dilatation coefficient. If the crystal is mechanically isolated, it simply adopts its new intrinsic state E 1 ; it is then free of stresses. This is not the case if the transformation affects only part of the system. We then have two extreme cases. The transformation is incoherent if it does not preserve any continuity between the crystal lattices of the transformed part and of its environment. If, on the contrary, lattice continuity is preserved at the interfaces, the transformation is coherent. This chapter deals with the second case. Let us call inclusion the volume that is transformed and matrix the untransformed part of the system (indexed by exponents I and M). Coherency is obviously incompatible with the adoption by the inclusion of its stress-free state E 1 , the matrix remaining unchanged. The system will thus relax, that is, suffer additional strains, which in general affect both inclusion and matrix. It is a strain relaxation in the following sense: if one imagines the inclusion having been transformed (for instance, heated) but remaining in its original reference mechanical state E 0 (which restores coherency, since the matrix has not been transformed from state E 0 ), it is subjected to stresses, since forces must be applied at its boundary to bring it from its new intrinsic state E 1 back to E 0 . With these stresses is associated an elastic energy. The coherent deformation of the whole system constitutes the elastic relaxation.

Figure 1.1 (a) Stress-free strain relative to the inclusion. (b) The three stages of an Eshelby’s process.

1.1 Basics of Elastic Strain Relaxation

This suggests a way to calculate relaxation, Eshelby’s method (Figure 1.1b) [1]: 1)

One applies to the transformed inclusion (state E 1 ) the strain eI* ij , which brings it back to state E 0 . This implies exerting on its external surface (whose external P I normal n has components nj ) the forces  3j¼1 sI ij nj per unit area, where s ij is the stress associated1) with the stress-free strain eI ij .

2)

3)

Having thus restored coherency between inclusion and matrix, one may reinsert the former into the latter. The only change that then occurs is the change of the surface density of forces applied at stage (1) into a body density fi , since the surface of the inclusion becomes an internal interface.2) The resulting state is not a mechanical equilibrium state, since forces fi must be applied to maintain it. One then lets the system relax by suppressing these forces, that is, by applying forces fi , while at the same time maintaining coherency everywhere. One thus has to compute the strain field, in the inclusion (eIr ij ) and in ), solution of the elasticity equations for body forces f , under the the matrix (eMr i ij coherency constraint, which amounts to equal displacements uIr ¼ uMr at the interface.

We may generalize this approach by not differentiating matrix and inclusion. The whole system experiences a transformation producing an inhomogeneous stressfree strain eij ðrÞ (defined at any point r) with respect to initial uniform state E 0 (perfect crystal). One then applies the body forces producing strain eij , namely, P fi ¼ 3j¼1 @s ij =@xj , where sij ðrÞ is the stress associated with strain eij . Finally, one calculates the relaxation field erij , the solution of the elastic problem with forces fi ðrÞ that preserves coherency everywhere (displacements must be continuous). It is important to specify the reference state with respect to which one defines the final state of the system. It is often easier to visualize the relaxed state relative to the uniform state E 0 ; strain is then simply erij. If, on the contrary, the elastic energy W stored in the system is to be calculated, we must take as a reference state for each volume element its intrinsic state after transformation (E 1 ), with respect to which the Ð P total strain is etij ¼ erij eij . Hence, W ¼ ð1=2Þ V 3i; j¼1 etij stij dV, where s tij is the strain associated with etij and where the integral is taken over the whole volume (in the reference state). In case of an inclusion (Figure 1.1), one may easily show that W¼

1 2

ð X 3 I i; j¼1

It eI ij s ij dv

ð1:1Þ

This is a fundamental result obtained by Eshelby [1]. In particular, the total elastic energy depends only on the stress in the inclusion. An example of application to an infinite system with a continuously varying transformation will be given in Section 1.2. Eshelby’s method may also be adapted to 1) Via the constitutive relations, for instance, Hooke’s law in linear elasticity. 2) In the present case (single inclusion), this density is nonzero only in the zero-thickness interface layer, so for a facet x ¼ x0 , one has fi ¼ s I ix dðxx0 Þ.

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other problems. In particular, if the interface between matrix and inclusion does not entirely surround the latter (which happens if the inclusion has a free surface), it is not necessary to apply strain eI ij to the inclusion at stage 1. It suffices to apply a strain that restores the coherency in the interface, which may make the solution of the problem simpler. An example is given in Section 1.4. 1.1.3 Methods of Calculation: A Brief Overview

The problem thus consists in determining the fields relative to stage 3 of the process. One has to calculate the elastic relaxation of a medium subjected to a given density fi of body forces. In addition to numerical methods, for instance, those based on finite elements, there exist several analytical methods for solving this problem, in particular, the Green’s functions method [2] and the Fourier synthesis method. In elasticity, Green’s function Gij ðr; r0 Þ is defined as the component along axis i of displacement at point r caused by a unit body force along j applied at point r0 . For a solid with homogeneous properties, it is a function Gij ðrr0 Þ of the vector joining the two points. One easily shows that for an elastically linear solid (with elastic constants Cjklm ), the displacement field at stage 3 is ui ðrÞ ¼ 

ð @Gij Cjklm elm ðr0 Þ ðrr0 Þd r0 @xk j;k;l;m¼1 3 X

ð1:2Þ

where the integral extends to all points r0 of the volume. Green’s functions depend on the elastic characteristics of the medium, but, once determined, any problem relative to this medium is solved by a simple integration. However, if the Green’s functions for an infinite and elastically isotropic solid have been known since 1882, only a few cases have been solved exactly. If the medium is not infinite in three dimensions, the Green’s functions also depend on its external boundary and on the conditions that are imposed to it. For epitaxy-related problems, the case of the half-space (semi-infinite solid with planar surface) is particularly interesting. These functions have been calculated for the elastically isotropic half-space with a free surface (no external tractions) [3, 4]. Mura’s book gives further details [2]. The method also applies to the relaxation of two solids in contact via a planar interface; in this case, this surface is generally not traction-free and the boundary conditions may be on these tractions or on its displacements. Pan has given a general solution in the anisotropic case, valid for all boundary conditions [5]. In the Fourier synthesis method, one decomposes the stress-free strain distribuÐ tion into its Fourier components: eij ðrÞ ¼ ~eij ðkÞexpðikrÞd k, where k is the running wave vector. In linear elasticity, the solution is simply the sum, weighted by the Fourier coefficients ~eij ðkÞ, of the solutions relative to each periodic wave of wave vector k, which are themselves periodic with the same wave vector. If the system is infinite, the elementary solution is easily determined (see Section 1.2.2). The only nontrivial point is then the integration. This method allows one to treat elegantly the

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

Figure 1.2 Comparison of the elastic relaxation of a truncated square-base pyramidal inclusion in an infinite matrix (a) and in a halfspace (b). Maps in the symmetry plane xz of strain component exx normalized to the intrinsic misfit e0 of the inclusion with respect to the

matrix. Analytical calculation by the Fourier method (see also Ref. [10]). Thick blue lines mark the inclusion contour and the free surface, and the intensity scale is the same for (a) and (b).

stress-free strain discontinuities, as happening at the interface between a matrix and a misfitting inclusion3) having the same elastic constants. The number of problems solved by these methods steadily increases. As for inclusions, let us only mention, in addition to Eshelby’s pioneering work on the ellipsoidal inclusion [1], the case of parallelepipedic inclusions in an infinite matrix [6] and in a half-space [7, 8] and that of the truncated pyramidal inclusions in an infinite matrix [9] and in a half-space [10]. These are important for being the shapes commonly adopted by semiconducting quantum dots. For a given inclusion, the elastic relaxation may be deeply modified by a free surface, on which the tractions must vanish (Figure 1.2).

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

As a first example of strain relaxation, we examine the common case of an alloy whose stress-free state (for instance, its lattice parameter) depends on its composition. If composition variations that preserve lattice coherency develop in an initially homogeneous alloy, internal strains and stresses appear. In this case, elucidating strain relaxation amounts to calculating these fields. We shall see that strains may deeply affect the stability of such an alloy. However, the stability is not determined only by elastic effects. To provide a term of comparison, we shall first consider alloys where compositions variations induce no strain (Section 1.2.1), before turning to the calculation of the elastic fields (Section 1.2.2) and to the way in which they alter the alloy thermodynamics (Section 1.2.3). Finally, Section 1.2.4 discusses how the presence of a free surface affects strain relaxation and hence stability. 3) We call misfitting an inclusion (respectively, a layer) having the same crystal structure as the matrix (respectively, substrate) but different lattice parameters.

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j 1 Elastic Strain Relaxation: Thermodynamics and Kinetics

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gm

gm

(a) h m

gm

(b)

(c) T = T1 ≤ Tc

T = T2 ≥ Tc

0

c

T1 T2

(d)

T1

1

−Τ sm

T T2

g md

d

g mh

h

0

ca

c

c1

M cb 1

0

Figure 1.3 (a) Mixing free enthalpy gm at two temperatures (full lines), with enthalpic and entropic contributions (dashed lines). (b) High temperature case. (c) Low temperature case; tangent construction, equilibrium

cα c 2 c 1s c1 c s2 c β

1 0

cα

I c 1s

M 1 c s2 c β

compositions ca , cb and spinodal compositions c1s , c2s corresponding to inflexion points (circles). (d) Boundaries of miscibility gap (full line) and spinodal gap (dashed lines).

1.2.1 Spinodal Decomposition with No Elastic Effects

Let us consider a bulk binary AB substitutional alloy with atomic concentrations ð1cÞ and c in species A and B.4) With each thermodynamic quantity is associated a mixing quantity, the difference of this quantity between the alloy and the same numbers of its atomic constituents taken as pure solids. In the regular solution model, the mixing free enthalpy (per atom) of the disordered alloy at temperature T is gm ¼ hm Tsm ¼ Vcð1cÞ þ kB T½c ln c þ ð1cÞ ln ð1cÞ, where V is an energy, the interaction parameter (here taken per atom), and kB is the Boltzmann’s constant (Figure 1.3a). If V > 0, the alloy tends to decompose, since alloying its constituents produces an energy hm. However, at high temperatures, entropy may prevent decomposition. The equilibrium state of the system at given temperature and pressure is found by minimizing gm at constant numbers of atoms. Depending on temperature T, gm adopts two different forms, with either a single or a double minimum (Figure 1.3a).5) At high T (Figure 1.3b), the mixing free enthalpy gmh of a homogeneous alloy of any composition c1 is less than that of any mixture of two homogeneous alloys of compositions ca and cb into which it might decompose (the latter, gmd , is given by the cord construction), so the alloy is stable. On the contrary, for T  Tc0, with Tc0 ¼ V=ð2kB Þ, the tangent construction gives two stable equilibrium compositions ca , cb , into which any alloy with intermediate composition tends to decompose (Figure 1.3c). However, two different behaviors are expected. For an alloy of composition c1 intermediate between the compositions cs1 ðTÞ and cs2 ðTÞ 00 corresponding to the inflexion points of the gm curve (i.e., such that gm ðc1 Þ ¼ ð@ 2 gm =@c 2 Þc¼c1 < 0), decomposition into two alloys is always energetically favorable, 4) This also applies to pseudobinary ternary alloys with two sublattices, one homogeneous and the other mixed, such as alloys of compound semiconductors. 5) In the regular solution model, which we retain in the following calculations, the curve is symmetrical about c ¼ 0:5. In the figure, we illustrate a slightly more general situation highlighting the salient feature of the curve, its double-well.

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

even if these two alloys differ infinitesimally in composition (curve gm is above its cord). Conversely, for an alloy composition such as c2 , between a stable composition 00 and an inflexion point (gm ðc2 Þ > 0), it is only decomposition into two compositions differing by a finite amount that is favored (curve gm is below its cord). The homogeneous alloy is unstable in the first case and metastable in the second. In the ðc; TÞ plane (Figure 1.3d), curves ca ðTÞ, cb ðTÞ enclose the miscibility gap, inside which the spinodal curve, locus of points cs1 ðTÞ, cs2 ðTÞ, separates the metastable (M) and unstable (or spinodal, I) domains. The process through which an alloy quenched below Tc0 starts decomposing via composition variations of arbitrary small amplitude is spinodal decomposition. The same results are obtained by linear stability analysis [11]. This general method consists in studying the instability of a system not against an arbitrary perturbation but against a Fourier component of the latter. If the problem is linear (a linear combination of solutions is a solution, the boundary conditions being themselves linear), global instability is equivalent to the existence of at least one unstable elementary perturbation. Here, we study the instability of an alloy of average composition c1 against a sinusoidal composition modulation of amplitude dc along the direction x: cðxÞ ¼ c1 þ dc sin kx. The mixing free enthalpy of the modulated alloy,Ð found by averaging gm over a modulation wavelength l ¼ 2p=k, is l 00 l1 0 gm ½cðxÞdx ¼ gm ðc1 Þ þ gm ðc1 Þd2c =4 at order 2 in dc . The excess of mixing 00 ðc1 Þd2c =4. We thus recover the previous enthalpy due to modulation is dgmc ¼ gm result, namely, an alloy of average composition c1 is unstable (dgmc < 0) at temper00 ðc1 Þ < 0.6) ature T against a composition modulation of vanishing amplitude if gm The foregoing analysis assumes that fractioning the alloy into two phases or modulating its composition produces no excess energy.7) It is however not generally the case, and this can be taken into account by adding to the energy density of the composition modulation a Landau-type phenomenological gradient term that opposes abrupt composition variations [12]. This produces a critical decomposition wavelength, lower wavelengths being energetically unfavorable. The gradient term may be of purely chemical origin, but, in addition, in alloys with size effects that decompose coherently, stresses appear between regions having different compositions and generate additional elastic energy. The next section describes the modifications of the previous results due to these stresses and to the way in which they relax. 1.2.2 Elastic Strain Relaxation in an Alloy with Modulated Composition

Let us assume, for simplicity, that the alloy is of cubic structure or elastically isotropic and that its intrinsic lattice parameter varies linearly with composition (Vegard’s law) aðcÞ ¼ a0 ð1 þ gcÞ with a0 ¼ að0Þ and g ¼ ½að1Það0Þ=að0Þ, the relative lattice mismatch between the pure constituents A and B. If the alloy composition lies within

00 6) In the regular solution model, gm ðc1 Þ ¼ 2V þ kB T=½c1 ð1c1 Þ and Tc0 is obtained for c1 ¼ 0:5.

7) In the first case, this corresponds to an interface energy.

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the metastable domain M, one may expect the nucleation of finite volumes of phases with distinct compositions, hence with different lattice parameters, and decomposition may be coherent or not, depending on the size and composition contrast. In the unstable domain I, the transformation into the stable phases may however occur via the amplification of composition modulations starting with zero amplitude (Section 1.2.1). One then expects the decomposition to be coherent, at least at the start of the process. This is the case we treat here. Two major questions then arise. What are the elastic strain fields induced by a given composition distribution? How does the associated elastic energy modify the stability of the alloy? To answer these questions, let us consider again a sinusoidal composition modulation cðxÞ ¼ c1 þ dc sin kx in direction x. From Vegard’s law, this induces a spatial modulation of intrinsic parameter aðxÞ ¼ a0 ½1 þ gc1 þ ec sin kx with amplitude ec ¼ gdc . As regards elasticity (Section 1.1.2), this corresponds to a stress-free strain modulation eij ðxÞ ¼ dij ec sin kx with respect to the homogeneous state (c1 ).8) From Section 1.1.2, the elastic relaxation with respect to the latter is obtained by P imposing a density of body forces fi ðxÞ ¼  3j¼1 @s ij =@xj . If the alloy is elastically 9) isotropic, fi ðxÞ ¼ dix ½E=ð12nÞec k cos kx. We look for a relaxation displacement field that is continuous (coherency condition) and has the same wavelength as the perturbation: uri ðxÞ ¼ dix ðA cos kx þ B sin kxÞ. The relaxation strain is then erij ðxÞ ¼ dix djx kðA sin kx þ B cos kxÞ10) and the associated stress srxx ¼ fEð1nÞ= ½ð1 þ nÞð12nÞgerxx ; sryy ¼ srzz ¼ ½n=ð1nÞs rxx ; s rij ¼ 0; if i 6¼ j. From the equilibrium between stresses and forces fi , one gets B ¼ 0 and A ¼ ½ð1 þ nÞ=ð1nÞk1 ec . The relaxation field is thus a tetragonal strain modulation along x in phase with the intrinsic modulation but amplified by factor ½ð1 þ nÞ=ð1nÞ [12]. The relaxed lattice parameter remains equal to aðc1 Þ in the directions normal to x. The elastic energy is that of the total strain fields (etij ¼ erij eij ) (Section 1.1.2). Its density is easily found to be dw ¼ ð1=2ÞE 0 e2c =ð1nÞ, where E 0 ¼ E=N is the Young’s modulus E normalized by Avogadro’s number N in order to obtain a density per atom. What is the meaning of relaxation here? With respect to the intrinsic (reference) state of each volume element of the modulated alloy, in which elastic energy is zero by definition, there is an increase of elastic energy due to the coherency between these elements. On the contrary, the energy is lower than that in the state where all elements (with different compositions) would adopt the average lattice parameter (virtual reference state at the end of stage 2; Figure 1.4). If, at least during the early stages of decomposition, the system prefers continuous composition variations and coherent relaxation, it is because the formation of finite domains with distinct compositions, which would all adopt their intrinsic lattice parameter, would induce interfacial defects whose cost in energy would be even higher (see also Section 1.4).

8) Such a problem, where the stress-free strain is a pure dilatation, is a thermal stress problem. 9) Its elastic behavior is then governed by two parameters, Young’s ratio n modulus E and Poisson’s o n, and P strains and stresses s ij o¼ ½E=ð1 þ nÞ eij þ ½n=ð12nÞdij 3m¼1 emm and n are linked by relations P eij ¼ ½ð1 þ nÞ=E  sij ½n=ð1 þ nÞdij 3m¼1 s mm . 10) The only nonzero element of the strain tensor is exx .

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

1.2.3 Strain Stabilization and the Effect of Elastic Anisotropy

The total excess of mixing free enthalpy dgm due to modulation is found by adding the elastic relaxation energy dw to the term dgmc calculated in Section 1.2.1   without 00 ðc1 Þ þ 2E 0 =ð1nÞ e2c =4. The taking stresses into account. One finds dgm ¼ g2 gm condition of instability of alloy with composition c1 against modulations of vanishing 00 amplitude thus changes from gm ðc1 Þ < 0 to 00 gm ðc1 Þ þ 2g2 E 0 =ð1nÞ < 0

ð1:3Þ

Since n < 1, the extra term is positive; so the condition for instability is more restrictive in terms of composition. If the mixing enthalpy at low T is a double-well curve (Figure 1.3c), we see that the domain of unstable compositions is reduced and the critical temperature lowered, an effect often called stress stabilization. In the regular solution model, the critical temperature decreases from Tc0 ¼ V=ð2kB Þ to TcC ¼ f1g2 E 0 =ðVð1nÞÞgTc0 . This reduction may reach several hundreds of kelvins in some metallic alloys [12]. The elastic contribution may even be large enough to render negative the calculated TcC , which means that the instability is then totally suppressed (no composition satisfies Eq. (1.3)). The previous calculations, based on isotropic elasticity, do not specify any favored direction of modulation. To extend the calculations to cubic crystals [13], it suffices to replace term g2 E 0 =ð1nÞ in Eq. (1.3) by a modulus Yk^ depending on modulation direction ^k ¼ k=k. In particular, Y100 < Y110 < Y111 if the elastic constants satisfy to 2C44 þ C12 C11 > 0. Modulations then tend to form in the soft directions of type h100i. These considerations are corroborated by the observation in spinodally decomposed alloys of a characteristic microstructure, manifested in the transmission electron microscopy images by a modulated contrast in the soft directions [11, 14]. More generally, in elastically anisotropic materials, the shape of inclusions and their relative disposition tend to be determined by elastic relaxation [11]. 1.2.4 Elastic Relaxation in the Presence of a Free Surface

The presence of a free surface may also deeply affect strain relaxation. In the infinite solid considered so far, the elastic energy of any composition distribution is the sum Ð 0 of those of its Fourier components, which do not interact since V eik  r eik  r dr ¼ 0 if k 6¼ k0 . This does not necessary hold in a solid bounded by a surface (see Section 1.5). However, this is still true for a planar half-space. Moreover, for a modulation with a wave vector parallel to the surface, the elastic energy may be considerably reduced with respect to the infinite solid. This reduction stems from the extra stress relaxation permitted by the free surface. The optimal modulations have an amplitude that is exponentially attenuated in the direction normal to the surface. Strain stabilization is thus less pronounced than in the bulk and the critical temperature increases accordingly [8]. In the regular solution model, one finds a new critical temperature

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TcI ¼ f1g2 E 0 =ð2VÞgTc0 , such that TcC < TcI < Tc0 [15, 16].11) Composition modulations in directions parallel to the substrate have indeed been observed in epitaxial layers of semiconducting alloys [17, 18]. More generally, the strain relaxation of misfitting inclusions may be strongly affected if they lie close to a free surface, typically within a distance on the order of their dimensions (Figure 1.2) [10].

1.3 Diffusion

Many transformations (such as considered in Section 1.1.1) require a redistribution of matter in the system to become effective. Hence, the realization of the instability is conditioned by kinetic processes, in particular, diffusion. In this sense, the elastic relaxation that accompanies the transformation is also conditioned by diffusion. However, one usually considers that the timescale for diffusion is much longer than that for the mechanical adjustment (relaxation) of the system to the instantaneous distribution of atoms, so the system is continuously in a state of mechanical equilibrium (but of course not in global thermodynamic equilibrium) during the transformation (quasistatic approximation). This is not to say that diffusion and strain relaxation are independent: we shall see in Section 1.3.2 that diffusion is affected by the elastic strain and stress fields. Before this, we briefly consider diffusion without elastic effects. 1.3.1 Diffusion without Elastic Effects

As an example of how diffusion conditions the realization of an instability, consider a bulk alloy AB subject to decomposition (as in Section 1.2), and first ignore the elastic effects. If the homogeneous alloy is quenched below its critical temperature Tc0 , it becomes unstable and tends to decompose (Section 1.2.1). The concentration c of, say, component B becomes nonuniform. According to Fick’s law, this induces a diffusive flux of B atoms, JdB ¼ DB rðc=vÞ, where DB > 0 is the diffusion coefficient of B [19] and v is the atomic volume in the homogeneous reference state of the alloy.12) Setting D0 B ¼ DB =v, we have JdB ¼ D0 B rc. Since D0 B > 0, this flux tends to smooth the concentration gradient, which should inhibit the formation of the composition modulation. It is because the B flux is not limited to Fick’s diffusive term that the modulation may actually develop. Indeed, generally, to the diffusive flux must be added a transport flux depending on the force FB exerted on each B atom [19]. If this force derives from a potential (FB ¼ rwB ), the Nernst–Einstein equation (demonstrated by canceling the total flux at equilibrium and by using the appropriate statistics) leads to 11) It is obtained for modulations with a z-dependent amplitude, such as will be considered in Section 1.5.5. 12) This appears here since we use, as in Section 1.2.1, the atomic concentration c rather than the number of atoms per unit volume.

1.3 Diffusion

JtB ¼ D0B =ðkB T ÞcFB ¼ D0B =ðkB T ÞcrwB . For instance, in the case of charged particles in an electric field, the transport flux is directly related to the electrostatic force exerted on each particle. Now, if the alloy  is ideal, the chemical potential of species B at concentration c is mB ¼ kB T ln c=ceq , where ceq is the equilibrium concentration at temperature T, so the diffusive flux becomes JdB ¼ D0B =ðkB T ÞcrmB and the total flux is JB ¼ JdB þ JtB ¼ D0B =ðkB T ÞcrðmB þ wB Þ ¼ D0B =ðkB T Þcr~ mB , ~B ¼ mB þ wB is a generalized potential, including external forces. where m In the case of the alloy, a similar approach applies, but the definition of the appropriate potential and the generalization of Fick’s law require some care. If the alloy is substitutional, A and B share the same crystal lattice and, in the absence of vacancies, any B atom leaving a site must be replaced by an A atom so that the fluxes of A and B are opposite: JB ¼ JA . The B flux then becomes JB ¼ DrðmB mA Þ ¼ DrM, where D is a phenomenological diffusion mobility, mA and mB are the local and concentrationdependent values of the chemical potentials of A and B, and M is the diffusion potential that replaces and generalizes the chemical potential. Since mB mA is the change of free enthalpy when an A atom is replaced by a B atom at the point considered, we have mB mA ¼ @g=@c, where g is the free enthalpy (per atom) of the alloy at composition c. Moreover, by definition of the free enthalpy of mixing gm introduced in Section 1.2, we have g ¼ cm0B þ ð1cÞm0A þ gm , where m0A , m0B are the position-independent chemical potentials of the pure elements. Hence, JB ¼ DrM ¼ Drð@gm =@c Þ ¼ Dð@ 2 gm =@c 2 Þrc. It now becomes clear that in the spinodal domain, which is precisely defined by @ 2 gm =@c 2 < 0 (Section 1.2.1), the diffusion flux of species B does not oppose the gradient (as follows from the sole consideration of Fick’s diffusive flux) but actually tends to amplify it. In other words, as expected in this domain, the alloy is unstable against composition modulations of vanishing amplitude. This calculation of the diffusion flux is actually much richer than simply confirming the static analysis carried out in Section 1.2.1. It opens the way to a study of the kinetics of decomposition (provided D is known), since the evolution of the composition profile with time t obeys the usual conservation equation div JB þ @c=@t ¼ 0. The interested reader is referred to, for example, Cahn’s publications [14]. This general approach also permits to take into account any effect that modifies the energy of the system (e.g., electric or magnetic fields) by simply adding its contribution to the free enthalpy g. In the next section, we show how the effect of elastic strain fields on diffusion can be calculated in this way. 1.3.2 Diffusion under Stress in an Alloy

The elastic contribution to the free enthalpy per atom is simply ½1=2vetij s tij, where etij ¼ eij eij is the strain corresponding to the transformation of a given volume element from its intrinsic state to its final relaxed state (Figure 1.1b and 1.4) and stij is the associated stress (see Section 1.1.2). Since s tij is defined only formally (etij being the strain between the independent stress-free volume elements and the final relaxed coherent state), it is preferable to express the density of elastic energy as a function of the elastic fields describing the transformation of the homogeneous alloy with composition c1 into the modulated and relaxed alloy (the extreme stages of Figure 1.4),

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ε ij* ( x + λ 2 )

a [c ( x + λ 2 )] − ε * ( x + λ 2 ) ij fi

x+λ 2

a (c1 ) = a1

ε ij* ( x ) (*)

homogeneous alloy continuum, no stress

x

a [c ( x ) ]

a1

− fi

ε ijr ( x )

(1), (2)

ε ij = ε ijr , σ ij

− fi

a1

− ε ij* ( x )

elements in their intrinsic state

ε ijr ( x + λ 2)

fi

(3) modulated alloy, not relaxed

ε

modulated alloy, relaxed

t ij

Figure 1.4 Eshelby process for a bulk alloy with modulated composition.

  namely, ½1=2v eij eij s ij . Elasticity introduces a complication, namely, nonlocal effects, since the stress and strain at a given point depend not only on local composition (as was the case with the chemical potentials) but also on the whole concentration distribution. Nevertheless, it is possible to express the effect of elasticity as an additional term in the diffusion potential M [20], so that the diffusion equation becomes JB ¼ DrM0 , with M0 ¼

X @Sijkl X @e*ij @gm v sij v sij s kl @c @c @c i; j i; j;k;l

ð1:4Þ

where Sijkl are the elastic compliances. If the latter do not depend on composition, the third term of Eq. (1.4) disappears. If this is the case and if the material is cubic or elastically isotropic, then e*ij ¼ gðcc1 Þdij , where g is the relative lattice mismatch between the pure constituents introduced in Section 1.2.2; so M0 ¼ P P @gm =@cvg k s kk , where k s kk is simply the local dilatation. Note that only nonhomogeneous stresses affect diffusion in this way, although, in addition, stress may affect the elastic constants and the diffusion coefficients. For more details on diffusion under stress, including surface diffusion, see Refs [21–23]. Spinodal decomposition may be studied in this way, considering a possible gradient energy of chemical origin (Section 1.2.1) plus the elastic energy. This leads to different growth rates for perturbations of different wavelengths, with a rather narrow peak centered around a fastest developing wavelength. Experimentally, spinodally decomposed alloys indeed tend to exhibit a microstructure with a fairly well-defined decomposition periodicity [14].

1.4 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers 1.4.1 Introduction

Knowing which amount of a given material can be deposited coherently on a mismatched substrate is of great importance in the field of epitaxy. For semicon-

1.4 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers

ducting materials in particular, the extended defects that form when plastic relaxation occurs often affect deleteriously the electrical and optical properties of heterostructure-based devices, and great care is usually taken to remain in the coherency domain during growth. In a simple equilibrium picture, the transition between elastic and plastic relaxation is governed by the total energy of the system. It is thus important to analyze elastic strain relaxation in the coherent case. In the present section, we consider the standard case of a layer of uniform thickness, before turning to possible thickness variations in Section 1.5. We consider mismatched heterostructures formed by depositing a homogeneous layer L of uniform thickness t (0 < z < t) onto a semi-infinite planar substrate S (z < 0), both being single crystals. In the spirit of Section 1.1.2, one may consider that in its intrinsic stress-free state, L results from an elastic distortion of S, described by the stress-free strain that transforms the intrinsic lattice of S into that of L [8, 24]. The methods of Section 1.1 may then be applied. We further assume that S and L have the same structure and differ only by the magnitude of their lattice parameters, so the stress-free strain eij is a pure isotropic dilatation, equal to the relative difference between the lattice parameters aL of the layer and aS of the substrate (or lattice mismatch), e0 ¼ ðaL aS Þ=aS (which is positive if the layer is compressed by the substrate).13) If the deposit L extends infinitely parallel to the planar interface (plane xy), the lattice mismatch makes it impossible for S and L to retain their intrinsic bulk stress-free states if the interface is coherent, simply because the spacings of the lattice planes that cross the interface are different for S and L. Hence the necessity of an accommodation of the lattice mismatch, which can take two extreme forms. If the coherency at the interface is preserved, thanks to a deformation of one or both materials, the accommodation is purely elastic. Conversely, accommodation may be realized plastically, via the formation of a network of misfit dislocations at the interface, which thereby becomes incoherent (section 1.1.1). Leaving aside plastic relaxation, the detailed discussion of which falls outside the scope of the present chapter, we shall briefly show how the methods of Section 1.1 give the solution of the elastic problem. 1.4.2 Elastic Strain Relaxation

If the deposit is much thinner than the substrate, one can safely consider (ignoring possible curvature effects) that only the former is strained while the latter retains its bulk lattice. Moreover, the layer has a free surface z ¼ h, so the substrate/layer interface z ¼ 0 does not entirely surround the layer. As mentioned in Section 1.1.2, it is then not necessary to apply strain eij ¼ e0 at stage 1 of the Eshelby process. It suffices to apply a strain that restores lattice continuity across the S/L interface, that is, ð1Þ ð1Þ such that exx ¼ eyy ¼ e0 . This may be achieved by applying forces along x and y on the elementary cubes that compose the layer, but not along z, so at this (modified) 13) The problem is then equivalent to heating or cooling a layer of thickness h of a stress-free half-space of material S.

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Figure 1.5 (a) Tetragonal strain of a unit cell of parameter a subjected to biaxial stress s 0 (here taken negative, corresponding to compression); e0 ¼  ð1nÞs0 =E. (b) Planar (dashed line) and

corrugated (full line) surfaces. (c) Schematics of strain at crests (c) and valleys (v) with respect to intrinsic (i) and tetragonally strained (q) unit cells.

ð1Þ

stage 1, s zz ¼ 0 (Figure 1.5a). Considering, to simplify, an elastically isotropic medium, the equations of elasticity (see footnote 9) indicate that one then has ð1Þ ð1Þ ð1Þ s xx ¼ syy ¼ Ee0 =ð1nÞ ¼ s0 and ezz ¼ 2ne0 =ð1nÞ (all nondiagonal strains and ð1Þ ð1Þ stresses are zero). Because the forces corresponding to sxx , s yy are uniform, they cancel mutually when the layer elements are reassembled at stage 2, so the corresponding density of body forces is zero. Moreover, since there are no forces along z, the layer is entirely free of applied forces at the end of stage (2), so that considering stage (3) becomes irrelevant. This adaptation of Eshelby’s method thus offers a particularly simple solution of the problem. To summarize, elastic relaxation affects only the thin layer, which adapts its parameter to that of the substrate in the interface plane and strains tetragonally (extending or contracting, depending on the sign of e0 ) in the normal (z) direction, with an amplification factor ð1 þ nÞ=ð1nÞ with respect to intrinsic strain e0 , since the ð1Þ dilatation along z with respect to the substrate is ezz þ e0 ¼ ½ð1 þ nÞ=ð1nÞe0 . The corresponding elastic energy per unit volume of layer is readily found to be ðE=ð1nÞÞe20 . 1.4.3 Critical Thickness

One of the major issues regarding strain accommodation in heterostructures is to find out which factors determine the mode of relaxation (elastic or plastic). It is usually observed that dislocations do not form until the growing layer reaches some critical thickness hc . Basically, such a critical thickness exists because the energies stored in the system per unit area scale differently with layer thickness h. In the coherent state (elastic relaxation), we have just seen that the energy is uniformly distributed in the layer, so the energy per unit area scales with h. On the contrary, in the plastically relaxed state, the density of dislocations of a given type that accommodates a given mismatch (ensuring that the material passes from the stress-free lattice parameter of the substrate to that of the layer across their interface) is fixed and inversely proportional to the relative mismatch [25]. In this case, as a first approximation, the layer is strained in the vicinity of the interface (because of the nonuniform strain fields of the dislocations) but quickly recover its stress-free parameter away from it, so the elastic energy per unit area does not depend on h.

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress

Actually, since the dislocations have a long-range strain field, the elastic energy increases with h, but only logarithmically, much more slowly than in the elastic case (it even saturates when the layer thickness becomes larger the dislocation spacing). In addition, the dislocation cores contribute a constant term to the energy. Hence, the energy stored is larger in the plastic case at low layer thicknesses and in the elastic case at high thicknesses. These considerations are at the origin of the most widely used criterion for calculating hc for a given couple of materials, which consists in comparing the total energies of a given heterostructure in the coherent and plastically relaxed states as a function of thickness and in defining hc as that thickness at which the energy in the former becomes larger than in the latter. This is an equilibrium criterion, equivalent to finding the thickness at which the misfit-induced force acting on a preexisting dislocation tends to pull and extend it into the S/L interface [26–28]. Other criteria are of a kinetic nature and deal with the nucleation of the misfit dislocations or with their motion toward the interface. The critical thickness decreases rapidly when the S/L mismatch increases [26]. In practice, for typical semiconductor materials, hc  10 nm for e0 ¼ 1% and hc  1 nm for e0 ¼ 4%.

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress 1.5.1 Introduction

Consider a homogeneous half-space z  0, subjected to a uniform biaxial stress q q (exponent q) s xx ¼ syy ¼ s0 in plane xy and with a planar traction-free surface q z ¼ 0 (s iz ¼ 0; i ¼ x; y; z). Assuming, for the sake of simplicity, that the medium is elastically isotropic, its response (relaxation) to this stress has been calculated in q q another context in Section 1.4.2:14) it is a uniform tetragonal strain exx ¼ eyy ¼ q e0 ; ezz ¼ 2ne0 =ð1nÞ, with e0 ¼ ð1nÞs 0 =E; the nondiagonal terms are zero (Figure 1.5a). It has been known for a few decades that if one abandons the arbitrary constraint that the free surface be planar (Figure 1.5b), the system can relax even more and adopt a different state, with lower elastic energy [29–31].15),16) Why such a morphological (planar ! nonplanar) transformation might reduce the elastic energy is easily understood: at the crests of the surface, the system may deform not only in direction z but also laterally Figure 1.5c), since there is no matter to prevent it from doing so [31]. To confirm the decrease of the total energy, we must however also examine the 14) Whether the stress is applied by a rigid substrate or externally is irrelevant. 15) In Section 1.5.1, we assume that the system remains chemically homogeneous. The coupling between morphological and compositional instabilities is treated in Section 1.5.5. 16) Note that we consider here a true redistribution of matter along the surface with respect to the planar state and not a simple elastic deformation of the planar surface.

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strain in the valleys and allow for the fact that the area of the corrugated surface is larger than that of the planar one. 1.5.2 Calculation of the Elastic Relaxation Fields

In the present case, elastic relaxation stands for the modification of the strain fields accompanying the planar ! nonplanar transformation. Since the latter implies an actual change of shape following a redistribution of matter, Eshelby’s method (which deals with the change of intrinsic state of a given volume) is not adapted. Instead, we directly solve the elastic problem, the boundary condition being that the corrugated surface remains traction-free. Since this problem has no exact solution for an arbitrary surface profile, we study the elastic response of the system to an elementary perturbation (in the spirit of the linear stability analysis, Section 1.2.1), namely, a sinusoidal modulation (hereafter, undulation) along x of the position h of the surface along z, hðxÞ ¼ D sin kx, measured with respect to the planar state h ¼ 0 (Figure 1.5b).17),18) Let us look for the fields esij , ssij that have to be added to the “q” fields to obtain the total (equilibrium) field. Given the symmetry of the problem, no quantity depends on y; hence esiy ¼ ð1=2Þ@usy =@xi . For the same reason, usy ¼ 0, so esiy ¼ 0 for i ¼ x; y; z. The solution of such a plane strain problem is known to derive from an Airy function xðx; zÞ, solution of differential equation @ 4 x=@x 4 þ 2@ 4 x=@x 2 @z2 þ @ 4 x=@z4 ¼ 0, via relations ssxx ¼ @ 2 x=@z2 , s sxz ¼ @ 2 x=@x@z, and s szz ¼ @ 2 x=@x 2 [33]. Setting xðx; zÞ ¼ jðzÞ sin kx, one finds that j must satisfy differential equation d4 j=dz4 2k2 d2 j=dz2 þ k4 j ¼ 0, the general solution of which is jðzÞ ¼ ðA þ BzÞekz þ ðC þ DzÞekz , with A, B, C, and D constants. Since the fields must remain finite for z ! 1, one has C ¼ D ¼ 0. Finally, s sxx ¼ kð2B þ kA þ kBzÞekz sin kx;

sszz ¼ k2 ðA þ BzÞekz sin kx

ð1:5Þ s sxz ¼ kðB þ kA þ kBzÞekz cos kx; ssyy ¼ 2knBekz sin kx   since esyy ¼ 0 and hence ssyy ¼ n ssxx þ sszz . A shear strain thus appears in plane xz. The undulated surface with normal ðnx ; 0; nz Þ remains traction-free under total field q s ij ¼ s ij þ s sij . Hence,  E s e0 nx þ ssxz nz ¼ 0; sxx  ssxz nx þ sszz nz ¼ 0 ð1:6Þ 1n

the stress being calculated in z ¼ 0. We now assume that the amplitude D of the undulation is small compared to its wavelength 2p=k (i.e., kD  1), and compute the fields at first order in kD (even in the case of a sinusoidal perturbation, there is no exact solution at finite amplitude). Then, nx ¼ kD cos kx, nz ¼ 1. From Eqs (1.6) 17) Even in this case, the problem is not linear, due to the boundary conditions. Indeed, the cancellation of the tractions applies to a different surface for each (D, k) couple. The solution for an arbitrary h profile is not found by summing the solutions for each Fourier component of h. 18) One may also consider localized wavelet surface perturbations [32].

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress

and (1.5), one gets A ¼ 0; B ¼ Eð1nÞe0 D. By using Eq. (1.5), one obtains the stresses, from which the relaxation strains derive via the appropriate relations (see footnote 9). The nonzero components of this strain field are as the following: 1þn kB½2ð1nÞ þ kzekz sin kx E

ð1:7Þ

eszz ¼ 

1þn kB½2n þ kzekz sin kx E

ð1:8Þ

esxz ¼ 

1þn kB½1 þ kzekz cos kx E

ð1:9Þ

esxx ¼

1.5.3 ATG Instability

The elastic energy W of the solid with undulated surface, per unit area of planar (reference state), is easily calculated from the total fields: W ¼ ð1=2Þl1    Ðsurface Ð hðxÞ l q q s s eij þ eij d z. It follows that the variation of elastic energy per 0 dx 1 s ij þ s ij unit area when the system transforms from planar to undulated state is dW ¼

1 2l

1 2l

ðl

ð hðxÞ  dx

0

(ð

ð0

l

dx 0

1

1

 q q sij esij þ ssij eij þ ssij esij d z

s sij esij d z þ

ðl

ð hðxÞ  dx

0

0

q sij esij

q þ ssij eij

)



ð1:10Þ

dz

the second equality being valid at lowest (second) order in kD. Replacing in Eq. (1.10) the “s” fields by their expressions calculated from Eqs (1.7)–(1.9) via the linear elasticity formulas, we get dW ¼ 

11þn1 2 Ee ðkDÞ2 2 1n k 0

ð1:11Þ

Since dW < 0, any surface undulation reduces the elastic energy of the system.19) This fundamental result specifies the driving force for the instability. However, an undulation also increases the effective area of the free surface and induces a surface strain. Here, we only consider the first effect, which translates into an excess energy per unit reference area, equal to cdA=A, where c is the surface free energy (assumed to be independent of the slight orientation changes of the surface) Ð l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and dA=A is the relative variation of area. The latter equals l1 0 1 þ ðdh=dxÞ2 dx,

19) This is not obvious. If indeed, with respect to the planar state, the elastic energy is reduced at the crests, it increases in the valleys (Figure 1.5c). It is only because the former contribute more than the latter that the global balance is favorable.

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that is, k2 D2 =4 at order 2 in kD. The total energy variation per unit area due to the undulation is thus  c 1þnE 1 2  e0 ðkDÞ2 dW 0 ¼ ð1:12Þ 4 1n 2 k The planar surface is unstable with respect to an undulation if dW 0 < 0. This is equivalent to the undulation wavenumber k ¼ jkj being less than the critical value kc ¼ 2ð1 þ nÞð1nÞs20 =ðEcÞ. In other words, there exists a critical wavelength lc ¼ 2p=kc , such that the planar free surface of the biaxially stressed half-space is unstable with respect to any undulation with wavelength larger than lc , with lc ¼ p

1 cE 1n2 s20

ð1:13Þ

The existence of a critical wavelength is due to the fact that the undulation-induced fields (which reduce elastic energy) penetrate the solid over a depth of the order of l (Eqs (1.5) and (1.7)–(1.9)), whereas the excess energy due to the increased area is independent of l. Equation (1.12) indicates that an undulation with a given wavelength is all the more easy to create that s 0 is high and all the more difficult that surface energy is high. This analysis can be generalized. The point is that the stress must be nonhydrostatic. The instability of a solid subjected to such stresses with respect to morphological perturbations of its surface is often called ATG (Asaro–Tiller–Grinfel’d) [29, 30]. It has been observed (with millimetric wavelengths) at the surface of 4 He crystals under uniaxial stress [34]. However, the ATG instability is particularly important for epitaxy. One indeed attributes to it the often-observed formation of undulations at the surface or at the interfaces of semiconducting layers mismatched with respect to their substrates [35, 36]. Indeed, as seen in Section 1.4, if the mismatch is e0 , the semi-infinite substrate exerts biaxial strain s 0 ¼ ½E=ð1nÞe0 .20) Hence, the free surface of the layer is unstable against surface undulations with wavelengths larger than the critical value given by Eq. (1.13). For typical strains on the order of a percent, this wavelength is only on the order of a few tens to a few hundreds of nanometers. This is a mode of strain relaxation that differs from the usual tetragonal distortion of uniformly thick layers (Section 1.4) by its morphology and its elastic fields and also from their plastic relaxation, since relaxation remains elastic (no extended defect appears). However, at the atomic scale, the surface undulation of a low-index surface corresponds to the modulation of the spacing of preexisting or newly created surface steps. One may indeed recover the instability by considering, for instance, a vicinal surface, slightly misoriented with respect to a high-symmetry orientation, and hence composed of facets separated by steps. It has been shown that under nonhydrostatic stress, the steps interact attractively and thus tend to accumulate in bunches [37]. 20) In the half-space case, the relaxation fields are attenuated over a distance of the order of l perpendicularly to the interface. Hence, the two problems become similar if h > l, so that the top  surface and the interface are elastically decoupled.

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress

1.5.4 Kinetics of the ATG Instability

Here also kinetics matter, since they condition the actual formation of the undulation. Srolovitz has treated in a simple fashion two mechanisms whereby a planar surface may undulate, namely, surface diffusion and evaporation/condensation [31]. In both cases, one obtains a mode that develops more rapidly than the others, with a wavelength on the order of lc (as in the case of spinodal decomposition; see Section 1.3.2). This explains simply why if Eq. (1.12) leads to a semi-infinite band ½lc ; þ 1 of unstable wavelengths, the experiments show a rather well-defined wavelength. In the case of epitaxy, one must also take into account the influx of matter from the fluid phase (molecular beams, gas or liquid) in addition to the transport of matter along the surface. This has been done by Spencer et al. [38] who, in addition to substrate rigidity, identify two kinetic factors that tend to inhibit the formation of the undulation, namely, a low temperature (which reduces surface diffusion) and a high growth rate (which buries the undulations before they can develop). 1.5.5 Coupling between the Morphological and Compositional Instabilities

Let us consider a half-space z  0 of a regular solution alloy (A, B) with V > 0 (Section 1.2.1). We know that if its free surface remains planar, it is unstable for T  TcI against composition modulations with arbitrary wavelengths, if gradient energy is ignored (Section 1.2.4). We also know that if it remains homogeneous, its planar surface is unstable against undulations with wave vectors k  kc , at any T (Section 1.5.3). In both cases, elastic relaxation is of primary importance: it determines TcI and it is the driving force for undulation. In this section, we consider briefly how a possible coupling between the two instabilities affects their respective domains of existence.21) To answer this question, we calculate the elastic relaxation of a layer z  0 of average mismatch e0 with a composition modulation along a direction parallel to the substrate/layer interface, but allowing a z-dependent amplitude; without loss of generality, we then write the modulation g1 ec yðkzÞekz with yð0Þ ¼ 1, so that its stress free  strain with respect to the substrate is eij ðx; zÞ ¼ dij e0 þ ec yðkzÞekz sin kx . Assume that its free surface is undulated: z ¼ D sin kx. As in the case of the purely morphological perturbation (Section 1.5.2), the elastic relaxation fields cannot be computed exactly; we limit ourselves to the first order in both kD at ec [24]. The excess of elastic energy (per unit area) of the undulated/modulated state with respect to the planar/homogeneous state is then quadratic in these two variables:   1þnE 1 J2 ð1:14Þ dW ¼ J12 e2c e20 ðkDÞ2 2J1 e0 ðkDÞec þ 1n 2 k 1þn 21) Such a coupling has been observed in mismatched epitaxial alloy layers.

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Figure 1.6 Half-space under biaxial stress. (a) Instability domains with respect to separate lateral composition modulation (C) or surface undulation (M), in plane (wave vector, temperature) (S: stability). (b) Domain of joint morphocompositional instability (MC).

Ð0 where Jn ¼ 1 ½yðuÞn e2u du. In Eq. (1.14), the terms in ðkDÞ2 and e2c correspond respectively to a pure undulation (Eq. (1.12)) and to a composition modulation in a planar half-space [24]. Whatever the sign of product J1 e0 , the cross-product becomes negative with an appropriate choice of the sign of D (the phase of the modulation). The elastic energy of the mixed perturbation is then less than the sum of the two perturbations taken separately. If we assume that yðkzÞ > 0 and that e0 and ec have the same sign, then for x l=4ðmod lÞ, the layer is more mismatched than on average, and it is interesting to have a larger relaxation there, hence also a crest (D > 0). The coupling vanishes if the layer is on average lattice matched to its substrate [15]. To obtain the total excess of free energy, we add to dW the excess free enthalpy of mixing and the excess surface energy, as done respectively in Sections 1.2.3 and 1.5.3 [24]. The conclusion is that not only the instability domains but also the very nature of the instability are modified. We must consider together the two parameters that in the case of uncoupled disturbances ðFigure 1.6a) have critical values separating stable and unstable domains, namely, T and k for the compositional (C) and morphological (M) instabilities, respectively. With coupling (Figure 1.6b), one finds in the ðk; TÞ plane an extended domain of morphocompositional (MC) instability. To each wavenumber k corresponds a critical temperature 1 1 þ n E0 2 k T~ c ðkÞ ¼ TcC þ g 4 1n kB kkc

ð1:15Þ

where TcC is the bulk critical temperature (Section 1.2.3). As usual, this thermodynamical analysis must be completed by a kinetic analysis that will decide if the instability will actually develop, depending on the matter transport mechanisms available [39–41].

1.6 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures

We have seen in Section 1.4 that one way to prevent the formation of dislocations during the growth of a mismatched epitaxial layer on a substrate is to keep the

1.6 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures

layer thickness below its critical value for plastic relaxation. This becomes impractical at high lattice mismatch e0 , since the critical thickness decreases rapidly when e0 increases (Section 1.4.3). In such cases, one may play on the dimensionality and dimensions of the deposit (and sometimes of the substrate) to prevent or hinder dislocation formation. Indeed, when the constraint of infinite lateral extension is lifted, the deposit may recover its intrinsic (stress-free) state even if the interface remains coherent. The lattice planes may then deform continuously from the spacing of the substrate toward the intrinsic spacing of the deposit over some distance from the interface. This is realized in quantum dots and nanowires. 1.6.1 Quantum Dots

Section 1.5 indicates that a mismatched epitaxial layer may reduce its total energy by developing a surface undulation. The same driving force leads to the nucleation of coherent islands (instead of a uniformly thick 2D layer) of a strongly mismatched epitaxial deposit in the Volmer–Weber (VW) or Stranski–Krastanov (SK) growth modes.22) Observed for more than 25 years [42], the SK growth of semiconductors spurred the spectacular development of quantum dot nanostructures.23) Although their origin is the same, the descriptions of the ATG and SK (or VW) instabilities differ in that the ATG instability refers to the development of a smoothly varying surface corrugation (typically realized by step bunching on a singular surface), whereas islands are bound by facets with orientations different from that of the substrate [43]. It is precisely these additional lateral free surfaces that allow an efficient strain relaxation. A similar but weaker effect occurs for quantum wires (1D nanostructures) parallel to the substrate. Elastic relaxation, which affects both island and substrate, may be calculated by the general methods introduced in Section 1.1, the transformed part of the system being the island. If analytical solutions exist for the island buried in a half-space (which is indeed relevant since, in practice, most quantum dots are capped after growth) (see Section 1.1.3 and Figure 1.2), this is not the case for uncapped islands. One then has to resort to numerical methods, such as the finite elements method [44, 45] or the atomistic valence force field method [46]. In addition, approximate analytical solutions have been proposed for the strains or the energy [43, 47–51]. Elastic relaxation of the substrate is also important. Indeed, for an island under compression, the substrate is dilated under the island and compressed in its vicinity. Hence, when a first island has nucleated, it is unfavorable to nucleate another one close to it. This is at the origin of the tendency of the islands to self-organize.

22) These modes differ by the absence (VW) or presence (SK) of a thin wetting layer on the substrate. 23) All dimensions of the islands are nanometric, hence the often-used name 0D nanostructures.

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1.6.2 Nanowires

The same kind of effect occurs in nanowires. Epitaxial NWs growing in a direction perpendicular to their substrate are nowadays fabricated from a large range of elemental and compound semiconductor materials [52]. Similarly to quantum dots, NWs have lateral dimensions ranging from a few nanometers to a few tens of nanometers. At variance with quantum dots, which have inclined facets and nanometric heights, freestanding NWs may extend perpendicularly to the substrate with a uniform diameter and over potentially unlimited lengths (in practice, several microns are easily reached). These 1D nanostructures have remarkable physical properties and many potential applications. For a misfitting layer at the top of a NW, lateral relaxation should be even easier than for a quantum dot, since the effective substrate has the same finite diameter instead of being laterally infinite. The strain relaxation for such relaxation has indeed been calculated [53]. As expected, it is very efficient. For instance, for the same misfitting S/L couple, in the NW, the elastic energy is already only a quarter of its 2D value (for the same volume of layer) when the layer thickness is only 10% of the NW diameter. As a consequence, the critical layer thickness, which is now radius dependent, may be much higher than the 2D value for small, but accessible, NW diameters. Moreover, there exists a critical NW radius under which the critical thickness becomes infinite. Both quantities have been calculated [53]. Finally, a similar effect, albeit somewhat less efficient, operates when a NW is grown on a mismatched substrate. Hence, NWs are nanostructures that allow one to associate with strongly mismatched materials in a way that would not be feasible in the 2D (or even the 0D) geometry.

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6

of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A, 241, 376–396. Mura, T. (1991) Micromechanics of Defects in Solids, Kluwer, Dordrecht. Mindlin, R.D. (1936) Force at a point in the interior of a semi-infinite solid. Physics, 7, 195–202. Mindlin, R.D. (1950) Nuclei of strain in the semi-infinite solid. J. Appl. Phys., 21, 926–930. Pan, E. (2003) Three-dimensional Green’s functions in an anisotropic half-space with general boundary conditions. J. Appl. Mech., 70, 101–110. Faivre, G. (1964) Deformations de coherence d’un precipite quadratique. Phys. Status Solidi, 35, 249–259.

7 Chiu, Y.P. (1978) On the stress filed

and surface deformation in a half space with a cuboidal zone in which initial strains are uniform. J. Appl. Mech., 45, 302–306. 8 Glas, F. (1987) Coherent stress relaxation in a half space: modulated layers, inclusions, steps, and a general solution. J. Appl. Phys., 70, 3556–3571. 9 Pearson, G.S. and Faux, D.A. (2000) Analytical solutions for strain in pyramidal quantum dots. J. Appl. Phys., 88, 730–736. 10 Glas, F. (2001) Elastic relaxation of truncated pyramidal quantum dots and quantum wires in a half space: an analytical calculation. J. Appl. Phys., 90, 3232–3241.

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Structural Transformations in Solids, John Wiley & Sons, Inc., New York. Cahn, J.W. (1961) On spinodal decomposition. Acta Metall., 9, 795–801. Cahn, J.W. (1962) On spinodal decomposition in cubic crystals. Acta Metall., 10, 179–183. Cahn, J.W. (1968) Spinodal decomposition. Trans Metall. Soc. AIME, 242, 166–180. Glas, F. (2000) Thermodynamic and kinetic instabilities of lattice-matched alloy layers: compositional and morphological perturbations. Phys. Rev. B, 62, 7393–7401. Ipatova, I.P., Malyshkin, V.G., and Shchukin, V.A. (1993) On spinodal decomposition in elastically anisotropic epitaxial films of III–V semiconductor alloys. J. Appl. Phys., 74, 7198–7210. Cheng, K.Y., Hsieh, K.C., and Baillargeon, J.N. (1992) Formation of lateral quantum wells in vertical short-period superlattices by strain-induced lateral-layer ordering process. Appl. Phys. Lett., 60, 2892–2894. Glas, F., Treacy, M.M.J., Quillec, M., and Launois, H. (1982) Interface spinodal decomposition in LPE InxGa1xAsxP1y lattice matched to InP. J. Phys. (Paris), 43, C5-11–C5-16. Philibert, J. (1991) Atom Movements Diffusion and Mass Transport in Solids, EDP Sciences, Les Ulis, France. Larche, F.C. and Cahn, J.W. (1982) The effect of self-stress on diffusion in solids. Acta Metall., 30, 1835–1845. Larche, F.C. and Voorhees, P.W. (1996) Diffusion and stress: basic thermodynamics. Defect Diffus. Forum, 129–130, 31–36. Philibert, J. (1996) Diffusion and stresses. Defect Diffus. Forum, 129–130, 3–8. Wu, C.H. (1996) The chemical potential for stress-driven surface diffusion. J. Mech. Phys. Solids, 44, 2059–2077. Glas, F. (1997) Thermodynamics of a stressed alloy with a free surface: coupling between the morphological and compositional instabilities. Phys. Rev. B, 55, 11277–11286.

25 Hirth, J.P. and Lothe, J. (1982) Theory of

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Dislocations, John Wiley & Sons, Inc., New York. Fitzgerald, E.A. (1991) Dislocations in strained-layer epitaxy: theory, experiment, and applications. Mater. Sci. Rep., 7, 87–142. Matthews, J.W. and Blakeslee, A.E. (1974) Defects in epitaxial multilayers. 1. Misfit dislocations. J. Cryst. Growth, 27, 118–125. People, R. and Bean, J.C. (1985) Calculation of critical layer thickness versus lattice mismatch for GexSi1x/Si strained-layer heterostructures. Appl. Phys. Lett., 47, 322–324. Asaro, R.J. and Tiller, W.A. (1972) Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion. Metall. Trans., 3, 1789–1796. Grinfel’d, M.A. (1986) Instability of the separation boundary between a non-hydrostatically stressed elastic body and a melt. Sov. Phys. Dokl., 31, 831–834. Srolovitz, D.J. (1989) On the stability of surfaces of stressed solids. Acta Metall., 37, 621–625. Colin, J., Grilhe, J., and Junqua, N. (1997) Localized surface instability of a nonhomogeneously stressed solid. Europhys. Lett., 38, 307–312. Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity, 4th edn, Dover, New York. Thiel, M., Willibald, A., Evers, P., Levchenko, A., Leiderer, P., and Balibar, S. (1992) Stress-induced melting and surface instability of 4He crystals. Europhys. Lett., 20, 707–713. Cullis, A.G., Robbins, D.J., Pidduck, A.J., and Smith, P.W. (1992) The characteristics of strain-modulated surface undulations formed upon epitaxial Si1xGex alloy layers on Si. J. Cryst. Growth, 123, 333–343. Ponchet, A., Rocher, A., Emery, J.-Y., Starck, G., and Goldstein, L. (1993) Lateral modulations in zero-net strained GaInAsP multilayers grown by gas source molecular-beam epitaxy. J. Appl. Phys., 74, 3778–3782.

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37 Tersoff, J. (1995) Step bunching instability

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of vicinal surfaces under stress. Phys. Rev. Lett., 75, 2730–2733. Spencer, B.J., Voorhees, P.W., and Davis, S.H. (1992) Morphological instability in epitaxially strained dislocation-free solid films: linear stability theory. J. Appl. Phys., 73, 4955–4970. Huang, Z.-F. and Desai, R.C. (2001) Instability and decomposition on the surface of strained alloy films. Phys. Rev. B, 65, 195421. Huang, Z.-F. and Desai, R.C. (2002) Epitaxial growth in dislocation-free strained alloy films: morphological and compositional instabilities. Phys. Rev. B, 65, 205419. Spencer, B.J., Voorhees, P.W., and Tersoff, J. (2001) Morphological instability theory for strained alloy film growth: the effect of compositional stresses and species-dependent surface mobilities on ripple formation during epitaxial film deposition. Phys. Rev. B, 64, 235318. Goldstein, L., Glas, F., Marzin, J.-Y., Charasse, M.N., and Le Roux, G. (1985) Growth by molecular beam epitaxy and characterization of InAs/GaAs strainedlayer superlattices. Appl. Phys. Lett., 47, 1099–1101. M€ uller, P. and Sa ul, A. (2004) Elastic effects on surface physics. Surf. Sci. Rep., 54, 157–258. Christiansen, S., Albrecht, M., Strunk, H.P., and Maier, H.J. (1994) Strained state of Ge(Si) islands on Si: finite element calculations and comparison to convergent beam electron-diffraction measurements. Appl. Phys. Lett., 64, 3617–3619. Ponchet, A., Lacombe, D., Durand, L., Alquier, D., and Cardonna, J.-M. (1998)

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Elastic energy of strained islands: contribution of the substrate as a function of the island aspect ratio and inter-island distance. Appl. Phys. Lett., 72, 2984–2986. Lin, Y.-Y. and Singh, J. (2002) Selfassembled quantum dots: a study of strain energy and intersubband transitions. J. Appl. Phys., 92, 6205–6210. Daruka, I. and Barabasi, A.-L. (1997) Dislocation-free island formation in heteroepitaxial growth: a study at equilibrium. Phys. Rev. Lett., 79, 3708–3711. Gippius, N.A. and Tikhodeev, S.G. (1999) Inhomogeneous strains in semiconducting nanostructures. J. Exp. Theor. Phys., 88, 1045–1049. Tersoff, J. and Tromp, R.M. (1993) Shape transition in growth of strained islands: spontaneous formation of quantum wires. Phys. Rev. Lett., 70, 2782–2785. Wang, Y.W. (2000) Self-organization, shape transition, and stability of epitaxially strained islands. Phys. Rev. B, 61, 10388–10392. Zinovyev, V.A., Vastola, G., Montalenti, F., and Miglio, L. (2006) Accurate and analytical strain mapping at the surface of Ge/Si(001) islands by an improved flat-island approximation. Surf. Sci., 600, 4777–4784. Dick, K.A. (2008) A review of nanowire growth promoted by alloys and nonalloying elements with emphasis on Au-assisted III–V nanowires. Prog. Cryst. Growth Ch., 54, 138–173. Glas, F. (2006) Critical dimensions for the plastic relaxation of strained axial heterostructures in free-standing nanowires. Phys. Rev. B, 74, 121302.

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2 Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity Pierre M€uller

2.1 Introduction

Understanding the relationship between the structure and the shape of a piece of matter and its mechanical properties has always been one of the primary goals of material science. Over the past decades, the rapid progress in the development of new materials with a size of a few nanometers has opened a new field of scientific and technological interest. From a fundamental viewpoint, one of the key features of nanosized material is their high surface/bulk ratio with the consequence that materials in small dimensions behave differently from their bulk counterpart. From a technological viewpoint, nanomaterials are promising building blocks in future devices, but the stresses they develop can be detrimental to their reliability or can be used to modify their physical properties. Thus, for both fundamental and technological applications, there is a growing interest in the study of the mechanics of nanosized objects. For instance, it has been found that the elastic modulus, which measures the proportionality between the applied stress and the measured strain, is size dependent for dimension smaller than roughly 200 nm. However, according to the authors, calculated or measured elastic moduli have been reported to increase or decrease with the size of the object. From an experimental viewpoint, this discrepancy may come from the difficulty of manipulation and the lack of characterization of such small objects, as well as from the uncertainty of the complex boundary conditions between a nanosized object and the micronic-sized tool used to measure its elastic properties. From a theoretical viewpoint, there are still some difficulties to perfectly describe elasticity at the nanoscale. Most of the models used for describing the mechanical properties of a body are based on the classical linear and infinitesimal theory of elasticity of continuum mediums [1, 2]. In such models, a macroscopic object is considered as a continuum medium, the surface of which is a simple boundary at which external forces are applied. It means that the classical macroscopic theory ignores surface effects since it does not attribute any specific elastic properties to the surface. However, because of the missing bonds, there is a redistribution of

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn.  2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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electronic charges close to the surface that modifies the local binding properties so that the surface layers cannot have the same elastic properties as the underlying bulk. This means that because of the reduced coordination of surface atoms, a small piece of matter cut in a homogeneous body is not homogeneous. The simpler way to describe this inhomogeneity is to divide the object into a bulk core and a surface zone with different elastic properties. We will show that when such specific surface properties are considered, the elastic behavior of a finite-size body results from a subtle interplay between its bulk and surface properties and thus depends on the absolute size of the object. Our goal in this chapter is to describe the elastic properties of nanoscale objects from the point of view of surface physicists. In other words, our purpose is to properly introduce surface effects on elastic classical theory.1) This chapter is divided into two parts. The first part emphasizes the theoretical background. It begins by a few results valid for the bulk (Section 2.2.1), followed by a description of the surface elastic properties (Section 2.2.2) in the Gibbs meaning (Section 2.2.3) and then in the twophase models (Section 2.2.4). The second part concerns applications. We will describe how surface stress may induce spontaneous deformation of nanoparticles (Section 2.3.1), may modify the effective modulus of freestanding thin films (Section 2.3.2), and may play a role for the static bending of thin films (Section 2.3.3) and nanowires (Section 2.3.4). A short conclusion will sum up the main effects the surfaces induce on the elastic properties of nanoscale objects. Again, our goal is not to review all recent results, since in particular the number of papers devoted to the study of the mechanical properties of nanosized objects is increasing exponentially, but essentially to underline the fundamentals of nanoelasticity.

2.2 Theoretical Background 2.2.1 Bulk Elasticity: A Recall

The classical theory of elasticity treats solids as continuum mediums in which the deformation is described by continuum fields such as the displacement vector ~ u ð~ rÞ and the strain tensor eð~ r Þ, while internal forces are described by the stress tensor sð~ r Þ. Within linear infinitesimal elasticity, the equilibrium deformation field is a solution of a second-order equation that has to be solved with boundary equations expressed in terms of external forces applied at the free surface of the body. In this section, we will recall some classical results valid for bulk elasticity but necessary as a prerequisite before studying surface elasticity in Section 2.2.2. 1) The limit of the description is that it is necessary to be able to discriminate surface from bulk. Such models thus cannot be used to describe the elastic properties of aggregates formed only by a few atoms, for which it is no more possible to attribute to the atoms a surface or a bulk character.

2.2 Theoretical Background

2.2.1.1 Stress and Strain Definition If a continuum medium is strained, a point in the body is displaced from its initial r þ~ u . The vector ~ up offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi components ui ¼ xi0 xi is called the position ~ r to ~ r0 ¼~ two points of displacement vector. The distance d‘ ¼ dx12 þ dx22 þ dx32 between pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi infinitesimal vicinity d~ x before deformation becomes d‘0 ¼ dx102 þ dx202 þ dx302 with dxi0 ¼ dxi þ dui after deformation. For small displacements, this distance can P be written as d‘02 ¼ d‘2 þ 2 i; j eij dxi dxj , where the quantity   1 @ui @uj eij ¼ þ ð2:1Þ 2 @xj @xi

is called the bulk strain tensor. When the body is strained, there develop forces tending to restore its initial configuration. These internal forces are transmitted across the surface that bounds the volume. The forces ~ F dS exerted on the faces of an elemental cube centered on the considered point can be written for a face i as [1, 2] X Fi ¼ sij nj ð2:2Þ j

where sij are the components of the so-called stress tensor and ~ n is a unit vector directed toward the exterior of the face i (see Figure 2.1). Notice that in the framework of infinitesimal elasticity, the stress and strain definitions do not make a distinction between the deformed and the nondeformed configuration of the body. This is not the case for large deformation [1, 2]. 2.2.1.2 Equilibrium State From Eq. (2.2), the Hforce component Fi (i ¼ 1; 2; 3) on a volume V bounded by a P surface S is Fi ¼ j s ij nj dS. It can be transformed into a volume integral

Figure 2.1 Action of the components s ij of the bulk stress tensor applied on the three front faces of an elementary cube cut in a piece of matter. Each face normal to xj axis bears a triplet

s ij . The first index i ¼ 1; 2; 3 gives the direction xi where the stress acts. On the back faces of the cube, there are identical stresses of opposite sign.

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ÐP Fi ¼ Ð j ð@s ij =@xj ÞdV, and so taking into account the Newton’s second law Fi ¼ rðd2 ui =dt2 ÞdV (valid for a body of density r), we get X @sij j

@xj

¼r

d2 u i dt2

ð2:3Þ

which is called the elastodynamic equation. At equilibrium, it reads X @sij j

@xj

¼0

ð2:4Þ

Note that we have neglected all forms of body forces.2) 2.2.1.3 Elastic Energy Under the action of external forces ~ F applied to the surface S of an elastic body of volumeHV, the surface points are displaced by d~ u, so the work of the external forces is Fd~ u dS. Using the expression of Fi given in Eq. (2.2), we have dW ¼ H ~ P dW ¼ j s ij nj dui dS. This surface integral can be transformed into a volume integral by means of the divergence theorem, so from Eqs (2.1) and (2.4), we have dW ¼

ðX

ð sij deij dV þ

i; j

r

d2 ui dui dV dt2

ð2:5Þ

The first part of (2.5) is the elastic contribution due to the elastic deformation and the second part is the variation of the total kinetic energy during the time dt. We will denote bulk elastic energy by ðX dW bulk ¼ sij deij dV ð2:6Þ i; j

2.2.1.4 Elastic Constants In linear elastic theory, each stress component is assumed to be proportional to each strain component. Taking into account the tensorial nature of both strain and stress fields, this linear dependence (Hooke’s law) can be written as X sij ¼ Cijkl ekl ð2:7Þ k;l

where the material constants Cijkl are called elastic (or stiffness) constants.

2) Body forces are proportional to the body mass. They include gravitational forces, magnetic forces, and inertial forces. They can be written as a volume integral over the body. In contrast, in the context of the classical elastic theory, surface forces result only from the physical contact of the body with another body and are thus expressed as a surface integral over the whole area of the body.

2.2 Theoretical Background

Using (2.7), after integration with respect to strain, Eq. (2.6) becomes ð 1 X dW bulk ¼ Cijkl eij ekl dV 2 i;j;k;l

ð2:8Þ

Depending on the crystal symmetry, one can further reduce the number of elastic constants. The maximum number of elastic constants is required for triclinic crystals, where it is 21. For isotropic bodies, it is 2. In this latter case, Hooke’s law reads ! X E n X sij ¼ l ekk dij þ 2meij or sij ¼ ekk dij ð2:9Þ eij þ 1þn 12n k k when using the Lame (l and m) or Hooke (E and n) constants, respectively. Note that the dilatation of a isotropic cube, uniformly loaded by pressure P, is X k

ekk ¼ 

3ð12nÞ P E

where K ¼ E=3ð12nÞ is the bulk elastic modulus. 2.2.2 How to Describe Surfaces or Interfaces?

In classical elastic theories of the previous section, surfaces solely serve to define boundary conditions but have no specific properties. In Section 2.2.3, we will see that as any extensive thermodynamic potentials (e.g., U, F, G, etc.), the elastic energy can be seen as a contribution from two homogeneous phases plus an interfacial term that contains all the elastic properties of the interfacial zone. The localization and the description of this interfacial zone are discussed in this section. Consider a slab of thickness e in which, because of the missing bonds at the surface, any extensive quantity must continuously vary with the altitude z (see Figure 2.2). The central problem is how to discriminate bulk from surface contributions.

Figure 2.2 A slab of thickness e in which an extensive quantity continuously varies with the altitude. The extensive quantity (let us say a stress parallel to the surface) is represented by a double array whose length varies when approaching each surface.

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Figure 2.3 At the interface between two materials, A and B, extensive quantities are modified from their bulk values by an excess amount gðzÞ (shaded) owing to the presence of

the interface (Figure 2.3a). In the Gibbs model, the whole excess quantity will be ascribed to the dividing surface separating two bulk materials (Figure 2.3b).

A first approach uses the concept of excess quantity [3]. More generally, let us consider two homogeneous phases A and B separated by a planar interface SAB and let us consider an extensive quantity G whose density gðzÞ varies across the interface (z is the axis perpendicular to the interface). In the bulk, far from the interface, g-profile is homogeneous. The interface modifies g by an excess quantity that corresponds to the shaded area in Figure 2.3a. The so-defined excess quantities is 2z 3 ðB ð2:10Þ g interf SAB ¼ SAB 4 gðzÞdzgA ðj0 zA ÞgB ðzB z0 Þ5 zA

where gi are the bulk density in each phase (i ¼ A; B), SAB the interfacial area, and g interf the interfacial excess quantity per unit interface area. This definition does not depend upon the values of zA and zB provided they are, respectively, lower and higher than jA and jB . Some well-known interfacial excess quantities are the number of particles in excess at the surface, the surface entropy, or even the mass excess of the interfacial zone. Gibbs [3] proposed, for a pure substance in contact with the vacuum, to choose the localization of the dividing surface j0 to the mathematical plane for which there is no surface excess of mass3) (Figure 2.4). In the case of a multicompound substance, the dividing surface can be fixed to the mathematical plane for which the surface excess mass of one species is zero. In this case, a surface excess remains for the other components j. Once the position of the Gibbs surface is defined, the system is modeled as a bulk material with unaltered properties up to the dividing surface at which is added some excess quantity completely assigned to the Gibbs dividing surface (Figure 2.3b and Figure 2.5a). In this ideal scheme, the volume of the interfacial zone between the ideal bulk and the vacuum is zero (Figure 2.3a). Notice that elastic deformations change the position of the Gibbs surface [4].

3) It is sometimes called the Gibbs equimolar dividing surface.

2.2 Theoretical Background

Figure 2.4 The location of the Gibbs dividing surface is chosen at theÐmathematical plane j0 for which the surface excess number of a given specie vanishes (Nsurf ¼ nðzÞdzNA NB ¼ 0).

An alternative choice is to attach the dividing surface to a given piece of matter (Figure 2.5b) and, for instance, to treat the surface layers as a coating thin film with elastic properties different from those of the underlying bulk. The thickness of the film is thus arbitrary defined by the position of the dividing surface. The two models must not be confused, even when the surface film is one monolayer thick. Indeed, these two idealized schemes (Figure 2.5a and b) correspond to different definitions of associated thermodynamics quantities. For instance, when using the ideal Gibbs dividing surface, the surface energy c and the surface stress sab components are perfectly defined as excess quantities, which is not the case for the two-layer models (surface film plus bulk-like phase) for which the energy and the stress of the surface layer, respectively, are Elayer ¼ Ebulk þ c and tab ¼ s ab þ sab , where Ebulk and sab , respectively, are the energy and the stress of a “bulk-like” monolayer. For very thin films, the two surface zones shown in Figure 2.2 may overlap, making it impossible to define excess quantities with respect to the (nonexistent) bulk. Thus, the idealization of the dividing surface is valid only if the body is much larger than several atomic layers.

Figure 2.5 Two approaches to distinguish surfaces from the bulk. (a) Gibbs model in which surface excesses are ascribed to a mathematical plane (the dividing surface). (b) The surface can also be modeled as a thin film of thickness e. These models represent the variation in Figure 2.2.

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2.2.3 Surfaces and Interfaces Described from Excess Quantities

Here, we will consider Gibbs’ model (Figure 2.5a). For the sake of simplicity, we will write x  x1 ; y  x2 ; z  x3 , with this last axis normal to the surface. Moreover, in what follows, Latin indices (i; j ¼ 1; 2; 3) describe bulk properties, while Greek indices (a; b ¼ 1; 2) describe surface properties. 2.2.3.1 The Surface Elastic Energy as an Excess of the Bulk Elastic Energy The elastic energy (Eq. (2.6)) is an extensive quantity that may be present in excess at the interface between two homogeneous materials as defined by Eq. (2.10). This excess quantity is defined as the difference between the elastic energy of the whole system and the sum of the elastic energies of the two homogeneous materials (see Figure 2.3): zðB

dW

interf

¼ zA

X

s ij ðzÞdeij ðzÞdV

ij

X

sAij deAij V A 

ij

X

sBij deBij V B

ð2:11Þ

ij

where sij ðzÞ and eij ðzÞ are the bulk stress and strain profiles across the interface, while s nij and s nij (with n ¼ A; B) are the homogeneous stress and strain tensors in the two homogeneous materials A and B separated by the interface (normal to the axis z). The interfacial quantity is the shaded area in Figure 2.3. Equation (2.11) can be simplified when considering two physical conditions. i) The first condition is mechanical equilibrium that states that in absence of body forces, Eq. (2.4) gives (since there is only one variable z ¼ x3 ) @s i3 ¼ 0; @x3

i ¼ 1; 2; 3 8 x3

ð2:12Þ

This means that the normal components of the stress tensor are homogeneous in the whole material: siz ðzÞ ¼ sAiz ¼ sBiz

ð2:13Þ

ii) The second condition is a nongliding condition4) that states that any infinitesimal change of the strain tensor components parallel to the interface must be the same in the whole material: deab ðzÞ ¼ deAab ¼ deBab

ða; b ¼ 1; 2Þ

Using conditions (2.13) and (2.14), Eq. (2.11) becomes [5] X X dW interf =SAB ¼ sab deab þ siz deiz ab

i

4) Glissile epitaxy will be treated in Section 2.2.3.7.

ð2:14Þ

ð2:15Þ

2.2 Theoretical Background

where sab

2z 3 ðB 1 4 ¼ AB sab ðzÞdVsAab V A sBab V B 5 S

ð2:16Þ

zA

and 2z 3 ðB 1 4 A A B B5 ei3 ¼ AB ei3 ðzÞdVei3 V ei3 V S

ð2:17Þ

zA

These equations suggest that we define the interfacial stress tensor 0 1 s11 s12 0 ½s ¼ @ s21 s22 0 A 0 0 0 and the interfacial 0 0 ½e ¼ @ 0 e31

strain tensor 1 0 e13 0 e23 A e32 e33

ð2:18Þ

ð2:19Þ

Notice that interfacial stress and interfacial strain are orthogonal tensors since sij eij ¼ 0. 2.2.3.2 The Surface Stress and Surface Strain Concepts Let us now consider the case where the phase B is vacuum so that s Bi3 ¼ 0 at mechanical equilibrium. In this case, Eq. (2.15) becomes X dW surf ¼ SAB sab deab ð2:20Þ ab

where sab

2z 3 ðB 1 4 ¼ A sab ðzÞdVsAab V A5 S

ð2:21Þ

zA

now defines the surface stress as an excess quantity that generally is nonzero even if there is no stress in the bulk. Surface stress components may be positive (tensile) or negative (compressive). Note that for a Hookean solid, the total elastic energy of a semi-infinite solid is the sum of a bulk elastic energy (Eq. (2.8)) and a surface elastic energy (Eq. (2.20)): ð ðX 1 X dW el ¼ dW bulk þ dW surf ¼ Cijkl eij ekl dV þ sab eab dS ð2:22Þ 2 i;j;k;l ab For illustrative purposes, Figure 2.6 shows the calculated bulk energy versus normal deformation ezz (Figure 2.6a) and then versus in-plane deformation exx

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Figure 2.6 (a) Elastic energy versus strain ezz for a bulk material (circles), an unrelaxed slab (squares), and a relaxed slab (diamonds). (b) Elastic energy versus strain exx for a bulk material (circles), an unrelaxed slab (squares),

and a relaxed slab (diamonds). Note that when the elastic relaxation of the slab is properly taken into account, the parabola DE ðexx Þ is shifted but not the parabola DE ðezz Þ. It means that the components siz ¼ 0.

(Figure 2.6b). Three calculations have been performed: for a bulk material, for an unrelaxed slab of n layers, and for an elastically relaxed slab (ab initio calculations by Wien code [6], cubic material). For the slab, two calculations have been performed: the first one is for an unrelaxed slab and the second one for a relaxed slab. Figure 2.6 shows that (i) for such small deformations, Hooke’s law is valid (quadratic shape of the bulk energy); (ii) for unrelaxed slab, the surface stress shifts the parabola of the bulk energy that, according to (2.22), now contains a quadratic term and a linear term and roughly scales Ce2 þ se; and (iii) for an elastically relaxed slab, the bulk parabola is shifted for the in-plane deformation but remains centered on zero for the normal deformation. This last point (iii) simply illustrates that at mechanical equilibrium, surface stress components siz must be zero, so surface stress can be considered as a degenerated two-dimensional second rank tensor of order 2 (in the referential of the flat surface):   s s ð2:23Þ ½s ¼ 11 12 s21 s22 Notice that Eq. (2.20) implies that the surface stress is the isothermal work per unit area against surface deformation at constant number of surface atoms. It must not be confused with the work done per unit area against surface creation at constant strain obtained by reversible cleavage, which is called surface energy (at least for pure materials). Figure 2.7 shows the main differences between surface stress and surface energy. The surface energy c is associated with breaking bonds. It can be calculated, at least at 0 K, by counting the broken bonds (and thus by adding interatomic potentials). The surface energy is a scalar quantity that scales as energy per unit area. The surface stress, which is a tensor, is associated with the work against surface deformation, and it is the result of forces acting at the material surface (and thus calculated as a sum of the first derivatives of the interatomic potentials). More precisely, if the surface is

2.2 Theoretical Background

Figure 2.7 Difference between surface stress and surface energy. The surface stress originates from an elastic deformation and thus from bond stretching. It is an excess of stress at the surface (double arrows). Surface energy originates from

a cleaving process and thus from breaking bonds. It is thus calculated as the sum of bonds (A is the surface area and ni the number of ith neighbors whose bonding energy ji are sketched by the arrows).

divided into parts by a curved boundary, where ~ n is the unit vector normal to the P boundary in the surface plane, the surface stress tensor gives the force fi ¼ j sij nj exerted across the boundary curve. 2.2.3.3 Surface Elastic Constants If a stress-free system is deformed in a direction parallel to the surface, a bulk stress will appear. For small deformations, Hooke’s law is valid, so the total energy of a piece of the deformed matter (with surface and bulk contributions) is (up to the second order in strain) E ¼ E0 þ A0 c0 þ A0

X

s0;ab eab þ

ab

V0 X Cijkl eij ekl 2 ijkl

ð2:24Þ

where E0 is the cohesive energy, A0 c0 the surface energy, and the last term the elastic energy shared in its surface and bulk contributions (see Eqs (2.8) and (2.20)). (A0 is the area of the free undeformed surface and s0;ab the surface stress components.) Equation (2.24) can be compared with the energy stored by an equivalent volume but without any surface: E bulk ¼ E0 þ

V0 X bulk C eij ekl 2 ijkl ijkl

Comparing Eqs (2.24) and (2.25), we obtain   A0 c0 ¼ EE bulk e¼0

ð2:25Þ

ð2:26Þ

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which defines the surface energy c0 A0 as the surface excess of the cohesion energy measured at zero strain. In the same way, we obtain   @E @E bulk A0 s0;ab ¼  ð2:27Þ @eab @eab e¼0 which defines the surface stress as the surface excess of the first derivative of the cohesion energy versus bulk deformation. Finally, the comparison of Eqs (2.24) and (2.25) gives  2  h i @ E @ 2 E bulk surf bulk A0 Cijkl ¼  ¼ V0 Cijkl Cijkl ð2:28Þ @eij @ekl @eij @ekl e¼0 surf which now defines the surface elastic constants Cijkl as the surface excess of the second derivative of the cohesion energy versus strain. The surface elastic constants measure how the surface stress changes with strain. Note that

i) From Eq. (2.28), the surface elastic constants can, in principle, be negative. It does not violate thermodynamic stability. Indeed, a surface cannot exist on its own, but must be supported by a bulk material. In other words, it is the total energy (surface plus volume) that ensures solid stability. ii) Since, at equilibrium, the components of the stress tensor perpendicular to the surface must be zero, every bulk elastic constants has not an excess at the surface. Table 2.1 lists the independent surface elastic constants for different plane point groups as reported by Shenoy [7] X sab ¼ s0ab þ Sabcd ecd ð2:29Þ cd

Table 2.1 Surface elastic constants.

Point group

Independent constants

Constraints

1, 2

All the 9 Sabcd

No

1 m, 2 mm

S1111 ; S1122 ; S1212 ; S2211 ; S2222

S1112 ¼ S1211 ¼ S1222 ¼ S2212 ¼ 0

4

S1111 ; S1112 ; S1211 ; S1212 ; S1122

S1122 ¼ S1211 ; S2211 ¼ S1122 S2212 ¼ S1112 ; S2222 ¼ S1111

4 mm

S1111 ; S1122 ; S1212

S1112 ¼ S1211 ¼ S1222 ¼ S2212 ¼ 0 S2211 ¼ S1122 ; S2222 ¼ S1111

3,6

S1111 ; S1112 ; S1122

S1211 ¼ S1112 ; 2S1212 ¼ S1111 S1122 S1222 ¼ S1112 ; S2211 ¼ S1122 S2211 ¼ S1122 ; S2222 ¼ S1111

3 m, 6 mm

S1111 ; S1122

S1112 ¼ S1211 ¼ S1222 ¼ S2212 ¼ 0 2S1212 ¼ S1111 S1122 S2211 ¼ S1122 ; S2222 ¼ S1111

Adapted from Ref. [7].

2.2 Theoretical Background Table 2.2 Relations between surface quantities.

Euler

Lagrange

Excess energy

AðeÞcE ðeÞ  A0 c0 þ A0 s0 e þ

Surface energy

cE ðeÞ  c0 þ ðs0 c0 Þe þ

Surface stress

 E sE ðeÞ  c0 þ @c @e 

e¼0

A0 C0surf e2 2

ðC0surf s0 Þe2 2



þ C0surf s0 e

A0 cL ðeÞ  A0 c0 þ A0 s0 e þ cL ðeÞ  c0 þ s0 e þ  E sL ðeÞ  @c @e 

e¼0

A0 C0surf e2 2

C0surf e2 2

þ C0surf e

Adapted from Ref. [5].

meaning Sabcd ¼

1 @ 2 W surf s0 dcd S @ eab @ecd ab



in agreement with the simplified expression s ¼ s0 þ C0surf s0 e, given in Table 2.2 for isotropic surface stress. In Table 2.2 are shown the most useful relations valid for isotropic stresses and strains written in Eulerian (the reference state is the deformed state) and Lagrangian (the reference state is the undeformed state) coordinates. 2.2.3.4 Connecting Surface and Bulk Stresses At mechanical equilibrium, the forces exerted by the surface to the underlying substrate must be equal and opposite to the distributed forces exerted by the bulk on the surface. It follows that the presence of surface stress results in nonclassical boundary conditions valid at equilibrium. Mechanical equilibrium is achieved on much smaller timescales than shape equilibrium that needs mass transport. Mechanical equilibrium can thus be derived at constant surface shape. In other words, mechanical equilibrium is achieved when the first variation of W el (given by (2.22)) with respect to a variation of displacement dui vanishes. The first variation is ðX ðX ðX @ekl dW el ¼  Cijkl dui dV þ Cijkl ekl dui nj dS þ sab eab dS @xj ijkl i; j;k;l ab ð2:30Þ

Note that the last integral can be written as5) ð  ½divS s þ ðs : kÞ~ n d~ u dS

ð2:31Þ

5) For this purpose, it is enough to replace in (2.30) the strain components by the expressions of e in terms of the derivative of the displacement field (see Eq. (2.1)) and then to use the divergence theorem.

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where for the sake of readability, we avoid index notation and use the surface divergence operator divS, the curvature tensor k, and: for tensor product. Finally,~ n is a unit vector normal to the surface. Since at equilibrium the first variation dW el must vanish for any arbitrary value of P d~ u , one obtains (with Eq. (2.7) and in absence of body forces) fi ¼ j @sij =@xj ¼ 0 in n at the surface or more precisely @sab =@xa ¼ sa3 the bulk and s ij nj ¼ divS s þ ðs : kÞ~ and s ij ni nj ¼ sab kab . It is more convenient to project the last equations on the threeaxis surface ~ x i to get 8 @s11 @s12 > > s13 ¼ þ ðaÞ > > @x1 @x2 > > > > < @s21 @s22 s23 ¼ þ ðbÞ @x1 @x2 > > > > > s11 s12 > > ðcÞ > : s33 ¼ R1 þ R2

ð2:32Þ

where Ri are the principal curvatures. Parts (a) and (b) in Equation (2.32) mean that, at the surface, the surface stress variation must be compensated by bulk shearing stresses. Moreover, note that the last equation is the Gibbs–Thomson equation, giving the Laplace overpressure in a curved crystal. Obviously, it reads s 33 ¼ 0 for a free planar surface. For completeness, Weissm€ uller and Cahn [8] have derived a general expression for the mean stress in microstructures due to interfacial stresses. 2.2.3.5 Surface Stress and Surface Tension Using Eq. (2.15), the excess of internal energy at the free surface of a body becomes dU surf ¼ T dSsurf þ

X ab

sab deab A þ

X

mi dNisurf

ð2:33Þ

i

where Ssurf is the surface entropy, Nisurf the number of particles in excess at the surface of chemical potential mi , and A the surface area [5]. Using the Euler integral of the surface excess energy [5] in the form P U surf ¼ TSsurf þ cA þ i mi Nisurf , we obtain the Gibbs–Duhem equation valid for a free surface [5] dc ¼ 

X X

Ssurf dT þ sab cdab deab  Cn dmn A n ab

ð2:34Þ

where Cn is the surface number density of particles n of chemical potential mn . One of the derivatives of c stemming from Eq. (2.34) gives the so-called Shuttleworth equation that connects surface stress and surface energy:  @c  sab ¼ cdab þ ð2:35Þ @eab m ;T;eiz n

2.2 Theoretical Background

Equation (2.35) is obtained using the deformed surface (Eulerian coordinates) as a reference state. If the undeformed state is used (Lagrangian coordinates), the Shuttleworth relation becomes  @c  sab ¼ ð2:36Þ @eab m ;T;eiz n

Bottomley et al. [9] discuss the compatibility of the Shuttleworth equation with the Hermann formulation of thermodynamics. In response to Bottomley, subsequent works [10–13] clarified and validated the Shuttleworth relation. 2.2.3.6 Surface Stress and Adsorption Another derivative of c stemming from (2.34) gives the so-called Gibbs adsorption isotherm  @c  ¼ Cn @mn T;eij

which shows how the surface energy varies with respect to the chemical potential mn . Using this expression, the Shuttleworth Eq. (2.35) gives the surface stress change with respect to the chemical potential as  @sab  @Cn ¼ Cn dab  ð2:37Þ @mn T;eij @eab For a Langmuir adsorption, the surface coverage q linearly depends on the pressure P by means of q P ¼ expðj=kT Þ 1q P1

ð2:38Þ

where P1 is a constant and j the interaction energy between the adsorbate and its substrate.6) In this case, using the Gibbs adsorption isotherm and Eq. (2.37), the surface stress and the surface energy change with coverage become [5] Dc ¼

kT lnð1qÞ and a2

Dsab ¼ Dcdab þ

q @j a2 @eab

ð2:39Þ

where 0 < q < 1. Some examples of the so-calculated surface stress change are shown in Figure 2.8a in case of Langmuir adsorption, where Eq. (2.39) is valid, and also in case of Bragg–Williams adsorption, where adsorbate–adsorbate interactions play a role (see Ref. [15]). The surface stress change induced by an adsorbate can be measured. Consider a sheet (or a cantilever), a face of which is exposed to foreign molecules. Owing to the 6) Notice that Cn ¼ Nn =A, where Nn is the number of adsorbed atoms and A the surface area, whereas qn ¼ Nn =N, where N is the number of surface sites. At the same, the chemical potential is defined as m ¼ kT lnðP=P1 Þ.

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Figure 2.8 Surface stress versus adsorption: (a) Surface stress variation calculated for Langmuir adsorption and Bragg–Williams adsorption. Adapted from Ref. [15]. (b) Surface

stress variation measured for C/Ni(1000) and Co/Pt(111). Adapted from Ref. [14]. (c) Surface stress variation versus coverage for Fe/W(001). Adapted from Ref. [18].

adsorption, the two sides of the sheet are no more in the same state of stress (see Eq. (2.39)), so the sheet spontaneously bends. The radius of curvature RðqÞ is simply related to the surface stress difference by means of the Stoney formula [16] valid for thin isotropic sheets characterized by their Young modulus E and Poisson ratio n: 1 6ð1nÞ ¼ DsðqÞ RðqÞ Eh2

ð2:40Þ

Since the surface stress varies with the adsorbed coverage q, the measurement of the coverage variation of the radius of curvature gives access to sðqÞ when the surface stress of the bare sheet (before adsorption) is known. Note that for very thin films, the Stoney formula needs to be corrected [17]. Many experimental results on surface stress change with adsorption have been published (for a review, see Ref. [14]). For example, Figure 2.8b shows the surface stress change induced by carbon atoms adsorbed on Ni(001) surface. Figure 2.8b shows the surface stress change experimentally recorded for Fe/W(110) by Sander et al. [18]. In this last case, the sudden variations of surface stress versus coverage are associated with the appearance of successive surface phases. The case of nonhomogeneous adsorption has been studied by Bar et al. [19]. 2.2.3.7 The Case of Glissile Interfaces To be complete, notice that for glissile epitaxies, the description of stresses at the interface requires two interfacial tensors. One reflects the straining at constant deformation in both phases, while the other reflects the alteration of the interface structure at constant average strain [20, 21]. 2.2.4 Surfaces and Interfaces Described as a Foreign Material

When using two-phase models (see Section 2.2.2), the surface is considered as a foreign thin film coating an underlying material. The basic equations consist of classical elastic equations valid for the bulk coupled to elastic equations valid for the film. Generally, the surface film cannot glide on the underlying bulk.

2.2 Theoretical Background

2.2.4.1 The Surface as a Thin Bulk-Like Film In unorthodox approaches, the system (solid þ surface) may be considered as a composite material constituted by two different materials.7) As an illustration, let us consider the flexural rigidity (defined as the force torque required to bend a system to a unit curvature). For a single material, the flexural rigidity depends on the elastic modulus and the second moment of inertia of the sheet. For a multilayered composite, the flexural rigidity can be written as a function of the flexural rigidity of each of its layers. In a similar way, the flexural rigidity of a piece of body separated by a bulk core surrounded by a surface zone can be calculated by attributing specific elastic properties to the surface considered as a layer film with specific elastic constants and a given thickness. Notice again that this layered material (body þ surface) simply is a pileup of bulk-like materials without any specific mechanical interfacial properties in the Gibbs meaning. Such models will be discussed in detail in Section 2.3.4. 2.2.4.2 The Surface as an Elastic Membrane The main difference from the previous case is that now the film modeling the surface is considered to be an elastic membrane, that is, a material of negligible thickness and negligible flexural rigidity8) bonded to a bulk substrate material. For the sake of simplicity, both materials (core þ membrane) are generally assumed to be isotropic and described by two Lame constants li ¼ E i ni =ðð12ni Þð1 þ ni ÞÞ and mi ¼ E i =ð2ð1 þ ni ÞÞ, with i ¼ surf for the membrane and i ¼ bulk for the bulk material. The stress tensor (of components tij ) characterizing the surface is defined by constitutive equations that ensure that the tractions transmitted by the membrane to the volume are equal and opposite to the force distribution in the bulk at which is added a displacement gradient which can be neglected when Lagrange and infinitesimal strains can be confused: [23–25] ( P surf 0 surf t þ t0 Þ k e t0 Þeab þ t0 ð@u a =@xb Þ ab ¼ t dab þ ðl kk dab þ ðm 0  t a3 ¼ t ð@u3 =@xa Þ

ð2:41Þ

where “” refer to the up and down surfaces that, because of the deformation, cannot be in the same state of stress (Figure 2.9). In our context, the Lame coefficients of the up and down surfaces are identical. The quantity t0, called residual surface stress, under unconstrained conditions [23–25] may originate from thermal or mechanical treatment and thus is unrelated to s or c. The previous definition (Eq. (2.41)) of t can be formally compared to the expression of s given in Eq. (2.29), meaning the sum of a stress in absence of deformation and a linear function of strain. Beyond the formal equivalence, let us note that the physical meaning of the various components of Eqs (2.29) and (2.41) are not the same since 7) One can thus distinguish three-phase model for plate (a core plus two opposite surfaces) from twophase models valid for cylindrical or spherical objects (a core plus one surface). 8) Steigman and Ogden [22] took into account the flexural rigidity of the surface layers. In this case, there will be additional equilibrium equations, meaning that bending couples must vanish on the edge of the surface.

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Figure 2.9 Sketch of the bulk stress variation (black arrows in the bulk) inside a curved sheet at which adds the surface stress (red arrows at the surface). Inset: Expression of the variation s xx ðzÞ versus the second derivative of the profile.

surface stress tensor ½s is a true excess tensor (in the Gibbs meaning), but not ½t except for some peculiar choice of the dividing surface.

2.3 Applications: Size Effects Due to the Surfaces

We will describe in turn how surface stress (i) induces spontaneous deformation of nanoparticles (Section 2.3.1), (ii) modifies the effective modulus of freestanding thin films (Section 2.3.2), and (iii) plays a role on the static bending of thin film (Section 2.3.3). At the same, we will consider the static bending of nanowires (Section 2.3.4). Generally, we will use approaches expressed in terms of surface stress, but in some cases (according the state of the art) we will also use two-phase models. 2.3.1 Lattice Contraction of Nanoparticles

Following Ref. [26], let us take a free cubic crystal A with a rectangular shape of basis ‘x ‘y and height ‘z . We will note sA and s0A , respectively, the surface stresses of the basal and lateral faces of the crystal. Let us now consider a virtual homogeneous deformation described by the bulk tensor eij. The elastic energy due to this deformation is W ¼ W bulk þ W surf

ð2:42Þ

where, according to (2.6) and (2.20), W bulk ¼

  E‘1 ‘2 ‘3 ð1nÞðe211 þe222 þe233 Þþ2nðe11 e22 þe11 e33 þe22 e33 Þ 2ðð1þnÞð12nÞÞ ð2:43Þ

and W surf ¼ 2s‘1 ‘2 ðe11 þe22 Þþ2s0A ‘3 ½‘1 ðe11 þe22 Þþ‘2 ðe22 þe33 Þ

ð2:44Þ

2.3 Applications: Size Effects Due to the Surfaces

The equilibrium strains are obtained by minimizing Eq. (2.42) with respect to strain. At equilibrium and for a square-shaped crystal ‘1 ¼ ‘2  ‘, ‘3 ¼ h, we get     1n 2sA 2s0A 13n 1n 4s0A 2sA 2n eq eq eq þ  e11 ¼ e22 ¼  ; e33 ¼  h ‘ 1n ‘ h 1n E E ð2:45Þ For a free cubic crystal, h ¼ ‘ and sA ¼ s0A , there is simply [26] eeq ¼ 4

12n sA E ‘

ð2:46Þ

A generalization of Eq. (2.46) valid for various shapes has been given in Ref. [27]. Injecting the equilibrium strain (2.46) into the total elastic energy (2.42) gives a negative value that for a freestanding cubic crystal reads W ¼ 6sA ‘2 eeq . The negative sign means that due to its own surface stress, a small crystal is spontaneously deformed. Thus, it has a different crystallographic parameter compared to a large crystal (the deformation scales as ‘1 ). If the surface stress is positive (respectively, negative), then the crystallographic parameter is smaller (respectively, greater) than the crystallographic parameter of the mother phase. The measurement of such size-induced lattice contraction has been used to determine a mean value of the isotropic surface stress of cubic crystal [28–30]. More recent works [31] consider crystalline spheres (diameter D) cut in cubic crystals of compressibility K. In this case, (2.46) becomes e ¼ 4KsA =3D. In the absence of reliable values for surface stresses, the authors use asymptotic behaviors of surface energy combined with the Shuttleworth relation (2.35). In this way, the asymptotic form sA ¼ ½ð3c0 aÞ=8K1=2 (where a is an atomic unit and c0 the usual surface energy of the material) is more or less justified. The so-calculated deformations e ¼ ð4K=3DÞ½ð3c0 aÞ=8K1=2 can be compared to experimental ones measured for various materials. The results are shown in Figure 2.10. Experiments and calculation clearly show the size effect. The deformations are negative, so the intrinsic surface

Figure 2.10 Lattice contraction (in %) of nanoparticles. Adapted from Ref. [31].

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stress of these metals is positive and leads to a crystal contraction with respect to the mother phase. Notice that at the nanometer scale, deformations of a small percentage can be reached. Obviously, for such values, linear elasticity may become questionable. Let us stress again that two main assumptions have been used, isotropic surface stress and homogeneous deformation of the nanocrystal, so that only mean values of surface stress can be obtained by such methods. 2.3.2 Effective Modulus of Thin Freestanding Plane Films

Consider a nonsupported film (‘2 ¼ ‘2 ! 1, ‘3 ¼ h0 ) submitted to its own surface stress. The film is assumed not to bend. For the sake of simplicity, we will consider cubic crystals so that, at equilibrium, from (2.45) eq

eq

e11 ¼ e22 ¼ e ¼ 

2sA h0 Y

and

e33 ¼

2nsA n ¼ e Eh0 1n

ð2:47Þ

where Y0 ¼ E=ð1nÞ is the usual elastic modulus defined as Y0 ¼ ð1=2Þðd2 =de2 Þ ðW Bulk =V0 Þ in absence of any surface stress (W Bulk ¼ Y0 V0 e2 is the bulk elastic energy). In other words, (2.47) implies that because of its own surface stress, the film is, at equilibrium, deformed with respect to its bulk counterpart, so that the thickness of the film is no more h0 but h0 ð1 þ e33 Þ and the initial surface area A0 becomes A0 ð1 þ eÞ2 . The total elastic energy per unit of nondeformed area (see Eq. (2.22)) of the system now becomes W  Y0 h0 e2 þ 4sA e A0

ð2:48Þ

where the quadratic term due to bulk elasticity and the linear term due to surface stress again appear. The factor of 4 rises for the two in-plane directions and the two surfaces. It is now possible to define an effective elastic modulus in presence of surface stress as Yeff ¼ ð1=2Þðd2 =de2 ÞðW=VÞ, where now V ¼ h0 A0 ð1 þ eÞ2 ð1 þ ezz Þ is the volume after spontaneous deformation and W is given by (2.48). It is found that at mechanical equilibrium and up to the first order in 1=h0 , Yeff  Y0 þ

2sA ð2gÞ h0

ð2:49Þ

where g ¼ n=ð1nÞ scales as 1/2 for usual values of n. Equation (2.49) states that, due to surface stress, one can define an effective elastic modulus, the value of which varies as the reciprocal of the film thickness. Freestanding thin films with positive (respectively, negative) surface stress have a larger (respectively, lower) effective modulus than the material from which they have been cut. This simple model has been extended and checked by Streitz et al. [32], who take into account higher order elastic effects by means of Y0 ðeÞ ¼ Y0 ð1BeÞ and

2.3 Applications: Size Effects Due to the Surfaces Table 2.3 Values determined from simulations of the (001) surfaces of various metals [32].

Y0 (eV/A3)

B

2g

sA (eV/A2)

s1 (eV/A2)

  s1 2sA B þ ð4g3Þ þ sA

0.968 1.542 0.623 0.628

13.10 14.70 13.73 14.61

1.25 1.08 1.36 1.58

0.056 0.047 0.050 0.077

0.14 0.07 0.19 0.38

1.13 1.44 0.97 1.51



Cu Ni Ag Au





sA ðeÞ ¼ sA þ s1 e, so the total elastic energy now becomes     W 2 3 B  4sA e þ ðY0 h0 þ 2sA þ 2s1 Þe2 þ Y0 h0 3 g þ 2s1 e3 A0 3 2 2

ð2:50Þ

Equation (2.49) then up to the first order in 1=h0 becomes Yeff  Y0 þ

  2sA s1 B þ ð4g3Þ þ h0 sA

ð2:51Þ

While in the linear approximation the elastic nature of the freestanding film (softer or stiffer than that of its bulk counterpart) depends only on the sign of the surface stress of the body (see Eq. (2.49)), it is no longer the case when nonlinear effects are taken into account (see Eq. (2.51)). More generally (for more complex shapes), it can be shown that the effective Young modulus depends on the third-order bulk and elastic constants [27]. Streitz et al. [32] used (2.51)9) to calculate the equilibrium strain and the effective modulus for various metals for which they calculate surface stress sA , the usual Young modulus Y0 , and the higher order elastic constants B and s1 . The results of their calculations are shown in Table 2.3. It appears that B is positive, scales as s1 =sA , and is greater than unity. It follows that the leading term in Eq. (2.51) is not the surface stress by its own, but it is the coupling between the higher order constants (s1 and B) and the surface stress sA by means of DY  2sA B=h0 . The strain (calculated from the formula given in footnote 9) and the effective modulus (calculated from Eq. (2.51)) with the data given in Table 2.3 are shown in Figure 2.11, in which the same quantities calculated by molecular dynamics simulations are also shown [32]. There is an excellent agreement but notice that the sizes at which the surface effect cannot be negligible are at the nanoscale. As a conclusion, the elastic modulus of a freestanding thin film is actually determined by nonlinear elastic properties of its bulk and surface (for a complete discussion, see Ref. [33]). 9) Note that the equilibrium strain is not given by Eq. (2.47), but by eeq  2sA =ð4h0 þ 2s1 þ h0 Y0 Þ. Moreover, note that (2.49) cannot be recovered by simply putting B ¼ 0 and s1 ¼ 0 in (2.51) because both expressions are not at the same order in strain.

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Figure 2.11 (a) Strain versus thickness calculated for various metals. Adapted from Ref. [32]. (b) Corresponding effective modulus. Adapted from Ref. [32].

2.3.3 Bending, Buckling, and Free Vibrations of Thin Films 2.3.3.1 General Equations We will now consider bending, buckling, and free vibrations of thin films when incorporating surface elasticity effects. The main formulation consists of adding surface forces and bending moments due to surface stress to the equations of the classical description of a plate. Indeed, let us consider a thin asymmetric, isotropic sheet of thickness e. The faces 1 and 2, respectively, bear surface stress s1ab and s2ab . These surface stresses generate two moments, a first moment (a force) and a second moment (a torque). Ð e=2 surf ¼ e=2 ½s1ab dðx3 þ e=2Þ þ s2ab dðx3 e=2Þdx3 , so using the The first moment is Nab sifting properties of the Dirac function dðxÞ, we get surf þ Nab ¼ s1ab þ s2ab ¼ Dsab

ð2:52Þ

þ This moment, which depends on the total surface stress Dsab , is equivalent to a resultant force per unit length applied on the medium plane. This results in the strength of the medium plane. Ð e=2 surf ¼ e=2 ½s1ab dðx3 þ e=2Þ þ s2ab dðx3 e=2Þx3 dx3 can also The second moment Mab be calculated: surf Mab ¼ ðs1ab s2ab Þe=2 ¼ Ds ab e=2

ð2:53Þ

which depends on the differential surface stress Ds ab . This moment is a torque that bends the sheet (Figure 2.12). Let us recall that because of the bending, the stresses are not constant in the bulk but vary with the altitude (see Figure 2.9). The total moments in the presence of bulk stress and surface stresses are the sum of the bulk and surface moments and can be written as

2.3 Applications: Size Effects Due to the Surfaces

Figure 2.12 Sketch of the decomposition of the differential of surface strain in the first moment applied in the medium plane and a second moment that is a couple that tends to bend the sheet.

e=2 ð surf Nijtot ¼ Nij þNab

e=2 ð

þ sij dx3 þDsab ;

¼

surf Mijtot ¼ Mij þMab

e=2

¼

s ij x3 dx3 þDs ab e=2

e=2

ð2:54Þ

The equations of motion for the body (see Eq. (2.3)) of the plate are @sij d2 ui þ fi ¼ r 2 @xj dt

ð2:55Þ

Equation (2.55) accounts for the body forces (e.g., gravity, see footnote 2) of components fi . Again, r is the density and ui the components of the displacement field. Equation (2.55) can be multiplied by dx3 or x3 dx3 and then integrated to get the equations of motion governing the resultant moments [34, 35]. @Nia up þ si3 s down þ i3 @xa

e=2 ð

e=2 ð

fi dx3 ¼ e=2

@Mib e up þ Ni3 þ si3 þ s down i3 @xb 2

r e=2

d2 ui dx3 dt2 e=2 ð

e=2 ð

fi x3 dx3 ¼ e=2

ð2:56Þ

r e=2

d2 u i x3 dx3 dt2

ð2:57Þ

up si3

where and s down are the stress at the up and down surfaces, respectively. i3 The mechanical equation valid for the upper and lower surfaces (respectively, labeled ( þ ) and ()) reads @s bi @xb

s u i3 ¼ r€ i

€ . Obviously, at equilibrium this latter equation is where we write d2 u=dt2 ¼ u Eq. (2.32). The previous mechanical equation can be inserted into Eqs (2.56) and (2.57) to obtain @ ðNia þ Dsbiþ Þ þ @xa

e=2 ð

e=2 ð

€ 2i Þ r:€ u i dx3 þ rð€ u 1i þ u

fi dx3 ¼ e=2

e=2

ð2:58Þ

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@ e Mab þ Ds ai Na3 þ @xb 2

e=2 ð

e=2 ð

fi x3 dx3 ¼ e=2

r:€ u a x3 dx3 þ e=2

e 1 € a € u 2a r u 2 ð2:59Þ

At equilibrium (no time dependence) and in absence of body forces, these equations become @ ðNia þ Dsbiþ Þ ¼ 0 @xa

ð2:60Þ

@ e Mab þ Ds ai ¼ Na3 @xb 2

ð2:61Þ

These equilibrium equations have to be solved with well-defined boundary conditions. 2.3.3.2 Discussion Most of the surface stress-induced modifications of the usual behavior of thin plates [34–41] have been studied with a surface modeled as a foreign membrane [23–25]. However, owing to the analogy between Eqs (2.29) and (2.39), the results are comparable to those obtained with the true surface stress in the Gibbs meaning. Usually two assumptions can be found in the literature. The first, the thin film assumption, is that s33 is zero inside the whole film.10) The second assumption is a þ þ s linear variation of s 33 in the film as, for instance, s33 ¼ ð1=2Þðs 33 33 Þ þ þ1  ð1=hÞðs33 s33 Þx3 . Here, we will illustrate only surface effects in the framework of the thin film approximation and Kirchhoff theory11) [34–38]. For this purpose, consider an isotropic material (bulk and surfaces) in which an infinitely long (in the x2 direction) sheet of finite width ‘ in direction x1 and thickness e is cut. The (dimensionless) results of Ref. [38] are shown in Figure 2.13 for the maximum deflection, the load at which the sheet buckles, and the frequency of vibration. The materials constants are E ¼ 5:625  1010 N=m2 , n ¼ 0:25, r ¼ 3  103 kg=m3 , l0 ¼ 7  103 N=m, m0 ¼ 8  103 N=m (the Lame coefficients of surfaces), and the residual surface stress t0 ¼ 110 N=m (more details can be found in Ref. [35]). It is easy to see in Figure 2.13 that the static as well as the dynamic response of a sheet depends on its surface properties. Obviously, the thinner the sheet, the proportionally greater the surface effects. Notice that the surface may help or oppose the external stress according to the sign of the stress ½t (given by Eq. (2.41)) at the surface. For the thinner sheets, surface stress effects can become so important that nonlinear 10) At mechanical equilibrium, s33 must vanish at the free surfaces. 11) In the classical thin plate description, two theories can be encountered: the Kirchhoff plate theory and the Mindlin plate theory that differs only by the assumptions on the asymptotic form of the displacement fields.

2.3 Applications: Size Effects Due to the Surfaces

Figure 2.13 Surface effect on bending, buckling, and free vibration of a sheet with edges at x1 ¼ 0 and x1 ¼ ‘ and t0 ¼ 110 N=m and h0 ¼ 106 m. Adapted from Ref. [35]. (a) P ¼ 100ð1n2 ÞP0 E versus H for a simply supported sheet. P ¼ P0 sinðpx1 =‘Þ is the transverse load and H the reduced sag (w=h0 ). (b) N ¼ 12ð1n2 ÞT=Eh versus log h0 (h0 in meter) for a simply supported sheet at both

edges submitted to a pair of compressive forces of amplitude T. A strong negative surface stress may the sheet buckled without any external stress. (c) Reduced frequency of vibration versus the reduced sag (w=h0 ) for a simply supported sheet submitted to vertical displacement, u3 ¼ U sinðpx1 =‘Þ. Notice that t ¼ 0 means no surface effects (t0 ; l0 ; m0 ¼ 0).

elasticity effects may play a role. However, the result is that the size-dependent behavior originates from the strain dependence of the surface stress, that is, from the surface elastic constants. In other words, the usual results of the thin plate are recovered for zero or constant surface stress. It is only when surface elastic constants are introduced that there can be a stiffening or a softening of the material, as more recently confirmed by the analytical calculations of Refs [42, 43]. Again, the surface effect is significant for film thinner than 10 nm. A simple proof of this conclusion is given in Ref. [44]. The energy of vibration (per unit length) of a rectangular beam (length L, height 2h, width ‘) prestressed by its surface stress can be written as the sum of its bulk and surface energies [1, 44] (s is the surface stress assumed to be isotropic): ðL ðL

1 002 02 W¼ EIy Fy d x þ ‘sy02 d x 2 0

ð2:62Þ

0

where E is the bulk Young modulus, I the moment of inertia, F a bulk compressive force that prestress the sample and, at equilibrium, balance the forces exerted by the surfaces, and yðx; tÞ the vertical deflexion (the primes denote the derivatives with respect to x). Since the equilibrium requires F ¼ 2s‘, the total energy (2.62) is independent of surface stress. When taking into account surface elasticity by means of s ¼ s0 þ Ssurf e, where Ssurf is surface elastic constant and e  hy00 the strain at the surfaces, a term Ð L a surf e2 dx adds to (2.62), so 0 ‘S ðL

1 EI þ 2Ssurf ‘h2 y002 d x W¼ 2 0

ð2:63Þ

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We can define an effective modulus Eeff ¼ E þ

2Ssurf ‘h2 I

ð2:64Þ

which depends on the shape (via I) and the surface elasticity (via Ssurf ). The frequency of vibration of the cantilever v2 ¼ v20 ð1 þ ð2Ssurf ‘h2 =EIÞÞ [44] does not depend on s, but only on the strain dependence of the surface stress, that is, on surface elasticity.12) Notice that more recently finite element method calculations have confirmed this result [45]. 2.3.4 Static Bending of Nanowires: An Analysis of the Recent Literature

Static and dynamic bending tests have been widely reported for nanowires [46–49]. The experiments are complex. In particular, it is generally hard to control the boundary conditions, so there is a wide scattering in the experimental data (see Section 2.3.5). However, most of the experiments show that the elastic behavior of a nanowire is different from the elastic behavior of its bulk counterpart (or mother phase), affecting the size-dependent modulus that can be softer or harder than the modulus of the bulk material. Again, this size effect essentially comes from the surface/bulk ratio. Notice that for rectangular wires, the presence of two pairs of free surfaces gives additional effects due to edges and corners. In the following section, we will study only the simpler case of cylindrical wires. 2.3.4.1 Young Modulus versus Size: Two-Phase Model For nanowires, most previous works use a two-phase model in which the wire is divided in a cylindrical core of radius r0 surrounded by a shell of thickness e, so that the diameter of the whole wire is D ¼ 2ðr0 þ eÞ. The core and the shell are, respectively, characterized by their elastic modulus E0 and Es . The flexural rigidity of the nanowire is written as EI ¼ E0 I0 þ Es Is , where I0 and Is are the moment of inertia of cross section of the core and the sheet. Using the expression of the inertia momentum, the effective modulus of the wire is [48]     Es E ¼ E0 1 þ 8 1 ðg3g2 þ 4g3 2g4 Þ E0

where g ¼ e=D. A fit of the experimental data gives the surface thickness e and the ratio Es =E0 . However, this phenomenological ratio cannot be simply interpreted in terms of elastic constants.

12) Some authors omit F and thus find that Eeff depends on s, which is clearly wrong [44].

2.3 Applications: Size Effects Due to the Surfaces

He and Lilley [40] have extended this approach to different boundary conditions and different geometries (rectangular or circular cross section). However, their approach is slightly different since they introduce an additional distributed transverse force due the stress jump across the surface Ds ij ni nj ¼ sab Kab (ni are the components of the unit vector normal to the surface and Kab the curvature tensor so that sab Kab simply is the Laplace overstress) written as s0 y00 ðxÞ for an isotropic surface stress (yðxÞ is the transverse displacement). The equilibrium equation EIy00 00 ðxÞ ¼ 2Ds0 y00 ðxÞ is thus solved for different boundary conditions: a cantilever (submitted to a concentrated load force at its end), a simply supported wire, and a clamped wire (submitted to a concentrated load force applied to the middle of the wire). The main result is that a positive surface stress s0 opposes concave curvature of the mean plane of the wire and enhances convex curvatures. For positive surface stress, the cantilever behaves as a softer material, while a simply supported or clamped sheet behaves as a stiffer material (compared to its bulk counterpart). This result could illustrate why quantifying the mechanical properties at the nanoscale is so challenging. Indeed, the effective modulus appears to depend on the boundary conditions! However, as shown in Section 2.3.3.2 and Ref. [44], this analysis neglects the prestress effect the surface stress (considered as a membrane) induces, so the true equilibrium equation should be EIy00 00 ðxÞ ¼ Fy00 ðxÞ þ 2Ds0 y00 ðxÞ, where at equilibrium the compressive axial force due to surface stress is F ¼ 2D‘s0 . This prestress effect thus should modify the surface stress effect in the absence of surface elasticity (see Section 2.3.4.3). 2.3.4.2 Young Modulus versus Size: Surface Stress Model Cuenot et al. [46] have reported effective modulus measured for ZnO nanowires: the smaller the diameter of a nanowire, the greater its effective modulus. To interpret these experimental results, the total bending energy due to a force F applied on the wire inducing a deflection d is [46] U ¼ Fd þ ð1=2Þkd2 þ spDDLð1nÞ, where the first term is the work of the applied force F, the second term the bulk elastic energy depending on the stiffness modulus k of the bulk material of the wire, the last term the work against surface deformation, where s is the surface stress (assumed to be isotropic), and DL ¼ ð12=5Þðd2 =LÞthe extension of the wire. For clamped wires, the total energy becomes U ¼ Fd þ ð1=2Þkeff d2 , where keff ¼ kþð24=5ÞsðpD=LÞð1nÞ. For the geometry of the wire, E ¼ ðL3 =192IÞk, where I ¼ pD4 =64 is the moment of inertia of the section, so Cuenot et al. [46] define an effective modulus Eeff ¼ E þð8=5Þsð1nÞðL2 =D3 Þ. Again, even if experimental data are well fitted by the model, the model neglects the prestress bulk induced by the surface stress (see footnote 12). 2.3.4.3 Prestress Bulk Due to Surface Stresses We have seen that most of the recent works neglect the bulk stress initially induced by the surface stresses. The effect of this prestress has been properly taken into consideration by Wang et al. [50] in the case of pure bending of nanowires. For this purpose, they consider an isotropic nanowire of thickness h, width b, and length ‘ and isotropic surface stress s. The authors (i) calculate the bulk stress (supposed to be

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homogeneous) due to the surface stress (as in Section 2.3.1), (ii) calculate the strain field in the bulk and at the surface, the (iii) the strain-dependent surface stress, and (iv) the bulk and surface elastic energies. Let us underline that the authors allow the nanowire to relax axially at its ends. They can thus define an effective Young’s modulus that can be expressed as    2  2 ! 6 2 s ‘ 2 b ð2:65Þ Eeff ¼ E þ Esurf  þ 2n þ2 h b h h h where Esurf , a surface Young’s modulus, is a combination of the surface elastic constants. This equation can be formally compared to Eqs (2.49) and (2.51) obtained for thin freestanding planar films (that are not allowed to bend) or to Eq. (2.64) obtained from vibration properties. The main conclusion is that Young’s modulus can be considerably affected by the surface properties. It is modified not only by the absolute size of the nanowire but also by its aspect ratio. More precisely, for positive elastic constants and positive surface stress, the effective Young’s modulus decreases with the decrease in the nanowire thickness or the increase in its aspect ratio. Let us underline that this result has been obtained for free boundary conditions. It could be different for clamped conditions (see the discussion in Section 2.3.5). Again, Eq. (2.65) predicts that stress effects play a role only at the nanoscale (roughly for thickness smaller that 20 nm) [50]. 2.3.5 A Short Overview of Experimental Difficulties

We cannot end this short review of surface effects on elastic properties of nanoscale objects without any analysis of the experimental uncertainties that concern the sample preparation, the sample characterization, the mechanical system used to excite the nano-object, and even the simple definition of the object geometry. We list below some of the experimental difficulties. Most of the methods used to fabricate micro- or nanoscale objects employed techniques that potentially affect the mechanical behavior by ions implantation, surface amorphization, surface roughness, or even dislocation implantation [51]. The mechanical probe is generally larger than the size of the object [47–60], so the mechanical properties of the whole system (probe þ object) should be checked. Furthermore, the boundary conditions (clamping, simply supported, free boundary) are difficult to define accurately and to reproduce from one experiment to another. It is all the more important that, as numerically shown by Park and Klein [61], effective Young’s modulus depends on the boundary conditions. For instance, they [61] found (for gold) that increasing the nanowire aspect ratio leads to an increase (respectively, decrease) in Eeff for clamped (respectively, free ended) nanowire. This can be easily understood since the clamped condition prevents the axial relaxation due to surface stress. The experimental difficulties due to the boundary conditions can be

2.4 Conclusion

partially overpassed by using acoustic methods that simply excite the vibrational modes [62, 63] without contact. It is sometimes difficult to accurately characterize the geometry of a nano-object (for instance, the diameter of a nanowire is not necessarily constant on its whole length). An error of 5% on the nanowire scales leads to an uncertainty of 25% on the effective Young modulus. Since plastic properties are also size dependent [52, 53], the knowledge of the dislocation density may be crucial. Indeed, the mechanical properties cannot be the same if the object already contains a dislocation that can move under the external force or if it is necessary to create the dislocation before activating it. All these limitations may be at the origin of the large dispersion of the available experimental results and even at the origin of several systematic errors. For instance, most of the theoretical calculations predict size effects at the nanometer scale, while a few experiments show size effects that already occur at some hundred of nanometers! May be some of these experimental results have not been obtained with a well-defined system. For instance, the Young modulus of a vibrating cantilever has been found to decrease [56], increase [64], or to be constant [60] according to the cantilever diameter,13) while Raman scattering of nanoparticles does not put in evidence any size dependence of the Young modulus [62, 63]. The analysis of the mechanical properties of cantilevers with varying width seems to be a necessary challenge [65].

2.4 Conclusion

A great deal of research has been done on the elastic properties of nanoscale materials. Like many other properties, the mechanical properties of small objects deviate from those of macroscopic objects of the same material. The overall elastic behavior of nanosized objects is size dependent. This size dependence originates from proportionally greater surface effects. Indeed, the surface has bond length and strength different from the volume, so a nanoscale object must, at least, be divided into a bulk core and a surface zone with different properties. The surface zone can be described in various ways. The best one consists of using the concepts of dividing surface and surface excess. The main advantage is that the so-defined quantities are perfectly defined from a thermodynamic viewpoint. Other choices are possible, but in all the cases the localization of the dividing surface should be specified before any calculation. In some cases, it is easier to split the dividing surface into two surfaces separated by a distance that defines the surface thickness. When specific elastic properties are attributed to surfaces, the elastic behavior of a finite-size body results from a subtle interplay between its bulk and surface properties and thus depends on the absolute size of the object. Such size dependence becomes significant when at least one dimension becomes smaller than, let us say, a few tens of nanometers.

13) But obviously the effect depends on the sign and the value of the surface stress of the material.

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Most of the mechanical characteristics of nanoscale objects can be inferred from models coming from surface physics in which the surface is characterized by an excess of stress in the Gibbs meaning. More precisely, it is possible to show with these models that because of surface stress, the crystallographic parameter, the Young modulus, the flexure rigidity, and so on of nanoscale structures differ from those of bulk (infinite) materials. However, surface stress alone is not enough to explain all the experimental occurrences. In particular, surface elastic constants have to be introduced since the stiffening effect is essentially due to the strain dependence of the surface stress. Moreover, at the nanoscale surface stress may induce huge bulk strain, so the whole mechanical behavior can no more be described by linear elasticity theory. In some cases, the surface stress effect, by its own, is less important than the nonlinear behavior it induces in the bulk [66]! Finally, notice that since surface effects are dominant at the nanoscale, the elastic properties also depend on the shape of the objects. It is clearly put in evidence in Ref. [27] in the case of Cu. For nanowires, the Young modulus increases as the wire becomes thinner, while for a freestanding film, it decreases. Again, let us underline that mechanical tests at the nanoscale are still a challenge. In particular, since most of the tools used for the stress measurements are at the macroscale, it is generally hard to control the boundary conditions. At the same, it is difficult to simply control the initial state (surface roughness, defects density, and even geometrical data as initial curvature or size) with a good accuracy. It results in a wide scattering in the experimental data, so efforts have to be made to access better data even if some statistical attempts to go round these intrinsic limitations have been explored [67]. From a theoretical viewpoint, let us notice again that the models coming from surface physics cannot be applied to the smallest sizes at which it is no more possible to divide the piece of matter into a core and a surface zone. In this case, it is necessary to resort to atomistic approaches expressed in terms of spring models, pseudopotentials, or ab initio calculations [66, 68] or may be to use nonlocal elasticity [69]. Moreover, let us notice that strained nanowires may be morphologically unstable. Such instabilities are beyond our scope of this book, but the reader can refer to Refs [70–72].

Acknowledgment

I wish to acknowledge Ezra Bussmann, Raymond Kern, and Andres Saul for fruitful discussions.

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3 Onset of Plasticity in Crystalline Nanomaterials1) Laurent Pizzagalli, Sandrine Brochard, and Julien Godet 1) This chapter is dedicated to Pierre Beauchamp, who has not only guided the authors into the complex and tortuous world of dislocations for the past 10 years but has also showed them a fair way to behave as a scientist.

3.1 Introduction

The past two decades have witnessed an amazing development in the fabrication of systems characterized by one or several dimensions in the nanoscale. These so-called nanostructures include, for instance, nanospheres, nanowires, and nanopillars. An intensive research is being done looking for potential applications in various domains such as mechanics, electronics, optoelectronics, photonics, phononics, and so on. The mechanical characterization of these nanomaterials constitutes an important part of this research, since it has been shown that size reduction is often accompanied by large variations in common properties such as strength, hardness, or ductility [1]. For instance, an interesting and intriguing phenomenon is the apparent increase of the ductility range for nanoscale systems compared to the bulk material. Many experiments suggest that nanomaterials could exhibit specific and unusual mechanical properties [2–5], which explains the current interest in the research community and the large number of dedicated studies. Several limits are commonly used to define the different regimes of the response of a material submitted to a mechanical stress. Among these, the elastic limit, also known as the yield strength, defines the point at which the deformation of the material becomes irreversible, that is, plasticity occurs. Figure 3.1 shows the variation of the yield strength as a function of size for different kinds of nanoscaled systems made of silicon. Obviously, the onset of plasticity is different here compared to bulk materials, with a dramatic increase of strength for smaller sizes. Several mechanisms have been proposed for explaining experimental observations [1].

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Figure 3.1 Variation of the yield stress (left) and true fracture strain (right) as a function of the size for different nanomaterials. Courtesy of W.W. Gerberich et al. [11]

The determination of the yield strength is difficult in general, since it is associated with the onset of plasticity. Usually, the elastic limit and the yield strength correspond to the deviation from linearity in the measured stress–strain curve, although other choices can be made depending on the material. In bulk materials, it depends on the sensitivity of the measurement. In fact, the irreversible displacement of dislocations present in the material may occur even for very low deformation, in the apparent elastic regime. It is therefore difficult to define the onset of plasticity in bulk materials from macroscopic measurements. Its determination is also difficult in the case of nanomaterials, since specific apparatus and techniques are needed for measurements. Additional insights can be obtained from several theoretical approaches, which do not suffer from the same limitations. The issue of the onset of plasticity has been largely examined in the framework of elasticity theory [6–9]. Numerical simulations at the atomistic level also allow reproducing mechanical tests, although they are generally restricted to systems with dimensions closer to tens of nanometers rather than micrometers. In this chapter, we describe the elastic and atomistic modeling of the onset of plasticity in nanomaterials. Note that we focused on systems including surfaces and for which one or several dimensions are in the nanoscale. This definition not only includes nanopillars and nanowires (1D) and nanospheres (0D) but also supported thin films (2D). Conversely, we did not consider systems characterized by interfaces such as embedded defects (aggregates of point defects, dislocations). Another important issue is the mono- or polycrystalline nature of the nanomaterial. In fact, it has been shown that the plastic properties of nanopillars could strongly depend on their internal structure [10]. For the sake of simplicity, here we focus on monocrystalline materials.

3.3 Driving Forces for Dislocations

3.2 The Role of Dislocations

The plastic deformation in crystalline materials can be associated with different mechanisms, all involving an irreversible displacement of the matter. The latter can result from the motion of individual atoms (diffusion) or from the collective displacement of atoms in dislocation form. For polycrystalline materials, the plastic deformation can also be due to the rotation and motion of grains. Here, we will consider only the plastic deformation through dislocation nucleation and displacement prior to the failure of the material (fracture). Useful basic information and theory of dislocations in bulk materials can be found in well-known textbooks [8, 9]. In usual bulk materials, the onset of plastic deformation is characterized by the displacement of the dislocations originally present. Depending on the conditions, the later stages of the deformation correspond to the generation of new dislocations, typically through multiplication. Several kinds of mechanisms have been identified [8, 12], the most common one being the Frank–Read and Bardeen–Herring sources, the multiplication by double cross-slip, or the nucleation from surfaces and interfaces. However, it is not clear whether the first mechanisms could be dominant in nanomaterials. In fact, these systems are characterized by one or several dimensions in the nanoscale, which is likely to impede such sources. Furthermore, these mechanisms require the presence of initial defects, usually in very low concentration in nanomaterials. As a consequence, although original multiplication mechanisms specific to nanomaterials have been postulated [13], dislocation nucleation is expected to be the main process in these systems, an assumption supported by experimental observations [14–16] and the obvious fact that the surface/volume ratio is much higher in nanomaterials than in bulk materials. Even in the case where dislocations are present in the system, it has been proposed that these dislocations are first annihilated by escaping to free surfaces, followed by the nucleation of new dislocations [17]. In the following, we focus on the mechanism of dislocation formation from an infinite surface. We first describe the two main forces that will act on a dislocation and the specific case of a dislocation in the vicinity of a surface. The elastic and atomistic modeling of the dislocation nucleation process is then described. Finally, we discuss how these results can be extended to more complex geometries (nanowires) and propose some perspectives.

3.3 Driving Forces for Dislocations

The elastic field associated with dislocations is slowly decaying from the dislocation core according to an inverse power law. Consequently, a dislocation in a real material is expected to interact with many defects (other dislocations, grain boundaries, point defects, surfaces, etc.), all of them exerting a force on the dislocation. However, for our purpose, we first focus on a single isolated dislocation in an otherwise perfect

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bulk material. In this case, there are two factors that may change the dislocation configuration in the crystal. 3.3.1 Stress

When a dislocation of Burgers vector b with the dislocation line j is displaced in the field of a local stress s, there is a change of energy proportional to the product between the stress and the area swept by the dislocation displacement. Following the definition of a force acting on a dislocation given by Hirth and Lothe [8], it is therefore possible to define a force Fs acting on an infinitesimal dislocation segment of length l, with the following expression: Fs ¼ ðsbÞ  j l

ð3:1Þ

which is well known as the Peach–Koehler formula (see Ref. [18] for instance for an explained derivation). This is the driving force for displacing dislocations due to the local stress, and in extenso in a material submitted to a mechanical action. 3.3.2 Thermal Activation

Temperature is the other important factor since it allows overcoming the energy barriers between two different dislocation configurations. Several kinds of thermally activated mechanisms, such as dislocation displacement and unpinning or dislocation core transformations, have been observed or postulated [19]. A usual framework for studying thermally activated mechanisms in materials science is the transition state theory, in particular in its harmonic approximation [20]. Such an approach has been shown to be valid as long as energy barriers are not too small or, equivalently, temperatures are not too high, which is generally the case for dislocations. It is essential to emphasize that thermal activation does not favor any specific direction, conversely to the mechanical driving force. The displacement of a dislocation under the sole action of temperature is equivalent to a random walk. Also, thermal activation is essentially restricted to spatially localized mechanisms, that is, involving a limited number of atoms. In fact, the probability to perform a specific collective displacement of atoms by thermal motion is rapidly decreasing as a function of the number of atoms. 3.3.3 Combination of Stress and Thermal Activation

Temperature and stress, combined together, will help to overcome the energy barrier associated with a change in the dislocation configuration, leading to a displacement or to a core transformation. The relative importance of each factor depends on the investigated mechanisms. The energy barrier can be overcome, thanks to stress.

3.4 Dislocation and Surfaces: Basic Concepts

For instance, the critical stress required for displacing a dislocation without any thermal activation, that is, at 0 K, is called the Peierls stress. The latter is an important quantity that is usually determined by numerical simulations or by interpolating measurements at finite temperature. Conversely, mechanisms leading to transformation of dislocation cores can be activated only by temperature. Several regimes can therefore be obtained for a single material, depending on the magnitudes of stress and temperature. Typically, stress is the main driving force in many systems such as metals for an isolated dislocation. But in materials with a high lattice friction such as covalent or geophysical systems, the weight of thermal activation in dislocation-related mechanisms grows. In the case of dislocation nucleation, we will see that both temperature and stress are important.

3.4 Dislocation and Surfaces: Basic Concepts

To study the formation of a dislocation from a surface, it is helpful to first examine the situation where the dislocation is already present in the system and still in interaction with the surface, that is, close enough. 3.4.1 Forces Related to Surface

The presence of a surface introduces an additional complexity in the study of dislocations. In fact, it is well known that there is a long-range interaction on the dislocation due to the surface [9]. This interaction can be understood by considering that the self-energy of a dislocation is decreasing when the dislocation is brought closer to the surface. Alternatively, one can consider that there is relaxation of the stress field of the dislocation by the surface, thus lowering the energy. Another possible subtle effect is the relaxation of the surface stress by the dislocation. The total energy change is equivalent to an attractive force between the dislocation and the surface. The interaction force between the surface and a dislocation can be derived in the most simple cases using the concept of an image dislocation located in a symmetric position relative to the surface. For a straight edge dislocation with a line parallel to the surface, this force is inversely proportional to the dislocation–surface distance d, and is equal to Fi Kb2 ¼ l 4pd

ð3:2Þ

where K is a function of the elastic constant parameters depending on the dislocation character. Due to this force, a dislocation segment of length l is drawn closer to the surface. In the specific case of interest here, we consider a half-loop dislocation whose two ends are in connection with the surface (Figure 3.2). This is equivalent to a pinning of the half-loop dislocation by the surface. Therefore, in addition to image interactions, there is an additional line tension force exerted on the half-loop

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Figure 3.2 Schematic representation of a circular half-loop dislocation of Burgers vector b nucleated from a surface and propagating in a slip plane (white), making an angle q with the surface normal n. Edge and screw components

of the Burgers vector are be and bs , relative to the dislocation front. Fs and Fs are forces acting on the dislocation associated with surface interaction and stress, respectively.

dislocation that tends to bring the dislocation closer to the surface. In the following, the total force resulting from the interaction of a half-loop dislocation and the surface is called Fs . 3.4.2 Balance of Forces for Nucleation

We now analyze the balance of forces exerted on a spherical half-loop dislocation of radius R, as represented in Figure 3.2. The force Fs due to the surface tends to bring the dislocation closer to the surface, thereby reducing R. Conversely, for the appropriate stress direction, the force Fs associated with stress relaxation tends to propagate the dislocation into the material, thus increasing R. Because these forces are opposed, there is no possible stable configuration. The energy of the half-loop dislocation as a function of R is schematically represented in Figure 3.3. Depending on the value of the stress applied on the dislocation, there are two possible regimes. For low or moderate stress, the energy first increases until it reaches a maximum, defining an unstable equilibrium configuration, and then decreases. This maximum, characterized by an energy Ea and a radius Rc , has to be overcome by thermal activation for nucleating a propagating dislocation. Otherwise, for stress higher than a given value s c , the stress contribution is large enough for the energy barrier to vanish, the dislocation formation process becoming athermal. s c is the critical stress for nucleating dislocation, comparable to the Peierls stress for displacing dislocation in bulk materials. 3.4.3 Forces Due to Lattice Friction

Up to now, we have discussed the balance of forces for dislocation nucleation from the surface without taking into account the lattice friction of the materials. In fact, to propagate into a material, a dislocation must overcome the lattice resistance, whose

3.4 Dislocation and Surfaces: Basic Concepts

Energy variation

σ < σc

Ea

Rc σ >σc

Half-loop dislocation radius Figure 3.3 Possible energy variations as a function of the radius of a nucleated half-loop dislocation, and definition of the activation parameters for stresses lower than the critical stress sc .

magnitude depends on the nature of the material itself. In fcc metallic systems, this resistance is very low and additional barriers in the energy variation can be safely neglected (Figure 3.4). Conversely, in covalent systems, the lattice friction may be very large and these energy barriers could be in the same range or even larger than the activation energy due to surface forces. Then, they have to be considered in the mechanism of dislocation formation from the surface of covalent materials (Figure 3.4). In any case, it is clear that the nucleation of a half-loop dislocation from a surface is possible only if stress and temperature reach values required for dislocation propagation in the bulk.

Energy variation

Covalent

fcc metal

Half-loop dislocation radius Figure 3.4 Examples of possible energy variations as a function of the radius of a nucleated halfloop dislocation, taking into account the lattice friction for fcc metallic and covalent materials.

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3.4.4 Surface Modifications Due to Dislocations

In addition to the above-discussed interaction, a dislocation will change the surface state when it is nucleated or when it leaves the material, which involves an energy variation. The most important (and often visible) surface modification is the creation of a step or a decrease/increase of the height of an existing step. This modification is bounded by the two surface points pinning the half-loop dislocation. The change of the step height is given by ðbe  nÞ, be being the edge component of the Burgers vector and n the surface normal (Figure 3.2). Defining the angle q between the surface normal n and the dislocation slip plane, the step height change is be cosðqÞ. 3.5 Elastic Modeling 3.5.1 Elastic Model

We aim at determining the energy of the configuration represented in Figure 3.2 in the case of an isotropic medium, relatively to the same system but without the dislocation. In early analyses of dislocation nucleation at surface, it was usually assumed that the self-energy of the half-loop dislocation is simply half the self-energy of a full circular dislocation loop [7–9, 21, 22]. A more accurate formulation has been proposed by Beltz and Freund, who introduced a correction factor m in the logarithmic part of the energy [23]. The self-energy of the circular half-loop is therefore given by     mb2 ð2nÞ 8maR R ln 2 U¼ 8ð1nÞ b

ð3:3Þ

In (3.3), m is the shear modulus, n is the Poisson coefficient, a ¼ b=r0 is a nondimensional factor defining the unknown dislocation core radius r0 , and m is a geometrical parameter that depends on the Poisson coefficient, the system geometry (angle between surface and slip plane), the shape of the loop, and the Burgers vector orientation. m is necessarily bounded by 0 and 1, but is generally not known. In their seminal work, Beltz and Freund proposed an expression for m in the case of a circular half-loop in a slip plane perpendicular to the surface [23]. The total energy of the system should also include the energy gained by enlarging the dislocation loop, that is, the work associated with stress relaxation. This quantity is proportional to the area swept by the dislocation that is pR2 =2 in the case of a circular half-loop, yielding 1 W ¼  pR2 sb 2

ð3:4Þ

3.5 Elastic Modeling

In the framework of linear isotropic elasticity theory, the latter quantity can also be expressed as a function of the applied deformation e. In the case of an uniaxial deformation, s ¼ 2mð1 þ nÞse, s being the Schmid factor, and W is now given by W ¼ mbð1 þ nÞpR2 s e

ð3:5Þ

A third possible contribution to the total energy is related to surface modifications after dislocation formation. Since the step height change is given by be cosðqÞ, the corresponding energy variation is Es ¼ 2Rbe cosðqÞcs

ð3:6Þ

cs is equivalent to a surface energy in the case of a high step. When a single step is created in coherence with the crystal structure, be cs is a step energy. Note that very small energy contributions due to dislocation pinning points at the surface are neglected here. Finally, we have to consider the situations where single partial dislocations are nucleated from the surface. In such case, the propagation of dislocation into the crystal is accompanied with the formation of a stacking fault. Defining the stacking fault energy cf , the additional contribution is 1 Ef ¼ pR2 cf 2

ð3:7Þ

Combining (3.3) and (3.4) or (3.5) and (3.6) and eventually (3.7) provides the total energy EðR; sÞ (or EðR; eÞ) associated with the formation of a half-loop dislocation of radius R from a surface for a stress s (or an applied deformation e). It is quite instructive to plot the different contributions to the total energy as a function of R, as shown in Figure 3.5. Here we have selected realistic values for the

Figure 3.5 Variation of the different energy contributions from the elastic model (described in the text) as a function of the radius of a nucleated half-loop dislocation, using realistic values for the different parameters. Inset shows these variations for a very small radius.

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different parameters entering into Eqs (3.3)–(3.7). These correspond to a nucleated partial dislocation with a leading 90 orientation, in aluminum, leaving a step on the surface. For large R, it appears that the step energy Es is negligible compared to the other contributions. The energy cost Ef for stacking fault creation in the case of partial dislocations, quickly increasing due to the R2 dependence, is also emphasized. The inset in Figure 3.5 shows the energy variation for small values of R. A local minimum is present in the total energy curve, because of the R lnR dependence on UðRÞ. This minimum occurs for R values typically lower than the core radius r0 , where the validity of elasticity theory is obviously questionable. Therefore, this minimum has no physical meaning within the framework of the elastic model. 3.5.2 Predicted Activation Parameters

Knowing the expression of the total energy, it is straightforward to determine the activation parameters Ea and Rc as a function of the stress or the deformation. Rc corresponds to the maximum energy, which is obtained when @EðRÞ=@R ¼ 0 or     mb2 ð2nÞ 8maRc 1 þ ðcf 2mbð1 þ nÞseÞpRc  be cosðqÞcs ¼ 0 ln b 8ð1nÞ ð3:8Þ

in the case of an applied deformation e. There is no analytical solution, and Rc has to be determined numerically. Once Rc ðeÞ is known, Ea ðeÞ ¼ EðRc ; eÞ is easily computed. Figure 3.6 shows the variation of Rc ðeÞ and Ea ðeÞ as a function of e using the same parameters as in Figure 3.5. In the regime of small deformation, both the predicted activation energy and the critical radius are large, suggesting that a thermally activated dislocation nucleation is highly unlikely. An increase of the applied stress leads to a sharp decrease of both quantities, approximately according to a 1=e relation. It is difficult to set a defined boundary, but one can reasonably consider that the onset of plasticity by half-loop dislocation nucleation would occur when Ea becomes lower than about 2 eV. In fact, a rough estimation of the time required to activate one event at room temperature for such an activation energy is on the order of the duration of a usual deformation experiment. For high applied stresses, both activation energies and critical radius are predicted to decrease. Finally, when the strain (or equivalently the stress) is larger than the athermal threshold, defined in Section 1.4.2, the activation energy vanishes. 3.5.3 What is Missing?

In order to use the elastic model, one has to determine two parameters: m that is a geometrical factor and a that is linked to the dislocation core radius. In the original

3.5 Elastic Modeling

Figure 3.6 Variation of the activation energy Ea (left scale) and critical radius Rc (right scale) as a function of the resolved shear strain se for similar parameters than in Figure 3.5. Open symbols are data obtained from atomistic calculations. For small deformations (left gray

area), a thermal activation of the nucleation mechanism is unlikely. For large deformations (right gray area), the validity of the elastic model is questionable. Inset recalls the definition of Ea and Rc , as in Figure 3.3.

formulation of Beltz and Freund, m was determined for a specific geometry and it varies from 0.5 to 0.6 for a Poisson ratio ranging from 0 to 0.5. However, the extent of variation in a general case is not known. Besides, it is not possible to determine accurately a, which is often set to values between 2 and 4 in elasticity studies. Unknowns m and a are not independent in (3.3), meaning that only the product ma has to be determined. Such a feat could be achieved using numerical simulations at the atomistic level, as will be shown in the following section. Moreover, this elastic model is built on several assumptions. For instance, in the Beltz–Freund derivation of (3.3), the half-loop dislocation is assumed to be perfectly circular. However, in the general case, a full dislocation loop is likely to adopt an elliptical shape, since edge and screw dislocation segments have different energies and mobilities. An elliptical half-loop dislocation could be taken into account in the elastic model, with the transformation R2 ! eR2 in (3.5) and (3.7), and R ! eR in (3.6). The parameter e defines the ellipticity of the loop. Other assumptions concern the determination of the resolved shear stress on the dislocation from the applied deformation, or conversely. There are two issues here. One is related to the use of linear isotropic elasticity theory, which could not be well suited for strongly anisotropic materials, or for large applied deformations. The other is the inhomogeneous character of the stress depending on the system geometry. In fact, the presence of an initial step on the surface obviously changes the stress distribution. As shown by several authors, there is a stress increase in the vicinity of the step [24, 25]. This is expected to favor dislocation formation and is not taken into account in the elastic model, which assumes an homogeneous stress distribution.

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Finally, one has to keep in mind that the validity of the elastic model becomes questionable when Rc is on the same order than the unknown dislocation core radius r0 . Therefore, it is doubtful that it could be used for determining sc , the critical stress for which the energy barrier is vanishing. In this particular case, it appears necessary to use atomistic modeling. The latter could also be necessary for investigating systems with a large lattice friction. Indeed, the modification of the atomic environment at the surface is expected to change this lattice friction, which may be locally higher than that in the bulk. Therefore, an additional energy barrier for the very first steps of dislocation nucleation, due to the surface (or the step), could be the critical factor. Additional effects like surface or step reconstructions are also expected to influence the nucleation process. These atomistic effects are clearly not described in the elastic model. 3.5.4 Peierls–Nabarro Approaches

It is possible to make more accurate investigations of the dislocation nucleation process by incorporating selected information at the atomic scale. The well-known Peierls–Nabarro approach allows solving some of the previous issues, however at the expense of an increased complexity. Hence, extensions of the original 1D Peierls–Nabarro model [26, 27] have been developed for dealing with similar 2D and 3D problems [28–30]. The dislocation nucleation from a surface has been investigated by Li and Xu using a general variational boundary integral formulation of Peierls–Nabarro model [31]. This framework allows to deal with complex system, but requires the use of numerical simulations like finite element calculations. Li and Xu showed that an increase of the step height leads to a large reduction in the activation energy and studied the influence of slip plane and step inclinations. Compared to the above elastic model, the stress inhomogeneity is taken into account in these calculations. Also, there are no limitations regarding the shape of the dislocation loop. Partial information on the dislocation core is included in the calculations through empirical models or from generalized stacking fault surfaces determined with atomistic simulations. Nevertheless, it is of interest to note that the dislocation core is still approximately described within these approaches, and that the influence on the nucleation process of atomistic details of the surface and step cannot be dealt with.

3.6 Atomistic Modeling

In order to remove the limitations reported in the previous section and to determine the input parameters in the elastic model, theoretical investigations can be made by using atomistic simulation methods. Within this framework, the description of the dislocation nucleation process is done at the atomistic level, which allows access to the very beginning of a half-loop dislocation formation. Generally, one can also expect

3.6 Atomistic Modeling

a better accuracy than with elasticity theory. Unfortunately, it is not all rosy. The first downside of these methods is the usually large computational effort, which results in strong limitations in the size of the considered system and in the timescale for dynamic simulations. This aspect is especially important for first-principles calculations, for which simulations typically include only few hundreds of atoms, with durations on the order of picoseconds. Other methods such as classical molecular dynamics allow to deal with much bigger systems with larger timescale, although the characteristic duration of such a simulation is typically in the nanosecond range. Classical simulations also generally imply an undefined loss of accuracy compared to first-principles calculations. The second downside of atomistic simulations is that the general nature of the elastic treatment is lost. In fact, it is more difficult to determine a general behavior since the investigated process can depend on the atomistic details of the input system, such as the structure of the surface or a step. More simulations are then required to study the possible configurations. 3.6.1 Examples of Simulations

Most of the few atomistic investigations of dislocation nucleation from surfaces are recent and are focused on simple materials. The aim appears to be a full understanding of the process rather than to numerically reproduce experiments made with real and complex systems. The nucleation of a half-loop dislocation in a ductile simple fcc metal like aluminum has been studied by the authors. Classical molecular dynamics simulations at RT of a stressed Al(100) slab showed the formation of a half-loop partial dislocation from steps initially built on the surface (Figure 3.7) [32]. The nucleation started at an imposed tensile strain of about 6% (with an orientation perpendicular to the surface step), the dislocation gliding in the (111) plane in the continuity of the surface step. Analyses of the dislocation revealed a 90 orientation as expected, since it corresponds to the largest Schmid factor. An equivalent result was obtained in copper by Zhu et al. [33], as they investigated the formation of a half-loop partial dislocation from the flat (001) surfaces of a h100i square section nanowire under compressive stress, and for Al and Ni in the case of dislocation nucleation from surfaces at crack tips [34]. Covalent systems have also been considered, aiming at a better understanding of the mechanisms relaxing the high stresses that may occur in the thin layers of semiconductor devices [35]. For instance, Izumi and Yip have studied the formation of a dislocation in silicon from a sharp corner, which is equivalent to a high and straight step [36]. The nucleated dislocation exhibits a half-hexagonal shape, which is characteristic of deep Peierls valleys as expected in covalent materials, and glides in the dense h111i planes. Similar results were obtained for smaller steps by the present authors (Figure 3.8). We have also shown that depending on the range of applied strain and temperature, both partial and perfect dislocations could be nucleated [37, 38], a situation equivalent to what is observed in bulk silicon [8, 39].

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Figure 3.7 Snapshot of a molecular dynamics run at RT showing the formation of a half-loop partial dislocation in a Al(100) slab, leaving a stacking fault connecting with a step on the surface. Only atoms in a nonbulk environment

are represented to ease the visualization. With the chosen color code, the dislocation core and stacking fault are represented by blue atoms, whereas atoms forming the surface step are red.

3.6.2 Determination of Activation Parameters

The usual and most appealing simulation framework for investigating thermally activated atomistic processes is molecular dynamics [40, 41], since it allows mimicking the dynamical behavior of the system within various conditions. However, due to computing limitations, the timescale of simulations is severely restricted, which

Figure 3.8 Snapshot of a molecular dynamics run at 600 K showing the formation of a half-loop dislocation in a Si(100) slab, starting from a (111) edge. Only atoms in a nonbulk environment are represented to make the visualization easier.

3.6 Atomistic Modeling

prevents an efficient exploration of the onset of plasticity. In fact, the latter occurs, thanks to a single event, the first nucleated dislocation being usually followed by many others. In an experiment, the probability to observe this initial process is not negligible at moderate stress due to macroscopic timescale. With molecular dynamics, very high stresses, close to the athermal threshold, have to be reached for making the activation possible in the simulations. As a consequence, only a very small stress range could be investigated, and this approach appears to be not suited for a quantitative determination of the activation energy. Alternative methods to investigate activated processes with high-energy barriers (i.e., rare events) are available. Among the many different flavors of transition-state determination techniques, a chain-like method such as nudged elastic band (NEB) [42] is a favorite nowadays since it is fast, easy to use, and implemented in several computational packages. In an NEB calculation, a set of image configurations allowing to transform a given system from an initial to a final state are first built, and the consecutive images are then linked by springs in configurational space. The relaxation of these images leads to the minimum energy path (MEP), from which the activation energy can be easily deduced. Note that the use of such a method for dislocation nucleation is somewhat tricky [43, 44]. Figure 3.6 shows the critical radii and activation energies determined by NEB calculations for four different applied strains, in the case of an (001) aluminum surface including a one-layer width step. These values have been used for fitting the elastic model described above, considering that the dislocation half-loop may be elliptical. The best fit is reached for ma ¼ 1 and an ellipticity factor e ¼ 1:05, both reasonable numbers. In fact, using m  0:5, a value close to the one given in the original paper from Beltz and Freund, the core radius factor a is found to be about 2. Besides, e is close to 1, justifying the use of the circular half-loop approximation in this specific case. This point is confirmed by the analysis of the shape of the half-loop dislocation, which can be accessed from NEB calculations. Finally, this result indicates that the developed elastic model is sound and captures most of the physical aspects underlying the nucleation of half-loop dislocation from surfaces, at least for fcc metals. It is worth to mention two shortcomings of the approach. First, although a constant applied stress is assumed in the elastic model, atomistic simulations have been performed with a constant applied strain. In the latter, the resolved shear stress is expected to decrease when a dislocation is formed and propagates through the system, a property that is not included in the elastic model. Second, an NEB calculation is intrinsically static, that is, it only allows computing the internal energy barrier and not the free energy barrier. Vibrational contributions, which are known to be important for many dynamical systems, and inertia effects are not accessible with the NEB method, unlike molecular dynamics. It is difficult to estimate the importance of both issues on the activation energy and the critical radius curves. 3.6.3 Comparison with Experiments

Ideally, the next step would be a thorough comparison of predictions given by the theoretical approaches with available experimental data. Unfortunately, such a feat is

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difficult to achieve. We have already mentioned that the true onset of plasticity is difficult to measure in nanoscaled systems. Furthermore, the modeling corresponds to an ideal material that can be relatively far from the real material. For instance, it is well known that most of the mechanical tests of nanopillars have been performed with samples prepared using a focused ion beam technique, which tends to leave a nonnegligible concentration of Ga atoms in the surface. Nevertheless, theoretical approaches are expected to provide the correct orders of magnitude when compared to experiments. Navarro et al. have recently examined the onset of plasticity from gold surfaces with nanoindentation [45]. The measured shear stress are 2.1 and 1.6 GPa for flat and stepped surfaces, respectively. In the case of Al investigated here, the computed values for the elastic limit correspond to yield strengths of few GPa, thus in the same range. Generally, it seems that theoretical data are overestimated compared to experiments, typically by a factor of 2 or 3. The most likely explanation for this discrepancy is the difference in space and timescales between numerical simulations and experiments. In fact, due to computer limitations, molecular dynamics calculations are usually limited to tens of nanoseconds. Accordingly, the strain/strain rates used in the numerical simulations are unrealistically high [33]. Since the nucleation of the initial dislocation is a stochastic event, the associated onset of plasticity is most likely to occur during an experiment time on the order of a second. As a result, higher stress is required in simulations for initiating plasticity. This issue is known for the investigation of dislocation mobility in bulk materials [46, 47] and has been recently examined for the nucleation of dislocations in nanomaterials [33, 43]. Another possible discrepancy origin is the difference in dimensions between simulations and experiments. For instance, periodic boundary conditions are mostly used for modeling infinitely long nanowires from a small system. The number of possible nucleation sites is, therefore, much smaller than that in a real sample, which makes the dislocation nucleation less likely in simulations. These aspects should be kept in mind when comparing modeling with experiments. 3.6.4 Influence of Surface Structure, Orientation, and Chemistry

In the elastic model described in the previous section, we have considered the general case of a surface, with a step of arbitrary height or without. Nevertheless, a step is generally used in atomistic studies, for it has been shown not only to decrease the amount of strain required for dislocation nucleation but also to localize the nucleation event. Now, since the elastic model has been fitted by considering stepped surfaces, the stress modification due to the step is taken into account, although it is not explicitly included in the model. Test simulations, performed for flat surfaces, suggest that this effect could amount to several tens of electron volts for large applied strains. Recently, Li and Xu have investigated the effect of step height and angle relative to the surface using a Peierls–Nabarro framework [31]. A significant reduction of the required stress is obtained when the step height increases, this effect being stronger for low angles. This result was recently confirmed by atomistic

3.6 Atomistic Modeling

simulations in metallic systems [43]. These investigations also revealed a nonmonotonous behavior between mono- and multilayer steps, which seems to be linked with the atomic structure of the step. In a covalent material like silicon with a strong directional bonding, such an effect is expected to be even more important. In these materials, dislocation cores are usually complex [39] and several step geometries are possible. Godet et al. have shown that the initial formation of the core would depend on the step geometry [48]. Other aspects such as the influence of kinks on the steps and of the surface chemistry have not been extensively studied. Atomistic simulations aiming at the formation of dislocations from steps with kinks suggested that kinks are not favorable nucleation sites [32]. This point is supported by experimental evidences that dislocation nucleation is easier from straight than irregular steps (B. Pichaud, private communication). The structure of the surface has also been shown to have a paramount importance for the nucleation process. In fact, first-principles simulations of silicon surfaces under stress [49] revealed that while dislocation formation succeeded from bare reconstructed surfaces, the nucleation is hindered when the surface is passivated with hydrogen (Figure 3.9). This result obviously calls for additional investigations. Finally, there have been very few investigations of the influence of the surface orientation regarding the dislocation formation process. Different orientations would mean the selection of different slip planes, as well as different step geometries. Although the atomic structure is not taken into account in their analysis, Li and Xu have shown that when the angle between the surface and the slip plane is smaller,

Figure 3.9 Successive steps (from left to right) of the first-principles relaxation of a stressed silicon slab. At the top, a single dislocation is nucleated at the step edge and propagates toward the opposite surface. At the bottom, the

presence of hydrogen atoms passivating the surface prevents the dislocation formation at the surface, which occurs by homogeneous nucleation into the slab.

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Figure 3.10 Snapshot of a thin buckled Al slab after the nucleation and propagation of several partial dislocations, resulting from the modeling of a delamination process. Only atoms in a nonbulk environment are shown for clarity. Courtesy of J. Durinck et al. [51]

dislocation nucleation becomes easier. In the different context of brittle-to-ductile transition, an atomistic study of dislocation emission from crack tip also pointed to a significant effect of the surface orientation [50].

3.7 Extension to Different Geometries

The elastic and atomistic modeling of the onset of plasticity described in the previous sections deals with flat surfaces, eventually containing steps, under the action of a unixial stress (strain), which is rather typical of epitaxial thin films. However, there are other configurations for which the dislocation nucleation from surfaces is expected to be the main plastic mechanism. For instance, molecular dynamics studies of the buckling of metallic thin films revealed the nucleation of many dislocations from the surface (Figure 3.10). Other cases include the nucleation of dislocation from crack tip [52]. The process of dislocation nucleation from surfaces has recently regained attention with the development of nanowire/nanopillar deformation tests [11, 53–57]. Most of the investigations performed with atomistic simulations focused on cylindrical nanowires with diameters usually lower than 10 nm. For fcc metals, a general result of these simulations is the nucleation of dislocations from the surface [58–60]. For such small diameters, the curvature of the surface wire is large and is clearly expected to play a significant role in the process of nucleation. This aspect is not included in the model described in the previous sections, and to our knowledge no attempts have been made to develop an appropriate framework. Nevertheless, experimentally investigated metallic nanopillars have much larger diameters, generally greater than 200 nm. In this case, the surface curvature is small and large flat terraces are likely to exist. The model of surface nucleation should therefore be appropriate. Atomistic simulations have also been performed for small nanowires made of covalent materials, with a focus on the size effect in the brittle-to-ductile transition [61, 62]. Conversely to metals, very thin covalent nanowires can be synthesized.

3.8 Discussion

Although almost spherical nanowires have been studied, reconstructed facets should be predominant for low-diameter nanowires. The influence of the nanowire section shape then becomes another factor to consider [63]. Note that in the case of a nonrealistic square Cu nanowire, the onset of plasticity has been predicted to occur from dislocation nucleation at the edge rather than at the surface [33].

3.8 Discussion

In this chapter, we have investigated the onset of plasticity in single-crystalline materials having one or several nanometric dimensions, in the framework of elasticity theory and atomistic simulations. The elastic modeling of the nucleation of a dislocation from a surface allows a qualitative description of the process. This model requires to be fitted on atomistic simulations, and one may wonder whether its use is judicious if these simulations have to be performed for each new system. However, it has been shown that only one parameter has to be fitted, and that the resulting value was close to what could be expected. Accordingly, one can tentatively assume that the proposed elastic model could be used for investigating the dislocation nucleation process in other metals like Au, Ag, Pb, or Ni, simply by using the correct physical data and the same fitted parameter. Conversely, for other families of materials such as bcc metals or covalent systems, further atomistic investigations are certainly required. We have discussed in the preceding sections several issues that tend to make harder the numerical determination of quantities such as the critical stress corresponding to the onset of plasticity in experiments. Yet, the effect of the large differences in timescale and space scale between experiments and simulations has been examined by several authors, and ways to solve the problems have been proposed. Therefore, it should be possible to use the model described here to compute critical stresses for different systems, apply the proposed corrections, and compare with available experiments. To our knowledge, such a systematic comparison remains to be made. Besides, yield stress estimations proposed in experimental papers are usually derived from crude assumptions. For the specific case of nanowires and nanopillars, the effect of size should be examined. In particular, for the smaller ones, it would be necessary to take into account the surface curvature. This is especially important for nanowires with midrange diameters, too large to be dealt with atomistic simulations, but for which these effects could be important. Finally, in experiments, the onset of plasticity due to the nucleation of an initial dislocation is often followed by the formation and propagation of other dislocations in adjacent planes. A similar avalanche mechanism, spanning a very short time, is seen in atomistic simulations. The formation of these successive dislocations seems to be easier, thanks to a dynamical and geometrical effect. Nevertheless, a full understanding of this process is still lacking, which would certainly help for a better comparison between experiments and simulations.

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Acknowledgments

This chapter is essentially based on research works made by the authors and by Pierre Hirel during his PhD. D. Rodney, B. Devincre, J. Bonneville, B. Pichaud, L. Kubin, J. Rabier, A. Pedersen, and J. Grilhe are gratefully acknowledged for fruitful discussions.

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4 Relaxations on the Nanoscale: An Atomistic View by Numerical Simulations Christine Mottet

Atomic structure of materials reveals usually significant deviations compared to perfect periodic bulk crystals. Condensed matter is made of defects, some of them being essential for physical properties such as doping in semiconductors (extrinsic defects), other ones such as vacancies, dislocations, or grain boundaries are key elements for mechanical properties of materials, especially in the plastic domain. Defects come from the elaboration mode and thermal treatment of the materials. When the dimensions of the system decrease, the surface and interfaces become important, and the structure can be modified in the vicinity of these extended defects. The cut bonds at the surface or the epitaxial relation at the interface may not only induce stress and strain in the neighborhood of the defect but also possible reorganization, both atomically and chemically in case of alloys. Such structural modifications have a stronger impact when the size of the system is reduced at the nanoscale, as in clusters or nanoparticles. Nowadays, experimental improvements make it possible to elaborate and characterize nanomaterials. In parallel, theory and especially atomistic models based on numerical simulations (semiempirical potentials or ab initio methods in the density functional theory, DFT) made significant progress in describing the relaxation processes leading to energy optimization. As a consequence, we know that such defects are associated with lattice deformations and reconstructions, or even chemical rearrangements minimizing the total energy of the system but not necessarily its stress. In case of pure systems such as surface or clusters, we show that energy minimization can lead to relatively high-stressed systems as in the case of the (1  2) reconstructed metallic surfaces or in the case of small icosahedral clusters with high compression in their core. In contrast, in alloy surfaces and nanoalloys, we usually find that the release of the stress coming from the size mismatch between the two components is a driving force for stabilization of the system. This chapter will develop different aspects of the atomistic approach by numerical simulations of relaxations at the nanoscale in five sections. After a short introduction, the theoretical models and basic numerical simulations will be described in Section 4.2. Then, the relaxations in surfaces and interfaces (surface reconstructions, alloy surfaces, and heteroepitaxial thin films) will be discussed in

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Section 4.3, before dealing with the relaxations in nanoclusters (unsupported pure clusters, supported clusters, and nanoalloys) in Section 4.4. The last section will stand for conclusions.

4.1 Introduction

In linear elasticity theory, the stress and strain relation described under the form of the Hooke’s law is based on the continuum concept for the matter that has been well represented using a tensorial notation, as introduced by the French mathematician Augustin Louis Cauchy (1789–1857). It has been able to describe most of the elastic deformation both in macroscopic and in microscopic systems. However, going to smaller and smaller systems sizes up to the nanometer scale, it becomes more evident that the description at the atomic level is more relevant. In particular, when the atomic structure is sensibly modified compared to the bulk periodic structure, we can expect that in the vicinity of the perturbations, the elastic theory fails. In the following, we will show, however, that the linear elasticity theory reproduces well the strain at a certain distance of the stressed region but not necessarily exactly in the center of the defect. So we find that the two aspects, continuum theory and atomistic description, are essentially complementary. The concept of “relaxation” in physics is not exactly the same as in current life. In the occidental way of life, the relaxation represents a process or state with the aim of recreation through leisure activities or idling, the opposite of stress or tension. In the oriental civilization, the relaxation could be associated with meditation as taught by Zen masters. In physics, the relaxation is a response to external (or internal) stress (or strain). It implies both the time-dependent process of this mechanical perturbation that we will rather call dynamical relaxation, involving the physical and atomistic mechanism the system uses to reach its relaxed state, and the final relaxed state that we will essentially consider here. When a load is applied to a system as in the traction experiment, the system is strained because of the external stress and the atomic positions may change. We will not consider the way they will move (the dynamics) but their final configuration in the final relaxed state, whatever the way this state has been reached. Even if a system is free of external stress, this does not mean it is free of strain. Indeed, because of defects such as surfaces, interfaces, grain boundaries, or dislocations, a system can be strained by itself, without any load applied. In such cases, the system presents some deviations compared to the bulk periodic structure. This will be described at the atomic level by numerical simulations. The strain pattern is controlled by the energy minimization. The system optimizes its energy by moving some of its atoms (optimizing the bond length as a function of the local environment and in particular the number of near neighbors) and by exchanging atoms of different species in the case of alloys. For example, some surfaces are reconstructed in order to compensate their lack of bonds, such as the famous (7  7) reconstruction of the Si(111) surface [1] or the Au(111) surface reconstruction [2].

4.2 Theoretical Models and Numerical Simulations

4.2 Theoretical Models and Numerical Simulations

Modelization of the energy of the system is the crucial point in order to perform realistic and reliable simulations. There are different approaches with different degrees of accuracy corresponding to different levels of description of the electronic structure from the first-principles methods using the density functional theory to semiempirical potentials fitted to experimental properties or, eventually, on ab initio calculations if experimental data do not exist. In parallel to the energetic model, we will illustrate the basic statistical thermodynamic methods in numerical simulations, that is, molecular dynamics and Monte Carlo simulations, to describe the equilibrium configuration corresponding to the minimum energy state in the fundamental state (at OK) or at finite temperature. 4.2.1 Energetic Models

According to the first principles, the Schr€odinger equation for a system of N electrons moving in the electrostatic field created by M atomic nuclei is written as HV ¼ EV

ð4:1Þ

where H is the Hamiltonian operator, E is the energy, and Vðr1 ; . . . ; rN ; R1 ; . . . ; RM Þ is the wave function, which depends on the electrons’ ri and nuclei Ri positions. The Born–Oppenheimer (adiabatic) approximation consists in separating the movement of the electrons from those of the nuclei, knowing the electron mass is much less than the nuclei one. Thus, the electrons in their fundamental state follow the nuclei in such a way that V can be written as V ¼ WðR1 ; . . . ; RM ÞYðr1 ; . . . ; rN Þ

ð4:2Þ

and the wave function of the electrons comes from the resolution of the Schr€ odinger equation in the field of the nuclei He YðrÞ ¼ Ee YðrÞ

ð4:3Þ

where the electrons’ Hamiltonian He is the sum of three contributions He ¼ T þ V þ U

ð4:4Þ

with the kinetic energy T T¼

1X rY* ðrÞrYðrÞdr 2

ð4:5Þ

the potential V that represents the interaction of the electrons in the potential of the nuclei X VðrÞ ¼ vðrÞY* ðrÞYðrÞdr ð4:6Þ

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and the Coulomb interaction of the electrons between each other U U¼

1X 1 Y* ðrÞY* ðr 0 ÞYðrÞYðr 0 Þdr dr 0 2 jrr 0 j

ð4:7Þ

The coupling of N electrons by the Coulomb repulsion and the exchange interaction coming from the Pauli exclusion lead to a complex system that is essentially impossible to resolve without approximation. There are two kinds of approximations: the Hartree–Fock method, where the N electrons’ problem is transformed into N coupled equations with one electron in an effective potential via a Slater determinant, and the density functional theory from the Hohenberg and Kohn theorem (1964) [3]. We will briefly describe the second method, the most used at present. According to the remarkable theorem of Hohenberg and Kohn [3], the energy of a free electron gas in interaction in an external potential vðrÞ is a functional F½nðrÞ of their density nðrÞ, with the minimal value corresponding to the fundamental state. The functional F½nðrÞ, independent of vðrÞ, is defined, according to Kohn and Sham [4], as F ½nðrÞ ¼ Ts ½nðrÞ þ

1 2

ðð

nðrÞnðr 0 Þ dr dr 0 þ Exc ½nðrÞ rr 0

ð4:8Þ

where Ts ½nðrÞ is the kinetic energy of an electron gas without interaction with density nðrÞ, the second term is that of Hartree (mean field approximation) to treat the Coulomb interaction, and the last term is that of exchange and correlation that takes into account all that has been neglected before. The variational problem of the Schr€odinger equation can be solved as a system of one electron in an effective potential   1  r2 þ Veff ðrÞ yi ðrÞ ¼ ei yi ðrÞ ð4:9Þ 2 with the effective potential ð nðr 0 Þ 0 Veff ¼ vðrÞ þ dr þ Vxc ðnðrÞÞ jrr 0 j

ð11Þ

where Vxc ðnðrÞÞ is the exchange and correlation contribution to the potential. This system of Kohn–Sham equations can be solved in an iterative way: starting with a density nðrÞ using monoelectronic functions yi ðrÞ on a given set of wave functions, we calculate the effective associated potential and we solve the Kohn–Sham equations by diagonalization of the matrix. The eigenvectors give the new wavefunctions yi ðrÞ and thus the new density nðrÞ. The energy is integrated numerically over all the k vectors of the Brillouin zone and the self-consistent process is followed up to the convergence of the energy. The two principal exchange and correlation functionals, widely used, are the local density approximation (LDA), which corresponds approximately to a homogeneous electron gas with a sum of local contributions. It is quite correct when the variations

4.2 Theoretical Models and Numerical Simulations

in the density are weak. The second one is the generalized gradient approximation (GGA), which is better in the case where the density can fluctuate as in the vicinity of defects (surfaces) or open systems (molecules). Anyway, ab initio methods in general are time consuming and can be used efficiently only on very small systems (less than hundreds of atoms) or low symmetry, and using a quite poor statistic or a quite short time of dynamics. For larger-scale systems, always keeping the atomistic description, we can use semiempirical potentials and in particular many-body potentials for metallic systems. We will describe briefly the tight binding second moment approximation (TB-SMA) model. In this model, the main features of the cohesion of transition metals are well reproduced by the bandwidth of the density of states [5], which we approximate by a rectangular shape with the same width [6]. By this way, the band energy term at site i can be written simply sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð4:11Þ Eib ¼  j2 e2qðrij =r0 1Þ j

where j is an effective hopping integral, rij is the distance between the atoms at sites i and j, r0 is the first-neighbor distance in the metal. The summation goes over all the neighbors up to the cutoff distance rc . The total energy of a system of N atoms is written as a sum of the band energy term (attractive part) and a repulsive term of the Born–Mayer type X Eir ¼ A epðrij =r0 1Þ ð4:12Þ j

where the parameters (j; A; q; p) are fitted to different experimental values: bulk cohesive energy (eB ), lattice parameter (a), and elastic constants (B, C44 , C0 ) [7] of the metal. This many-body potential is comparable to other well-known methods such as the embedded atom method (EAM) [8] or the corrected effective medium (CEM) theory [9, 10]. Although not as precise as the ab initio methods [11, 12], the TB-SMA approach describes quite correctly, at least qualitatively if not quantitatively, the relaxation and/or some of the possible reconstructions of the low-index surfaces [13]. 4.2.2 Numerical Simulations

The archetypal methods used in numerical atomistic simulations are essentially of two kinds, the molecular dynamics (MD) and the Monte Carlo (MC) simulations, which are well described by Frenkel and Smit [14], and Allen and Tildesley [15]. A lot of other methods have been developed, mainly to overcome the time limitation of the MD simulation to describe real time-dependent processes. The idea of these methods (accelerated dynamics [16], ART [17], NEB [18], or dimer method [19]) is to accelerate the dynamics by the study of the whole energy landscape [20] and in particular the determination of the saddle points in order to cross the energy barriers more efficiently and in an acceptable simulation timescale. We will not describe all these

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more sophisticated methods but give here only the essential elements to give a first insight into the basic atomistic simulation methods (MD and MC simulations). Moreover, in the large majority of the examples given hereafter, we will concentrate on the final relaxed states, obtained from a local relaxation of the atomic positions starting from a configuration that is near the final one. This does not require to perform a full dynamical relaxation process because we are less interested by the relaxation mechanism than by the description of the final relaxed state. In the case of the nanoalloys, some of the atomic structures are coming from global optimization methods [21] that are efficient methods to find the lowest energy structure (global minimum) in the whole energy landscape [20]. In MD simulations, the Newton equations Fi ðtÞ ¼ m

dEpot d2 ri ðtÞ ¼ drij dt2

ð4:13Þ

give, by integration using the Verlet algorithm [22], the atomic positions ri ðtÞ ri ðt þ dtÞ ¼ 2ri ðtÞri ðtdtÞ þ

Fi ðtÞ 2 dt þ eðdt4 Þ m

ð4:14Þ

and the atomic velocities vi ðtÞ ¼

ri ðt þ dtÞri ðtdtÞ 2dt

ð4:15Þ

We can deduce from the velocities, the kinetic energy Ecin ¼

N X 1 i¼1

2

mi vi ðtÞ2

ð4:16Þ

and the temperature T of the system thanks to the equipartition principle N X 3 1 NkT ¼ mi vi ðtÞ2 2 2 i¼1

ð4:17Þ

where k is the Boltzmann constant. We can let the system evolve in the microcanonical ensemble keeping constant the total energy (kinetic þ potential), the number of particles, and the volume of the system. Looking for the equilibrium state, we can perform quenched MD in order to search for the potential energy minimum: the velocity of an atom i is set to zero each time Fi vi < 0. Equilibrium state at finite temperature can be reached if the MD simulation is sufficiently long in time in order to explore the phase space according to the ergodicity principle. The MC simulations are the other great family of statistical simulation methods to determine the equilibrium state of one system in terms of atomic structure and chemical arrangement. Based on the Metropolis algorithm [23], it makes successive random trials modifying the atomic configuration of the system following a Markov

4.2 Theoretical Models and Numerical Simulations

chain. The probability of acceptance of a given configuration is defined by a Boltzmann canonical distribution PðCÞ / eEC =kt

ð4:18Þ

and the probability of transition from one initial configuration C to a final one C0 is defined by   n o PðC0 Þ ð4:19Þ WðC ! C0 Þ ¼ min 1; ¼ min 1; eðEC0 EC Þ=kT PðCÞ which means that the new configuration C0 is accepted if its energy is lower than the energy of the initial configuration. If its energy is higher, the new configuration is accepted with the probability eDE=kT . Performing average of the energy over a large quantity of sampled configurations allows to characterize the equilibrium state of the system. 4.2.3 Definitions of Physical Quantities

The pertinent physical quantities used in order to describe the state of the system are essentially the defect energies extended to surfaces, interfaces, adatoms, vacancies or impurities, the surface stress, and the local pressure or stress on an atomic site. The defect energy (surface, interface, and adsorption) is defined by the cost in energy of the system with the defect in the final state compared to the equivalent system without defect in the initial state, normalized by the number of atoms concerned with this kind of defect (Nnorm ). The initial and final states are represented schematically in Figure 4.1. The defect energy is written as Edefect ¼

Efinal Einitial Nnorm

ð4:20Þ

The term “equivalent” means notably that we keep the same number of atoms and the same nature of atoms in the final and the initial state. In that way, the surface energy is defined by the cost in energy to separate a bulk in two pieces creating a surface on each side. It is given by c¼

Eslab NEcoh 2Nsurf

ð4:21Þ

where Eslab is the energy of a system of N atoms with periodic conditions in two directions and no periodic condition in the third direction (see Figure 4.2). The slab is constituted of p layers of Nsurf atoms on each, which leads to two surfaces of Nsurf atoms. Ecoh is the cohesion energy. The interface energy writes b¼

A B Eslab N A Ecoh N B Ecoh A B Nsurf þ Nsurf

ð4:22Þ

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Figure 4.1 Schematic representation of the creation of surfaces, interface, and adatoms adsorption on a surface in final state compared to an equivalent system in initial state. Nnorm gives the relevant number of atoms to normalize the defect energy. A B A where Ecoh and Ecoh are the cohesion energies of the two materials and N A (Nsurf ) and B B N (Nsurf ) are the number of A or B atoms in total (or at the surface). The adhesion energy is slightly different from the interface energy as the initial state is composed of two slabs of A and B with their respective surfaces. Then, it is written as

W¼

A B Eslab Eslab Eslab A B Nsurf þ Nsurf

ð4:23Þ

The adsorption energy of Nads adatoms, for a coverage y ¼ Nads =Nsurf , is written as Eads ¼

y y¼0 Eslab Eslab mNads Nads

ð4:24Þ

where m is the atomic potential (energy in the gas phase).

z

y x

Figure 4.2 Schematic view of a slab used in numerical simulation to represent a surface (in fact, two surfaces).

4.3 Relaxations in Surfaces and Interfaces

Then, we can also determine the vacancy energy in a system of N atoms: ðN1Þ

Ev ¼ Eslab 

ðN1Þ ðNÞ Eslab N

ð4:25Þ

And finally the solution energy of an atom B in a matrix of A is written as AðBÞ

Esol

AðBÞ

A ¼ Eslab þ mA Eslab mB

ð4:26Þ

where the initial state is a slab of A and one atom of B and the final state, the same slab where one A atom has been replaced by a B atom and the A atom in gas phase. The surface stress is a quantity related to the stress tensor in bulk but which concerns the surface. A thermodynamic definition is given by M€ uller and Sa ul [24] on the basis of an interfacial excess quantity. Choosing the z-axis as the normal of the surface, the surface stress tensor is symmetric and comprises only two nonzero components: s xx and s yy . with the The Shuttleworth’s equation links the surface stress component s surf ij surface energy c ssurf ¼ ij

1 @ðcAÞ @c ¼ c0 @ij þ A @eij @eij

ð4:27Þ

where A is the surface area and c0 the surface energy of the initial surface (before the deformation). The local pressure, in an atomistic model, represents the pressure localized on each atomic site. It corresponds to the trace of the stress tensor P¼

1X sij 3 i

ð4:28Þ

It derives from the energy of the system using the definition of the hydrostatic pressure as given by Kelires and Tersoff [25]: Pi ¼ 

rij dEi dEi ¼ d ln Vi 3 drij

ð4:29Þ

where Ei is the energy of the system at site i, Vi is the atomic volume, and rij is the distance between the atom at site i and their neighbors j.

4.3 Relaxations in Surfaces and Interfaces

Surfaces and interfaces are the place of atomic relaxations because of the modification in the local environment of the atoms located at the surface (broken bonds) or at an interface (change in the nature and the distances of near neighbors). As a consequence, the surfaces and interfaces can reorganize their structure in order to minimize their energy, leading to surface reconstruction or interfacial dislocations.

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We will consider in the following examples of surface reconstruction, surface with foreign adsorption (alloy surfaces), and heteroepitaxial thin films. 4.3.1 Surface Reconstructions

Because of the broken bonds at the surface, the interatomic distances of the surface atoms are modified as compared to the ones in the bulk. For example, it is well known that the metallic surfaces undergo an inward relaxation that is well reproduced by many-body potentials [26] at the difference of the pair potentials that give essentially an outward relaxation. Sometimes, the surface atoms can even reorganize their structure in order to minimize the surface energy. It is the case for the well-known gold(111) herringbone reconstruction [27, 28] as displayed by scanning tunneling microscopy (STM) pictures in Figure 4.3. Such reconstruction is well explained using a 2D Frenkel–Kontorova model by a spontaneous formation of stress domains including long-range elastic interactions [28]. We will concentrate here on another surface reconstruction, the (1  2) missing row reconstruction that concerns the (110) surface of the 5d transition metals. In the S. Olivier’s thesis, directed by A. Saùl in Marseille [29], the author studied theoretically the influence of the stress on metallic surface reconstruction. The author first calculated in ab initio the surface energy gain to reconstruct the (1  2) (110) surface into the missing row (1  2) structure. The author found as illustrated in Figure 4.4 that only the 5 d Ir, Pt, and Au metals reconstruct (Dc < 0) in good agreement with the experiments. Then, the author wanted to determine a criterion, beyond the energetic one that supposes to know the final reconstructed structure, in order to predict if the surface is susceptible or not to undergo the reconstruction. The author performed a detailed description of the surface stress tensor of the unreconstructed surface as compared to the reconstructed one. As mentioned before, the z

Figure 4.3 Figures reproduced from Barth et al. [27].

4.3 Relaxations in Surfaces and Interfaces

100

Δγ(1x2 – 1x1)

50

US 0

mJ/m2

TM

–50

–100

–150

Pt

Au

Ir

Pd

Ag

Figure 4.4 Surface energy difference between the (1  2) (110) reconstructed surface and the (1  1) (110) surface for the late-transition metals calculated using the density functional theory and the PWSCF (Plane-Wave

Cu

Rh

Ni

Self-Consistent Field) code [30] with a slab of 12 atomic planes (diamonds) or a slab of 18 atomic planes (circles and continuous line). Reproduced from Ref. [29].

components are necessarily vanished by definition at the surface and there are only the s xx and s yy that are nonzero. Looking for a general criterion to separate the 3d and 4d metals (which do not reconstruct) from the 5d metals (which do reconstruct), the author first considered general tendencies as observed on other types of surfaces. For example, the Au(111) surface has the tendency to densify in the herringbone reconstruction, in order to compensate the lack of bonds. But this is clearly not the case here as the surface suppresses itself some rows leading to an even more open surface. Then, the author researched if the reconstruction was a way to release the surface stress as compared to the unreconstructed surface. The results show no clear difference in behavior between the 5d metals and the others, and even some possible increase in some surface stress components on the reconstructed surface. So, such criterion, which is not necessarily sufficient, but can be sufficient in some cases as we will see hereafter, is not a good criterion in that case. The best criterion the author obtained comes from the derivation of the surface energy as a function of the strain; in other words, the Shuttleworth equation as expressed before. In the graph of Figure 4.5, we see how the 5d metals are well separated from other transition metals, in order to discriminate the metals with tendency to reconstruct the (110) surface. Finally, it has also been checked that the driving force for this reconstruction comes from the electronic structure and in particular the relativistic effects in what concerns the 5d transition metals.

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100 75

Ni

50

Δγ(1×2 – 1×1)

25

Rh

Cu

0

Pd

Ag

–25 –50

Ir Au

–75 –100 –125

Pt

–150 –175 –200 –1000 –500

0

500

1000

1500

2000

2500 3000

δγ/δε××, 1×1 Figure 4.5 Same as Figure 4.4 for a slab of 18 atomic planes but plotted as a function of the derivative of the surface energy as a function of the deformation in the x-direction, in the unreconstructed (1  1) structure. Reproduced from Ref. [29].

4.3.2 Surface Alloys: a Simple Case of Heteroatomic Adsorption

The adsorption of Ag atoms on a surface of Cu(111) leads to interesting superstructures because of the strong size mismatch. The first modelization of an Ag monolayer deposited on Cu(111) substrate by quenched molecular dynamics simulation in the TB-SMA potential (as described in Section 1.1.2) showed that the system adopts a p(10  10) superstructure in good agreement with low-energy electron diffraction (LEED) experiments [31]. The atomistic simulations show beyond the periodicity an original motif with a strong corrugation of the surface layer extended in the substrate on the first 10 surface layers [32]. This corrugation is well illustrated by the local pressure map as depicted in Figure 4.6a and b, where the gray scale represents the pressure scale from tensile zones (in white) to black zones (in black). Such picture is directly comparable to the atomic elevation map of the Ag surface atoms (see Figure 4.7a) and the STM image [33] (Figure 4.7b). However, the STM images suggest other possible structures because of the bright atoms inside the dark triangles. Such STM picture is better represented with a structure where four or five atoms have been removed in the Cu layer in order to release the local pressure in compression to the Cu layer (Figure 4.6b without Cu vacancies: we observe black zones in compression that disappear with the introduction of Cu vacancies in Figure 4.6d). In that case, the stress release by the introduction of Cu vacancies consisting in the formation of partial dislocation loops is a way not only to reduce the local pressure but

4.3 Relaxations in Surfaces and Interfaces

Figure 4.6 Local pressure maps in the Ag (a) and (c) and Cu (b) and (d), layers for the (10  10) superstructure of Ag/Cu(111), without (a) and (b) and with five Cu vacancies (c) and (d). The gray scale is chosen such that the white/black atoms are the most tensile/

compressed ones with the following extreme values: In the Ag layer, Pmin ¼  88 kbar and Pmax ¼ 19 kbar; and in the Cu layer Pmin ¼ 77 kbar and Pmax ¼ 84 kbar. Adapted from Ref. [32].

Figure 4.7 Height elevation maps in the Ag layer of the Ag/Cu(111) (10  10) superstructure (a) and with five vacancies (c). Black and white colors are associated with the

deepest and highest elevations with the typical extreme values: Hmin ¼ 0.033 nm, and Mmax ¼ 0.075 nm; adapted from Ref. [32] In (b) we recall the STM image from Ref. [33].

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also to minimize the surface energy leading to the stabilization of a new superstructure in perfect agreement with STM observations. 4.3.3 Heteroepitaxial Thin Films

The last example of extended surface and interface is taken from a thin metallic film epitaxially grown on a MgO(100) substrate. Obviously, we do not consider the growth mode but only the structure of the film in a state we will consider as an equilibrium state. Such system has been both experimentally [34, 35] and theoretically studied [36], and here again the confrontation of the two approaches is stimulating. In that case, we also compared the atomistic approach with the continuous elastic theory to see to which extent the analytical elastic theory can be used. In that system, as previously, the size mismatch leads to a deformation of the deposit that prefers to rearrange by introducing interfacial dislocations in order to release the elastic deformation. The difference with the preceding case is that the MgO(100) substrate is rigid (an approximation of our model), which prevents any deformation (or even vacancy formation) in the substrate. However, because of the different nature of the two materials (a metal and an oxide), such hypothesis is legitimate in a first approximation, but we should go beyond in further studies using a new energetic model for the oxide and the metal–oxide interaction. The metal is modeled within the TB-SMA as in the preceding case and the oxide is kept rigid, as mentioned above. The metal–oxide interaction is modeled via a potential energy surface approach fitted on ab initio calculations [37]. When considering a thin film of Ag on epitaxy on the MgO(100) substrate, the minimum energy structure corresponds to a partially relaxed film with interfacial dislocations, whose periodicity depends slightly on the film thickness. Figure 4.8 illustrates the propagation of the stress across a 20 monolayer (ML) film starting from the interface with the MgO(100) substrate where are located the core of the dislocations, up to the surface that we can see with a top view. We can distinguish the compressive zones in red and tensile zones in blue. Such motif at the surface is susceptible to change as a function of the film thickness, as shown in Figure 4.9, where thinner films display smaller motifs. In this figure, we also compare the stress profile using an elastic model and the atomistic model. In conclusion, we notice that the two models reach their best agreement for the thicker film (20 ML), where the elastic model is almost as precise as the atomistic model. For 7 and 5 ML, the agreement is still good, far from the dislocation core, but we clearly see that the elastic model fails to describe the stress on top of the dislocation core, which is not so surprising taking into account the approximations of the model. Finally, let us mention the experimental study by grazing incidence small-angle X-ray scattering [34] that shows the self-organization of Co nanoparticles on a silver surface patterned by a buried dislocation network, which is a nice confirmation of what we have obtained in theory. Moreover, by calculating the adsorption energy of one Co atom on top of different sites on the nanostructured Ag surface, we show a clear correlation between the stress corrugation and the amplitude of the adsorption

4.3 Relaxations in Surfaces and Interfaces

Figure 4.8 Top view and side view in the (010) plane of the atomic stress map of the Ag nanostructured film. Color code from red (dark) to blue (light) corresponds to compressive to

tensile atomic sites. Because of its exponential decay, the stress field at the top surface is barely visible and, hence, a different color scale has been used. Adapted from Ref. [34].

Figure 4.9 Top panel represents the map of the surface atomic stress (same color code as Figure 4.8) for different film thicknesses. Bottom panel gives the corresponding stress

graph along the [100] direction (x ¼ y). Circles for the atomistic calculations and full lines for the elastic theory calculations. Adapted from Ref. [36].

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1 20 ML 10 ML

σsurf (GPa)

0.5 0 −0.5 −1 0.04

0

0.2

0

0.2

0.4

0.6

0.8

1

0.8

1

ΔE

Co ads

(meV)

0.02 0.00 −0.02 −0.04 −0.06

0.4 0.6 (x=y)/Λ along [100]

Figure 4.10 Surface atomic stress (top graph) and adsorption energy of a Co atom for different film thicknesses: 10 ML (circles) and 20 ML (squares) along the [100] direction (x ¼ y). L is the length of the nanostructure.

energy (Figure 4.10), which is a good indication of the probable nucleation and growth of Co nanoparticles on it.

4.4 Relaxations in Nanoclusters

Because of their surface, nanoparticles undergo a surface stress in tension that is compensated by a core stress in compression leading to a total pressure equal to zero, as long as the nanoparticles are free from interaction with the exterior (under ultravacuum conditions). Their equilibrium shape respects the Wulff theorem up to quite small sizes if we include the edges and the corner energies in the Wulff description, as we have checked by comparing the results given by atomistic calculations and the thermodynamic approach [38]. As a function of their size, when the size becomes sufficiently small, they can change in structure in order to minimize the surface energy when the internal core pressure is not so high compared to the surface energy gain. The main result of the comparative study using thermodynamic and atomistic approach [38] is that not only edge effects but also mainly surface stress effects leading to a Laplace overpressure inside the particle have to be taken into account in order to validate the Wulff theorem for sizes smaller than 10 nm.

4.4 Relaxations in Nanoclusters

j99

In the following, we will see the example of Pd-free nanoclusters and the possibility of stress release by vacancy introduction. As most of the nanoparticles are deposited on a support or included in a matrix, we will study the effect of the cluster environment on the structure and morphology of nanoparticles. We concentrate here on the case of metallic nanoparticles supported on a MgO(100) substrate. Finally, we will consider the alloy effect on bimetallic nanoclusters (called “nanoalloys”) where the misfit between the two elements is at the origin of the stabilization of new structures that does not exist in pure systems. 4.4.1 Free Nanoclusters

It is well known that nanoparticles, and especially metallic nanoparticles, undertake new structures and notably the fivefold symmetry as their size decreases [39, 40]. This change in structure at small size results from an optimization of their surface by minimization of the surface energy forming only pseudo (111) facets in the icosahedral structure, whereas the fcc truncated octahedron, which is the optimized morphology taking into account the Wulff theorem, displays necessarily both (100) and (111) facets (see Figure 4.11). As the (111) facet is more dense, its surface energy is lower than more open (100) facet. However, the structural transformation from face-centered cubic (FCC) structure to fivefold symmetry structure is accompanied by an internal strain as illustrated in Figure 4.11. In this figure, the three morphologies are displayed with the two FCC ones and the icosahedron one. The first FCC structure

Cuboctahedron

Wulff polyhedron

Vacancy formation energy

Atomic shells pressure

Energy

3

1.7

1.6

1.5 0

5

10

15

20

Ev(core)/Ev(bulk)

800

Pd

Pressure (Kbar)

Energy (eV/at.2/3)

Icosahedron

Pd 600 400 200 0 200 0

N1/3 Figure 4.11 Energy, atomic shell pressure, and vacancy formation energy of Pd clusters under three different structures schematically represented: the cuboctahedron, the Wulff polyhedron, and the icosahedron. Their energy is compared with open blue circles for

5

10

N1/3

15

20

Pd 2 1 0 –1 –2 0

5

10

N1/3

cuboctahedra, open black squares for the Wulff polyhedra, and red diamonds for icosahedra. The local pressure and vacancy formation energy are calculated for cuboctahedra and icosahedra only because they have exactly the same number of atoms to be compared.

15

20

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is the cuboctahedron that has the same number of atoms as the icosahedron, each (100) facet is transformed in two (111) facets leading to 20 pseudo (111) facets in the icosahedral structure. This structure has six fivefold symmetry axis crossing two vertices. The second FCC structure is called the Wulff polyhedron because it optimizes the extension of the facets according to the Wulff construction that links the surface energy to the distance of the facet to the center of the particle. The extension of the (111) facets and the reduction of the (100) ones in the Wulff polyhedron compared to the cuboctahedron makes the Wulff polyhedron much more stable. The energy per cluster is reported in Figure 4.11 and we notice that the icosahedron is stable only at very small size. The critical size of transition from one structure to another is very sensitive to the metal itself and the icosahedron is stable on a larger range of size in Cu [40]. Looking at the local pressure on each concentric layer of the icosahedron as compared to the cuboctahedron (Figure 4.11), we clearly see that the icosahedron is highly compressed in its core whereas the equivalent cuboctahedron undergoes only a slight tension/compression at the vicinity of the surface, with an oscillating profile in order to cancel the total pressure. In order to attenuate the very high stress on the central site of the icosahedron, it is possible to remove the atom on the central site [41]. This leads to the next graph in Figure 4.11 where we see that the vacancy formation energy on the central site of the icosahedra can be negative for a size larger than 100 atoms, whereas with no surprise it is positive and equal to the bulk vacancy formation energy in the case of cuboctahedra. 4.4.2 Supported Nanoclusters

When the nanoclusters are in contact with a support, their structure is susceptible to be modified by their interaction with the substrate. This is the case of metallic clusters deposited on the MgO(100) surface. We first checked that the Wulff–Kaishew theorem that is an extension of the Wulff theorem but on supported systems (analogous to the Young–Dupre relation for a liquid droplet on a solid substrate) also works for nanoparticles. As for free clusters, this macroscopic thermodynamic approach can be extended to small sizes at the condition we integrate the variations in the adhesion energy with cluster size [42]. Indeed, the adhesion energy is one of the energetic ingredients in the Wulff–Kaishew equation where the ratio ci 2cA W ¼ hi H

ð4:30Þ

is a constant. ci and cA are the facet energy of facets i and A, A being the top facet; W is the adhesion energy; and H is the height of the particle. In the epitaxial relation with the MgO(100) substrate, due to the lattice mismatch of 8% between the Pd deposit and the MgO(100) surface, the Pd deposit is strained in order to accommodate the lattice of the substrate. The metal atoms have a preferential adsorption site on top of the oxygen atoms so that adhesion energy is optimal when the maximum of Pd atoms are adsorbed on top of oxygen atoms (as illustrated in Figure 4.12). As a consequence,

4.4 Relaxations in Nanoclusters

j101

1 (a) 0.95 W (J/m2)

interfacial dislocations 0.9 2

oxygen ads. sites (%)

0.8

1

3

5

7 (b)

80

70

60

50

reentrant layer

1

3

5 edge size (nm)

7

mean stress per atom (GPa/at.)

0.85 1.5 interfacial dislocations

1

0.5 reentrant layer 0

−0.5

1

3

5 edge size (nm)

Figure 4.12 Adhesion energy (W), percentage of metal atoms adsorbed on top of oxygen sites, and average stress per atom and per particle as a function of cluster size. Adapted from Ref. [43].

the nanoparticle is highly stressed, as shown in Figure 4.12 where we can follow the total stress per atom and per particle as a function of the cluster size. This stress is fully relaxed by the introduction of the first interfacial dislocation, which corresponds to a Vernier rule between the relaxed nanoparticle and the substrate. Before the introduction of the second dislocation, the cluster is partially relaxed. Such interfacial dislocations have been evidenced in Ni nanoparticles on MgO (100) surface by high-resolution electron microscopy [44]. 4.4.3 Nanoalloys

Nanoalloys are typically systems referring to bulk alloys but with a finite size on the order of one to a few nanometers scale. As in the surface alloys mentioned in the previous section, the misfit between the two elements induces some strain and stress that are interesting to analyze in order to understand the kind of structure adopted by the system. In very small nanoalloys, we found a strong relation between the structure and the misfit that leads to the formation of what we have called “new magic polyicosahedral core–shell clusters” [45]. The term “magic” is generally attributed to highly stable structures. It is concerned first with the simple metal clusters with magic number of atoms corresponding to the completion of electronic shells. Then, more generally, it is concerned with the geometrical completion of atomic layers at the surface. Here, we propose a new criterion related to the composition of the bimetallic cluster. Indeed, in the particular cases of fivefold symmetry structures that are stabilized at small sizes, it is possible to release partially the internal stress by

7

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replacing the core atoms by smaller atoms. They form core–shell structures with polyicosahedral symmetry (see Figure 4.13). These nanoalloys present a particularly high stability, related not only to their low energy compared to others but also to their high melting point compared to pure clusters of equivalent sizes [45]. Beside the stress relaxation, the driving forces responsible of such structures are the same as the ones leading to surface segregation and core/shell structures in nanoparticles that are the surface segregation of one of the elements (Ag in case of Ag–Cu and Ag–Ni) and the tendency to phase separation in bulk alloys. Stress relief can have also surprising effects on the melting behavior of nanoalloys. It is illustrated by the single impurity effect on the melting of nanoclusters [46]. The lattice mismatch between the impurity and its matrix seems to play a decisive role in the melting property of a 55-atom icosahedron of Ag as illustrated below. When the lattice parameter of the impurity is the same as the Ag one, the melting behavior is the same as the pure Ag cluster. The melting temperature is shifted toward higher temperature when the lattice parameter of the impurity is smaller than the one of Ag, and the amplitude of the shift is well correlated with the lattice misfit (see Figure 4.14). Even if the relation between the stress and the thermodynamic stability is not trivial to

Figure 4.13 Different views of magic core–shell polyicosahedra with Cu or Ni atoms in the core (yellow color) and Ag atoms on the shell (gray color) with different sizes and compositions. Adapted from Ref. [45].

4.5 Conclusions

Caloric Curve

5.0

525K

Ag55

Ag54Ni1

600K E–E0–3(N–1)kT (eV)

400K

4.0

j103

Ni impurity Cu impurity Pd impurity Au impurity Pure Ag

3.0

2.0

1.0

0.0 450

550

650

750

Temperature (K)

Ag54Au1

Metal misfit with Ag : Metal

Figure 4.14 Representation of pure 55 atoms Ag icosahedra at different temperatures and the same with an impurity of Ni or Au. While the pure and Au-doped clusters melt at 600 K, the Ni-doped one is still solid at this

Ni

Cu

Pd

Au

%Misfit 14

12

5

0

temperature. Besides, the caloric curves relative to different impurities in this cluster are plotted and a correlation is made between the melting transition and the lattice parameter mismatch. Adapted from Ref. [46].

characterize, this is an evidence of the implication of the stress release in the higher stability of nanoclusters.

4.5 Conclusions

The potentiality of the atomistic simulations in the description and analysis of the strain and stress of a nanoscale system is promising, in particular the analysis of their possible relaxations at the atomic scale in unsolicited systems. Such systems that are free of external stress present nevertheless local or extended defects compared to the bulk solid giving rise to quite complex atomic structures, as we have seen in the different examples presented here. This opens up a wide domain of research in what concerns systems under external stress. Many questions arise when considering mechanical properties at the nanoscale because the points of the contact are already a vast domain to be investigated. We can think about the experiments performed using an STM tip touching the surface and, by a slow backward movement, being able to form a quasiatomic metallic wire of gold or platinum. Possessing interesting transport properties with the quantization of the current, the mechanical properties of these metallic nanowires should also be interesting, as an extreme case of the wellknown nanopillars. However, the difference in size, from one nanometer or less in

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one case to some tens or hundreds of nanometers in the other case makes the atomistic description not always appropriate in all the situations. This is why it is necessary to have a constant feedback to the macroscopic thermodynamic description (notably in terms of the elastic theory) in order to validate or extrapolate if possible the results of the atomistic modelization.

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10 Stave, M.S., Sanders, D.E., Reaker, T.J.,

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A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys., 111, 7010. Wales, D.J. (2003) Energy Landscapes, Cambridge University Press. Ferrando, R., Fortunelli, A., and Johnston, R.L. (2008) Searching for the optimum structures of alloy nanoclusters. Phys. Chem. Chem. Phys., 10, 640. Verlet, L. (1967) Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules. Phys. Rev, 159, 98. Metropolis, N., Metropolis, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E. (1953) Equation of state calculations by fast computing machines. J. Chem. Phys., 21, 1087. M€ uller, P. and Sa ul, A. (2004) Elastic effects on surface physics. Surf. Sci. Rep., 54, 157–258. Kelires, P.C. and Tersoff, J. (1989) Equilibrium alloy properties by direct simulation: oscillatory segregation at the Si–Ge(100) surface. Phys. Rev. Lett., 63 (11), 1164–1167. Desjonqueres, M.C. and Spanjaard, D. (1996) Concepts in Surface Physics, 2nd edn, Springer, Berlin. Barth, J.V., Brune, H., Ertl, G., and Behm, R.J. (1990) Scanning tunneling microscopy observations on the reconstructed au(111) surface: atomic structure, long-range superstructure, rotational domains, and surface defects. Phys. Rev. B, 42, 9307–9318. Narasimhan, S. and Vanderbilt, D. (1992) Elastic stress domains and the herringbone reconstruction on au(111). Phys. Rev. Lett., 69, 1564–1567. Olivier, S. (2004) R^ole des Contraintes Dans les Reconstructions de Surfaces Metalliques. PhD thesis, Universite de la Mediterranee, Faculte des Sciences de Luminy, Marseille. Baroni, S., Corso, A.D., de Gironcoli, S., and Giannozzi, P, http://www.pwscf.org. Bauer, E. (1967) Multiple scattering versus superstructures in low energy electron diffraction. Surf. Sci., 7, 351–364.

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Aufray, B., and Legrand, B. (1999) Ag/cu (111) structure revisited through an extended mechanism for stress relaxation. Phys. Rev. B, 59, 10910–10917. Aufray, B., Gothelid, M., Gay, J.M., Mottet, C., Landmark, E. et al. (1997) Ag/ cu(111): incommensurate reconstruction studied with STM and surface X-ray diffraction. Microanal. Microstruct., 8, 167–174. Leroy, F., Renaud, G., Letoublon, A., Lazzari, R., Mottet, C., and Goniakowski, J. (2005) Self-organized growth of nanoparticles on a surface patterned by a buried dislocation network. Phys. Rev. Lett., 95, 185501. Robach, O., Renaud, G., and Barbier, A. (1999) Structure and morphology of the Ag/MgO(001) interface during in situ growth at room temperature. Phys. Rev. B, 60, 5858–5871. Ouahab, A., Mottet, C., and Goniakowski, J. (2005) Atomistic simulation of Ag thin films on MgO(100) substrate: a template substrate for heterogeneous adsorption. Phys. Rev. B, 72, 035421. Vervisch, W., Mottet, C., and Goniakowski, J. (2002) Theoretical study of the atomic structure of Pd nanoclusters deposited on a MgO(100) surface. Phys. Rev. B, 65, 245411. M€ uller, P. and Mottet, C. (2007) Equilibrium nanoshapes: from thermodynamics to atomistic simulations. J. Comput. Theor. Nanosci., 4, 316–325. Baletto, F. and Ferrando, R. (2005) Structural properties of nanoclusters: energetics, thermodynamics, and kinetic effects. Rev. Mod. Phys., 77, 371–424. Baletto, F., Ferrando, R., Fortunelli, A., Montalenti, F., and Mottet, C. (2002) Crossover among structural motifs in transition and noble-metal clusters. J. Chem. Phys., 116, 3856–3863. Mottet, C., Treglia, G., and Legrand, B. (1997) New magic numbers in metallic clusters: an unexpected metal dependence. Surf. Sci., 383, L719–L727.

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42 Mottet, C. and Goniakowski, J. (2007)

Influence of epitaxial strain on supported metal cluster shapes via atomistic simulations. J. Comput. Theor. Nanosci., 4, 326–334. 43 Vervisch, W., Mottet, C., and Goniakowski, J. (2003) Effect of epitaxial strain on the atomic structure of Pd clusters on MgO(100) substrate. Eur. Phys. J. D, 24, 311–314. 44 Sao-Joao, S., Giorgio, S., Mottet, C., Goniakowski, J., and Henry, C.R. (2006)

Interface structure of Ni nanoparticles on MgO (100): a combined HRTEM and molecular dynamic study. Surf. Sci., 600, L86–L90. 45 Rossi, G., Rapallo, A., Mottet, C., Fortunelli, A., Baletto, F., and Ferrando, R. (2004) Magic polyicosahedral core–shell clusters. Phys. Rev. Lett., 93, 105503. 46 Mottet, C., Rossi, G., Baletto, F., and Ferrando, R. (2005) Single impurity effect on the melting of nanoclusters. Phys. Rev. Lett., 95, 035501.

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Part Two Model Systems with Stress-Engineered Properties

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j109

5 Accommodation of Lattice Misfit in Semiconductor Heterostructure Nanowires Volker Schmidt and Joerg V. Wittemann 5.1 Introduction

By far the most important trait of semiconductor nanowires is that, relative to their volume, they have a large surface. Many essential properties, such as the thermodynamics of growth or the electrical properties, are affected by the presence and the quality of the nanowire surface [1]. With the surface area to volume ratio being inversely proportional to the nanowire diameter, the physical properties of nanowires are naturally more severely affected when the nanowire is thinner. The fact that nanowires possess a large free surface also has substantial influence on their mechanical properties, in particular considering heterostructure nanowires combining materials with different lattice constants. The coherent epitaxial growth of such two materials would cause one or both materials to be strained by the misfit. What makes misfit-strained heterostructure nanowires such appealing objects is that contrary to heteroepitaxial layers on planar substrates, which cannot expand or shrink in lateral direction, a nanowire possesses a free surface that can adjust according to the strain within it. By changing its diameter and—suppose that the nanowire has a free end—also its lengths, a heterostructure nanowire can elastically relax a part of the strain energy induced by the misfit. Owing to its enhanced possibilities for partial elastic relaxation, the onset of other strain relaxation mechanisms such as dislocation formation or a roughening of the surface can be suppressed or at least deferred. It will be shown in the following to which extend this is indeed the case. Generally, one can distinguish between two main types of nanowire heterostructures: (a) axial heterostructures, in which cylindrical pieces of distinct materials contact each other with their flat ends, creating a circular interface and (b) core–shell heterostructures, in which a cylindrical piece of one material is radially enwrapped by the second material. Both types, axial heterostructures and core–shell nanowire heterostructures, have been experimentally realized [2–9]. In addition to these two types of heterostructures, there are two main strain relaxation mechanisms that are usually tried to be avoided. The first is dislocation formation. Besides destroying the coherency of the heterostructure interface,

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j 5 Accommodation of Lattice Misfit in Semiconductor Heterostructure Nanowires

110

(b)

DISLOCATIONS

(c)

ROUGHENING

(a)

DISLOCATIONS

dislocations may have a negative influence on the electrical properties of the nanowires due to carrier recombination at the dislocation core. Another reason why dislocation formation is mostly undesired is that the misfit strain affects the band structure alignment of the materials of which the heterostructure is composed. Dislocations now locally render the strain field, so the band structure alignment is also disturbed. The second basic mechanism by which a misfit-strained layer or structure can relieve part of the strain energy is through surface roughening. During the growth of a strained layer on a substrate of a different lattice constant, the layer is inherently unstable with respect to modulations of the growing surface. The reason for this is that the overall strain energy of a system having a strained but undulated surface is smaller compared to a system with a flat and coherently strained layer; and this reduction in strain energy is what is driving the instability. Ultimately, this instability, the so-called Asaro–Tiller–Grinfeld instability [10, 11], may lead to the formation of clusters on the surface (see Figure 5.1c). This effect is also employed for the selforganized synthesis of quantum dots, and in this context it has received considerable attention in the past (see, for example, the excellent review by Stangl et al. [12] and references therein). Concerning the synthesis of heterostructure nanowires, this instability of the surface is rather thought of as a nuisance than as an advantage, as it would prevent the synthesis of heterostructure nanowires strained in a well-defined manner. For clarity, the two strain relaxation mechanisms (dislocation formation and roughening) are separately dealt with. Applied to the two main types of heterostructures (axial and core–shell), this leaves us with four separate scenarios. However, as strain-induced roughening during the synthesis of axial heterostructures completely prevents the synthesis of an axial nanowire heterostructure, this possible scenario is not considered. First, we will take a look at dislocation formation in axial heterostructure nanowires in the following section, as schematically indicated in

Figure 5.1 The different strain relaxation scenarios. (a) Dislocation formation in case of axial heterostructure nanowires. (b) Dislocation formation in core–shell nanowires. (c) Roughening of the shell in case of core–shell nanowires.

5.2 Dislocations in Axial Heterostructure Nanowires

Figure 5.1a, then dislocation formation in core–shell nanowires (Figure 5.1b) will be discussed, and finally surface roughening of core–shell nanowire heterostructures will be considered.

5.2 Dislocations in Axial Heterostructure Nanowires

As already mentioned, the two main strain relaxation mechanisms are the formation of dislocations and the roughening of the surface. Considering the growth of strained layers on planar substrates, Tersoff and LeGoues [13] could elegantly show that the interplay of these two relaxation mechanisms is such that at small lattice misfits the formation of dislocations is favored, whereas large misfits rather promote a roughening of the surface. In the latter case, this does not exclude that dislocations will be formed, but a roughening of the surface would still precede the formation of dislocations. In the following, we will always refer to a roughening of the surface, even if island formation would in some cases be the more appropriate description of the surface morphology. Concerning the growth of axial heterostructure nanowires, it should be mentioned that these nanowires are in most cases synthesized by the vapor–liquid–solid [14] mechanism. This means that a liquid catalyst droplet promotes the growth of nanowires by acting as a preferential site for adsorption of precursor molecules from the gas phase, supplying the semiconductor material. The balance of such a liquid catalyst droplet at the nanowire tip is delicate. Changes in the growth parameters, for example, can easily lead to a kinking of the nanowires. A roughening of the catalyst–nanowire interface presumably perturbs the growth of nanowire to such an extent that a well-defined synthesis cannot take place anymore. A roughening of the catalyst–nanowire interface, therefore, has to be prevented at all costs. Since there are presently to best of our knowledge no experimental studies on the interplay of the vapor–liquid–solid mechanism with a potential roughening of the catalyst–droplet interface, this problem will not be considered. Still, one should be aware that in particular for the synthesis of axial nanowire heterostructure with large lattice misfit, interface roughening might turn out to be a serious obstacle. Instead, we concentrate in this section on dislocation formation, keeping in mind that according to the analysis of Tersoff and LeGoues [13], dislocation formation is the more critical mechanism at smaller misfits. The analysis presented in this section closely follows the work of Ertekin et al. [15], who at first recognized the potential of nanowires for the synthesis of dislocation-free strained heterostructures. In fact, the calculation presented represents a simplified version of Ertekin’s work, assuming equal Young’s modulus E and Poison ratio n for both materials. The system considered is a single nanowire heterostructure interface (cf. Figure 5.1a) with the two materials having a (positive) misfit m. The two material segments are assumed to have infinite length and a common equilibrium radius R at infinite distance from the interface. That means that the relative difference in radius, which is equal to the misfit, is neglected in the analysis as it would render the outcome only slightly. Considering only the radial component of the strain tensor

j111

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112

(in cylindrical coordinates), the strain energy density, en, at the interface is given by   2 E m ð5:1Þ en ¼ 1n 4 Ertekin et al. [15] assumed that the strain energy density decays exponentially with the distance from the interface and that the characteristic length for this exponential decay is equal to a  R. They found that a ¼ 0:1875 indeed describes the behavior well. Integrating over the length of the nanowire, the total strain energy En0 stored in a dislocation-free nanowire is then given by   apR3 E m2 En0 ¼ ð5:2Þ 2 1n In a rather crude approximation, it is assumed that the formation of a dislocation reduces the misfit to meff ¼ m

b 2R

ð5:3Þ

with b being Burger’s vector of the dislocation. In addition to reducing the strain energy of the system by reducing the misfit, the intrinsic energy of the dislocation, that is, the energy of the dislocation core and the strain field of dislocation, should be taken into account. Taking R and b=4 as the upper and inner cutoff limits for the strain field of the dislocation and accounting for the reduced effective misfit, the strain energy of a nanowire having a single dislocation at the interface can be approximated as        apR3 E b 2 Rb2 E 4R En1 ¼ þ m log ð5:4Þ 2 2pð1 þ nÞ 1n 1n 2R b If it is further assumed that dislocations are only introduced once it turns out to be energetically favorable; that is, if the energy En1 is smaller than En0 , a criterion for a critical misfit mcrit can be derived by equating Eqs (5.3) and (5.4). One can easily find that   b b 4R ð5:5Þ þ mcrit ¼ log 4R ap2 Rð1 þ nÞ b Note that the critical misfit does not depend on Young’s modulus E of materials. This model, based on energetic considerations, neglects several important aspects such as dislocation nucleation and gliding of dislocations, and the outcome therefore should be considered as a rough estimate of the onset of dislocation formation. A calculation on dislocation formation in axial nanowires in which nucleation aspects are taken into account has been performed by K€astner and G€ osele [16]. Although their result differs from the one of Ertekin et al. [15], the general diameter-dependent behavior is similar. The critical misfit mcrit , according to Eq. (5.5), at which the formation of dislocations should set in is shown in Figure 5.2 as a function of the nanowire diameter 2R, assuming a Burger’s vector b ¼ 0:3 nm. As long as the misfit

5.3 Dislocations in Core–Shell Heterostructure Nanowires 0.14 mcrit after equation (5) ref. E. Ertekin et al. 2005

0.12

CRITICAL MISFIT mcrit

edge, ref. F. Glas et al. 2006

0.10

60°, ref. F. Glas et al. 2006

0.08

DISLOCATIONS

0.06

0.04 NO DISLOCATIONS

0.02

0.00 0

10

20

30

40

50

60

70

80

NANOWIRE DIAMETER [nm] Figure 5.2 Critical misfit mcrit for b ¼ 0:30 nm and n ¼ 0:25 using Eq. (5.5). As reference, the data of Ertekin et al. [17] (*) and Glas [18] D, &) are shown.

m at a certain radius does not exceed the critical misfit mcrit , the nanowire heterostructure interface should remain dislocation free. One can see in Figure 5.2 that the critical misfit, at which the formation of dislocation should theoretically set in, is quite substantial. At the nanowire diameter of 35 nm, the critical misfit is already 4.5%, which should theoretically suffice to synthesize dislocation-free Si–Ge heterostructure nanowires, and owing to the dominant 1=R dependence, the critical misfit becomes huge for even smaller diameters. Since the calculation presented presumably oversimplifies the complexity of the real situation, the outcomes of the more elaborate calculations of Ertekin et al. [17] and Glas [18] (considering both edge and 60 dislocations) are also shown as reference. In particular, the data for 60 dislocation seem to be in reasonable agreement with experimental observations [18], so one may conclude that the critical misfit as given by Eq. (5.5) rather underestimates the true value.

5.3 Dislocations in Core–Shell Heterostructure Nanowires

Synthesizing dislocation-free core–shell nanowires in which the shell material has a large misfit with respect to the core is an interesting task because one can in principle achieve very high strains in the material, in particular in the core. Such high strains could induce changes in the band structure, which is appealing from an electronics device point of view as it may increase the charge carrier mobility [19]. In the case of a

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114

core–shell structure, such a mobility increase could be further enhanced by carefully adjusting the core–shell band alignment and the doping profile [8], which together with the possibility of realizing a surround-gate architecture [20] could be an interesting approach to high performance devices. Yet, there are two main obstacles that need to be overcome, and these are the formation of dislocations and the roughening of the shell. The point at issue is whether, and for which parameters, the onset of these two strain relaxation mechanisms can be avoided or suppressed for such a strained core–shell nanostructure. In this section, the question of dislocation formation will be discussed following the work of Liang et al. [21]. Let us consider a core–shell heterostructure nanowire as shown in Figure 5.1b, having a core radius R1 , a shell radius R2 , and a misfit m at the core–shell interface. Liang et al. [21] investigated the effect of two different types of dislocations: edge dislocations and dislocation loops. They found that dislocation loops are less critical than edge dislocations; we concentrate here on the latter case. According to their calculation, the formation of an edge dislocation reduces the strain energy stored in the core–shell system by an amount DEn ¼

bGmð1 þ nÞR1 ðR22 R21 Þ ð1nÞR22

ð5:6Þ

with G ¼ E=2ð1 þ nÞ being the shear modulus and n the Poisson ratio. Both constants are assumed to be the same for the core and shell materials, respectively. Furthermore, Liang et al. [21] approximate the energy of the edge dislocation as  2 2   2 ! b2 G R R R1 3 ð5:7Þ Endis ¼  ln 2 1 þ R2 rc R2 4pð1nÞ 2 where b again denotes Burger’s vector of the dislocation and rc is the inner cutoff of the dislocation strain field. Equations (5.6) and (5.7) then directly lead to the critical misfit that the core–shell system can sustain without forming edge dislocations. mcrit

 2 2   2 ! bR22 R R R1 3 ¼  ln 2 1 þ R2 rc R2 2 4pð1 þ nÞR1 ðR22 R21 Þ

ð5:8Þ

Note that similar to (5.5), Eq. (5.8) does not depend on the shear modulus of the materials involved. For a Burger’s vector of b ¼ 0:3 nm and an inner cutoff radius rc ¼ b=4, the critical misfit mcrit according to Eq. (5.8) is shown in Figure 5.3a as a function of the shell thickness for various core radii R1 . As long as the misfit m is smaller than the value given for a specific shell thickness and core radius, the system should remain dislocation free. Considering the curve for R1 ¼ 5 nm, one can see that mcrit decreases with increasing shell thickness until it reaches a minimum value (about 0.016) at a shell thickness of about 10 nm. Thus, if shell thicknesses of less than 10 nm are aimed at, a misfit larger than 0.016 would be tolerated by the system without forming edge dislocations. If, however, the intended shell thickness is larger than 10 nm, it is necessary that the misfit be chosen below the minimum value. At a misfit smaller than the value at the minimum, the system will

5.4 Roughening of Core–Shell Heterostructure Nanowires R1 = 3 nm

CRITICAL MISFIT mcrit

0.035 0.030 0.025

R1 = 5 nm 0.020 0.015 R1 = 10 nm 0.010 R1 = 20 nm 0.005

R1 = 100 nm

(b) 0.030

MIMIMAL CRITICAL MISFIT mcrit

(a) 0.040

j115

0.025

0.020

0.015

0.010

0.005

0.000 0.000 1

10

100

Figure 5.3 (a) Critical misfit mcrit that a core–shell nanowire can sustain without forming edge dislocations given here as a function of the shell thickness for various core

4

10

40

CORE DIAMETER [nm]

SHELL THICKNESS [nm]

radii R1 . The Burger vector b is taken to be 0.3 nm and rc ¼ b=4. (b) Critical misfit at the minimum of mcrit (as shown in (a)) as a function of the core diameter.

not tend to form dislocations, irrespective of the shell thickness. Therefore, it is instructive to take a look at how this minimum of mcrit behaves as a function of the core diameter 2R1. This is shown in Figure 5.3b. As one can see in this graph, the minimum of mcrit is strongly diameter dependent and increases to a value of about 0.03 at a core diameter of 4 nm. This outcome is somehow similar to that presented in the previous section, in the sense that reducing the diameter shifts the misfit limit for dislocation formation to greater values. Thus, reducing the diameter is advisable when dislocations in core–shell nanowires are to be prevented.

5.4 Roughening of Core–Shell Heterostructure Nanowires

The second major obstacle that complicates the synthesis of misfit-strained core–shell nanowires is the potential roughening of the shell. During the growth of the shell, the system tends to partially relax strain by developing a modulation of the surface. This may in the end lead to the creation of islands or notches on the surface, as schematically indicated in Figure 5.1c. As already mentioned, this relaxation mechanism can be expected to be particularly critical at large misfit strains [13]. The creation of islands or notches is caused by the so-called Asaro–Tiller–Grinfeld instability [10, 11], a strain-induced morphological instability of the surface. The driving force for this instability is that an undulation of the surface leads to a redistribution of the strain energy, such that it reduces the overall strain energy of the system. On the other hand, this surface waviness comes with an increase in the overall surface area. Nevertheless, for some types of modulations, the energetic costs of increasing the surface area are smaller than the gain of reducing the elastic energy,

100

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116

so the growth of a modulated surface is energetically favored. For the growth of strained films on planar substrates, this instability has been the subject of extensive investigations [22–25]. However, these results may not be directly applicable to misfit-strained core–shell nanowires, as a bulk substrate is much less susceptible to a strained layer than a nanowire. For a misfit-strained core–shell nanowire, one can expect that a large portion of the elastic energy is stored in the core, whereas in case of a bulk substrate, it is the strained film that basically contains the whole elastic energy. In this section, the misfit-driven morphological instability of core–shell nanowires is analyzed by performing a linear stability analysis, closely following the work of Schmidt et al. [26]. Instead of assuming unidirectional stress along the centerline of the wire [27, 28], the misfit at the interface will be properly accounted. Furthermore, nonradially symmetric modulations will also be taken into account, which is necessary as the presence of islands usually breaks axial symmetry [29]. As pointed out previously, the effects of surface stress on the instability are consistently incorporated into the model. The system under consideration is a cylindrical, completely dislocation-free, core–shell nanowire. The shell material has a misfit m with respect to the core that causes both core and shell to be elastically strained. For simplicity, the shear modulus G, the Poisson ratio n, the surface stress t, and the surface free energy c are taken to be isotropic quantities. The linear stability analysis follows the works of Mullins [30] and Spencer et al. [25]. The idea behind it is that a core–shell nanowire in reality is not perfectly cylindrical; instead, it will show local deviations of cylindrical geometry. These local deviations can be also described by a broad distribution of sinusoidal surface modulations characterized by their wavenumber q in axial direction (the ^z-direction) and the mode number n in circumferential direction. Thus, if the outer radius of the unperturbed core–shell nanowire is R2 and d is the amplitude of the perturbation, then the actual surface radius Rs can be expressed as Rs ¼ R2 þ d cosðqzÞcosðnwÞ

ð5:9Þ

These modulations or perturbations of the shell thickness change the strain energy distribution that in turn leads to a variation in the chemical potential at the nanowire surface. The inhomogeneity of the chemical potential can drive surface diffusion, leading to an increase or decrease of the initial perturbation amplitude. In this way, a positive or negative feedback is created; the question arises whether there are modes for which a positive feedback exists (i.e., perturbations that exponentially increase in amplitude), what characteristics these modes have, and how different parameters such as the misfit m or the core radius R1 affect the stability of the system with respect to the growth of these modes. To answer this question, first one should figure out how the strain distribution changes as a consequence of a surface modulation of amplitude d. This is done perturbatively by analytically calculating the stress and strain distributions to first order in d. After some definitions, the zeroth order, that is, the stress/strain distribution of a perfectly cylindrical misfit-strained core–shell nanowire, is calculated. All corresponding zeroth-order quantities, such as the displacement vi , the

5.4 Roughening of Core–Shell Heterostructure Nanowires

 ij , are marked by an overbar. In the ij , the elastic strain eij , or the stress s strain u following section, the first-order changes to the stress/strain distribution are determined. All first-order contributions are marked by a tilde. Furthermore, for brevity, an index a is introduced, which is equal to 1 if a quantity refers to the nanowire core and 2 for a corresponding shell quantity. 5.4.1 Zeroth-Order Stress and Strain

To determine the zeroth stress/strain distribution, we assume that the system is in equilibrium, which means that in the absence of external body forces, the displacement vector u has to fulfill the equations of equilibrium [31] ð12nÞDu þ rðr  uÞ ¼ 0

ð5:10Þ

By solving the above equations and imposing proper boundary conditions, one can arrive at an analytical solution for the displacement vector u. Using cylindrical ^ þ uz ^z, the strain tensor uij can be derived from the coordinates, that is, u ¼ ur ^r þ uw w displacement vector u [31]: urr ¼ @r ur

ð5:11Þ

uzz ¼ @z uz

ð5:12Þ

uww ¼ r 1 @w uw þ r 1 ur

ð5:13Þ

urz ¼ ð@z ur þ @r uz Þ=2

ð5:14Þ

uwz ¼ ðr 1 @w uz þ @z uw Þ=2

ð5:15Þ

urw ¼ ð@r uw r 1 uw þ r 1 @w ur Þ=2

ð5:16Þ

ij These equations are analogously valid for the zeroth- and first-order quantities u ~ij , respectively. To account for the lattice misfit m defined as and u m¼

l2 l1 l1

ð5:17Þ

with l1 and l2 being the lattice constants of the core and the shell, respectively, it is necessary to introduce the elastic strain tensor eij. eaij ¼ uaij mdij da2

ð5:18Þ

with dij and da2 being Kronecker deltas. The idea behind introducing this additional tensor is that one can elegantly incorporate the misfit into the calculation this way. It is the elastic strain tensor eij that reflects the strain in the core–shell system. According to the above definition, compressive stresses/strains are negative

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118

and tensile stresses/strains are positive. From the elastic strain tensor, one can then obtain the stress tensor [31] saij ¼

 2Ga  ð12na Þeaij þ na eall dij 12na

ð5:19Þ

with Ga being the shear modulus and na the Poisson ratio of the core or shell. Together with Eq. (5.18), this leads to the following relation: saij ¼

 2Ga  ð12na Þuaij þ na uall dij ð1 þ na Þmdij da2 12na

ð5:20Þ

Similar relations hold for the zeroth-order quantities eaij ¼ u aij mdij da2  aij ¼ s

 2Ga  all dij ð1 þ na Þmdij da2 ð12na Þ uaij þ na u 12na

ð5:21Þ ð5:22Þ

and to the first-order contributions ~eaij ¼ u ~aij ~ aij ¼ s

ð5:23Þ

 2Ga  ~all dij ð12na Þ~ uaij þ na u 12na

ð5:24Þ

Let us come back now to the task of calculating the stress/strain distribution of an unperturbed, perfectly cylindrical, misfit-strained core–shell nanowire. Owing to radial symmetry and translational invariance with respect to translations in ^z-direction, the equations of equilibrium (5.10) reduce to ar þ @r2 u

1 1 a ar  2 u  ¼0 @r u r r r

az ¼ 0 @z2 u

ð5:25Þ ð5:26Þ

with the corresponding solutions ar ¼ aa r þ ba u az ¼ ca z u

1 r

ð5:27Þ ð5:28Þ

where aa , ba , and ca are six so far undetermined constants—or better five unknown constants as b1 has to be equal to zero to obtain solutions that are finite at r ¼ 0. The remaining unknown constants (a1 , a2 , b2 , c1 , c2 ) can be determined by imposing proper boundary conditions. As already mentioned, surface effects play a crucial role in the physics and properties of nanowires. Considering the elastic properties of nanowires, surface stress needs to be considered, in particular if nanowires of very small radius are taken into account. But one should be careful not to mix up surface stresses with surface

5.4 Roughening of Core–Shell Heterostructure Nanowires

free energies because, as first pointed out by Gibbs [32], the surface free energy c of a solid does not necessarily equal the surface stress t (see, for example, the work of Shuttleworth [33]). The difference between these two quantities is that the surface free energy c is related to the work of creating new area, for example, by splitting or dewetting, whereas the surface stress t is related to the work of increasing the surface area by elastically deforming the solid [34]. In general, this deformation work should be characterized by introducing a second-rank tensor tij, the so-called surface stress tensor. For isotropic surfaces, however, this tensor tij ¼ tdij reduces to a scalar, the surface stress t. To find a reasonable estimate for the magnitude of t turns out to be difficult as the surface stresses often exhibit a pronounced anisotropy [35–37], depend on the type of surface reconstruction [38–41], and are furthermore altered by the presence of adatoms [42, 43]. Surface stress values, for example, calculated by Meade and Vanderbilt [42] for a Si(111) surface, range from 0.7 to 2.4 N/m depending on the specific surface configuration. In view of these ambiguities, a surface stress value of t ¼ 1 N/m will be used in the course of the calculation. This seems a fair estimate considering the values given in Refs [38–43]. One of the boundary conditions to be imposed is that there are no net forces acting normal to the surface. Including the effect of surface stress, this corresponds to sij nj þ P f ni þ tkni ¼ 0

ð5:29Þ

with Pf being the pressure in the surrounding fluid, nj the outward-pointing surface normal, t the surface stress, and k the sum of the principle curvatures of the surface. In our case, we can neglect the pressure Pf of the surrounding fluid as it is typically orders of magnitude too small to affect the elasticity problem. The boundary conditions to be imposed are as follows [26]: 1) Equal displacements at the core–shell interface (coherent interface), which gives ð1Þ ð2Þ ð1Þ ð2Þ u r jR1 ¼ u r jR1 and u z jR1 ¼ u z jR 1

2) Zero net normal force at the core–shell interface. With the outward normal ð1Þ n ¼ ^r of the core–shell interface, this leads to j  rrð2Þ jR1  ð1Þ s rr jR1 ¼ s

3) Zero net force in ^z-direction. Considering the force is created by the surface stress leads to 2 2 ð2Þ  ð1Þ R21 s szz ¼ 2tR2 zz þ ðR2 R1 Þ

ð2Þ 4) Zero net normal force at the surface. With n ¼ ^r, this gives j  ð2Þ s rr jR2 ¼ 

t R2

If it is further assumed that core and shell have the same shear modulus and Poisson ratio, that is, G1 ¼ G2 ¼ G and n1 ¼ n2 ¼ n, the five unknown constants become

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120

a1 ¼

mð13nÞðR22 R21 Þ tð13nÞ  2Gð1 þ nÞR2 2ð1nÞR22

ð5:30Þ

a2 ¼

mð2R22 ð1nÞR21 ð13nÞÞ tð13nÞ  2Gð1 þ nÞR2 2ð1nÞR22

ð5:31Þ

b1 ¼ 0

ð5:32Þ

b2 ¼

mð1 þ nÞR21 2ð1nÞ

ð5:33Þ

c1 ¼

mðR22 R21 Þ tð1nÞ  Gð1 þ nÞR2 R22

ð5:34Þ

c2 ¼ c1

ð5:35Þ

r and u z , which then Using Eqs (5.27–5.28), one can derive the displacements u ij (using (5.11–5.16), the elastic strain tensor eij leads to the strain tensor u  ij (using (5.22)). (using (5.21)), and the stress tensor s 5.4.2 First-Order Contribution to Stress and Strain

Let us now examine how adding a sinusoidal perturbation changes the stress/strain ~ ij is, in distribution. To arrive at expressions for the first-order contributions ~eij or s principle, straightforward; instead of solving the equations of equilibrium (5.10) for the displacement vector, they are solved for the so-called Papkovich–Neuber poten^ þ Yz ^z being a vector potential and j tials [44, 45] Y and j, with Y ¼ Yr ^r þ Yw w being a scalar potential. From the Papkovich–Neuber potentials, one can then derive the displacement using the following relation [46]: u ¼ 4ð1nÞYrðR  Y þ jÞ

ð5:36Þ

with R ¼ r^r þ z^z. The main advantage of the Papkovich–Neuber potentials is that they reduce the equations of equilibrium (5.10) to independent Laplace’s equations DY ¼ 0

ð5:37Þ

Dj ¼ 0

ð5:38Þ

Considering radial symmetry, the corresponding solutions are easily found: ja ¼ ½da I n ðqrÞ þ ea Kn ðqrÞcosðqzÞcosðnwÞ

ð5:39Þ

yar ¼ ½fa I n þ 1 ðqrÞ þ ga I n1 ðqrÞ þ ha Kn þ 1 ðqrÞ þ ia Kn1 ðqrÞcosðqzÞcosðnwÞ

ð5:40Þ

5.4 Roughening of Core–Shell Heterostructure Nanowires

yaw ¼ ½fa I n þ 1 ðqrÞga I n1 ðqrÞ þ ha Kn þ 1 ðqrÞ ia Kn1 ðqrÞcosðqzÞsinðnwÞ yaz ¼ ½ja I n ðqrÞ þ ka Kn ðqrÞcosðqzÞcosðnwÞ

ð5:41Þ

ð5:42Þ

with I n ðqrÞ and Kn ðqrÞ being the modified Bessel functions of order n. The values of the constants da , ea , fa , ga , ha , ia , ja , and ka should be determined by imposing boundary conditions. Three of these constants can be found directly. By demanding that the solution has to be finite at r ¼ 0, the constants e1 , h1 , and i1 have to be equal to zero. ~ð2Þ ~ð1Þ Moreover, it turns out to be useful to introduce the normal vectors n i and n i of the core–shell interface and the surface, respectively. The core is assumed to be ^ þ gz ^z, ~ ð2Þ ¼ ^r þ gw w ~ ð1Þ ¼ ^r. The normal to the outer surface n cylindrical, so that n with gw ¼ d

n cosðqzÞsinðnwÞ R2

gz ¼ dqsinðqzÞcosðnwÞ

ð5:43Þ ð5:44Þ

Moreover, the curvature k of the nanowire surface is k¼

 2  1 n 1 2 þd þ q cosðqzÞcosðnwÞ R2 R22

ð5:45Þ

With this, the boundary conditions then become 1) Equality of displacements at the core–shell interface, leading to ~ð2Þ ~ð1Þ u r jR1 ¼ u r jR 1 ~ð1Þ ~ð2Þ u z jR1 ¼ u z jR 1 ~ð1Þ ~ð2Þ u w jR1 ¼ u w jR 1

2) Zero net normal force at the core–shell interface. This gives three conditions: ~ ð2Þ ~ ð1Þ s rr jR1 ¼ s rr jR1 ~ ð2Þ ~ ð1Þ s rw jR1 ¼ s rw jR1 ~ ð1Þ ~ ð2Þ s rz jR1 ¼ s rz jR1

3) Zero net normal force at the surface: ð2Þ ð2Þ ð2Þ ~j jr¼R2 ¼ ~ sij n ni tk

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This finally leads to the last three conditions   n2 1 2 ~ ð2Þ cosðqzÞcosðnwÞ s rr jR2 ¼ dt q þ R2 ~ ð2Þ s rw jR2 ¼ d

~ ð2Þ s zr jR2

  n t  ð2Þ s cosðqzÞsinðnwÞ j þ ww R2 R2 R2

  t ð2Þ  zz jR2 þ ¼ dq s sinðqzÞcosðnwÞ R2

This set of equations can be solved for the nine unknown constants, which is simplified by the fact that for G1 ¼ G2 ¼ G and v1 ¼ v2 ¼ G, the boundary conditions are fulfilled if e2 ¼ h2 ¼ i2 ¼ 0, d1 ¼ d2 , f1 ¼ f2 , and g1 ¼ g2 . Thus, in this case, it suffices to determine d2 , g2 , and h2 . Having these expressions at hand, one can then determine the Papkovich–Neuber potentials, derive the displacement, and from the displacement the strain, the elastic strain, and the stress. These first-order contributions can then be combined with the corresponding zeroth-order results to obtain the full stress and strain distributions to first order in d. 5.4.3 Linear Stability Analysis

Having calculated the stress/strain distribution to first order in d for an arbitrary surface perturbation, defined by its wavenumber q and its mode number n, one has to determine how surface diffusion affects the amplitude of the perturbation, that is, to figure out whether surface diffusion amplifies or attenuates a specific perturbation. Following the work of Spencer et al. [24], the diffusion-induced surface flux Js can be expressed as [47–49] Js ¼

Ds C rs M v kT

ð5:46Þ

with Ds being the surface diffusion constant, C the area density of lattice sites, and rs the surface gradient; kT has its usual meaning and the diffusion potential, Mv , is given by [24]   1 Mv ¼ V ck þ sij eij 2 r¼Rs

ð5:47Þ

where c is the surface energy density, k the curvature of the strained system, and V the volume per atom. The ð1=2Þs ij eij term represents the energy density of the stress/ strain field. In an attempt to model the synthesis of the shell, the deposition of atoms onto the nanowire surface is assumed to proceed at a fixed rate Q in atoms per unit

5.4 Roughening of Core–Shell Heterostructure Nanowires

area and unit time. Considering the continuity equation, one can show that to first order in d the radial component of a vector Rs to a point on the surface will change with time as [26]   Ds CV2 1 ð5:48Þ R_ s ¼ VQ þ Ds ck þ sij eij kT 2 r¼Rs Here, Ds denotes the surface Laplacian. Since all terms in parentheses in Eq. (5.48) are at least of order d, it is sufficient to use the zeroth-order approximation for the Laplacian Ds [26]. D0s ¼

1 2 @ þ @z2 R22 w

ð5:49Þ

Using the same argument, that is, that we are only interested in first-order contributions, one can evaluate the expression in parentheses at R2 instead of Rs . Equation (5.48) then becomes Ds CV2 0  ij ~eij Þr¼R2 R_ s ¼ VQ þ Ds ðck þ s kT

ð5:50Þ

Furthermore, one can show that ~eij is proportional to d cosðqzÞcosðnwÞ [26]. Inserting the explicit forms of k, performing D0s , and separating the terms proportional to the cosines from those that are not, one can find that R_ 2 ¼ VQ

ð5:51Þ

Ds CV2 c Sd d_ ¼ kT

ð5:52Þ

So the radius R2 of the shell grows at a rate that is equal to the deposition rate as expected. The time dependence of the amplitude d of a perturbation is governed by a simple differential equation indicating an exponential time dependence   Ds CV2 c dðtÞ ¼ d0 exp St ð5:53Þ kT with d0 being the amplitude of the perturbation at t ¼ 0. The stability parameter S is given by  S¼

n2 þ q2 R22

   ij ~eij 1n2 2 s q  c R22

ð5:54Þ

and it is this parameter that determines the magnitude and sign of the exponent. This exponential time dependence according to Eq. (5.53) leads to an increase of d if S > 0. That means that the surface is unstable with respect to this particular perturbation, as the amplitude of the perturbation will grow exponentially. For S < 0, the initial amplitude of perturbation decays exponentially with time and the cylindrical surface is said to be stable with respect to this particular perturbation. The upcoming task is to

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sort out those perturbations with the greatest positive value of S, as they are most critical for the roughening of the shell. 5.4.4 Results and Discussion

We have seen that the exponential behavior of d depends on the sign and magnitude of the stability parameter S. To give an impression of how the stability parameter S generally behaves, S is shown in Figure 5.4 as a function of q for the first six modes of a Ge nanowire of 10 nm radius covered by a 1 nm thick Si shell. The surface free energy is assumed to be c ¼ 1:5 J=m2 and the surface stress t ¼ 1:0 N=m [50, 51]. The first thing to notice in Figure 5.4 is that (going from left to right) the stability parameter S of n ¼ 0 is zero in the limit q ! 0, then exhibits a maximum, and finally becomes negative for large values of q. This behavior is quite analogous to what has been found by Spencer et al. [25] considering semi-infinite substrates. For this mode and wavenumbers q < 0:37 nm1 , S is positive, so the amplitude of the perturbation would grow exponentially, corresponding to a roughening of the surface. Just as an aside, it is interesting to note that the n ¼ 0 mode exhibits positive valuespofffiffiffi S even for vanishing misfit. In this case, the maximum of S is located at about 2p 2R2 , which corresponds to the wavelength of the classical, surface stress-driven Plateau–Rayleigh instability [52, 53].

0.004

STABILITY PARAMETER S

0.003

n=0 n=1 n=2 n=3 n=4 n=5

0.002 0.001 0.000 –0.001 –0.002 –0.003 –0.004 0.0

0.1

0.2

0.3

0.4

0.5

-1

WAVENUMBER q [nm ]

Figure 5.4 Stability parameter S as a function of the wavenumber q for R1 ¼ 10 nm, R2 ¼ 11 nm, G ¼ 46 GPa, n ¼ 0:26, m ¼ 0:043, t ¼ 1:0 N/m, and c ¼ 1:5 J/m2.

5.4 Roughening of Core–Shell Heterostructure Nanowires

j125

Furthermore, one can see in Figure 5.4 that all modes with n  4 do have positive values of S for certain wavenumbers q, ranging from q ¼ 0 to about q  0:3 nm1. Thus, the surface of such a core–shell nanowire would be unstable for the first four modes and wavelengths (in z-direction) longer than roughly the diameter of the nanowire. Owing to the exponential time dependence of d on S, the perturbation with the greatest S value will grow the fastest. In case of the nanowire considered in Figure 5.4, this is the n ¼ 3 mode that exhibits a maximum at about q  0:16 nm1 (marked by a circle in Figure 5.4) corresponding to a wavelength of about 4R2 . This will be called the fastest growing mode. This fastest growing mode is characterized by its wavenumber qfg, wavenumber nfg, and the stability parameter Sfg. Owing to exponential time dependence of d, one may assume that the fastest growing mode will dominate the other perturbation modes after a while, so that the fastest growing mode determines the final morphology of the shell. Therefore, in the following discussion, we will mainly concentrate on the properties of the fastest growing mode and, in particular, on the dependence of Sfg and nfg on parameters such as the core radius or the shell thickness. First, let us examine the properties of nfg , shown in Figure 5.5a, as a function of the core radius R1 for various shell thicknesses ranging from 1 to 7 nm. The misfit parameter is taken to be m ¼ 0:044, corresponding to the misfit of a Ge-core–Sishell nanowire. One can see in Figure 5.5a that the mode number nfg of the fastest growing mode increases strongly for increasing core radius and decreasing shell thickness. So one can expect that roughening, in particular for the initial phase of shell growth on rather thick nanowires, will show a quite complex behavior. At larger shell thicknesses, modes with lower mode number should become dominant. It is also interesting to note that the question which modes become dominant strongly depends on the sign of the misfit. Assuming a positive surface stress t and a negative misfit m, one can show that due to the surface stress, modes with large surface (b)

22 m=-0.044

20

0.01

Shell thickness = 1 nm

18 Shell thickness: 1 nm 2 nm 3 nm 4 nm 5 nm 6 nm 7 nm

16 14 12 10 8 6

Stability Parameter Sfg

Mode Number nfg (fastest growing mode)

(a)

4

1E-3

1E-4

2 0 -2 0

5

10

15

Core Radius R1 [nm]

20

25

Misfit m 0.045 0.040 0.035 0.030 0.025 0.020 0.015

1E-5 0

5

10

15

20

Core Radius R1 [nm]

Figure 5.5 (a) nfg as a function of the core radius R1 for m ¼ 0:044 and various shell thicknesses, t ¼ 1:0 N/m. (b) Stability parameter ðSfg Þ as a function of the core radius R1 for various misfit parameters m, t ¼ 0 N/m. G ¼ 46 GPa, n ¼ 0:26, and c ¼ 1:5 J/m2.

25

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126

curvature, that is, large wavenumber n, are favored (in the sense of greater S) [26], whereas for positive values of m, the opposite is the case. For positive misfits, surface stress reduces the magnitude of the stability parameter of high-n modes. Thus, one can expect that for a positive surface stress, thin Ge-core–Si-shell nanowires will show a more complex roughening behavior than Si-core–Ge-shell nanowires. For positive m, it is in fact often the n ¼ 0 mode that grows the fastest. The dependence of the stability parameter Sfg of the fastest growing mode on the core radius R1 is displayed in Figure 5.5b for various misfits m and a fixed shell thickness of 1 nm. It is evident that Sfg depends quite strongly on the misfit m as expected. Furthermore, what is interesting to note is that for large misfits, for example, m ¼ 0:045, the curve exhibits a minimum at a core radius of about 6 nm. This means that from a surface stability point of view, it is advantageous to choose such a core radius if smooth, nonroughened core–shell nanowires are to be synthesized. Figure 5.6a and b shows Sfg as a function of the core radius for various shell thicknesses. Again, surface stress is t ¼ 1:0 N/m. Figure 5.6a, with a misfit m ¼ þ 0:044, approximately corresponds to the case of a Si-core–Ge-shell nanowire (Figure 5.6b), with m ¼ 0:044 to a Ge-core–Si-shell nanowire. The first apparent feature of Figure 5.6 is the strong dependence on the shell thickness. In both cases, m ¼ þ 0:044 and m ¼ 0:044, an increase in the shell thickness from 1 to 7 nm leads to a decrease in Sfg by more than one order of magnitude—more or less independent of the core radius. Concerning the overall stability, one therefore has to conclude that the initial phase of growth, where the shell is thinnest, is most critical with respect to a roughening of the surface. For experimentalists, this signifies that extreme care should be taken in the initial phase of shell growth. To reduce the roughening tendency, it might be advisable to synthesize a graded junction in the initial phase of shell growth. This can be, considering a Ge-core–Si-shell nanowire, for example, be easily done be steadily reducing the Ge content in the initial phase of 0.01

(b)

m=+0.044

1E-3

1E-4

STABILITY PARAMETER Sfg

STABILITY PARAMETER Sfg

(a)

Shell thickness: 1 nm 2 nm 3 nm 4 nm 5 nm 6 nm 7 nm

0.1 m=-0.044

0.01

1E-3

1E-4

Shell thickness: 1 nm 2 nm 3 nm 4 nm 5 nm 6 nm 7 nm

1E-5 1E-5 –5

0

5

10

Core Radius R1 [nm]

15

20

25

–5

0

5

10

15

20

25

Core Radius R1 [nm]

Figure 5.6 Stability parameter ðSfg Þ for various shell thicknesses as a function of the core radius R1 using G ¼ 46 GPa, n ¼ 0:26, t ¼ 1:0 N/m, and c ¼ 1:5 J/m2; (a) m ¼ þ 0:044 (Si-core–Ge-shell) and (b) misfit m ¼ 0:044 (Ge-core–Si-shell).

References

shell growth. The most interesting feature, visible in Figure 5.6a and b, is the pronounced minimum one can find at a core radius of 5 nm. This means that one way of reducing the tendency for roughening is to use nanowires with diameter of about 10 nm as a starting material for the core–shell structure. By reducing the diameter from 50 to 10 nm, R1 , the stability parameter of a Ge-core–Si-shell nanowire (see Figure 5.6b), is reduced by more than one order of magnitude, which means a significant increase in surface stability. Within the framework of the model, it could be shown that the surface is most unstable when the shell thickness is of the order of 1 nm or less. Consequently, it is the initial phase of shell growth that would be most sensitive to a roughening of the surface. In addition, the model shows that for m > 0:03, there exists a core radius of maximum stability, which means that synthesizing cylindrical core–shell nanowires should be easiest for core diameters of about 10 nm.

5.5 Conclusion

The main conclusion of considering dislocation formation in both axial and core–shell nanowires is that the critical misfit, up to which dislocation-free nanowires can be synthesized, is strongly radius dependent and it increases with decreasing radius. This is mainly a scaling effect. By reducing the radius of the nanowires, the strain energy stored in the system changes proportionally to the volume of the system. Moreover, if the volume is small enough, at some point the energy stored will not suffice anymore to induce the formation of a dislocation. Therefore, reducing the radius helps to prevent dislocation formation. Concerning the strain-induced roughening instability of shell, one must conclude that, in contrast to dislocation formation, this relaxation mechanism does not vanish at smaller radii. Nevertheless, reducing the radius might still be beneficial to increase the stability of the system.

Acknowledgements

This work was supported by the joint Fraunhofer–Max Planck project nanoSTRESS.

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3 Redwing, J.M., Lew, K.K., Bogart, T.E.,

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6 Strained Silicon Nanodevices Manfred Reiche, Oussama Moutanabbir, Jan Hoentschel, Angelika H€ahnel, Stefan Flachowsky, Ulrich G€osele, and Manfred Horstmann

6.1 Introduction

The history of solid-state devices began with the invention of the transistor by Bardeen, Brattain, and Shockley in 1947 [1]. Only a few years later, the first integrated circuits were realized by Texas Instruments (1958) and Fairchild Camera in 1961, assigning the first stages of the development of the microelectronics industry. Since its inception, the industry has experienced five decades of unprecedented explosive growth driven by two factors: Noyce and Kilby inventing the planar integrated circuit [2, 3] and the advantageous characteristics that result from scaling (shrinking) solid-state devices. Scaling devices have the peculiar property of improving cost, performance, and power. As a result, the microelectronics industry has driven transistor feature size scaling from 10 mm to about 30 nm during the past 40 years. However, starting with 90 nm technologies, the performance enhancements of complementary metal–oxide–semiconductor (CMOS) devices started to diminish through standard device scaling such as shrinking the gate length and thinning the gate oxide due to several physical limitations in miniaturization of metal–oxide–semiconductor field-effect transistors (MOSFETs). For example, thinning the gate oxide requires a reduction in the supply voltage and an increase in the gate tunneling current occurs. Furthermore, raising the dopant concentration in the substrate is substantial to suppress short-channel effects that decrease both the carrier mobility and the drive current. Thus, new channel structures and materials, which mitigate the stringent constraints regarding the device design, have recently stirred a strong interest. These socalled technology boosters [4] include strained silicon channels, ultrathin silicon on insulator (SOI), metal gate electrodes, multigate structures, ballistic transport channels, and others. Among them, strained silicon channels have been recognized as a technology applicable to near term technology nodes [5–7]. The mobility enhancement obtained by applying appropriate strain provides higher carrier velocity in MOS channels and drive current, respectively, at the same supply voltage and gate

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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oxide thickness. It implies that thicker gate oxides and lower supply voltage need to be used for a fixed drive current, leading to the mitigation of the trade-off relationship among drive current, power consumption, and short-channel effects.

6.2 Impact of Strain on the Electronic Properties of Silicon

The change of the electrical resistance of a material due to applied stress is generally known as piezoresistive effect. In electrodynamics, this is expressed by the material equation ^ ^J ¼ s ^E

ð6:1Þ

^ is the tensor of the electric field. where ^J is the tensor of the current density and E ^ on Applying mechanical stress results in a dependence of the conductivity tensor s the tensor of mechanical stress. The components s ij of the conductivity tensor can be written as [8] ð e2 @f0 ðeÞ sij ¼  3 tðeÞWi Wj d3~ k; i; j 2 fx; y; zg ð6:2Þ 4p @e 1 ^ Þ=@ki is the ith component of the Here e is the electron charge, Wi ¼ h @eð~ k; X ^ is the group velocity of charge carriers, e is the carrier energy, ~ k is the wave vector, X elastic stress tensor, f0 is the equilibrium distribution function, and t is the relaxation k), Eq. (6.2) describes the conductivity of the unstrained silicon. If time. At e ¼ eo(~ strain is applied, that is, the energy of the free carriers receives some additional De, ^ which is a function of the wave vector ~ k and the stress tensor X,

^Þ e ¼ eð~ kÞ þ Deð~ k; X

ð6:3Þ

then the linear addition of the components of the conductivity tensor is [8, 9] 2 3 ð e2 @f ðeÞ 0 k si;j ¼  3 D4tðeÞ Wi Wj 5d3~ 4p @e 8 # # 9 " " ð ð = e2 < @ @f0 ðeÞ @j ðeÞ @De @De 0 De ¼ 3 tðeÞ Wi Wj Wi Wj d3~ k þ 1h tðeÞ k Wj þ Wi d3~ ; 4p : @e @e @e @ki @kj ð6:4Þ

Using the first-order piezoresistance coefficients determined by the relation [10] pijkl ¼ 

^ Þ  @sij ðX 1  ðX^ ¼0Þ ; ^ sij ðX ¼ 0Þ @Xkl

i; j; k; l 2 fx; y; zg

the conductivity tensor for silicon could be written as [8] X ^ Þ ¼ s0 ðdij  sij ðX p X Þ kl ijkl kl

ð6:5Þ

ð6:6Þ

6.2 Impact of Strain on the Electronic Properties of Silicon

with dij as the Kronecker delta. Caused by the cubic symmetry of silicon (space group m3m), the fundamental coefficients of the piezoresistance tensor are reduced to 3, that is, p11  pxxxx ;

p12 ¼ pxxyy ¼ pyyxx ;

p44  2pxyxy

where p11 are the longitudinal (current and field are in the direction of the strain), p12 the transverse (current and field are perpendicular to the strain direction), and p44 the shear piezoresistance coefficients. The piezoresistance coefficients of silicon and germanium were measured for the first time by Smith [11] in 1954. The data are valid only for bulk material and are widely applied to configure micromechanical elements such as pressure sensors [12]. For devices such as MOSFETs, however, surface confinement effects must be taken into account. Coleman et al. [13] measured the p-coefficients in p-type inversion layers for (100), (110), and (111) surface orientations. More recently, Chu et al. [14] published data for p-coefficients for n- and p-type silicon with different surface orientation and channel directions. It was shown that the p-coefficients are significantly different from the bulk values if surface confinement effects are regarded. The coefficients strongly depend on doping density, electric field, and channel direction even for nMOSFETs on (100) surfaces. The dependence of the p-coefficients on the channel length of MOSFETs prepared on SOI material was studied by Chang and Lin [15] for h110i-oriented, submicrometer n- and p-type channels on (100) substrates. In the presence of strain in a cubic semiconductor, the reduced degree of symmetry gives rise to significant changes in the band structure. Based on group theory and k–p perturbation calculations, several theoretical investigations have been carried out on the valence and conduction bands in silicon. Most of these investigations are based on the deformation potential theory. In this theory, the effect of the deformation is represented by the deformation potential Hamiltonian HDP, which is linear in the components of the strain tensor. To calculate the change of the band structure, HDP is usually treated as a small perturbation. A more generalized formulation for homogeneous strain is the Pikus–Bir Hamiltonian [9]. The deformation potential theory was originally developed by Bardeen and Shockley in 1950 [16] to calculate the components of the relaxation time tensor in terms of the effective mass, elastic constants, and a set of deformation potential constants. The deformation potential theory was generalized by Herring and Vogt [17] to model carrier transport in strained multivalley semiconductors and summarized a set of independent deformation potentials to characterize the conduction band valleys. The deformation potentials describe the shifts of the extremum energy of a particular valley depending on the magnitude of stress and its direction with respect to the ~ k vector of the valley. The original model of Herring and Vogt was refined over the last decades. Balslev [18] determined the dilatation (Jd ) and shear deformation potential constants Ju . The author also measured the deformation potentials a, b, and d defined by Ref. [9], characterizing the effect of strain on the valence band edge. While Balslev described the deformation potentials in the case of uniaxial strain, Hinckley and Singh [19]

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assigned deformation potentials for biaxially strained layers. Furthermore, Nakayama [20] modified the theory of Herring and Vogt in combination with the results of Pikus and Bir (compare Ref. [9]) for stronger deformation, where the periodicity of the deformed crystal is different from that of the initial, undeformed state and the ordinary perturbation theory cannot be applied. Deformation potentials are applied in current models of the effects of uniaxial or biaxial strain on the band structure of silicon and the consequences on the electron and hole mobility [21–23]. The effect of strain on the band structure of silicon and on the carrier mobility takes place mainly due to (a) the reduction of the carrier conductivity effective mass, and (b) the reduction in the intervalley phonon scattering rates. The conduction band of unstrained bulk silicon has six equivalent valleys along the h100i direction of the Brilloun zone, and the constant energy surface is ellipsoidal with the transverse effective mass mt ¼ 0.19 m0 and the longitudinal effective mass ml ¼ 0.916 m0 [24]. In the inversion layer on a (100) surface, these six valleys are split into twofold degenerate valleys with ml perpendicular to the Si/SiO2 interface and the fourfold degenerate in-plane valleys with mt. If strain is applied, the energy of the conduction band minima of the fourfold valleys on the in-plane h100i axes rises with respect to the energy of the twofold valleys on the h100i axes perpendicular to the plane. As a consequence, the electrons prefer to populate the lower valleys, which are energetically favored. This result in an increased electron mobility via a reduced inplane and an increased out-of-plane electron conductivity mass. Numerical simulations of the increased electron mobility were done using the Kubo–Greenwood formula for mobility, which are in good agreement with the experimentally measured data [25]. For a given strain, quantifying the effective mass reduction and comparing it to the enhanced mobility reveals that mass reduction alone explains only part of the mobility enhancement. Vogelsang and Hofmann [26] first suggest that the suppression of intervalley scattering rate is also important. Electron scattering is reduced due to the conduction valleys splitting into two sets of energy levels, which lowers the rate of intervalley phonon scattering. The effect of phonon scattering on the relaxation time t was discussed by Fischetti and Laux [21]. Both types of scattering (acoustic and optical phonons) as well as their dependence on the density of states (DOS) in each valley were considered. The effect of the different scattering mechanisms depends on the strength of the electric field. Optical phonon scattering dominates at higher electric fields, while Coulomb, surface roughness, and acoustic phonon scattering are significant for low electric fields. Many types of strain increase the electron mobility such as in-plane biaxial and uniaxial tensile and out-of-plane uniaxial compressive strain. Uchida et al. [27] first described the effect of biaxial and uniaxial strains parallel to h100i and h110i, respectively, on the electron mobility and their behavior on the electrical performance of MOSFETs. It was shown that lower uniaxial strain parallel to h110i results in a larger enhancement of the electron mobility than the biaxial strain. For holes, the valence band structure of silicon is more complex than the conduction band. Hasegawa [28] as well as Hensel and Feher [29] used band

6.3 Methods to Generate Strain in Silicon Devices

structure calculations to systematically study the valence band effective masses and deformation potentials in strained silicon. They revealed that the hole mobility is mainly affected by band splitting and warping, mass change, and consequently by changes of the DOS, which alters band occupation and phonon scattering. A theoretical description of the hole mobility enhancement by strain was recently published by Fischetti et al. [30] using kp perturbation calculations. The band warping is responsible for the fact that different types of strain (biaxial tensile and uniaxial compressive) behave differently. For unstrained silicon at room temperature, holes occupy the top two bands: heavy and light hole bands [30]. Applying strain, the hole effective mass becomes highly anisotropic due to band warping and the energy levels become mixtures of the pure heavy, light, and split-off bands. Thus, the light and heavy hole bands lose their meaning, and holes increasingly occupy the top band at higher strain due to the energy splitting. Important to achieving high hole mobility is a low in-plane conductivity mass for the top band. In addition to a low in-plane mass, a high density of states in the top band and a sufficient band splitting to populate the top band are also required. Uniaxial compressive strain on both (100) and (110) surfaces, for instance, create a high density of states in the plane of the MOSFETs.

6.3 Methods to Generate Strain in Silicon Devices 6.3.1 Substrates for Nanoscale CMOS Technologies

There are generally two different methods to introduce strain in the channel region: biaxial strain and uniaxial strain. Biaxial strain is also referred to as global strain and is introduced by epitaxial growth of Si and SiGe layers where the lattice mismatch between Si and SiGe causes a tensile strain. The increase of the electron mobility in channels of MOSFETs prepared on such substrates was extensively studied [31–33]. Furthermore, a variety of new substrates such as the strained Si/relaxed SiGe layers are formed on buried oxides (SiGe on insulator (SGOI)) and the strained silicon on insulator (SSOI), where strained silicon layers are directly bonded to buried oxides, have been demonstrated. The latter combines the advantages of SOI and biaxially strained silicon layers in a single substrate. In order to combine the different mobilities of electrons and holes, hybride-oriented substrates (HOTs) were introduced by IBM [34]. Uniaxial strain is generated by local structural elements near the channel region. Since these process modules that cause uniaxial strain are part of high-performance CMOS processes, uniaxial strain is also referred to as process-induced strain (PIS). The application of PIS does not require specific substrates. It can be applied on bulk wafers or SOI substrates. Owing to the relative ease of integrating process-induced strain modules in conventional CMOS processes, strain-enhanced scaling has relied on the development of new advanced methods of PIS. The application of local strain

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elements, however, is limited by further scaling making some of them ineffective or unfeasible. 6.3.2 Local Strain

Electron and hole mobilities respond to mechanical stresses in different ways. For MOSFETs with the [110] channel orientation on (001)-oriented silicon substrates, tensile strain along the [110] direction improves electron mobility but degrades hole mobility. Therefore, to improve both the electron mobility in n-channel MOSFETs and the hole mobility in p-channel MOSFETs, different approaches for strain inducement in the p- and n-channel transistors are needed. Incorporating local strain to enhance MOSFET performance was first introduced by Ito et al. [35] and Shimizu et al. [36], who used etch-stop nitride, and by Gannavaram et al. [37], who used SiGe source/drain regions. Advanced CMOS processes include different process-induced stressors. These are mainly stressed contact layers, embedded source/drain stressors, stress memorization techniques, and stressed contact and metal gates (Figure 6.1). Contact layers are typically stressed nitride layers deposited after salicidation on the top of devices. Tensile contact layers are deposited on nMOSFETand compressive contact layers are deposited on pMOSFET. The strain in the channel region depends on the intrinsic stress of the layer, thickness of the layer, and device dimensions. Using tensile and compressive contact layers (dual stress liner (DSL)), a significant hole and electron mobility enhancement was achieved [38]. Furthermore, important parameters of SOI CMOS devices, such as effective drive current enhancement, were proved in different 45 nm technologies.

Figure 6.1 Schematic representation of local stressors (process-induced stressors) in SOI CMOS (a). Tensile strain in nMOSFET is obtained by tensile contact layers and various stress memorization techniques. Compressive

strain in pMOSFET is induced by compressive contact layers and embedded SiGe. XTEM of SSOI nMOSFET without process-induced stressors (b), and SSOI pMOSFET with embedded SiGe and compressive overlayer (c).

6.3 Methods to Generate Strain in Silicon Devices

Figure 6.1 (Continued)

Stress memorization techniques (SMT) typically involve a preamorphization, a nitride capping layer, and an additional annealing process. SMT increases the nMOSFET drive current and degrade the pMOSFET device performance. Embedded SiGe layers (eSiGe) deposited by epitaxial growth of SiGe alloys in cavities etched into the source/drain areas are used in pMOSFET transistors. Due to the larger lattice constant of SiGe compared to silicon, a compressive strain is induced in the channel of the pMOSFETs. This technique can be optimized and improved by combining SiGe and SiC layers for CMOS devices [39].

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The impact of the increase of the carrier mobility in the channel of MOSFETs is an increase of the drain current. The correlation between the drain current (ID) and the effective carrier mobility (meff) is given in the linear region of the drain current–source/drain voltage (ID–VDS) characteristics of a MOSFET by the relation ID ¼

W m Cox ðVGS VT ÞVDS L eff

ð6:7Þ

where W and L are the channel width and length, respectively, VGS is the gate/source voltage, VT is the threshold voltage, and Cox is the capacitance of the gate oxide. Measurements of the strain-induced mobility change Dm and the corresponding change in linear drain current, however, show clear differences for n- and pMOSFETs [40]. A change of 50% in mobility results only in an increase of the drain current by 20%. In contrast, for pMOSFETs, a 40% increase in the drain current is obtained for the same change in mobility. The differences may be caused by the reduced source–drain resistance by the embedded SiGe channel in pMOSFETs originated from the lower SiGe valence band offset. Different scattering mechanisms reduce the effect of strain on the mobility enhancement even in short-channel devices [41]. Here, the carrier mobility is expressed by an effective mobility meff that is related to the carrier velocity v in the presence of low lateral fields Elat by the equation v ¼ Elat  meff

ð6:8Þ

In short-channel devices, carriers do not reach vsat instantaneously; instead, in saturation regime, the carrier transport is more and more governed by ballistic transport [42]. Nevertheless, meff at low lateral fields and v are correlated in scaled MOSFETs, although there is no general agreement about this correlation [43]. According to Lundstrom [44], the change in the saturation drive current DID,sat is DID;sat ¼ Dmð1BÞ

ð6:9Þ

where B is the ballistic efficiency and the saturation drive current ID,sat in the ballistic regime is given by [45] Id;sat ¼ WvT Cox ðVGS VT Þ

where Cox(VGS  VT) is the charge density and sffiffiffiffiffiffiffiffiffiffiffiffi 2kB TL vT ¼ pmt*

ð6:10Þ

ð6:11Þ

is the unidirectional thermal velocity [46] that depends on the strain-induced effective mass mt* . In Eq. (6.1), kB is the Boltzmann constant and TL is the temperature. Figure 6.2 shows the dependence of DID,sat on the change in mobility Dm. It can be seen that all data measured for n- and pMOSFETs are on a single curve with a similar slope for all strain techniques. The extracted value for B is about 0.61 for nMOSFETs and about 0.63 for pMOSFETs. From Eq. (6.9), it follows that an improvement of 50% in mobility causes an enhancement of the saturation current ID,sat by about 20% for n- and pMOSFETs.

6.3 Methods to Generate Strain in Silicon Devices

50

TOL SMT sSOI

∆ID,sat (%)

40

COL eSiGe

30 y = 0.31x + 4.7

20 10 y = 0.30x + 4.5

0

0

20

40

60

80

100

∆µ (%) Figure 6.2 Correlation between carrier mobility change Dm and the change in saturation drain current DID,sat for different local strain techniques for n- and pMOSFETs. TOL:

tensile overlayer; COL: compressive overlayer; SMT: stress memorization technique; eSiGe: embedded SiGe channel; SSOI: strained silicon on insulator.

6.3.3 Global Strain

Global strain on wafer level is induced by the epitaxial growth of a Si layer on a Si1xGex buffer. Because the lattice constant of Si1xGex (0  x  1) alloys varies between 0.5431 nm for silicon (x ¼ 0) and 0.5657 nm for germanium (x ¼ 1), tensile strain is induced in the silicon layer epitaxially grown on top of the SiGe. The strain is generally biaxial. Furthermore, uniaxially strained layers can also be obtained by mechanical straining. 6.3.3.1 Biaxially Strained Layers Various heterostructure substrates have been applied to realize biaxial strain and high-mobility channel materials [47]. Figure 6.3 illustrates various heterostructure substrates that have been employed to biaxial strain and high-mobility channel materials. Epitaxially grown Si1xGex layers on Si bulk wafers are generally applied acting as substrate for a strained silicon layer grown on top (bulk materials). In order to reduce the defect density in the strained silicon, a relaxed Si1xGex buffer is required grown on a graded Si1xGex layer (Figure 6.3a). Because the Ge concentration x increases continuously by about 10% per micrometer, the thickness of the graded buffer is several micrometers. An alternative is the relaxation of a thin pseudomorphic SiGe layer ( x (Figure 6.3b) and heterostructures on bulk using a second strained silicon layer

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Figure 6.3 Schematic illustration of various heterostructure substrates produced by epitaxial growth on bulk substrates (bulk materials) and by transfer of the strained layers to oxidized substrates (strained silicon on insulator (SSOI), strained Si/SiGe on insulator (SGOI)) [46, 48].

(Figure 6.3c) [49]. Layer stacks of the different types have been applied as virtual substrates for the preparation of SSOI and SGOI wafers. The realization of SSOI wafers from bulk materials is a complex process combining wafer bonding, layer transfer, and etch-back methods. The SSOI technologies provide a pathway to implementing mobility enhancement in partially or fully depleted devices, in ultrathin-body (UTB) MOSFETs, or nonplanar (double-gate) MOSFETs. Mobility enhancement in SSOI was reported in Refs [50, 51] for the different SSOI configurations. Furthermore, long-channel devices (Lg  1 mm) show clearly improvements of the device characteristics. For instance, drive current (ID,sat) improvements of 80% at the same gate-to-drain leakage (Ioff) have been measured. Improvements on the same order of magnitude were obtained only for short-channel devices if modified CMOS processes were applied in order to avoid interaction with process-induced stressors reducing the effect of the biaxial strain [18]. An electron mobility enhancement of about 40% and significant improvements of the Ion–Ioff characteristics were obtained for nMOSFETs prepared in a 45 nm technology node on standard SSOI material (prepared on Si0.78Ge0.22 virtual substrates), which increases to about 60% if highly strained silicon layers are used (prepared on Si0.69Ge0.29 virtual substrates) [52]. Devices were prepared for these investigations

6.3 Methods to Generate Strain in Silicon Devices

(a)

(b)

1e-7

IOFF [nA/μm]

IOFF [μA/μm]

1e-5

65%

1e-8

40

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SOI SSOI

50

60

70

80

IDSAT [μA/μm]

Figure 6.4 Ioff versus ID,sat curves for n- and pMOSFETs prepared on SSOI and SOI substrates. The biaxial strain (SSOI) was combined with local stressors for pMOSFETs (SiGe: embedded SiGe channel; COL:

-11%

1e-6

57%

1e-7 SSOI+SiGE+COL SSOI+COL SOI+SiGE+COL SOI+COL

1e-8 90

400

500

600

700

800

900

1000

IDSAT [μA/μm]

compressive layer). An increase of ID,sat of 65% was obtained for long-channel (W ¼ L ¼ 4.5 mm) nMOSFETs (a), while the effect of different overlayers on short-channel (L ¼ 45 nm) pMOSFETS is shown in (b).

using a fully depleted SOI (FDSOI) technology with a TiN/HfO2 gate stack. The thickness of the Si device layer was 9 nm. It was demonstrated that the mobility enhancement also depends on the channel orientation. A mobility enhancement of 135% was found for nMOSFETs on highly strained silicon having h110i-oriented channels. In addition, the hole mobility enhances in the same time for pMOSFETs with channel orientations parallel to h100i. The analyses have also shown that a sufficient amount of strain is preserved in MOSFETs fabricated in technology nodes below 45 nm. A mobility enhancement for nMOSFETs up to 40% is attainable in a 15 nm technology [51]. This means that applications of SSOI wafers require modifications of existing CMOS processes. The combination of biaxially strained SSOI and optimized uniaxial stressors (dual-stress nitride capping layer and embedded SiGe) was demonstrated resulting in ID,sat improvements of 27% and 36% for n-channel MOSFETs and p-channel MOSFETs, respectively, in sub-40 nm devices [53]. In addition, the gate leakage current was also reduced by 30%. Figure 6.4 shows that a further adjustment of the CMOS process results in additional improvements. An increase of ID,sat of 65% was obtained for nMOSFETS. For pMOSFETs, the strain of the SSOI substrate is tensile in both the longitudinal (with current) and perpendicular directions. The longitudinal component degrades hole mobility, while the perpendicular strain enhances hole mobility. The net result is a decrease in hole mobility. This is shown in Figure 6.4b, where the longitudinal tensile strain counteracts the compressive overlayer, causing a 11% degradation in pMOSFET drive current. However, the addition of a cavity etch for embedded SiGe effectively relaxes this longitudinal tensile strain and allows pMOSFET performance to be in line with standard SOI. All investigations suggest that the combination of biaxially strained SSOI and uniaxial strain by process-induced stressors is the optimum way for future requirements [50, 52].

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6.3.3.2 Uniaxially Strained Layers A concept to realize uniaxial, tensile strain on wafer level was first published by Belford [54]. An approach of uniaxial compressive strain on wafer level based on wafer direct bonding of prestressed wafers was also published [55]. Two wafers were bent over a cylinder, thereby creating a curved or bowed wafer with a strained state induced. The bending direction was parallel to [110]. The curved wafers are brought into contact via direct wafer bonding and the covalent bonds across the bonded interface form upon annealing in the bent state. By combining the process with hydrogen-induced layer splitting, thin strained layers were transferred. The process can generally be used to realize strained layers of either tensile or compressive strain. The strain introduced by this technique is significantly lower as for biaxially strained layers. Depending on the radius of curvature, strain values between about 0.08% and 0.04% were obtained for a radius of curvature ranging from 0.5 to 1 m. The concept of uniaxial strain on wafer level was originally applied to show the different behavior of uniaxial and biaxial strains on MOSFET performance [56].

6.4 Strain Engineering for 22 nm CMOS Technologies and Below

Numerous experiments over the past years proved the applicability of biaxially strained silicon, uniaxially strained silicon, and local strain elements, or combination of these in state-of-the-art device processes. A strained Si on SGOI substrate, for instance, was used to realize n- and p-MOSFETs with channel lengths up to 25 nm [57]. Mobility enhancements of 50% for electrons (with 15% Ge) and 15–20% for holes (with 20–25% Ge) have been demonstrated. Other investigations refer to a mobility enhancement of 46% for electrons and 60–80% for holes using SGOI with the same Ge concentration. An alternative process to attain SiGe layers with higher Ge content, that is, higher strain in the upper strained silicon layer, is the Ge condensation method [58]. The process consists of epitaxial growth of a strained SiGe layer with a low Ge fraction on a SOI substrate and successive high-temperature oxidation. SiGe layers having Ge fractions of more than 0.5 and large strain values over 1% were realized. A hole mobility enhancement by a factor of 10 was measured. Recently, however, variations of the threshold voltage are obtained on SGOIMOSFETs fabricated by the Ge condensation process. It was verified that the variation of the threshold voltage can be attributed to the variation of strain in the Si channel layers. This variation was found to be correlated with the variation of the lattice spacing in the SGOI layer, which is caused by the nonuniform lattice relaxation in the SGOI layers during the condensation process [59]. Therefore, strained silicon on insulator without a buried SiGe layer appears to be more favorable. This is also true with respect to the process integration. Associated with the presence of the SiGe layer below the channel, issues such as Ge segregation at the Si/SiO2 interface, enhanced As or P diffusion, limited thermal budget, and rapid dopant diffusion along misfit dislocations would be largely eliminated [49]. Most of the measurements reported until now show enhanced mobility data and

6.4 Strain Engineering for 22 nm CMOS Technologies and Below

Figure 6.5 TEM cross-section image of a SSOI wafer. The thickness of the strained silicon (sSi) layer is 6.5 nm.

drive current improvements for both long- and short-channel devices having gate lengths of 60 nm and below. The sustainment of enhanced mobility and improved Ion/Ioff characteristics for short-channel devices is mainly achieved by modifications of the CMOS process [50]. Process integration of SSOI materials in CMOS technologies is, therefore, an important issue [60]. The advantage of SSOIis the scalability to thinner device and insulator (BOX) layers, respectively. This allows the combination of strain with benefits of ultrathin-body MOSFETs [61]. Figure 6.5 shows the transmission electron microscope (TEM) image (cross-section image) of a SSOI wafer. The thickness of the strained silicon layer transferred by wafer bonding is only 6.5 nm. UTB SOI is an attractive option for device scaling, because it can effectively reduce the short-channel effect and eliminate most of the leakage paths. The ultrathin SOI thickness requirement for short-channel effect control in single-gate FETs can be relaxed by using a more complex double-gate FET that offers improved electrostatic gate control of the body. Numerical simulations indicate that scalability of double-gate FETs improves by a factor of 2.5–3 [62]. The reduction of the thickness of the silicon layer required for UTB MOSFETs also contributes to the strain conservation in the silicon layer. Figure 6.6a shows the dependence of the in-plane stress on the size (channel length) of devices for different thicknesses of the strained silicon layer (height in Figure 6.6a). It can clearly be seen that strain (or stress) relaxation is a function of the silicon layer thickness. Decreasing the layer thickness conserves the strain even for small structure sizes. This means that with the strained silicon layer at thickness below 10 nm (required for UTB SSOI MOSFETs) and gate length in the subnanometer range, the whole tensile strain is preserved. Varying the ratio of the structure length L to the thickness of the

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layer (height H) causes different stress distributions in the structures (Figure 6.6b). While at H/L ¼ 1, a sufficient (tensile) stress is located only at the Si/SiO2 interface, the stress is preserved in the whole layer at H/L ¼ 0.13. Furthermore, a compressive stress is obtained close to the surface at H/L ¼ 0.45. This means that under specific conditions, devices with different stress behavior (tensile or compressive) within one device may simultaneously exist by applying biaxially strained SSOI wafers. The results of the simulation were experimentally verified by Raman measurements [63]. In addition, Raman and TEM analyses also proved the transformation of the biaxial strain into uniaxial strain by increasing the ratio of the width to the length of the structures. Figure 6.7 shows the results of finite element (FEM) simulations of the strain distribution in channels with different width/length (W/L) ratios. The thickness of the strained silicon layer was 20 nm and W/L was varied between 1 and 20 at a constant width of 50 nm. The initial biaxial strain in the unpatterned layer was e ffi 0.6%. At a square-like channel geometry (W/L ¼ 1), the strain is more or less relaxed. A detectable strain of e ffi 0.3% exists only at the bottom of the structure, that is, close to the interface to the buried oxide. Most of the initial strain is preserved in the longitudinal direction by increasing W/L. At W/L ¼ 10, the initial strain value of e ffi 0.6% is obtained for about two-thirds of half of the longitudinal direction and is

Figure 6.6 (a) Results of FEM simulations of the in-plane stress (sxx) as a function of the gate length L (a). The height H (or thickness) of the strained silicon layer was in the range of 5–100 nm. (b) Dependence of the in-plane stress on the height/length ratio of individual structures.

6.4 Strain Engineering for 22 nm CMOS Technologies and Below

Figure 6.7 Remaining strain in the longitudinal direction of strained silicon channels (SSOI) having W/L ratios of 1 (a), 10 (b), and 20 (c). The strain was calculated at the

top, middle, and bottom of the layer. Constant values of the channel width W ¼ 50 nm and silicon layer thickness of 20 nm are used. FEM simulations of half of the channel length.

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relaxed in the perpendicular direction. A more homogeneous strain profile and a lower strain relaxation at the end are found for W/L ¼ 20. Because the strain is preserved in the longitudinal direction and relaxed in the orthogonal direction, the initial biaxial strain is altered to uniaxial strain. Therefore, biaxially strained layers, especially SSOI, are promising candidates to introduce strain in Si channels of sub-30 nm MOSFETS, where conventional local strain techniques cannot be applied.

6.5 Conclusions

Strained silicon channels are one of the most important technology boosters for further Si CMOS developments. The mobility enhancement obtained by applying appropriate strain provides higher carrier velocity in MOS channels, resulting in higher drive current under a fixed supply voltage and gate oxide thickness. Process-induced stressors including tensile or compressive contact layers and various stress memorization techniques are already integrated in today’s highperformance technologies. The application of local strain elements, however, is limited by further scaling. On the other hand, one of the most important advantages of globally strained SSOI is the scalability to thinner device and insulator layers. This allows the combination of strain with benefits of ultrathin-body MOSFETs. The reduction of the thickness of the silicon layer contributes to the strain conservation. Furthermore, varying the relation between channel length and layer thickness can introduce both types of strain and the modification of the biaxial into uniaxial strain.

Acknowledgments

We are thankful to S. Hopfe for the sample preparation and F. Naumann for some of the FEM simulations. This work was financially supported by the German Federal Ministry of Education and Research in the framework of the DECISIF project (contract no. 13 N 9881) and the Max Planck Society/Fraunhofer Society joint research project NANOSTRESS.

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37 Gannavaram, S., Pesovic, N., and

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€ urk, C. (2000) Low temperature Otzt€ (800  C) recessed junction selective silicon-germanium source/drain technology for sub-70 nm CMOS. IEDM Tech. Dig., 437–440. Yang, H.S., Malik, R., Narasima, S., Li, Y., Divakaruni, R., Agnello, P., Allen, S., Antreasyan, A., Arnhold, J.C., Bandy, K., Belyansky, M., Bonnoit, A., Bronner, G., Chan, V., Chen, X., Chen, Z., Chidambarro, D., Chou, A., Clark, W., Crowder, S.W., Engel, B., Harifuchi, H., Huang, S.F., Jagannathan, R., Jamin, F.F., Kohyama, Y., Kuroda, H., Lai, C.W., Lee, H.K., Lee, W.-H., Lim, E.H., Lai, W., Mallikarjunan, A., Matsumoto, K., McKnight, A., Nayak, J., Ng, H.Y., Panda, S., Rengarajan, R., Steigerwalt, M., Subbanna, S., Subramanian, K., Sudijono, J., Sudo, G., Sun, S.-P., Tessier, B., Toyoshima, Y., Tran, P., Wise, R., Wong, R., Yang, I.Y., Wann, C.H., and Su, L.T. (2004) Dial stress liner for high performance sub-45 nm gate length SOI CMOS manufacturing. IEDM Tech. Dig., 1075–1077. Yeo, Y.-C. (2007) Enhancing CMOS transistor performance using latticemismatched materials in source/drain regions. Semicond. Sci. Technol., 22 (1), S177–S182. Flachowsky, S., Wei, A., Illgen, R., Herrmann, T., H€ontschel, J., Horstmann, M., Klix, W., and Stenzel, R. (2010) Understanding strain-induced drive current enhancement in strainedsilicon n-MOSFET and p-MOSFET. IEEE Trans. Electron Devices, 57 (6), 1343–1354. Fleury, D., Bidal, G., Cros, A., Boeuf, F., Skotnicki, T., and Ghibaudo, G. (2009) New experimental insight into ballisticity of transport in strained bulk MOSFETs. 2009 Symposium on VLSI Technology, Digest, Japan Society of Applied Physics, Tokyo, pp. 16–17. Lochtefeld, A. and Antoniadis, D.A. (2001) Investigating the relationship between electron mobility and velocity in deeply scaled NMOS via mechanical stress. IEEE Electron Device Lett., 22 (12), 591–593. Fischetti, M.V. and Laux, S.E. (2000) Performance degradation of small silicon

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7 Stress-Driven Nanopatterning in Metallic Systems Vincent Repain, Sylvie Rousset, and Shobhana Narasimhan

7.1 Introduction

The desire to gain a fundamental understanding of the physical properties of nanostructures down to the atomic limit, as well as a host of practical applications, drives the need for achieving uniform arrays of nanostructures. Since standard lithography techniques are intrinsically limited in resolution, new physical methods have to be used. Epitaxial growth has been shown to be an interesting alternative, as thermodynamic or kinetic processes can be involved to get uniform nanostructures on surfaces [1]. From a kinetic point of view, the self-assembly of nanoislands by nucleation on surfaces [2, 3] has proved to be a solution to randomly grow nanostructures with an adjustable size and density. However, the size distributions are always broad. To overcome this drawback, the use of prestructured templates has been proposed that effectively provides an ordered growth regime under adequate flux and temperature conditions [4, 5]. One of the difficulties of such a technique is the ability to realize and control a regular surface pattern at nanometer scale. Fortunately, the presence of surface stress, and the processes that take place to minimize its contribution to the total energy, can lead to such natural surface patterning at nanometer length scales, either in homoepitaxial or in heteroepitaxial systems. Once again, the interest of such surfaces for applications comes from their ability to serve as templates for the growth of ordered nanostructures. By such a bottom-up technique, state-of-the-art arrays of monodisperse nanoparticles, as small as tens of atoms each, have been realized in the past two decades and subsequently used for the study of their physical properties. In this chapter, we review such phenomena, restricting ourselves primarily to metallic systems, including alloys. The study of such self-organized systems is basically divided into two different parts. In the first one, where surface stress plays a major role, the goal is to understand and predict the equilibrium morphology of surfaces at the nanometer scale. In the second one, this nanopatterning is exploited to grow nanostructures; their size, shape, and density are generally driven by a combination of local thermodynamic and kinetic

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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processes. At present, a variety of experimental and theoretical techniques are available to study such phenomena. However, from an experimental point of view, the invention of scanning probe microscopies, and more particularly of scanning tunneling microscopy, has been a crucial step in the systematic study of metallic crystalline self-organized surfaces and nanostructures. During the 1990s, pioneering groups discovered periodic patterns such as the herringbone reconstruction of Au (111) [6] and the CuO stripes on Cu(110) [7] and showed their ability to serve as templates for subsequent ordered growth [8]. To describe such phenomena theoretically, there is a hierarchy of possible approaches. The most accurate approach would be to perform ab initio density functional theory calculations. Surface stress can be calculated reliably and efficiently, making use of the Nielsen–Martin stress theorem [9]. However, the disadvantage of this approach is that one is restricted to fairly small length scales because of issues of computational cost. At larger length scales, one can make use of semiempirical potentials such as those obtained using the embedded atom model [10], effective medium theory [11], or the glue model [12]; these are considerably cheaper than first-principles calculations but not always reliable. Finally, to extend to large length scales, one can go over to descriptions using continuum elasticity theory. One very useful classical model, which helps one understand qualitatively many of the stress-driven phenomena that occur at surfaces, is the Frenkel–Kontorova model [13] (and subsequent generalizations of it). A successful modern approach has been to use this time-tested model, however, with the parameters in it obtained from ab initio density functional theory calculations. In many cases, the results thus obtained have been in remarkably good agreement with experiments.

7.2 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales 7.2.1 Surface Stress

The concepts of surface energy and surface stress date back to the pioneering work of Gibbs [14], developed further by Shuttleworth [15] and Herring [16]. The surface stress tensor s ab is defined by sab ¼ cdab þ

@c @eab

ð7:1Þ

where eab is the strain tensor, dab is the Kronecker delta, and c is the surface free energy per unit area. For a more detailed discussion of the surface stress, we refer the reader to Chapter 2. Even for a situation where the surface energy is minimized, the surface stress is in general not zero and may be tensile or compressive, indicating that the surface atoms would like to increase or decrease their density [17]. In general, the stress at a bulk-truncated metallic surface is tensile; this is because surface atoms have lost their neighbors in the layers above and would therefore like to come closer to

7.2 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales

their neighbors in the surface layer so as to be embedded in an optimal ambient electronic density. For heteroepitaxial metallic systems, the surface stress may be compressive or tensile, depending on the size mismatch between overlayer and substrate atoms. If the size of overlayer atoms is smaller or larger than that of the substrate, then one would expect the surface stress to be tensile or compressive, respectively. However, one should note that the “size” of an atom is not necessarily a simple concept; in particular, the effective size of an atom at the surface may differ considerably from that of a bulk atom. If the magnitude of the surface stress is large, it can serve as a driving force for reconstruction or alloying. In the former case, the density of atoms in the surface layer changes, thereby reducing the surface stress. In the latter case, atoms in the overlayer mix with those of the substrate, or two overlayer species mix on the substrate (even, in some cases, if they are bulk immiscible), thus reducing the magnitude of the surface stress. All these phenomena lead to patterning at nanometer length scales, as we will describe later. 7.2.2 Surface Reconstruction and Misfit Dislocations 7.2.2.1 Homoepitaxial Surfaces For simplicity of discussion, let us first consider a homoepitaxial system and assume that the surface stress is tensile, as is generally the case for metallic systems. As discussed previously, as a result of this tensile stress, surface atoms would like to increase their density. However, this tendency is opposed by the presence of the substrate. These two opposing factors are accounted for by the two main terms in the Frenkel–Kontorova model. In its original form, this model consists of a linear chain of atoms (the surface layer) connected by harmonic springs, sitting in a sinusoidal potential that represents the effect of the substrate (Figure 7.1). There are two competing periodicities in the system: the preferred length of the springs (the favored nearest-neighbor spacing of the surface layer alone) is less than the distance between neighboring minima of the substrate potential (which corresponds to the lattice spacing of the substrate). The advantage of the original form of the Frenkel–Kontorova model is that it can be solved analytically; however, it is simple to generalize it to two dimensions and other forms of potentials, in which case the model can be solved

Figure 7.1 Schematic diagram of the Frenkel–Kontorova model. The sine potential of depth W and period a (bulk lattice spacing) represents the substrate atoms. The surface atoms are connected by springs of natural length b (surface lattice spacing) and spring constant m.

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numerically so as to obtain the ground state configuration. In cases where the springs are sufficiently stiff to overcome the effects of the substrate potential, the surface layer reconstructs typically in a pattern such that regions where the surface layer remains in registry with the substrate are separated by misfit dislocations where the density is locally increased. Other possibilities are Moire patterns where the surface grid has a more or less uniformly increased density relative to that of the substrate; it may also happen that the symmetry of the overlayer grid may be different from that of the substrate—for example, the substrate may have square symmetry while the overlayer may have hexagonal symmetry. The best-known example of a system that displays such a reconstruction, driven by stress relief, is the Au(111) surface. There have been several calculations of the stress on the unreconstructed Au(111) surface [18]. Although there is some variation in the numbers obtained, all the calculations agree that there is a considerable tensile stress on this surface. As an example, in a recent calculation, we obtained a value of 3.6 N/m (we note that ab initio calculations of surface stress are notoriously difficult to converge, both with respect to basis size and with respect to the number of layers used in simulating a surface slab, explaining a relative dispersion of values obtained in different calculations) [19]. There are two lowenergy stacking sites for Au atoms in the topmost layer: face-centered cubic (fcc) and hexagonal close packed (hcp). Domains of the two types of stacking are separated by misfit dislocations where surface atoms occupy the bridge sites that lie between the fcc and hcp sites [20–22]; these show up clearly as light stripes in scanning tunneling microscopy images. As a result of alternating between occupying the two types of stacking sites, the surface layer densifies by 4%, thereby reducing the tensile stress. The resulting pattern has a repeat distance of 22 nearest-neighbor spacings, which is  about 63 A. It is worth noting that the Au(100) surface also displays a surface reconstruction with a large unit cell. In this case, the substrate has square symmetry, whereas the overlayer has triangular symmetry, resulting in a (26  48) unit cell [20]. Another homoepitaxial system that displays a reconstruction driven by stress relief is Pt(111). While this is similar to Au(111) in that Pt and Au are both fcc metals, the reconstruction differs in two ways: (i) the Au(111) surface nearly always displays a reconstruction, but Pt(111) displays a reconstruction only at high temperatures [23] or in the presence of a supersaturated Pt vapor [24]; (ii) in the case of Au(111), the densification takes place along only one direction in the surface plane, whereas in the case of Pt(111), there is an increase in density along three equivalent directions in the surface plane. In Figure 7.2, we have shown simulated STM images that show how the repeat distance and the pattern change on increasing the densification in the top layer relative to the substrate; it is possible to control this parameter by tuning the temperature or the chemical potential [25]. Mansfield and Needs mapped real surfaces onto the exactly solvable one-dimensional Frenkel–Kontorova model in a simple way and showed that a dimensionless parameter that involved combinations of the surface stress, surface energy, surface lattice constant, stacking fault energy, and stiffness of nearest-neighbor bonds (these parameters can all be estimated from ab initio density functional theory calculations) could be used to predict whether a metallic surface will reconstruct [26]. Subsequent authors have shown that this simple model works surprisingly well for the (111) surfaces of fcc metals [25, 27].

7.2 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales

Figure 7.2 Simulated STM images of a reconstructed Pt(111) surface. Between (a), (b), and (e), the enhanced density in the surface layer, relative to the substrate, is progressively increased from 2.9% to 21%. In (a), a rotation of

the surface layer relative to the substrate is not permitted, and in all other cases, it is permitted. The white/black line in each image represents a  length of 50 A , and note that the patterns have nanometer spacing.

7.2.2.2 Heteroepitaxial Systems Heteroepitaxial systems display reconstruction phenomena similar to those observed on homoepitaxial systems, with the difference that the stress can be tensile or compressive, and therefore the reconstruction can be compressive or expansive, respectively. The stress builds up with the number of overlayers deposited. Typically, with the deposition of the first monolayer, the stress is not sufficient to trigger a reconstruction, and the system remains pseudomorphic (i.e., the overlayer remains in registry with the substrate). Progressively more “drastic” reconstructions are observed as further layers are deposited, for example, if one considers Cu/Ru(0001): here, the lattice spacing of the overlayer, Cu, is 5.5% smaller than that of the substrate, Ru. The first monolayer goes down pseudomorphically, the bilayer shows a striped reconstruction similar to that observed on Au(111), the trilayer shows a triangular pattern (similar to Figure 7.2d or e),

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while for four or more Cu layers, a Moire pattern (similar to Figure 7.2f) is observed) [28]. A similar sequence of patterns has also been observed in theoretical modeling of this system [29]. The Ag/Ru(0001) system differs in that the lattice constant of Ag is larger than that of Ru, and the unreconstructed surface therefore exhibits a compressive stress. This system also relaxes through the formation of misfit dislocations, similar to those observed on Au(111), with the difference that in this case these misfit dislocations correspond to regions of rarefaction rather than densification [30]. Another compressively stressed system that shows stress-driven reconstruction is Ag/Pt(111). Once again, the first monolayer remains pseudomorphic, while the second monolayer displays a reconstruction. At first, there is a metastable striped reconstruction, similar to that observed on Au(111) (except, of course, that it corresponds to rarefaction rather than densification); upon annealing, it is transformed into a trigonal network of domain walls [31]. 7.2.3 Stress Domains

It is important to keep in mind that the surface stress is a tensor and not a scalar. When the surface reconstructs by densifying or rarefying along only one direction in the surface plane (as is the case for the various striped reconstructions described above, such as that of Au(111)), stress is relieved (partially) along only that direction, resulting in an anisotropic surface stress tensor. Note that in the case of Au(111), the surface plane has threefold symmetry, and therefore this densification could have occurred along any one of three equivalent directions. When there is an anisotropic surface stress tensor, and a degeneracy of directions along which it can be oriented, it has been shown by the theory of “stress domains” of Marchenko [32] and Alerhand et al. [33] that the lowest energy situation will always correspond to a regular alternation between domains of different orientations; it is possible to estimate the domain width from continuum elasticity theory. Indeed, this is what appears to happen in the case of Au(111): the striped pattern alternates periodically between two out of the possible three orientations, resulting in the famous “herringbone” or “chevron” reconstruction (Figure 7.3a) [6, 34, 35]. At alternating “elbows,” where there are bends in the soliton walls that follow the lines of the misfit dislocations, there are point dislocations, where the atoms have five to seven coordination instead of the sixfold coordination characteristic of the fcc(111) surface. These serve as nucleation sites for the growth of overlayers and constitute a regularly spaced twodimensional grid, since the herringbone pattern is periodic. Thus, the herringbone pattern constitutes a convenient template for the growth of self-organized nanostructures and has frequently been used as such. The same “stress domains” physics can appear in the case of adsorbates on surfaces. Among many examples, Figure 7.3b shows the self-organization of a partially oxygen covered Cu(110) surface [7]. Such a system is simply realized by a controlled adsorption of O2 onto the surface and a subsequent annealing. The natural anisotropy of the (110) surface leads here to a onedimensional pattern with a nanometer-scale period. On a more isotropic surface,

7.2 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales

Figure 7.3 STM images of self-organized surfaces with periodic nanometer-scale patterns. (a) Herringbone reconstruction of the Au(111) surface. The typical periodicity is 30 nm. (b) CuO stripes on a Cu(110) surface. Courtesy of Peter Zeppenfeld. The periodicity is around 6 nm. (c) CuN islands on a Cu(100) surface. The periodicity is around 5 nm. (d) Reconstruction of

the Au(788) vicinal surface that defines a twodimensional pattern with step edges of dimensions 7.2  3.8 nm2. (e) Reconstruction of a Au vicinal surfaces with 8 nm wide terraces. The interplay between step edges and the strain-relief pattern induces a complex morphology. (f) Faceted Au(455) vicinal surface. The typical periodicity is around 200 nm.

such as the Cu(100) surface, two-dimensional self-organization can be observed, as shown in Figure 7.3c with the checkerboard arrangement of 5 nm CuN islands [36]. 7.2.4 Vicinal Surfaces

It has been shown previously that some flat surfaces of homoepitaxial or heteroepitaxial metallic systems can display strained-relief patterned substrates such as dislocation networks that can be subsequently used for the growth of uniform nanoparticles. However, intrinsic defects such as step edges prevent the achievement of long-range ordered arrays of nanostructures since the presence of atomic steps generally limits the coherence of the surface dislocation networks. Steps can also act as nucleation sites for the growth of uncontrolled elongated clusters. Since macroscopic averaging methods are still commonly used for measuring physical properties of nanostructures, these defects are major drawbacks of such bottom-up techniques. To overcome these difficulties, a novel approach has been investigated using strainedrelief vicinal patterned substrates to achieve a high degree of both local and longrange order for the growth of nanostructures. Vicinal surfaces, also called stepped surfaces, display a one-dimensional network of atomic steps separating terraces. The choice of the miscut angle of such a surface compared to a flat surface allows one to control the mean width of terraces. The narrowness of the terrace width distribution is mainly driven by the elastic step–step interaction strongly related to the value of

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surface stress, as demonstrated both theoretically and experimentally on various systems [19] (for a detailed description of this interaction, refer to Chapter 13). Generally, vicinal surfaces offer a one-dimensional pattern that can serve as a template for the growth of one-dimensional nanostructures down to atomic wires [37]. However, the combination of the atomic step network and strain-relief patterns such as dislocation networks can lead to highly uniform two-dimensional template surfaces. Well-studied examples of such systems are the Au(111) vicinal surfaces. As discussed previously, the Au(111) surface is one of the famous examples of natural strain-relief patterns with its herringbone reconstruction. Its vicinal surfaces also show a variety of surface reconstructions with the particular features of very good long-range order and an extremely low density of defects. Figure 7.3d and e show STM images of two of these surfaces that have been shown to be useful for the growth of regular nanostructures. Among them, the Au(788) surface has been the most studied, with the demonstration of the growth of regular nanoparticles of Co, Fe, Ag, and C60 [41, 49–51]. The shapes of strain-relief patterns on vicinal surfaces are generally complex to simulate, due to an interplay between step edge and surface energies. In the case of Au(111) vicinal surfaces, basic arguments have been put forward to explain the shape of certain surface reconstructions and their influence on the stability of such surfaces [38, 39]. An example of an unstable vicinal surface is shown in Figure 7.3f. This faceted Au(455) surface constitutes once again an example of a self-organized system with a 200 nm period driven by the difference of surface stress between the two facets.

7.3 Nanopatterned Surfaces as Templates for the Ordered Growth of Functionalized Nanostructures 7.3.1 Metallic Ordered Growth on Nanopatterned Surface 7.3.1.1 Introduction Nowadays, naturally patterned surfaces are successfully used as templates to grow ordered nanostructures from metallic nanodots [40] to molecular assemblies [41] with controlled size and density. As shown previously, a broad range of materials can display mesoscopic periodic surface patterns and can therefore be suitable for ordered growth. Most of the early studies have focused on metallic surfaces [40] with the well-known Au(111) reconstructed surface [8], but spectacular results have also been demonstrated on semiconductors [42, 43], on insulating layers [44], and more recently on supported graphene layers [45, 46]. This last system can display a single domain Moire pattern over at least several micrometers scale, irrespective of surface defects such as step edges [47]. This improves the long-range order and the uniformity of nanostructures, which is necessary to study their physical properties measured by averaging techniques [48] and to predict possible technological applications. As explained previously, the use of reconstructed vicinal surfaces has also been

7.3 Nanopatterned Surfaces as Templates for the Ordered Growth of Functionalized Nanostructures

shown to increase the long-range order of these arrays of nanostructures over macroscopic scale [49]. For example, the Au(788) surface has been used for the growth of two-dimensional lattices of Co [49], C60 [41], Ag and Cu [50], Fe [51], and so on. The atomistic mechanisms responsible for ordered growth have been studied in detail in specific cases [40, 52] and the temperature dependence of the growth on patterned substrates has been analyzed [53]. In the following, we explain the basic concepts leading to ordered growth and our actual understanding of such phenomena. It is worth noting that an important condition to obtain ordered growth is that the surface patterns should remain unaffected by the growth process. Considering the complexity of surface science phenomena, this latter point is not trivial and generally difficult to predict. This hypothesis will be assumed to be true in the following. However, a few examples of self-organized surfaces modified during the growth of nanostructures have been reported [54, 55]. 7.3.1.2 Nucleation and Growth Concepts Nucleation and growth of islands on surfaces has been extensively studied for many years and has been reviewed in several articles and books [40, 56]. Atoms are deposited from a vapor pressure onto a surface, such as in the common case of solid on solid model. In the case of adatoms moving on a homogeneous substrate (which is called homogeneous growth), the process is well described by mean field theory and is essentially determined by atomistic parameters for surface diffusion and binding energies of adatoms to clusters. Values for these parameters may be determined by comparing scaling predictions with suitable experimental measurements [57]. One usually distinguishes three regimes versus the coverage of deposited atoms: the nucleation regime where the density of stable islands is increased, the growth regime where the density is almost constant but the size of islands increases, and the coalescence regime where the density of islands decreases since neighboring islands start to coalesce. The maximum cluster density versus the temperature can be determined from variable temperature STM experiments. In the regime of complete condensation generally relevant for metal on metal growth, re-evaporation of adatoms from the substrate into the vapor is negligible. The maximum cluster density nc is given by nc ¼ gðD0 =FÞ1=3 expðEdiff =3kB TÞ

ð7:2Þ

where g is a prefactor related to capture numbers, F is the deposition rate (flux), D0 is the diffusion prefactor, and Ediff is the diffusion energy. This expression is valid in the case of stable dimers on the surface, that is, critical cluster size i ¼ 1 (i is defined as the size of the biggest unstable cluster). In the case of i > 1, Eq. (7.2) should be modified and involves the binding energy of the critical cluster. In the simple i ¼ 1 regime, it is worth noticing that the slope of nc versus T in an Arrhenius plot gives Ediff . At higher temperature, the critical nucleus size increases and this leads to a higher slope. Such behavior is also found by using kinetic Monte Carlo (KMC) simulations. The advantage of a KMC simulation is that it goes beyond the mean field approximation that is known, for example, to overestimate the island density.

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7.3.1.3 Heterogeneous Growth What happens now when all atomic sites onto the substrate are not equivalent, such as on the self-organized surfaces described in Section 7.2? Some sites can act as preferential nucleation sites, such as, for example, the point dislocations located at the elbows of the Au(111) herringbone reconstruction. These sites can be described in a mean field model as traps for adatoms [58, 59]. Such a model had some success in the past by reproducing the nucleation and growth on surfaces with point defects [60, 61]. We show here how it can be applied nicely to growth on selforganized surfaces [52]. A typical theoretical curve of the critical cluster density versus the temperature is shown in Figure 7.4b [53, 56]. For the lowest temperature, no variation is found: the cluster density is constant with temperature. This corresponds to a low diffusion regime called “postnucleation” [40] when adatoms hardly diffuse on the surface and are stable. Between 45 and 80 K, a linear decrease in the cluster density with temperature in an Arrhenius plot is found. At such low temperatures, the mean free path of adatoms on the surface is lower than the mean

Figure 7.4 (a) STM image of an ordered growth of Co nanodots on a Au(788) surface. (b) Arrhenius plot of the island density (in islands/atomic sites) versus temperature calculated by numerical integration of rate equations in a point defect model. The analytical asymptotic regimes are also plotted (cf. text). (c) Size distributions calculated by KMC for growth on a homogeneous substrate at 90 and 120 K

(y ¼ 0:1 ML). The solid line shows the Amar and Family model. (d) Size distributions calculated by KMC for the growth on a heterogeneous substrate with a periodic array of atomic traps at 90 and 120 K (y ¼ 0:1 ML). The solid line shows the binomial distribution associated with the Vorono€ı area of the trap lattice.

7.3 Nanopatterned Surfaces as Templates for the Ordered Growth of Functionalized Nanostructures

distance between traps. This regime is identical to the homogeneous growth, and the slope of the Arrhenius plot is Ediff . Above the temperature threshold T0 , the system displays the ordered growth regime. The maximum cluster density is constant, equal to the density of traps. T0 is the temperature at which the adatoms’ mean free path determined by Eq. (7.2) is equal to the distance between traps. As a consequence, the parameters that determine T0 are Ediff and the trap density nt. Ordered growth occurs as long as the typical energies of the trapping mechanisms are sufficient to stabilize adatoms in the traps. We call Te the highest temperature for which an ordered growth is observed. The crucial parameter, which determines Te , is the trap energy Et. Above Te , the critical island density decreases dramatically with temperature. The slope is higher than a simple homogeneous growth regime. Such a high value is mainly due to the long time spent by adatoms in traps. The effect of traps is then to reduce the effective diffusion of adatoms [53]. Eventually, the mean field calculations including traps give a qualitative understanding of the ordered growth. Rapid adatom diffusion and strong trapping are the main ingredients needed to get an ordered growth over a large temperature range. We now focus on the other key point of ordered growth, which is the achievement of narrow size distributions. Unfortunately, the mean field approach of the previous mean field model cannot give any idea about the island size fluctuations during the nucleation and growth processes. Although a phenomenological model has been proposed for homogeneous growth by Amar and Family [62], very little is known for the growth on heterogeneous substrates, especially for ordered growth. To obtain some information on these size distributions, KMC simulations can be performed [53]. Some results of these simulations are shown in Figure 7.4c and d, which correspond to the case of an homogeneous surface and a surface prestructured with a rectangular array of traps, respectively. The homogeneous growth size distributions are perfectly reproduced by the Amar and Family model [62] and show typical full widths at half maximum (FWHM) of 110% whatever the temperature. In the case of the growth on the prestructured surface, when nucleation occurs on the traps (T0 < T < Te ), the size distributions are narrower and almost constant with temperature in this range. The FWHM for nt ¼ 1=200 and y ¼ 0:1 MC is typically 50%. Interestingly, these size distributions are very well fitted by simple binomial distributions pðkÞ ¼ Ckn yk ð1yÞnk , with k the island size (in number of atoms) and n the number of atomic sites of the Vorono€ı area in the trap lattice (i.e., n ¼ 1=nt ). This fact has already been pointed out for the ordered growth of Ag/Ag(2 ML)/Pt(111) [4] but is less trivial for growth induced by point defect nucleation. This means that during the growth regime, the probability that an adatom deposited on the surface does not stick to its closest island is vanishingly small, whatever the coverage or the temperature between T0 and Te , at least for the parameters used in our simulations. Moreover, simulations on randomly distributed atomic traps show a broadening of the island size distribution due to the distribution of traps Vorono€ı area. This confirms that the analysis of size distributions in terms of the Vorono€ı area distribution is always pertinent. Therefore, the FWHM of the size distribution is limited to the perfectness of the traps array for a given coverage. In addition, even for a perfect trap array, there is an intrinsic statistical limit to the size distribution width. As for the binomial distribution,

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the variance is s ¼ ð1yÞ=ny, the only way to improve the size distribution quality is to increase the coverage and use a surface with a lower trap density (larger Vorono€ı cells).

7.4 Stress Relaxation by the Formation of Surface-Confined Alloys

Stress effects play also an important role in the process of alloying. Already since ancient times, it has been known that alloying two metals can result in the formation of a new material with superior properties to that of the constituent metals. However, not all combinations of metals will form stable alloys; the empirical rules governing alloy formation were formulated by Hume-Rothery et al. [63]. The first of these rules states that if the size mismatch between the two metals is greater than 15%, then alloy formation is disfavored; this is because otherwise too much elastic strain builds up in the material. The surprising discovery in recent years has been that bulk-immiscible metals may, however, form surface-confined alloys. The driving force behind such surface alloying is generally believed to be stress relief, though, as we will discuss later, in some cases, such as when one of the constituents is a magnetic element, we have reason to believe that other mechanisms also play an important role. 7.4.1 Two-Component Systems

As an example, Au and Ni are immiscible in the bulk. However, when Au is deposited on Ni(110), surface alloying is observed [64]. This has been confirmed by calculations, making use of potentials derived from effective medium theory. There are many other such examples, such as Na/K on Al(111) and Al(100) [65], Ag on Pt(111) [66], and Sb on Ag(111) [67]. It has subsequently been shown by Tersoff [68] that one may expect to see such a phenomenon, in general, in systems dominated by size mismatch and in which, therefore, elastic stress plays a major role. The remarkable thing is that the same feature, atomic size mismatch, that prevents intermixing in the bulk actually promotes mixing at the surface. In some cases, such surface alloys are disordered; however, in other cases, they form ordered structures; long-range order is of interest for certain applications. Surface alloys, in general, offer possibilities as novel catalytic materials and (when one or more of the constituent metals is magnetic) materials for applications in nanomagnetism. 7.4.2 Three-Component Systems

One can also consider another type of surface alloy: one can codeposit two bulkimmiscible metals, A and B, on a substrate C of intermediate spacing. If A has a lattice constant smaller than that of C, while B has one larger than that of C, then an unreconstructed monolayer of A on C would display tensile stress, while a monolayer

7.4 Stress Relaxation by the Formation of Surface-Confined Alloys

of B on C would exhibit compressive stress. One might thus expect that stress relief might provide a driving force for mixing in such a system and one may obtain a surface alloy of A and B on C. Motivated by such considerations, Thayer et al. considered Co and Ag codeposited on a Ru(0001) substrate [69, 70]. Since the nearestneighbor distance of Ag is larger than that of Ru by 8%, while that of Co is smaller than that of Ru by 7%, one might expect to find such surface alloy formation in this system. However, one does not find atomic-level mixing. Instead, for submonolayer films, one finds Ag droplets surrounded by a Co matrix. For larger Ag concentrations, the system forms a phase consisting of dislocated Ag/Ru(0001) and alloy droplets. The reason that there is no atomic level mixing is because elastic interactions alone do not determine the surface structure, one also has to take into account chemical interactions. The droplet size is determined by a balance between energy reduction due to stress reduction and the energetic cost of forming unfavorable chemical bonds. A combinatorial study of small magnetic metals M alloyed with large nonmagnetic metals N on the Ru(0001) surface has recently been carried out using density functional theory techniques [71]. Among the results that emerged from this study, we mention that (i) several bulk-immiscible M-N pairs were found to atomically mix on the surface, (ii) atomic sizes at the surface were sometimes significantly different from bulk sizes, (iii) both elastic and chemical interactions were found to be important, and (iv) no simple size-dependent criterion emerged in predicting mixing; this is because unlike the bulk case, in the surface case the phase segregated forms contain large elastic energy. As a result of this study, Fe-Au/Ru(0001) was identified as a promising candidate to observe a long-range ordered surface alloy, even though Fe and Au are bulk immiscible. Indeed, using STM and low-energy electron diffraction (LEED) experiment, we have discovered a new ordered surface alloy made out of Fe and Au, deposited on a Ru(0001) substrate. The alloy films were prepared by depositing one metal, annealing, and repeating the same procedure for the second metal; in this way, large islands were obtained. Au was deposited from an e-beam heated Mo crucible at the rate of 0.04 ML/min and Fe was deposited from an e-beam heated Fe rod at the rate of 0.07 ML/min. Evaporation rates were determined by analyzing the STM images giving rise to a typical error bar for the concentration x of around 5%. It is worth noting that the final results were found to be independent of whether Fe or Au was deposited first, demonstrating that we indeed reached the equilibrium configuration. For alternate deposition where the Fe fraction x is close to 0.33, we obtain a periodic structure as shown in Figure 7.5d. Although we can observe some local defects due to an imperfect 1 : 2 stoichiometry, there is relatively good long-range pffiffiffi pffiffiffi order, giving rise to a clear LEED diffraction pattern, characteristic of a 3  3 unit cell. This structure is the simplest two-dimensional ordered phase on a hexagonal lattice for the 1 : 2 stoichiometry. Simulation of constant height STM image calculated from ab initio data strongly supports the observed structure of the alloy, showing that Fe is imaged lower than Au, in good agreement with the experiments. To get a deeper understanding of the driving forces that stabilized these pseudomorphic alloyed phases, we have performed spin polarized ab initio density functional theory calculations for several different configurations and compositions of

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Figure 7.5 Ab initio results for (a) the xx component of the surface stress, (b) the yy component of the surface stress, and (c) the enthalpy of mixing DH, for FexAu1x/Ru(0001), as a function of Fe concentration x. Forty-three different structures, containing up to six surface atoms per unit cell, have been considered. The

blue line represents the convex hull and the blue dots represent stable alloy phases that lie on the convex hull. (d) 2.1  2.1 nm2 STM image of 0.9 ML coverage of Fe0.33Au0.67 deposition, after annealing at 600 K. Fe atoms are seen in darker dots located at the center of hexagons made up of lighter dots that are Au atoms.

this system. These calculations have confirmed that mixing is favored in this system and that the stress is indeed reduced on mixing (Figure 7.5a and b). Note that the stress for a pure monolayer of Au on Ru(0001) is compressive, while for a pure monolayer of Fe/Ru(0001) is tensile and for intermediate compositions, the surface stress is reduced, following an almost linear trend. However, these results hint at the fact that surface stress may not, surprisingly, be the dominant mechanism in this system, since the enthalpy of mixing DH shown in Figure 7.5c (which determines alloy stability) is lowest for an Au-rich phase, and not for Fe-rich phases, whereas the surface stress appears to go to zero under Fe-rich conditions at x ’ 0:8. Moreover, the positions of the blue dots in Figure 7.5a and b indicate that the lowest energy configuration at a given value of Fe concentration x does not necessarily correspond to that with the lowest surface stress x. In fact, by performing additional calculations with the spin polarization suppressed, we have found that the principal driving force for mixing in this system is magnetism pffiffiffi rather pffiffiffi than stress relief. This surprising result explains the stability of the 3  3 structure of the long-range ordered surface alloy of Fe0.33Au0.67, which has been observed by STM and electron diffraction [72].

7.5 Conclusion

The realization and study of metallic nanostructures on surfaces is of particular importance for improved and innovative performances in various fields such as optics, magnetism, catalysis, and so on. We have shown in this chapter that a great

References

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Acknowledgment

Coworkers Cyril Chacon, Yann Girard, Mighfar Imam, Jerome Lagoute, Madhura Marathe, Shruti Mehendale, Stanislas Rohart, and David Vanderbilt who have contributed to this chapter are acknowledged. We acknowledge also funding from the Indo-French Center for the Promotion of Advanced Research, the Region Ile-de-France (SESAME), and the French Ministry of Research.

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8 Semiconductor Templates for the Fabrication of Nano-Objects Jo€el Eymery, Laurence Masson, Houda Sahaf, and Margrit Hanb€ ucken

8.1 Introduction

Owing to the continuing miniaturization in nanotechnology applications requested by the semiconductor roadmap, important works are undertaken to develop novel and complementary ways of creating semiconductor templates allowing the growth of objects on the nanometer scale. Basically, two conceptually different approaches have so far emerged as fabrication methods. Both can achieve reproducible and well-controlled fabrication of arrays consisting of well-defined nanostructures on various semiconducting substrates. The first approach is called top-down or descending and the second one, which is more recent, is called bottom-up or ascending. Top-down nanofabrication techniques were inherited from microelectronic development through lithography and etching to meet the expectations related to the constant miniaturization of optoelectronic devices. Bottom-up methods allow nanofabrication and nanopatterning through the spontaneous reorganization of the substrate atoms induced by the intrinsic properties of the material. Subsequent controlled deposition of atoms and molecules then leads to the formation of self-organized assemblies of various materials. The third approach combines these two methods by adding a pattern of artificial perturbations, such as holes or stressors, via a top-down process onto a substrate with defined intrinsic properties. In this chapter, some basic ideas concerning the growth of highly ordered arrays of well-defined and identical nano-objects on a substrate with extended dimensions will be described. Growth can be controlled by choosing the respective adsorbate and substrate materials and strain can be used to drive the adsorption of the deposited atoms or molecules at specific sites. Some of the most frequently used semiconductor templates and several deposited systems and architectures will be presented in this chapter.

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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8.2 Semiconductor Template Fabrication

Many strategies have been developed to pattern semiconductor surfaces for their uses in nanosciences [1, 2]. Emphasis will be placed in this section on techniques that use mechanical strain, specifically for nanostructuring surfaces. The usual morphological patterning that may induce local strain will be first briefly reminded. Then, we will point out elastic stress engineering using buried stressors either on flat surface or through lithography followed by etching. Finally, the use of intrinsic properties of the surface such as surface reconstructions and step edges on vicinal surfaces will be stressed. The combination of both approaches, artificial and natural patterning, can be also employed, but will not be presented in this section for the sake of simplicity. 8.2.1 Artificially Prepatterned Substrates 8.2.1.1 Morphological Patterning Etching techniques are commonly used to transfer the hole patterns of a mask into the surface of the sample with individual dimensions that may be less than 50 nm. Three etching methods are commonly employed in the laboratories and the microelectronics industry (Figure 8.1): physical etching, chemical etching, and reactive ion etching (RIE). Physical etching consists of bombarding the sample with high-energy ions (few keV). This method may produce anisotropic patterns due to the particle momentum transfer and the mask may suffer from some damages. Chemical etching is performed by immersing the sample in a chemically active bath. This method can be isotropic, highly chemically selective, and sometimes strain sensitive. Finally, RIE tries to combine the advantages of the last methods by using chemically reactive plasmas to obtain highly selective and anisotropic etching. The plasma, fitted to the materials under study (e.g., fluorine and chlorine based for Si and GaAs,

Figure 8.1 Schematic illustration of the three etching methods using masks commonly employed in laboratories and microelectronics industry.

8.2 Semiconductor Template Fabrication

respectively), is generated by an electromagnetic field under low pressure. Highenergy ions bombard the sample surface and react with the material. Nanoimprint lithography is used to transfer nanometer-scale patterns with molds on resists. This technique is already included in the international technology roadmap for semiconductor for the 22 nm node. It is also possible to apply maskless lithography techniques such as e-beam and interferometric and holographic lithography [3]. Interference lithography is a laboratory-scale patterning technique that uses the interference of coherent laser beams to write patterns over large areas. This technique can produce patterns with feature dimensions approaching 20 nm, but is limited to performing periodic patterns with micron to submicron periodicities only. Pit- or groove-patterned semiconductor substrates are commonly used to grow ordered arrays of islands. Such topographic templates allow uniform shape, size, and periodicity control of the grown islands. The investigation of growth kinetics and postgrowth treatments is important and optimized annealing and etching can enable the tailoring of the island properties. The evolution of the morphology, chemical composition, homogeneity, size uniformity, and coarsening can thus be tuned. On nominally flat substrates, the nucleation of strained islands is mainly governed by the competition between surface energy and strain energy [2, 4, 5]. On a lithographically engineered, patterned surface, nucleation will be governed by the local surface curvature and local strains. The nucleation mechanisms will thus be modified compared to those on a flat surface [6]. 8.2.1.2 Silicon Etched Stripes: Example of the Use of Strain to Control Nanostructure Formation and Physical Properties The advancement of silicon integrated circuits pushes lithography technology to the limits for making devices with smaller length scales. In the top-down approach, strain has been widely used to improve the carrier transport and the drive current of field effect transistor devices [7]. One solution among others consists of etching biaxially strained silicon on insulator layers obtained by wafer bonding [8]. For the stripe geometry (i.e., wires having a strain imposed by interfacial boundary conditions), the elastic relaxation with respect to the initial state depends not only on the width and thickness, which are now very thin in aggressive technologies (20 and 10 nm, respectively), but also on the technological steps necessary to integrate these active materials inside the transistor stacking (source–drain–gate materials). The measurement of the buried active layers (here strained silicon) has been performed to improve the physical understanding of the strain that drives the enhancement of the transport properties [9]. 8.2.1.3 Use of Buried Stressors A localized strain on the surface can be provided by the propagation of buried strain fields that may result from specific nanostructure growth, for example, quantum dots (as explained later) or localized implantations [2]. It has also been proposed to control the lateral ordering of nanostructures at the wafer scale to use periodic dislocation networks obtained by wafer bonding [10]. Grain boundaries and dislocation networks

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were known a long time ago in wafer bonding (see, for example, the early works dedicated to Au or Si), but recent improvements stand in the localization of the grain boundaries near the surface and in the control of their periodicity to tune finely the elastic fields [11, 12]. Network periodicity is controlled by the disorientation angle between the substrate and the bonded film, and the distance between the stressors and the surface adjusts the magnitude of the surface stress. This array is formed artificially by the dislocations appearing at the interface of two clean crystals bonded with a crystallographic disorientation. In general, the dislocation networks have to accommodate the lattice mismatch, the flexion and rotation angles of the two crystals. They also depend on the surface symmetry and on the interactions that may occur between different networks: for example, a high temperature annealing minimizes the total energy of interacting dislocations by changing the line arrangement [13]. For the hydrophobic bonding (without oxide layer) of identical silicon crystals, the interfacial structure has been studied by transmission electron microscopy (TEM) to determine the nature of dislocations and their interactions. For two (001) surfaces, twisted by 0.4 and having a negligible tilt, Figure 8.2a shows a square network of dissociated screw dislocations localized at the bonded interface. The dislocation network periodicity L is related to the twist angle a and Burgers’ vector modulus b ¼ a/2h110i by Frank’s relation L ¼ b=½2 sinða=2Þ. A triangular network is obtained for the bonding of two Si(111) surfaces within similar conditions (see Figure 8.2c) [13]. The strain field of slightly buried dislocation networks is used to drive the nucleation of Ge dots on a nearly flat surface [14]. Another more efficient way to induce the positioning of semiconductors [11, 15] and metals [13] islands consists in transferring the symmetry of the dislocation lines to the surface by suitable wet etching [13]. In this case, significant trench depths are obtained (see the STM profiles in Figure 8.2b and d) and the positioning is also controlled by the local curvature.

Figure 8.2 Transmission electron microscopy images of pure 0.44 twist boundaries of two— (a) Si(001) and (c) Si(111)—surfaces [13]. (b) and (d) Scanning tunneling images of the

same samples after chemical etchings (for details see Ref. [13]). The height variations along the blue line profiles are, respectively, 4 and 2.5 nm.

8.2 Semiconductor Template Fabrication

8.2.2 Patterning through Vicinal Surfaces 8.2.2.1 Generalities A vicinal surface is obtained by cutting a crystal along a direction close to a highsymmetry direction with a disorientation angle between 0 and 15 . Annealing a vicinal surface leads to a rearrangement of matter and gives rise to a new surface consisting of a succession of steps separated by terraces. When fabricating a surface by cutting a crystal in a nominal direction or with a disorientation, the surface atoms adopt a different lattice parameter than the one in the bulk in order to minimize the free surface energy. Depending on the polar (tilt) and azimuthal (twist) miscut angles, surfaces can display different reconstructions and step–edge energies. The difference between the surface and the bulk lattice parameter induces an intrinsic stress in the surface layer that is directly related to the surface free energy. This stress is an important parameter to take into account when nanostructuring surfaces on a large scale. Steps are the most common defect in nominal crystalline surfaces, but vicinal surfaces can be intentionally offcut from a specific plane in a controlled manner to favor anisotropic growth of various nanostructures [16, 17]. The structural properties of the most frequently used semiconductor surfaces, Si(111) and Si(100), will be presented. 8.2.2.2 Vicinal Si(111) Vicinal Si(111) surfaces transform under thermal treatment into two very distinct morphologies, depending on the azimuthal miscut direction. A vicinal Si(111) surface disoriented toward the ½112  azimuthal direction reorganizes into a series of terraces and steps. In this case, terraces with dimensions of several tens of nanometers are separated by step bunches [18]. When vicinals are cut toward the opposite azimuthal direction ½1 12, they rearrange into a pattern of smaller terraces separated by steps that are mono- or triple-layer high. An example of these two vicinal Si(111) surfaces is given in Figure 8.3. Step arrays with spacings of only a few nanometers can be made by using higher tilt angles such as on Si(557) 3  1 [19]. Here, the regularity of the step arrays is controlled by the step–step interactions, facet energies, and lateral extension of the 7  7 reconstruction. The periodicity is directly related to the Si lattice constant. By carefully choosing the thermal treatment, surfaces with both miscut directions can be prepared in a controlled manner and highly regular arrays can be achieved. These can subsequently be used as templates to grow various arrays of nano-objects. In Section 8.3, the use of a vicinal Si(111) surface as a template for the ordered growth of Au–Co nanoparticle array will be presented. 8.2.2.3 Vicinal Si(100) Owing to the atomic configuration of Si(100), the vicinal surfaces of this orientation show very distinct reconstructions. The Si(100) surface displays a square symmetry with a surface lattice parameter of 0.384 nm. The surface is unstable because

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Figure 8.3 Scanning tunneling microscopy images of Si(111) vicinals: (a) miscut in the ½1 12 direction showing an array of single steps separated by (7  7) reconstructed terraces and (b) miscut in the ½112 direction showing a step bunch.

each atom of the surface has two dangling bonds. The surface bonds dimerize in a (2  1) reconstruction resulting in (2  1) and (1  2) domains or for some conditions in c(4  2) and p(2  2) reconstructions. We conventionally define two kinds of steps: Sa and Sb, where the upper terrace dimerization direction is perpendicular or parallel to the edge, respectively [20]. On vicinal surfaces with misorientation angles lower than 1.5 , the step heights are monoatomic and both kind of steps Sa and Sb alternate. As the misorientation angle is increased, the surface steps merge into double-height steps Db so that for angles higher than 8 , the whole surface consists of biatomic steps Db. There is a chemical equilibrium between two phases Sa þ Sb and Db while the misorientation angle is increased, leading to a transition Sa þ Sb ! Db. This step-height transition has been extensively studied theoretically [21] and experimentally [22–24]. Changing the azimuthal misorientation gives rise to an even more complex surface reconstruction as shown in Figure 8.4a and b.

Figure 8.4 Scanning tunneling microscopy images of a Si(100) surface tilted by 2.7 (a) with azimuthal misorientation varying from the left to the right side of the image by 10 ; (b) shows

an enlarged view of (a). The scanning directions are [010] and [001] and the scanned areas are, respectively, 120  120 and 12  12 nm2.

8.3 Ordered Growth of Nano-Objects

8.3 Ordered Growth of Nano-Objects 8.3.1 Growth Modes and Self-Organization

When depositing an adsorbate on a substrate, the balance of their free energies (dadsorbate, dsubstrate) and their interface free energy d interface determines their growth mode. Three main growth modes are distinguished as follows: .

.

dsubstrate < dadsorbate þ dinterface In this case, the adsorbate will form three-dimensional islands on the substrate. This mode is called the Volmer–Weber mode (Figure 8.5). It is commonly observed if a reactive material is deposited on an inert substrate, for example, a transition metal on a noble metal or an oxide. dsubstrate > dadsorbate þ dinterface In this situation, the first layer of the adsorbate will wet completely the substrate. For subsequent growth, two situations are possible. The atoms arriving at the surface will form further layers and the growth is then called the Frank–van der Merwe mode (layer by layer), or the free energy of the substrate is reduced enough (and the energy due to the parametric mismatch between the elements and the energy of dislocation formation comes into play) and the growth continues in the form of three-dimensional dots on the first wetting layer. This mode is called the Stranski–Krastanov (SK) mode (see Figure 8.5). This mode is widely used in

Figure 8.5 Three major growth modes for the heteroepitaxy of 0, 1, and 2 equivalent monolayers.

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the case of semiconductors and is essential when performing spontaneous selforganization of quantum boxes. From the standpoint of the lattice parameter mismatch, we distinguish three situations linked to the presence of dislocations: coherent growth of the adsorbate on the substrate (pseudomorphic growth), totally relaxed growth of the adsorbate, and partially relaxed growth. 8.3.2 Quantum Dots and Nanoparticles Self-Organization with Control in Size and Position

The spontaneous self-organization is an efficient way to fabricate nanostructures such as quantum dots. Two routes can be distinguished to produce these nanostructures. First, spontaneously patterned surfaces can be used for self-organized growth. It proceeds by the preferential nucleation of species on regularly spaced surface traps [25] and occurs in a range of temperature where the diffusion length of adsorbed species is sufficient to be able to reach the preferential nucleation sites. The second way to grow the dots proceeds by strain-induced self-organization, which is also very efficient to produce ordered arrays of nanostructures [26]. In that case, the energy minimization with its strain component will drive the morphological evolution of the system. An alternative route is to combine the strain-induced self-organization on naturally or artificially prepatterned substrates. In all these direct bottom-up approaches, a high density of periodic nanostructures with a remarkably uniform size distribution can be obtained at large scale with a single growth step. They provide very promising ways to integrate nanodevices in the semiconductor technology process. 8.3.2.1 Stranski–Krastanov Growth Mode As previously discussed, the SK growth mode is one the three modes generally used to classify heteroepitaxial growth using thermodynamics arguments. Also known as “layer-by-layer plus island growth,” this growth mode occurs for a system with small interface energy but large lattice misfit between the substrate and the grown film. In the early stages of growth, the film grows with the lattice parameter of the substrate and a layer-by-layer strained wetting film growth is observed. As the elastic energy increases with film thickness, the film may become unstable at a critical thickness after the growth of a certain number of smooth layers. The accumulated strain can be relieved by the formation of tridimensional (3D) islands [27]. The increase in the surface energy associated with the formation of clusters can be compensated by a decrease in the energy strain due to misfit dislocation nucleation underneath the islands. Islands that may be fully relaxed at the top, that is, with the bulk lattice parameter of the film material, are then formed. In some cases, strain relief occurs through the formation of nanoscale dislocation-free islands, also called coherent strained islands [28]. The relaxation of islands toward their bulk lattice parameter arises from local deformation of near-surface layers in the substrate. These defectfree islands are of great interest in nanoscale semiconductor technology to access to quantum properties. They are also used in the superlattice geometry [29], where

8.3 Ordered Growth of Nano-Objects

the vertical correlation between islands in successive layers may improve the island size and spacing uniformities. This self-organization phenomenon takes advantage of the elastic coupling in between the dots [30, 31] through the spacing layer. Two examples will be discussed to illustrate the SK growth mode. 8.3.2.2 Au/Si(111) System Gold grows on Si(111) substrate via the SK mechanism to form individual Au-silicide islands on an intermediate two-dimensional layer. This wetting layer corresponds to a Si-rich Au silicide. In a recent work [32], a detailed study of the structure of the nanoparticles at the atomic scale has been performed by a combined scanning tunneling microscopy (STM) and TEM analysis. Au was deposited by means of molecular beam epitaxy in an ultrahigh vacuum chamber. Depending on the deposition temperature and the Au coverage, islands with facets or hemispherical shape can grow on the substrate. The characterization of the bulk structure by TEM and the surface structure by STM reveals that islands present several crystalline structures that were all identified as metal-rich Au silicides. We point out that the entire islands exhibit the silicide structure and no dislocations have been observed. An example of the atomic structure characterization of Au–Si islands is given in Figures 8.6 and 8.7. By controlling the growth parameters, Rota et al. [33] have shown that arrays of high-density Au-rich silicide islands can be obtained on vicinal Si(111) substrates. Under thermal treatment in ultrahigh vacuum, vicinal Si(111) surfaces, misoriented by 1.5 toward the ½112 direction, form a regular array of bunches of several steps separated by terraces of 70–100 nm width. This self-organized step-bunched

Figure 8.6 Scanning tunneling microscopy image in the derivative mode of a faceted Au–Si nanoisland on Si(111), 3.5 nm in height. Added red balls on the facet correspond to the surface mesh.

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Figure 8.7 High-resolution transmission electron microscopy image of two different hemispherical Au–Si nanoislands on Si(111). Cross-sectional views (insets) in which atomic planes are visible.

Si(111) surface can be used as a template for the self-organized growth of Au-silicide islands aligned in a 1D lattice, taking advantage of a preferred nucleation at step edges. Figure 8.8 shows the dependence of the island distribution on the surface with the growth temperature for a gold coverage of 3.5 ML (1 ML corresponds to 7.84  1014 atoms/cm2). At 340  C and above, the gold atoms can reach the step bunches due to the adatom diffusion length value and all islands nucleate at step edges. Remarkably, a high density (4  1010 cm2) of nanoislands of about 20 nm in diameter is well aligned along step bunches for the 340  C temperature. As expected in self-organized epitaxial growth, the islands have a narrower size distribution compared to “classical” epitaxial growth on homogeneous surfaces. As the metal-rich Au-silicide nanoislands are supported on a Si-rich Au-silicide layer, a chemical patterning of the substrate is thus obtained through the deposition of Au. Subsequent deposition of Co leads to a magnetic patterning of the surface. The 2D and 3D gold silicides formed on the substrate react in a different way with Co: a nonmagnetic silicide is formed on terraces, while the 3D nanoparticles covered with Co exhibit magnetic properties at room temperature. These magnetic properties have been characterized by magneto-optical Kerr effect and alternating gradient force magnetometry [32, 34]. This magnetic functionalization of the surface can find applications in high-density magnetic data storage.

8.3 Ordered Growth of Nano-Objects

Figure 8.8 Scanning tunneling microscopy images (1000 nm2) of a vicinal Si(111) surface after the deposition of 3.5 equiv Au monolayers at (a) 300  C, (b) 340  C, (c) 400  C, and (d) 430  C. The Au–Si nanoislands are shown in yellow.

8.3.2.3 Ge/Si(001) System The self-organization of Ge quantum dots on Si(001) has been extensively studied in the literature [35]. The nanostructures can be first fabricated by elastic strain relaxation without applying any patterning technique. Misfit lattice strain of SiGe materials deposited on Si substrates can relax by bunching of atomic surface steps with SiGe agglomeration at the step edges or by nucleation of Ge-rich islands in the SK growth mode. Their formation mechanisms have been studied in detail [36], as well as the self-alignment and coarsening of quantum dots embedded in silicon [37]. SiGe islands may also be grown on pit-patterned Si(001) substrates, which pins the island position and suppresses lateral motion [38]. The size distribution of the objects can be advantageously decreased by stacking (and ordering) along the vertical direction arrays of SiGe/Si(001) islands [39]. 8.3.3 Wires: Catalytic and Catalyst-Free Growths with Control in Size and Position

Wires, rods, and pillars1) constitute a new type of materials to control the positioning of nanostructures. They provide new opportunities to get original heterostructures with low-dimensionality and quantum effects. The properties of homogeneous wires 1) The nano prefix is usually used to stress a small diameter (10). Rods and pillars often refer to intermediate situations. For the sake of simplicity, the generic objects will be called wires in this chapter.

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Figure 8.9 (a) Radial InGaN/GaN multiple quantum wells at the top of GaN wire templates. (b) The wire cross section with five periods [48]. (c) Longitudinal InAs/InP superlattices inserted in InAs wires. (d) A magnification of the structure [66].

have been studied intensively for both individual objects and assemblies at random or selected positions, and they can also be used as templates for heterostructure growth, making possible longitudinal or radial stackings at the top or on the edges of the wires by a careful engineering of growth conditions [40]. This has been mainly evidenced in semiconductor systems and two examples exhibiting wires grown by metal organic chemical vapor deposition with radial GaN/InGaN and longitudinal InAs/InP superlattice insertions are shown in Figure 8.9. This geometry largely facilitates the integration of objects in devices [41] and also allows fabricating 3D wire networks with controlled diameters/arrangements and effective interwire connectivity. Their specific strain tensor—directly related to the free surface and interface relaxations—plays a key role in tuning the physical properties [42] and strain engineering has been used in a large number of applications. The most striking examples come again from the semiconductor field where an impressive number of wire-based demonstrations have been achieved in electronics (transistors, capacitors, etc.), optics (emitters, detectors), thermal (thermoelectricity), mechanical (NEMS), and chemical devices (sensors) [43, 44]. Many recent advances in synthesis have been achieved using several methods: chemistry, laser ablation, molecular beam epitaxy, and chemical vapor deposition, each of them involving very different growth mechanisms (see, for example, Ref. [45] for the molecular beam epitaxy growth). In most cases, the wire growth process utilizes a metal catalyst (usually gold) defining directly the wire diameter and the so-called vapor–liquid–solid (VLS) method, which was applied for the first time more than four decades ago by Wagner and Ellis [46]. In this mechanism, a catalytic liquid alloy phase can rapidly adsorb a vapor to saturation levels and crystal growth occurs from seeds nucleated at the liquid–solid interface [47]. Depending on materials and applications, the catalyst incorporation may harm the intrinsic electrical, optical, or chemical properties of the wires. These drawbacks have motivated the catalyst-free growth of semiconductor wires, which have been demonstrated, for example, in GaN [48–50] and ZnO [51] wires. Self-organized processes requiring no ex situ surface preparation before growth have been used; they generally need the deposition of a very thin mask or

8.3 Ordered Growth of Nano-Objects

additive layers that may be spontaneously thinned or rearranged locally to allow the epitaxial growth of wires (see some examples in Refs [48, 52, 53]). Thicker masks are also very effective in localizing the growth of semiconductor nanostructures. They are obtained by various techniques, such as interferometric lithography for regular arrays, nanoimprint, and e-beam lithography. For GaN materials (Figure 8.10), the use of thick patterned mask has been essentially developed in metal organic vapor phase epitaxy (MOVPE) with dielectrics (SiO2 or Si3N4) to control the position and size of nanostructures as stripes, pyramids [54], and wires [55–57] and more recently in molecular beam epitaxy using Ti patterned masks [58]. 8.3.3.1 Strain in Bottom-Up Wire Heterostructures: Longitudinal and Radial Heterostructures Like in conventional planar heterostructures, the structural quality of the wire surface is a critical feature necessary to produce epitaxial or fully coherent layers without defects. The strain energy can be relieved in planar films, for instance, by bending thin substrates, by introducing misfit dislocations at the interface, by forming new crystalline phases and interdiffusion, by surface instability and roughness, or by a transition from 2D film growth to 3D island growth (i.e., Stranski–Krastanov mechanism). These ingredients may also be present in wire heterostructures, but for this geometry and especially for small enough diameter, the main relaxation phenomenon can occur from a strong elastic relaxation coming from the surface and interfaces. The partitioning of the strain energy is therefore strongly modified, which allows accessing to new material combinations that overcome the usual 2D limitations in terms of critical film thickness [59]. In wire heterostructures, analytical models have extended the critical thickness approach of Matthews that explains equilibrium and coherency limits in planar heterostructures to take into account the lateral relaxations occurring at the boundaries [60, 61]. For radial heterostructures (so-called core/shell wires, see Figure 8.9a and b), assuming that the system exhibits radial symmetry and no displacement in the tangential direction, the application of continuum elasticity theory to misfitstrained core/shell structures gives very simple expressions of the displacement fields for coherently strained materials [60–62]: un ðrÞ ¼ An r þ Bn =r; un ðzÞ ¼ Cn z þ Dn , where z(r) is along the length (radius) of the wire and An ; Bn ; Cn ; Dn (n ¼ c, s for the core and shell, respectively) are eight coefficients expressed as a function of Rn radii and elastic coefficients. The corresponding strain energy—deduced from the displacement fields—is generally lower than an analogous planar heterostructure. The critical values giving the limit of coherently strained structures can be calculated for a given type of defect that is assumed to nucleate first (single dislocation, dislocation loop, etc.). The stability diagrams as a function of the Rn radii have been evaluated for several core/shell systems, for instance, cubic Si/Ge [60] and hexagonal GaN/InGaN and GaN/AlGaN [62]. The validity of these models has not been tested extensively by experiments due to the difficulty of fabricating homogeneous assemblies of core/shell heterostructures, but strain has been measured in single objects by electron microscopy to confirm the expansion of the coherent stability domain and study the nature of the interfacial defects [63–65]. X-ray diffraction techniques are beginning to be applied

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Figure 8.10 Atomistic valence force field simulations of nanowire heterostructures [66, 69]. (a) In-plane deformation for a 30 nm GaN core/3 nm AlN shell wire. Hydrostatic strain for a longitudinal heterostructure 10 nm InAs/20 nm InP: (b) axial and (c) radial views.

to gain the strain tensor of wire assemblies [66, 67] and more recently on single objects [68]. These strain mapping measurements can be directly compared to finite element calculations and atomistic calculations. For compound semiconductors, semi-empirical potentials have been used [69, 70]. Figure 8.10a shows the strain profile of a hexagonal GaN/AlN core/shell heterostructure denoting a high strain relaxation at the edges with a change of sign. This last method also gives the atomic positions of anions and cations necessary to calculate electronic or transport properties in wires: this point is illustrated in Ref. [69] through electronic and optical properties of quantum dots and tunnel barriers of InAs/InP nanowire heterostructures. The number of possible coherent structures for a given pair of materials can also be increased by using longitudinal wire heterostructures. Figure 8.9c and d shows a InAs/InP longitudinal wire growth and corresponding calculated elastic relaxations with atomistic potentials are shown in Figure 8.10b. For diameter in the nanometer range, longitudinal heterostructures allow controlling quantum dot formation with electronic confinement in the three directions [71]. This geometry

8.3 Ordered Growth of Nano-Objects

enables a straightforward coupling of the quantum dots along the growth axis to tune their properties [72]. A new approach also consists of controlling the change of structures (e.g., to switch between cubic zinc-blende and hexagonal wurtzite structures [73]) or nucleating twins in chemically homogeneous wires to get original electronic band structures [74]. As in core/shell structure, the calculations of insertions or superlattices of mismatched materials along wires have been studied with strain-partitioning arguments [69, 75–78]. Such calculations predict the existence of a critical radius below which the heterostructure is expected to remain coherent for any thickness of axial epilayer, depending on the magnitude of the Burger’s vector b of the dislocation that forms (see numerical applications for several systems in Ref. [75] for b in the 0.1–0.3 nm range). In each case, experimental observations have been shown to be in agreement with the models [76] although the observation of single dislocation nucleation remains a difficult task. 8.3.3.2 Wires as a Position Controlled Template Bottom-up growth coupled with conventional technology process [79] is often used to get a collective or homogeneous behavior of the wire assembly (see Figure 8.11 for the example of GaN wires), for example, in photonic crystals and plasmonic devices. But the wire itself with well-chosen orientations of the surface facets can be also considered as a template. It allows growing quantum dots and multiple quantum wells on defect-free crystals that are sometimes difficult to get with other techniques. For example, in GaN wurtzite wires oriented along the h11 20i direction, the growth of n-GaN/InGaN/GaN/p-AlGaN/p-GaN multicolor light emission devices has been demonstrated on the f1100g and f0001g surfaces [80] and similar structures are under development on the f1010g plane edge facets of c-axis wires. These studies are motivated by the bandgap engineering of optoelectronic devices that must take into account internal and piezoelectric fields imposed by the crystallographic structures [81]: semipolar and nonpolar surface orientations may favor the quantum efficiency of the emission process, that is, the radiative recombination of electrons and holes. The integration of such devices is obtained preferentially by direct growth of vertical wire arrays to benefit from parallel integration. For sensor applications [82], where it is better to use the whole surface, a bridging method between two electrodes can also be applied. So far, horizontally aligned wires have been mostly used for research purpose by contacting them individually to inject the current or to control the tension. However, many new improvements may benefit to future device mass

Figure 8.11 GaN wires (oriented along the c-axis) grown on patterned c-sapphire substrates. Selective growth is obtained with a Si3N4 mask. (a) Partial filing of the sites and (b) complete filing.

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production in this horizontal geometry [83]. Indeed, it has been shown that wires can be transferred by alignment techniques with fluid flow in microchannel, by interactions with chemically patterned surfaces, by Langmuir–Blodgett technique, by electric and magnetic fields assisted orientation, and so on.

8.4 Conclusions

This chapter has illustrated the large variety of avenues to fabricate semiconductor templates for the growth of nano-objects. The realization of artificially prepatterned surfaces is very versatile and takes advantage of the continuous improvement in many lithography and etching techniques. These approaches have been enriched by the use of strained materials and buried stressors and by the spontaneous patterning of vicinal surfaces. The broad area of solutions with top-down, bottom-up, and mixed approaches provides solutions to build new materials for nanosciences and applications. The challenge remains to get a better control of size and positions of the nano-objects to define more accurately their electronic and optical properties. Quantum dots and nanoparticles have taken advantage of the patterning of planar surfaces. A tridimensional patterning is being developed using wires as templates. But up to now no one method has a monopoly, and suitable solutions are continuously proposed.

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the misorientation angles in direct wafer bonding. Appl. Phys. Lett., 80 (5), 793–795. Leroy, F., Eymery, J., Gentile, P., and Fournel, F. (2003) Controlled surface nanopatterning with buried dislocation arrays. Surf. Sci., 543, 211–219. Eymery, J., Buttard, D., Fournel, F., Moriceau, H., Baumbach, G.T., and L€ ubbert, D. (2002) Dislocation strain field in ultrathin bonded silicon wafers studied by grazing incidence X-ray diffraction. Phys. Rev. B, 65, 165337. Bavard, A. (2007) Nanostructuration de surface de plaques de silicium (001) par revelation d’un reseau de dislocations enterrees pour l’auto-organisation a longue distance de nanostructures. PhD dissertation, Universite Joseph Fourier, Grenoble. Leroy, F., Eymery, J., Gentile, P., and Fournel, F. (2002) Ordering of Ge quantum dots with buried Si dislocation networks. Appl. Phys. Lett., 80 (17), 3078–3080. Pascale, A., Gentile, P., Eymery, J., Meziere, J., Bavard, A., Sch€ ulli, T.U., and Fournel, F. (2006) Ge quantum dots growth on nanopatterned Si(001) surface: morphology and stress relaxation study. Surf. Sci., 600, 3187–3193. Gonzalez, C., Snijders, P.C., Ortega, J., Perez, R., Flores, F., Rogge, S., and Weitering, H.H. (2004) Formation of atom wires on vicinal silicon. Phys. Rev. Lett., 93, 126106. Sekiguchi, T., Yoshida, S., and Itoh, K.M. (2005) Self-assembly of parallel atomic wires and periodic clusters of silicon on a vicinal Si(111) surface. Phys. Rev. Lett., 95, 106101. Viernow, J., Lin, J.-L., Petrovykh, D.Y., Leibsle, F.M., Men, F.K., and Himpsel, F.J. (1998) Regular step arrays on silicon. Appl. Phys. Lett., 72, 948. Himpsel, F.J., McChesney, J.L., Crain, J.N., Kirakosian, A., Perez-dieste, V., Abbott, N.L., Luk, Y.Y., Nealey, P.F., and Petrovykh, D.Y. (2004) Stepped silicon surfaces as templates for one-dimensional nanostructures. J. Phys. Chem. B, 108, 14484. Chadi, D.J. (1987) Stabilities of singlelayer and bilayer steps on Si(001) surfaces. Phys. Rev. Lett., 59, 1691.

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j189

Part Three Characterization Techniques of Measuring Stresses on the Nanoscale

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j191

9 Strain Analysis in Transmission Electron Microscopy: How Far Can We Go? Anne Ponchet, Christophe Gatel, Christian Roucau, and Marie-José Casanove

Nanostructures are characterized by dimensions in the range of 1–100 nm in at least one direction; this definition includes 2D structures (quantum wells, thin layers), 1D structures (quantum wires, nanowires, etc.), and the so-called 0D structures (quantum dots, nanoparticles, etc.). Because of the presence of surface, interface, and different sources of inhomogeneities, these nanostructures often undergo internal stresses associated with elastic strains. Due to the size reduction and the fabrication process that are generally conducted far from equilibrium, very high stresses can be localized in small volumes. Stresses of a few GPa and strain of a few percentages are thus commonly reached. Transmission electron microscopy (TEM) is a powerful tool for structural analysis at the nanometric scale. Its originality is to combine analysis in both direct and reciprocal space. Imaging mode allows localization at the nanoscale and also gives direct and very local information on interfacial morphology and extended defects, while precise measurement of the lattice parameters is achievable through electron diffraction. This chapter will present how the structural information accessible by TEM can be transformed into stress or strain quantities. Actually this supposes to consider the strain as a variation of lattice parameters. It will be shown that one has to build mechanical models to obtain the full strain or stress tensor from these experimental data. The hypotheses made in this frame and the limits of validity of these hypotheses will be discussed. After some brief recalls on elasticity and on TEM principles, we will introduce in Section 9.1 one of the main issues encountered in TEM analysis, the thin foil relaxation effect. Sections 9.2 and 9.3 will be devoted to two methods where the thin foil relaxation can be efficiently exploited as a probe of stress or strain, the TEM curvature and the convergent beam electron diffraction (CBED); in both of these methods, the sample thicknesses, in the range of 100–500 nm, can be measured accurately. Section 9.4 will discuss some issues specific to strain determination from high-resolution electron microscopy (HREM) image analysis, which can be applied to very thin samples.

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j 9 Strain Analysis in Transmission Electron Microscopy: How Far Can We Go?

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9.1 Introduction: How to Get Quantitative Information on Strain from TEM 9.1.1 Displacement, Strain, and Stress in Elasticity Theory

In a continuous description of the matter, the strain e is a tensor of rank 2 expressing the derivative of a displacement field U with respect to a reference state. In the firstorder approximation, the strain components in a given basis are eij ¼

  1 @Ui @Uj þ @xi 2 @xj

ð9:1Þ

where Ui and Uj are the components of U in this basis. Internal forces can be represented by the means of a tensor of rank 2, the stress tensor s, which has the dimension of a force per area unit (N/m2 or Pa). In the frame of the linear elasticity, the strain and stress are linearly linked by the elasticity (or stiffness) tensor C (Hooke’s law): s ¼ Ce

ð9:2Þ

In crystals, which are anisotropic by nature, C is a tensor of rank 4 containing 3–21 independent elastic coefficients, depending on the crystal symmetry. It ensues from these definitions several important consequences on strain measurements: . .

.

Any passage from strain to stress (and conversely) supposes that the elastic coefficients of the material are known. Determining the strain state of a material supposes to define a “reference state” or “relaxed state,” the one that the material would adopt without internal or external stresses. This relaxed state is often assimilated to the bulk state, which can be problematic in case of a metastable material that does not exist in the bulk state. The continuous concepts defined here are valid only at a scale larger than the lattice spacing. When the nanocrystal is formed by a few atomic planes, the lattice distortions are better described by atomic models.

9.1.2 Principles of TEM and Application to Strained Nanosystems

Crystalline materials can be explored in both diffraction and image modes (Figure 9.1). Due to the objective lens, the electron beams that have gone through the sample can be focused. If the incident beam is parallel, the diffracted and transmitted beams can be focused in the focal plane of the objective lens. By using a suitable intermediate lens, the diffraction pattern formed in the focal plane can be projected onto the screen (diffraction mode). The main modes of image formation are schematically represented in Figure 9.1b:

9.1 Introduction: How to Get Quantitative Information on Strain from TEM

Figure 9.1 Principles of TEM. (a) Diffraction by a crystal. (b) Principle of the image formation from the transmitted beam (bright field), from a single diffracted beam (dark field), or from interference of several beams (HREM mode).

.

.

Conventional imaging: The image is formed here either by the transmitted beam (bright field) or by one of the diffracted beams (dark field). The chosen beam is selected by an aperture located in the focal plane. The contrast in the image is due to variations of diffraction conditions in the sample. HREM imaging: The contrast here results from the interference of several beams, diffracted and transmitted, that produces a phase contrast. If an (hkl) family of planes parallel to the beam has an interplanar distance dhkl larger than the point resolution of the microscope (typically 0.1–0.2 nm), the image present parallel lattice fringes with a spacing corresponding to dhkl. Most often, the electron beam is chosen parallel to a crystallographic direction with a high symmetry, called zone axis. If more than one family of lattice fringes are observed, the image looks like a 2D projection in this direction of the atomic columns of the 3D crystal.

The strain, as defined by the elasticity theory in Section 9.1.1, is never directly measured in a TEM experiment. In fact, to get information on strain, one assimilates strain components to variations of lattice parameters that can be deduced either from a direct analysis of electron diffraction patterns or from analysis of high-resolution images. 9.1.3 A Major Issue for Strained Nanostructure Analysis: The Thin Foil Effect

Transparency to electrons depends on the electron wavelength and on the extinction distance, specific to the material and the diffracting planes. Conventional imaging (dark and bright fields) and diffraction are possible up to a few hundreds of nanometers. HREM imaging requires even smaller thickness, below 50 nm. Various techniques of thinning, including mechanical polishing, chemical etching, ion milling, and focused ion beam, are used to achieve the electron transparency. Alterations like surface amorphization or partial annealing are possible, but are likely to be limited by a careful use of these thinning methods.

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Figure 9.2 Sample geometry: the direction of observation is z in plane view (left) and x in cross section (right).

More annoying for the purpose discussed here is the creation of free surfaces with a major consequence: a change in the strain state. Stress relaxation in periodic superlattices has been the object of a theoretical work in 1985 by Treacy and Gibson [1, 2]. The considered system was a [001]-oriented strained superlattice in a cubic material. Using Fourier series, the authors have analytically calculated the strain gradient generated by the cross-sectional thinning along one [100] direction of the interface plane. The main conclusions of this early study can be qualitatively generalized to a single layer: the level of relaxation is fixed by the ratio between the thickness of the foil along the thinned direction, tx, and the thickness of the strained layer, hF (Figure 9.2). Schematically, if tx is much larger than hF, the stress remains biaxial; as tx decreases, the symmetry of stress and strain tensors is reduced and one tends to a full relaxation of the stress along the direction of observation and uniaxial stress along the perpendicular direction [1, 2]. Inhomogeneous strain fields and a reduction of the average strain in agreement with the theoretical model [1, 2] have been experimentally observed in InAlAs superlattices [3]. Single layers also exhibit experimental values of strain smaller than that expected [4, 5]. Finite element modeling (FEM) is now used to calculate numerically surface relaxation effect when analytical models are not suitable [6, 7]. In the following sections, we will review the main manifestations of the strain relaxation and some possible strategies to achieve the initial strain state from a thinned sample.

9.2 Bending Effects in Nanometric Strained Layers: A Tool for Probing Stress 9.2.1 Bending: A Relaxation Mechanism

Bending is a well-known mechanism of relaxation that occurs in any bilayer system with a finite size, independent of the origin of stress (Figure 9.3a). The TEM samples do not escape this phenomenon. Figure 9.3b is a scanning electron microcopy (SEM) observation of a 10 nm Ga0.8In0.2 As layer epitaxially grown on a GaAs substrate, after it has been thinned from the substrate side for plane-view TEM observation. The foil is characterized by a strong bending of the thinned zones. Cleavage also occurred spontaneously during the thinning. In this example, bending is a clear manifestation of the relaxation of the epitaxial stress in the layer that is due to elastic accommodation

9.2 Bending Effects in Nanometric Strained Layers: A Tool for Probing Stress

Figure 9.3 Curvature of a bilayer system with a finite size. (a) Schematic representation in section. (b) SEM observation of a 10 nm Ga0.8In0.2As layer epitaxially grown on a (001) GaAs substrate after thinning by the substrate side for plane-view TEM observation.

of lattice mismatch with the substrate. This can be generalized to internal stresses of any origin. Using bending of TEM specimen as a probe of stress appears highly attractive; it requires . .

a reliable relation between the internal stress and the curvature, and a means of measurement adapted to TEM specimens.

9.2.2 Relation between Curvature and Internal Stress

Let us first consider the stress in the layer for an infinite substrate. The stress tensor can be expressed in a (x, y, z) orthogonal basis, where z is the direction normal to the surface (Figure 9.2). From mechanical considerations, it comes that each component of stress along z is null (the surface being free): 0

sxx

B s¼B @ sxy sxz

sxy

sxz

1

0

sxx

syy

C B B syz C A ¼ @ sxy

syz

szz

0

sxy

0

1

syy

C 0C A

0

0

ð9:3Þ

In addition, if the stress is isotropic in the (x, y) plane, the stress tensor is entirely characterized by a single term, the in-plane component s 0: 0

s0 s ¼ @0 0

0 s0 0

1 0 0A 0

ð9:4Þ

In crystalline layers, this situation can occur if z is a direction of high-crystalline symmetry and the stress in the (x, y) plane is biaxial. For a finite size of the substrate (Figure 9.3a), the analytical model of Timoshenko [8, 9], based on the equilibrium of the forces and of the moments, establishes a linear relation between the curvature k and the “initial” stress s 0:

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1 ES hS2 Fðd;gÞ; ¼R¼ 6ð1nS ÞhF s0 k where Fðd;gÞ ¼

1þ4gdþ6gd2 þ4gd3 þg2 d4 EF ð1nS Þ hF ; g¼ ; and d ¼ 1þd hS ES ð1nF Þ ð9:5Þ

Here R is the radius of curvature, ES is the Young’s modulus, nS is the Poisson’s ratio, and hS is the thickness of the substrate; EF is the Young’s modulus, nF is the Poisson’s ratio, and hF is the thickness of the thin film. The function F(d, g) reflects the transfer of the stress from the layer toward the substrate associated with the specimen bending [8, 9]. When the layer thickness hF is negligible with respect to the substrate thickness hS, that is, when F(d, g) tends to 1, Eq. (9.5) is known as the Stoney’s formula [10]. This calculation can be generalized for multilayered structures [11]. Determining the stress, in a layer from the substrate bending is a well-known procedure; X-ray diffraction or optical reflectometry are well-suitable techniques for measuring radius of curvature in the range of 1–100 m and are generally applied to substrates whose thickness is some hundreds of micrometers [12, 13]. Examination of Eq. (9.5) indicates a huge exaltation of the bending effect due to thinning in plane-view TEM specimens. For instance, a radius of 1 m for a 100 mm thick substrate becomes 1 mm after thinning to 100 nm. 9.2.3 Using the Bending as a Probe of the Epitaxial Stress: The TEM Curvature Method

Conventional TEM provides reliable and accurate methods to measure the curvature in the range of 10–200 mm and the foil thickness in the range of 100–500 nm [14]. In bright field (Figure 9.1b), there is a deficit of the amplitude of the transmitted beam in the places where an (hkl) family of crystal planes is in Bragg position. Due to the foil curvature, these regions are in symmetrical positions with respect to the incident beam (Figure 9.4a) and dark lines called bend contours appear in the image (Figure 9.4b). Applying the Bragg law, the radius of curvature R is very directly linked to the distance D0 between the bend contours, following R ¼ D0 =2 sin y ¼ D0 dhkl =l

ð9:6Þ

Figure 9.4 Bend contours in a curved bilayers. (a) Formation of the bend contours. (b) Bright field image in plane-view; the diffracting planes are (220) (same sample as in Figure 9.3b).

9.2 Bending Effects in Nanometric Strained Layers: A Tool for Probing Stress

where y is the Bragg angle, dhkl is the interplanar distance of the diffracting planes (hkl), and l is the electron wavelength [14]. A precision of about 2% on R is achieved. The foil thickness tz (Figure 9.2) can be measured from diffraction contours in dark field with a method derived from that used for CBED [15] and adapted to take into account the curvature [14]. This measure is very precise (better than 2%) for thicknesses in the range of a few hundreds of nanometers. An example of stress determination in a nanometric layer using the curvature is given in Figure 9.5. The thinning process has produced a slow variation of hs, which allows to fit the Timoshenko’s relation over the range of 100–300 nm. The unique adjustable parameter is the in-plane component of the stress s 0 [14]. 9.2.4 Occurrence of Large Displacements in TEM Thinned Samples

The analytical approach described above uses the simplified theory of elasticity with the hypothesis of small displacements. A simple criterion for small displacements is that the deflection, that is, the total displacement perpendicularly to the sample, is small compared to the radius of curvature. Break of the linearity between stress and curvature is predicted in case of large deflections [9] and has been observed in micrometric samples [16]. It turns out that TEM foils are much more bent than in the classical curvature method. It is thus legitimate to question the validity of the Timoshenko’s approach [17]. The occurrence of large displacements is related to the lateral extension of the

Figure 9.5 Stress determination in a 10 nm thick Ga0.8In0.2As layer on a GaAs substrate. Inset: The cleavage occurred during the planeview thinning has resulted in a rectangular lamella (here in bright field). Curve: Radius of curvature Rx and Ry, measured in bright field in

two directions, reported as function of the substrate thickness hs, measured in dark field. One obtains s 0, the in-plane component of epitaxial stress before thinning, by fitting the experimental points with Eq. (9.5) (full line). Here s 0 ¼ 1.30  0.16 GPa.

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Figure 9.6 (a) FEM of a 10 nm GaInAs layer on a GaAs substrate. The substrate thickness hs varies linearly from 80 to 220 nm. (b) Projection in the (x–y) plane of the radius of curvature R.

Top to bottom: Analytical calculation (Timoshenko’s formula); FEM when the lamella width is 2 mm; FEM when the lamella width is 4 mm (R is shown in the two main directions).

sample [9], which is not considered in Eq. (9.5). An answer can be given case-by-case using a numerical method like the finite element method. So, for a rectangular lamella with a width W, the curvature presents different regimes as the dimensionless parameter W2/(Rhs) increases [17] (independent of the intrinsic characteristics of the strained layer): . .

.

Small displacements: The Timoshenko’s relation is applicable and the curvature is isotropic (Figure 9.6). Intermediate situation (when W2/(Rhs) increases): A reduction of curvature compared to the Timoshenko’s relation and a dissymmetry of the two main curvatures appear (Figure 9.6). Very large displacements: The curvature becomes cylindrical (Figure 9.7).

Figure 9.7 FEM of an ultrathin lamella (hs ¼ 50 nm). A full cylindrical curvature is achieved. Cylindrical rollers comparable to this modeling can be observed experimentally in Figure 9.3b.

9.3 Strain Analysis and Surface Relaxation in Electron Diffraction

9.2.5 Advantages and Limits of Bending as a Probe of Stress in TEM

The TEM curvature method appears as particularly suitable to measure stresses on the order of magnitude of a few GPa in epitaxial layers [14, 18]. The principle of this method makes it a very reliable approach from the mechanical points of view [17]: . . .

. . .

The mechanical behavior of the thinned sample is completely controlled. The thin foil relaxation effect is not seen as a problem, rather it is the basis of the stress determination principle. One measures an element of stress tensor (or the force exerted by the layer on the substrate), while CBED and HREM (being sensitive to atomic positions) measure elements of strain tensor. The complete determination of the stress tensor is achieved from this single element, provided that the stress is uniform and biaxial in the film plane. The TEM curvature method does not need a reference zone in the substrate. The knowledge of the relaxed state of the strained layer is not necessary.

Due to the problematic of large displacements, attention should nevertheless be paid to the theoretical frame used to link the curvature with the stress. FEM is an essential tool to determine in which limits the analytical models can be applied [17]. The main drawback is the spatial resolution in the layer plane, on the order of magnitude of the distance between the contours of extinction (typically 1 mm) [14].

9.3 Strain Analysis and Surface Relaxation in Electron Diffraction 9.3.1 CBED: Principle and Application to Determination of Lattice Parameters

Using a convergent beam for electron diffraction experiments (CBED) instead of a parallel beam offers the combined advantages of very small probes and remarkable sensitivity to small variations of the lattice parameters (down to 104). Such properties are due to the basic principle of CBED: . .

The incident electron beam forms a cone converging onto the sample with a probe size as small as 1 nm. The sample is probed at the same time by beams having different incident angles so that many lattice planes families are simultaneously in Bragg position.

All the beams having the same angle of incidence with a given lattice plane belong to a same plane. When the incident angle is a Bragg angle for the lattice plane, all these beams are diffracted and intercept the focal plane along a line (excess line) (Figure 9.8a). As a consequence, a deficiency line parallel to the excess line is formed in the transmitted beam. In fact, due to the conical shape of the beam, the CBED

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Figure 9.8 Principle of CBED. (a) Diffraction in convergent beam. (b) Experimental diagram (filtered) at 200 kV, in a [230]-oriented silicon crystal, and simulated pattern simulated with JEMS software.

pattern is formed by disks instead of spots. The transmitted (or central) disk thus shows numerous dark lines (deficiency lines), each of them corresponding to a particular lattice plane family (Figure 9.8a). All these lines make precise angle between them and their intersection highly depends on the six unit cell parameters (a, b, c, a, b, c). Without going further into the details of the technique, let us just mention that a large part of the information included in a CBED pattern comes from the high-order Laue zones (HOLZ) of the reciprocal lattice, which means that the pattern contains 3D information. Thus, most of the deficiency lines displayed in the central disk (all the thin ones) are in fact HOLZ lines. CBED is highly valuable for investigating the structure in a given area of a specimen: this goes from local variation of the lattice parameters to complete determination of the space group. The lattice parameters are determined through the comparison with simulated patterns (Figure 9.8b). The adjustable parameters required for the simulation are as the following: .

. .

The energy of the electron beam (or microscope voltage) that can be accurately measured from a CBED pattern taken in an unstrained region of a reference crystal. The specimen thickness that can be precisely determined from the analysis of the fringes displayed in ZOLZ lines (zero-order Laue zone) of the CBED pattern [15]. The six unit cell parameters.

Note that suitable patterns require thickness in the range of 200–500 nm. Besides, energy filtering of the recorded pattern in order to suppress inelastic contributions gives enhanced precision. An example is displayed in Figure 9.8b, which shows an experimental pattern together with a simulated one.

9.3 Strain Analysis and Surface Relaxation in Electron Diffraction

9.3.2 Strain Determination in CBED

CBED patterns can be analyzed in terms of elastic strain by comparing the six lattice parameters introduced in the model to fit the experimental pattern with the ones of the relaxed material. The sensitivity can be as good as 104. In addition, the local variation of the lattice parameters in different regions of a specimen can be recorded with a high spatial resolution of a few nanometers. Such analyses have been successfully carried out in cross-sectional [19–21] as well as in plane-view specimens [22]. A more complicated situation arises when we try to investigate nano-objects and, in particular, strained epilayer with nanometer thickness. In such cases, only crosssectional investigations are likely to produce suitable patterns. However, it comes out that the CBED pattern in the epilayer is completely blurred [20]. More precisely, the CBED pattern evolves from a “classical” pattern [23], observed in the substrate at a distance far from the layer/substrate interface, to more complex patterns in which the HOLZ lines progressively change from single to multiple lines with complex profiles (Figure 9.9a). The HOLZ lines thus widen in bands, while the probe approaches the interface, until the pattern is fully blurred. To understand what happens in this case, it is useful to model the whole specimen (epilayer and substrate) with suitable thickness. Such model is displayed

Figure 9.9 [230] CBED central disks of a crosssectional Si0.8Ge0.2/Si specimen; the foil thickness is 300 nm. (a) Experimental patterns taken in the substrate at respectively 450, 250,

and 150 nm from the interface with the epilayer. (b) TDDT simulated patterns taking into account the inhomogeneous strain field calculated by FEM [23, 26].

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Figure 9.10 FEM of the inhomogeneous strain field in a cross-sectional Si0.8Ge0.2/Si specimen. (a) Map of the Uz displacements. (b) Shape of the Uz profile in the substrate at different distances from the interface.

in Figure 9.10, after free surface relaxation has been calculated through finite element modeling. Clearly, the thin foil effect results in both slight bending of the specimen free surfaces and inhomogeneous strain states in the substrate, until depth is as high as 300 nm from the interface. As the convergent beam probes the 3D reciprocal lattice, it encounters regions with different states of strain while crossing the specimen thickness at a position close to the interface. It has been shown that the complex line profiles (also called HOLZ line splitting) come from such strain inhomogeneities [24]. Fortunately, several authors [23, 25–27] have developed new approaches to simulate the interaction of the convergent beam with this kind of specimens. Such methods hence provide accurate determination of the initial state of strain in the epilayer from the analysis of the substrate at different positions below the interface, as shown in Figure 9.9b in which the simulations were performed using our home-made software [26] based on the time-dependent dynamical theory (TDDT) approach [28]. In plane view, CBED patterns obtained from an epilayer on a substrate are also affected by the inhomogeneous strain due to the bending described in Section 9.2; in such case, the thinning method has been found to influence the degree of bending [29], making the comparison with simulated patterns even more complex. 9.3.3 Use and Limitations of CBED in Strain Determination

In principle, CBED allows to explore the lattice parameters in the three directions from a single diffraction pattern. When the CBED pattern is not blurred by surface relaxation effects, a direct determination of the strain is possible with a sensitivity of 104 and a localization at the scale of a few nanometers. In the more complex situation of heavily strained regions undergoing surface relaxation effects, quantitative modeling of CBED patterns can nevertheless be performed. This requires a complete model, including the initial strain field, a good knowledge of the thinned sample geometry, the displacement field in the thinned

9.4 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils

sample, and a good knowledge of the electron probe position on the sample. Application of this principle to strained epilayers has allowed using surface relaxation in the substrate as a probe of the initial strain undergone by the epilayer, with a precision of about 10%. CBED contains some other limitations and drawbacks: it requires defect-free specimens. This is not so obvious at the scale of TEM investigations. For instance, semiconductors and eventually oxides are much better specimens for CBED than metals that contain many defects due to lower fault energies. Even more important, CBED does not avoid the need for a reference state, as described in Section 9.1.1. 9.3.4 Nanobeam Electron Diffraction

In principle, lattice parameters can be determined simply from the position of the spots in electron diffraction diagrams performed with a parallel beam. However, the large illumination of the sample in the diffraction mode inhibits its use for local strain determination. Nowadays the nanobeam electron diffraction (NBED) mode allows to achieve beams almost parallel to a probe as small as a few nanometers. Strain variation of about 103 can be detected at the scale of 10 nm [30]. Recently a resolution of about 2.7 nm and a sensitivity of 6.104 have been reported [31]. As CBED, NBED necessitates a comparison of experimental and simulated strain profiles to take into account the thin foil relaxation phenomena [31] and it does not escape the use of a reference state.

9.4 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils 9.4.1 Principle

An HREM image can be considered as a sum of individual lattice fringes, each set of fringes displaying its own spatial frequency g. Strain determination consists in measuring precisely the lattice fringes spacing in an HREM image and determining a 2D displacement field u compared to a reference area in the image, displaying lattice fringes of well-known spacing (Figure 9.11). Two techniques are used: . .

The position of each “atomic column” can be determined directly in the image using a peak finding procedure [32, 33]. The variation in the fringe spacing from one region to another can also be measured in the Fourier space, through Fourier filtering around the spatial frequency g of interest (geometrical phase analysis (GPA)) [34]. The inverse Fourier transform after Fourier filtering produces a complex image, whose phase

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Figure 9.11 HREM analysis of a strained InAs quantum well inserted between two Ga0.47In0.53As barriers with a nominal lattice mismatch of 0.032 [5]. (a) h110i zone axis image and map of uz, the displacements in the growth direction z (GPA method). (b) Profile of uz in the growth direction z.

Pg displays the local variation Dg of the spatial frequency (phase shift) in each point of the image. The local displacement u of the fringes is directly related to the phase shift. Pg ¼ 2pg  u

ð9:7Þ

When two different sets of fringes are selected, each having its own spatial frequency g1 and g2, two components of displacement (u1, u2) can be determined. It is thus obtained a 2D displacement field U2D, whose derivative with respect to the spatial positions xi, xj (in an orthogonal basis) defines a 2D “strain“ tensor e2D: ! 2D 1 @Ui2D @Uj 2D ð9:8Þ eij ¼ þ @xi 2 @xj Strain determination thus consists in assimilating this 2D “strain” field, measured from the image, to the 2D projection of the strain tensor e occurring in the crystal, as defined in Section 9.1.1 (or assimilating the 2D displacement field U2D, measured from the image, to the 2D projection of the 3D displacement field U occurring in the crystal). The smallest displacement that can be measured is around 1 pm in optimal conditions [35] (in practice, it can be larger due to different sources of noise). The spatial resolution is not limited by the point resolution of the microscope (that determines which lattice fringes can be analyzed), but rather due to the image analysis principle. For peak finding procedures, averaging over neighboring columns can be necessary to improve the signal-to-noise ratio, which impacts the spatial resolution. For GPA, the spatial resolution is intrinsically limited by the use of a mask selecting the g frequency in the Fourier space of the image [34]. Using the largest filter, the resolution cannot be better than two times the distance between the analyzed fringes (in a semiconductor, this corresponds to about 0.6 nm when (002) planes are analyzed).

9.4 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils

9.4.2 What Do We Really Measure in an HREM Image?

The very good resolution made HREM highly attractive for strain determination in nano-objects. Nevertheless, despite its apparent simplicity, this is not a straightforward method. Actually, issues of different nature are hidden in the approach consisting in assimilating the 2D strain field e2D determined in the image to the 2D projection of the actual 3D strain field. 9.4.2.1 Image Formation Independent of the strain issue, considering the lattice fringes as the 2D projection of the atomic planes throughout the whole crystal is an approximation. In fact, two main sources of artifacts are possible when analyzing image contrasts: .

.

Inhomogeneities in the specimen: Due to the nature of the contrast in HREM, any change in the beam phase, as those due to thickness gradients or bending in the thinned sample, shifts the position of the lattice fringes in the image. Transfer function of the objective lens: Spherical aberration of the lens is specifically responsible for delocalization effects near interfaces (the lattice fringes in the images are shifted with respect to the exact projection of the lattice planes in the crystal).

These points can be fully or partially controlled by a careful preparation of the thinned sample, the use of new generation of TEM as those equipped with a corrector of spherical aberration, and the help of image simulation [6, 7]. 9.4.2.2 Reconstruction of the 3D Strain Field from a 2D Projection Let us express the strain tensor by the following matrix in a (x, y, z) orthogonal basis, where x is the direction of observation (Figure 9.2): 0

exx e ¼ @ exy exz

exy eyy eyz

1 exz eyz A ezz

ð9:9Þ

The components eyy, ezz, and eyz are, in principle, related to the components of the strain tensor e2D determined from the HREM image, as described in Section 9.3.1 (Eq. (9.8)), but the other components (along the direction of observation) are missing data. To reconstruct the full strain tensor, several steps are generally necessary. To illustrate this reconstruction step-by-step, let us consider the simple and usual situation displayed in Figure 9.11 of a [001]-oriented layer in a cubic system, grown on a cubic substrate with a lattice mismatch f. We suppose that the in-plane symmetry is preserved by the epitaxial growth (for a more general approach including any orientations, see Ref. [36]). y is the in-plane direction perpendicular to x and z is the growth direction. .

Removing the reference for zero strain: Measurements in the image are carried out by comparing the lattice fringes spacing in the area of interest with their

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spacing in a reference area supposed unstrained. In practice, this reference is not always the studied material in its relaxed state, but it is most often another material. To deduce the strain tensor e, one has to first remove a contribution due to this reference. The reference is often the substrate; in such case, its contribution is expressed in terms of the lattice mismatch f between the relaxed material and the substrate: eyy ¼ e2D yy f

and ezz ¼ e2D zz f

ð9:10Þ

It is worth noting that f is one of the unknown values that one intends to measure. . Elaborating a physical model of deformation: To go further, one needs a model of deformation including some hypotheses. Here, a simple model is that the layer is fully strained so that the in-plane strains are imposed by the lattice mismatch (exx ¼ eyy ¼ f ) and the stress is biaxial (s xx ¼ syy ¼ s 0) as in Eq. (9.4). . Use of a law of elastic behavior: A law of elastic behavior, here the Hooke’s law (9.2) relating the strain and stress through the elastic coefficients, is also necessary. By application of Eq. (9.2), it comes ezz ¼ 2

C12 exx C11

and f ¼

e2D zz 12ðC12 =C11 Þ

ð9:11Þ

The problem, that is, determining of both the relaxed and strained states of the layer from the image analysis, is now completely solved, on the basis of several hypotheses: . .

. .

The experimental reference of “zero” strain is supposed perfect and unstrained. Here, a full elastic accommodation of the lattice mismatch has been supposed. Other models, including interfacial dislocations or anisotropic accommodation, are possible. The law of elastic behavior necessitates the knowledge of the elastic properties of the strained layer (here the Cij). The status of the x-direction in the model is probably the most delicate point, as detailed in the next section.

9.4.3 Modeling the Surface Relaxation in an HREM Experiment 9.4.3.1 Full Relaxation (Uniaxial Stress) In the model developed above, surface relaxation was ignored. A full relaxation implies s xx ¼ 0 and leads to a different solution. In case of a h110i direction of observation, exx

¼

2 C11 ðC11 þ C12 2C44 Þ2C12 2 f C11 ðC11 þ C12 þ 2C44 Þ2C12

eyy

¼

f

ezz

¼

4C12 C44 f 2 C11 ðC11 þ C12 þ 2C44 Þ2C12

ð9:12Þ

9.4 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils

With typical values of Cij in semiconductors, the ezz component and the measured strain e2D zz are now reduced by about 50% and 30%, respectively, compared to the biaxial case. 9.4.3.2 Intermediate Situations: Usefulness of Finite Element Modeling Intermediate situations can be evaluated either analytically in some cases as proposed by Treacy and Gibson [1, 2] or numerically using atomistic modeling or FEM [5–7]. Figure 9.12a displays, for instance, displacement profiles across a 5 nm InAs strained layer embedded in two GaInAs buffers lattice matched to InP. The nominal lattice mismatch is 0.032. The profiles are calculated by FEM for two foil thickness tx (45 and 13 nm) and compared with the nominal displacement without relaxation [5]. The two main features are as the following: . .

The transfer of stress from the layer to the buffers, which is attested by the occurrence of negative displacements within the buffers. The decrease of the strain within the layer.

This decrease reaches 10% when the foil thickness is 20 times the layer thickness (Figure 9.12b). This indicates that the situations where the relaxation can be neglected are very limited. The reduction of 50% for a full relaxation is in agreement with the analytical approach above. 9.4.3.3 Thin Foil Effect: A Source of Incertitude in HREM This approach encounters nevertheless some limits: .

While the methods based on extinction contours are very efficient and precise to determine “large” thicknesses (in the range of 100–500 nm for semiconductors),

Figure 9.12 FEM of surface relaxation in a 5 nm InAs layer embedded in two Ga0.47In0.53As buffers [5]. The nominal lattice mismatch is 0.032. (a) Out-of-plane displacements for two foil thicknesses tx (45 and 13 nm) compared

with the theoretical profile without relaxation. (b) Normalized out-of-plane strain within the layer as a function of the ratio foil thickness (tx) over layer thickness (hF).

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Figure 9.13 Out-of-plane displacements (same sample as in Figure 9.12) modeled with various boundary conditions allowing thin foil effect in one, two, or three directions [5]. (a) Perfect relaxation, only in the x-direction (the sample has a finite size tx of 13 nm along x and is infinite along y). (b) Additional relaxation

.

is allowed, thanks to a free surface in the y-direction, at a distance ty of 400 nm. In addition, the out-of-plane displacements at the substrate side are either forbidden (full line) or free (dashed line). (c) Same as (b), with ty of 30 nm.

they fail below a few tens of nanometers. Unfortunately, this corresponds to the range of thicknesses suitable for HREM. In many cases, the foil thickness can just be estimated, which has serious consequences on the precision of the measured strain. Due to the very small thickness, it is difficult to fully control the shape of an HREM foil. The examined zone can be “lacework-like,” cleaved, with an inhomogeneous thickness, and so on. The calculated displacements are sensitive to any of these parameters. Figure 9.13a–c shows, for instance, modifications of the displacements when additional thin foil effects are permitted in addition to the main relaxation along the direction of observation. So, the concept of “perfect relaxation” generally adopted in the modeling should be questionable [5].

9.4.4 Conclusion: HREM is a Powerful but Delicate Method of Strain Analysis

HREM image analysis is until now the most local TEM method available to explore strain at the nanoscale and it presents a good sensitivity to displacements. Use of new generation of TEM, like those with a corrector of spherical aberration, makes it even more attractive. Nevertheless, it is not a direct method to measure strain. On the contrary, it implies a careful mechanical analysis to transform the experimental data – the relative distortions of lattice fringes in a 2D image – into components of a volume strain tensor. Among the various hypotheses chosen for this purpose, the surface relaxation along the thinning direction is particularly susceptible to degrade the reliability of the absolute measurement. Comparison of experimental and simulated displacements is thus necessary to evaluate the precision of the measured strain. Also note that the boundary conditions adopted in modeling are not necessary representatives of the actual situation. So, the concept of “perfect relaxation” generally adopted in the modeling should be questionable.

9.5 Conclusions

9.5 Conclusions

With TEM, we have at our disposal a variety of methods to explore internal strain in nanometric crystalline objects. TEM combines a good sensitivity with a fine spatial resolution. Technical improvements in various fields (lenses, energy filtering, image analysis, etc.) have recently upgraded the performances of quantitative TEM; new promising techniques like nanobeam electron diffraction [31] and dark field holography (HoloDark) [37] are in development to explore strain with resolution as good as 3–4 nm. TEM is nevertheless not a direct method for strain measurement. For this objective, the ability of TEM to explore crystalline materials is exploited: the physical quantities that are experimentally determined are essentially related to lattice parameter variations. To go from these experimental data to the more abstract concept of strain, one has to build mechanical models. They include some or all of the following components: 1) 2) 3) 4) 5)

A law of elastic behavior relating strain and stress in the material under study, including a relaxed state of the material. A model of mechanical loading reproducing the physical origin of stress in the sample. A geometrical model of the thinned sample. Boundary conditions. An experimental reference of zero strain.

The validity and the precision of strain or stress assessment depend on the hypothesis made at each step. While the first two components involve physical choices describing the studied system, the others (specimen geometry, boundary conditions, and reference of zero strain) fully depend on the thinning required for TEM experiment. Indeed, in this approach, one of the most critical points is that TEM measurements are performed on thinned samples, which implies that the strain state has been modified by surface relaxation (thin foil effect). Fortunately, if the sample geometry and boundary conditions are known with precision, the stress or strain before thinning can be calculated with a good reliability, using numerical calculation. Surface relaxation mechanisms can even be used as tools probing the stress before thinning. If the sample geometry and boundary conditions are not known with precision, surface relaxation constitutes one of the main weaknesses of TEM quantitative analysis of strain. In this sense, quantitative analysis of strain methods using relatively thick foils with well-characterized shapes (CBED, TEM curvature, and HoloDark) are more reliable than those using very thin foils (HREM). Nevertheless, HREM remains a highly valuable method, a unique one allowing the strain profile analysis at the scale of the atomic planes. Keeping in mind these limitations, TEM appears as one of the most powerful techniques for a local investigation of stressed crystalline nanostructures. It offers

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various approaches especially well adapted to strained nanometric objects like quantum dots or epitaxial layers.

Acknowledgment

The authors are very pleased to thank N. Combe for critical reading of this chapter.

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2

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4

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7

The effects of elastic relaxation on transmission electron-microscopy studies of thinned composition-modulated materials. J. Vac. Sci. Technol. B, 4, 1458–1466. Treacy, M.M.J., Gibson, J.M., and Howie, A. (1985) On elastic relaxation and long wavelength microstructures in spinodally decomposed InxGa1xAsyP1y epitaxial layers. Philos. Mag. A, 51, 389. Seitz, H., Seibt, M., Baumann, F.H., Ahlborn, K., and Schroter, W. (1995) Quantitative strain mapping using highresolution electron-microscopy. Phys. Status Solidi A, 150, 625–634. Robertson, M.D., Currie, J.E., Corbett, J.M., and Webb, J.B. (1995) Determination of elastic strains in epitaxial layers by HREM. Ultramicroscopy, 58, 175–184. Gatel, C., Tang, H., Crestou, C., Ponchet, A., Bertru, N., Dore, F., and Folliot, H. (2010) Analysis by highresolution electron microscopy of elastic strain in thick InAs layers embedded in Ga0.47In0.53As buffers on InP(001) substrate. Acta Mater., 58, 3238. Tillmann, K., Lentzen, M., and Rosenfeld, R. (2000) Impact of column bending in high-resolution transmission electron microscopy on the strain evaluation of GaAs/InAs/GaAs heterostructures. Ultramicroscopy, 83, 111. Rosenauer, A., Gerthsen, D., and Potin, V. (2006) Strain state analysis of InGaN/ GaN: sources of error and optimized imaging conditions. Phys. Status Solidi A, 203, 176–184.

8 Timoshenko,

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S. (1925) Analysis of bi-metal thermostats. J. Opt. Soc. Am., 11, 233–255. Freund, L.B., Floro, J.A., and Chason, E. (1999) Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations. Appl. Phys. Lett., 74, 1987–1989. Stoney, G.G. (1909) The tension of metallic films deposited by electrolysis. Proc. R. Soc. Lond. A, 82, 172–175. Feng, Z.C. and Liu, H.D. (1983) Generalized formula for curvature radius and layer stresses caused by thermal strain in semiconductor multilayer structures. J. Appl. Phys., 54, 83–85. Chason, E., Yin, J., Tetz, K., Beresford, R., Freund, L.B., Gonzalez, M.U., and Floro, J.A. (2000) In situ measurements of stress relaxation during strained layer heteroepitaxy, in Self-Organized Processes in Semiconductor Alloys, vol. 583 (eds D. Follstaedt, B. Joyce, A. Mascarenhas, and T. Suzuki), Materials Research Society, pp. 167–175. Sander, D., Enders, A., and Kirschner, J. (1995) A simple technique to measure stress in ultrathin films during growth. Rev. Sci. Instrum., 66, 4734–4735. Ponchet, A., Cabie, M., and Rocher, A. (2004) TEM measurement of the misfit stress by a curvature method in semiconducting epitaxial system. Eur. Phys. J., 26, 87. Kelly, P.M., Jostsons, A., Blake, R.G., and Napier, J.G. (1975) Determination of foil thickness by scanning transmission

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electron microscopy. Phys. Status Solidi A, 31, 771–780. Finot, M. and Suresh, S. (1996) Small and large deformation of thick and thin-film multi-layers: effect of layer geometry, plasticity and compositional gradients. J. Mech. Phys. Solids, 44, 683–721. Cabie, M., Ponchet, A., Rocher, A., Durand, L., and Altibelli, A. (2005) Geometrical criteria required for the determination of the epitaxial stress from the transmission electron microscopy curvature method. Appl. Phys. Lett., 86, 191901. Cabie, M., Ponchet, A., Rocher, A., Paillard, V., and Vincent, L. (2004) Transmission electron microscopy and Raman measurements of the misfit stress in a Si tensile strained layer. Appl. Phys. Lett., 84, 870. Toda, A., Ikarashi, N., and Ono, H. (2000) Local lattice strain measurements in semiconductor devices by using convergent-beam electron diffraction. J. Cryst. Growth, 210, 341–345. Toh, S.L., Li, K., Ang, C.H., Er, E., Redkar, S., Loh, K.P., Boothroyd, C.B., and Chan, L. (2004) High spatial resolution strain measurement of deep sub-micron semiconductor devices using CBED. IPFA 2004: Proceedings of the 11th International Symposium on the Physical & Failure Analysis of Integrated Circuits, pp. 143–146. Senez, V., Armigliato, A., De Wolf, I., Carnevale, G., Balboni, R., Frabboni, S., and Benedetti, A. (2003) Strain determination in silicon microstructures by combined convergent beam electron diffraction, process simulation, and micro-Raman spectroscopy. J. Appl. Phys., 94, 5574. Cherns, D., Kiely, C.J., and Preston, A.R. (1988) Electron diffraction studies of strain in epitaxial bicrystals and multilayers. Ultramicroscopy, 24, 355. Houdellier, F., Roucau, C., Clement, L., Rouviere, J.L., and Casanove, M.J. (2006) Quantitative analysis of HOLZ line splitting in CBED patterns of epitaxially strained layers. Ultramicroscopy, 106, 951. Chou, C.T., Anderson, S.C., Cockayne, D.J.H., Sikorski, A.Z., and

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Vaughan, M.R. (1994) Surface relaxation of strained heterostructures revealed by Bragg line splitting in LACBED patterns. Ultramicroscopy, 55, 334. Clement, L., Pantel, R., Kwakman, L.F.T., and Rouviere, J.L. (2004) Strain measurements by convergent-beam electron diffraction: the importance of stress relaxation in lamella preparations. Appl. Phys. Lett., 85, 651. Houdellier, F., Altibelli, A., Roucau, C., and Casanove, M.J. (2008) New approach for the dynamical simulation of CBED patterns in heavily strained specimens. Ultramicroscopy, 108, 426. Alfonso, C., Alexandre, L., Leroux, C., Jurczak, G., Saikaly, W., Chara€ı, A., and Thibault-Penisson, J. (2010) HOLZ lines splitting on SiGe/Si relaxed samples: analytical solutions for the kinematical equation. Ultramicroscopy, 110, 285. Gratias, D. and Portier, R. (1983) Time-like perturbation method in high-energy electron diffraction. Acta Crystallogr. A, A39, 576. Houdellier, F., Jacob, D., Casanove, M.J., and Roucau, C. (2008) Effect of sample bending on diffracted intensities observed in CBED patterns of plan view strained samples. Ultramicroscopy, 108, 295. Usuda, K., Mizuno, T., Tezuka, T., Sugiyama, N., Moriyama, Y., Nakaharai, S., and Takagi, S. (2004) Strain relaxation of strained-Si layers on SiGe-on-insulator (SGOI) structures after mesa isolation. Appl. Surf. Sci., 224, 113–116. Beche, A., Rouviere, J.L., Clement, L., and Hartmann, J.M. (2009) Improved precision in strain measurement using nanobeam electron diffraction. Appl. Phys. Lett., 95, 123114. Jouneau, P.H., Tardot, A., Feuillet, G., Mariette, H., and Cibert, J. (1994) Strain mapping of ultrathin epitaxial ZnTe and MnTe layers embedded in CdTe. J. Appl. Phys., 75, 7310. Bierwolf, R., Hohenstein, M., Phillipp, F., Brandt, O., Crook, G.E., and Ploog, K. (1993) Direct measurement of local lattice distortions in strained layer structures by HREM. Ultramicroscopy, 49, 273.

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34 Hytch, M.J., Snoeck, E., and Kilaas, R.

36 Yang, K., Anan, T., and Schowalter, L.J.

(1998) Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy, 74, 131. 35 Hytch, M.J., Putaux, J.L., and Penisson, J.M. (2003) Measurement of the  displacement field of dislocations to 0.03 A by electron microscopy. Nature, 423, 270.

(1994) Strain in pseudomorphic films grown on arbitrarily oriented substrates. Appl. Phys. Lett., 65, 2789–2791. 37 Hytch, M., Houdellier, F., Hue, F., and Snoeck, E. (2008) Nanoscale holographic interferometry for strain measurements in electronic devices. Nature, 453, 1086–1089.

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10 Determination of Elastic Strains Using Electron Backscatter Diffraction in the Scanning Electron Microscope Michael Krause, Matthias Petzold, and Ralf B. Wehrspohn

10.1 Introduction

The understanding and management of stress and strain is of fundamental importance in the design and implementation of novel materials and manufacturing processes in a wide range of applications. While stress due to external loads can be calculated with a certain degree of accuracy, residual stress, which remains in a body that is stationary and at equilibrium with its surroundings, is much more difficult to predict. To date, a large number of residual stress measurement techniques are available. Some of these techniques are destructive, while others can be used without significantly altering the component; some have excellent spatial resolution, whereas others are restricted to near-surface stresses or to specific classes of materials. However, even if these methods, for instance, X-ray diffraction, Raman spectroscopy, or TEM, have proved to provide acceptable measurements of the residual stress in certain cases, there is still a strong demand for alternative techniques that can provide a means of determining local strains with high spatial resolution on the one hand and high strain sensitivity on the other hand. In this chapter, we will show how electron backscatter diffraction (EBSD) can be used to determine the local state of stress and strain in a region of a material. Developed as an additional characterization technique to a scanning electron microscope (SEM), the technique has passed through various stages of development and has experienced rapid acceptance in research and industry in the past decade. Nowadays, EBSD is most widely used for orientation determination [1–3], discrimination of unknown crystalline phases [4–6] (see also Figure 10.1a), and orientation mapping (Figure 10.1b) on the surfaces of bulk polycrystals. Since this technique is based on recording electron backscatter diffraction patterns (EBSP)1) that represent all angular relationships in the crystal, it is obvious to use it for the determination of local elastic strains.

1) In literature, the terms “electron backscatter diffraction pattern (EBSP),” “backscatter Kikuchi pattern (BKP),” and “backscatter electron Kikuchi pattern (BEKP)” are often used interchangeably. Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Figure 10.1 (a) EBSD phase map and orientation map (inverse pole figure representation) of intermetallic compounds formed in a gold–aluminum bond wire

interconnect measured with a step size of 50 nm; (b) orientation map (inverse pole figure representation) of a polycrystalline Si wafer surface measured with a step size of 5 mm.

10.2 Generation of Electron Backscatter Diffraction Patterns

In order to understand the principles of strain determination using electron backscatter diffraction, it is necessary to model the physical processes that lead to the formation of the characteristic diffraction features: Kikuchi lines and bands. Although the dynamical theory of electron diffraction is needed to explain the exact intensity distribution in an electron backscatter diffraction pattern [7–10], a simplified model based on the kinematic approximation can also be used to explain the geometrical relations in the observed network [11]. This model will be presented briefly. An EBSP is generated on a detector screen by backscatter diffraction of a stationary beam of high-energy electrons that is focused on a small area of a crystalline solid. As the beam enters the sample, the electrons are subject to a diffuse inelastic scattering in all directions. This means that there must be always some electrons that impinge on a particular set of parallel lattice planes at the Bragg angle yB and undergo elastic scattering to give a reinforced beam. Since diffraction of the electrons through the Bragg angle is occurring in all directions, the locus of the diffracted radiation is the surface of two cones that extend symmetric around the normal of the reflecting atomic planes, separated by twice the Bragg angle (Figure 10.2b). If some sort of recording medium is positioned so as to intercept these diffraction cones, a pair of parallel conic sections results in hyperbolas that are essentially seen as two straight parallel lines, known as Kikuchi lines. The distance between a pair of Kikuchi lines is a function of the Bragg angle that is inversely proportional to the interplanar spacing dhkl . The geometry of the EBSP can be interpreted as a gnomonic projection of the

10.3 Strain Determination Through Lattice Parameter Measurement

Figure 10.2 (a) Schematic of a typical EBSD setup, showing the pole piece of the SEM, the tilted specimen (in this case, using a pretilted sample holder), and the detector. (b) Diffraction cones with respect to a reflecting plane separated by twice the Bragg angle.

crystal lattice on the flat detector screen with the center of the projection given by the point of impingement of the primary electron beam. The instrumentation for generating and capturing electron backscatter diffraction patterns usually consists of three main parts: the SEM, the pattern acquisition device, and the software (Figure 10.2a). The EBSP is commonly formed on a transparent phosphor screen (approximately 4 cm in diameter) that is positioned parallel to the primary beam and the tilt axis of the specimen. To enhance the proportion of backscattered electrons that are able to undergo diffraction and to escape from the specimen surface, the sample and the incident electron beam draw an angle of typically 20 . Nowadays, the pattern is recorded through a lead glass window from outside the specimen chamber using a high-sensitivity CCD camera that can produce binned images on the order of 100  100 pixels at a rate of more than 600 images per second. In most EBSD systems, the acquisition device is mounted on a retractable stage. The phosphor screen is generally matched to the spectral response of the CCD sensor and is covered by a thin, conductive coating (aluminum or tin–indium–oxide). This coating enhances the brightness of the phosphor by reflecting light back toward the camera, absorbs low-energy electrons, and reduces charging of the phosphor screen.

10.3 Strain Determination Through Lattice Parameter Measurement

As already mentioned, EBSD is widely used to determine local grain structure and texture of polycrystalline materials. In contrast, elastic strain measurement from EBSD patterns (EBSP) is not yet a standard routine. Nowadays, there is much activity in this area and thus many different approaches to measure elastic strains can be found in the literature. In general, elastic strains can be obtained using two distinct

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approaches: the direct measurement of lattice parameters on the one hand and determination of changes in the crystal structure with respect to a known reference position on the other hand. Due to the comparatively poor angular resolution of typical EBSD systems and the absence of higher-order diffraction lines and the low contrast of the Kikuchi bands, the direct measurement of the Kikuchi bandwidth from EBSPs, in order to determine the lattice plane spacing, cannot attain the precision required to enable strain measurement [12]. Moreover, it has been shown that the determination of changes of interplanar angles, which are of course measured in the conventional automated analysis of EBSD patterns for crystal orientation analysis, would not lead to a sufficient strain sensitivity for the same reasons [12]. To overcome the limitations of strain determination using direct measurement of Kikuchi band position mentioned above, Maurice and Fortunier developed an interesting approach that accounts for the hyperbolic nature of the Kikuchi band edges and thus allows greater precision in locating the bands [13]. They proposed to extend the standard procedure for the detection of Kikuchi bands based on the Hough transform that is widely used in commercial EBSD software to a three-dimensional space using a generalized 3D Hough transform. This transform is obtained from the derivative of an EBSP by taking into account the wave vector of the diffracted electron beam and the reciprocal vector associated with the particular (hkl) plane. By this means, the Bragg angle to which this pixel contributes is calculated and the intensity of each pixel is added to a 3D voxel. The corresponding 3D Hough space has axes composed of the base parameters r and y, but with an additional ordinate axis proportional to the width of the band, that is, sinðyÞ. Assuming that the crystal orientation is known, a maximum intensity search in fairly small volumes of the 3D Hough space is performed to obtain the three parameters describing the Kikuchi line hyperbola and thus its exact position. In a proof of concept, the authors demonstrated a strain sensitivity of 2  104 in case of geometrically calculated diffraction patterns. In general, the band detection accuracy is impaired by image distortions due to the camera optics and accurate knowledge of the position of the point source is required. Even if no measurements on real patterns are reported so far, the methodology shows promise though a reduction in strain sensitivity due to intensity asymmetries across the bands could be expected.

10.4 Strain Determination Through Pattern Shift Measurement 10.4.1 Linking Pattern Shifts to Strain

Beside attempts to detect intense elastic strain gradients through a blurring of the EBSD patterns [14, 15], strain measurement using the cross-correlation-based pattern shift analysis could be seen as the most mature one. Introduced by Troost et al. [16] and Wilkinson et al. [17], the approach relies on the fact that elastic strains and small rotations cause small shifts in features within the Kikuchi patterns. These

10.4 Strain Determination Through Pattern Shift Measurement

Figure 10.3 Schematic showing how a strain and a rotation in the crystal lattice can be related to a shift of an EBSD pattern on the detector screen (reference crystal: cubic; strained crystal: tetragonal distorted and rotated).

shifts can be measured using cross-correlation techniques and be related to the strain and the rotation tensor. Due to its high application potential, this methodology will be described in the following section in detail. The general principles of strain analysis using the “pattern shift” approach are as follows: One pattern from the series to be analyzed is chosen as a reference pattern, whereas this reference pattern should come from a point on the sample for which the strain is known or is assumed to be zero. A set of square subsections is defined across the patterns and shifts relative to positions in the reference pattern are determined for all the subsections/patterns in the series using 2D cross-correlation functions. To link the measured pattern shifts to the present strain state, Wilkinson et al. [18] used the displacement gradient tensor A ¼ ru describing the deformation mapping r onto r0 (Figure 10.3) so that r0 ¼ Ar ð10:1Þ (u-displacement at position x). The displacement of the crystal lattice Q caused by strain and rotation is related to A by Q ¼ r0 r ¼ ðAIÞr ð10:2Þ (I is unit matrix). Considering the fact that EBSD measures only the projection of Q perpendicular to r denoted by q, it follows q ¼ Qlr ¼ ½Aðl þ 1ÞIr

ð10:3Þ

(l is unknown scalar). Measurement of q for a given r gives three equations from which l can be eliminated. Expanding and transposing Eq. (10.3) yields the following two simultaneous equations: 2 3 @u @u @u1 @u1 2 @u3 @u3 1 35 þ r2 r3 r1 r3 4  þ r32 r r1 r2 ¼ r3 q1 r1 q3 @x1 @x3 @x2 @x3 1 @x1 @x2 ð10:4Þ 2 3 @u @u @u @u @u @u 2 3 2 2 3 3 2 2 5 þ r1 r3 r2 r3 4  þ r3 r1 r2 r ¼ r3 q2 r2 q3 @x2 @x3 @x1 @x3 @x1 2 @x2

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Measuring q for four widely spaced directions r allows an exact solution to be determined for the system of linear equations, giving eight of the nine degrees of freedom within the gradient displacement tensor A. The remaining ninth degree corresponds to hydrostatic dilatation that cannot be obtained using this method since a simple change in the widths of Kikuchi bands generates no change in the interplanar angles and thus does not cause any shift of a pattern feature in the EBSP. In order to improve the statistic of shift measurement for each pattern, shifts could be measured at more than four positions. Then, accordingly overdetermined system of equations has to be solved for an approximate solution in a least square of error sense. The last degree of freedom can be determined by imposing the conditions of a plane stress. Due to the fact that EBSPs are generated very close to the surface of a sample, it is reasonable to assume that the normal stress s 33 perpendicular to the free surface approximates to zero [19]. We can write s33 ¼ 0 ¼ C33kl ekl

ð10:5Þ

where Cijkl are the elastic constants referred to the sample axis system. By this means, an additional equation is available allowing separation of all three normal strains. Since the displacement gradient tensor A contains information about strains and rotations, it has to be decomposed into a symmetric part and an antisymmetric part using the transpose of A denoted by AT . It follows 1 1 A ¼ ðA þ AT Þ þ ðAAT Þ ¼ e þ v 2 2

ð10:6Þ

Thus, we can describe strain by 2

2

e11 e ¼ 4 e12 e13

e12 e22 e23

 3   @u1 1 @u1 @u2 1 @u1 @u3 þ þ 6 @x @x1 @x1 7 2 @x2 2 @x3 6 1 7 6 7 3 6  7    e13 6 1 @u1 @u2 7 @u 1 @u @u 2 2 3 6 7 þ e23 5 ¼ 6 2 @x þ @x 7 2 @x @x @x 2 1 2 3 2 6 7 e33 6 7 6  7    6 1 @u1 @u3 7 1 @u @u @u 2 3 3 4 5 þ þ @x1 @x2 @x3 2 @x3 2 @x3

while the lattice rotation can be described using the rotation tensor  3   2 1 @u1 @u2 1 @u1 @u3 0   6 2 @x2 @x1 2 @x3 @x1 7 7 6 7 2 3 6 7 6     0 v12 v13 6 1 @u2 @u1 7 1 @u @u 2 3 7 6 0   4 5 v ¼ v12 0 v23 ¼ 6 2 @x @x 2 @x3 @x2 7 1 2 7 6 v13 v23 0 6 7 6  7    6 1 @u3 @u1 7 1 @u @u 3 2 5 4  þ 0 2 @x1 @x3 2 @x2 @x3

10.4 Strain Determination Through Pattern Shift Measurement

10.4.2 Measurement of Pattern Shifts

Detecting the shift of a digital image or certain features in a series of similar images is a well-studied topic. Among all the different methods that could be found in literature, cross correlation and phase correlation are the conventionally used criteria. The use of cross correlation for template matching is motivated by the distance measure (squared Euclidean distance) X

dE2 ðr; sÞ ¼

ðIðr þ i; s þ jÞRði; jÞÞ2

ð10:7Þ

ði;jÞ2R

(Rðu; vÞ, reference image; ðr; sÞ, shift of the reference image; and Iðu; vÞ, arbitrary gray scale image). In order to determine the best match between Iðu; vÞ and Rðu; vÞ, it is sufficient to minimize dE2 ðr; sÞ that can be written as dE2 ðr; sÞ ¼

X

ðIðr þ i; s þ jÞRði; jÞÞ2

ði;jÞ2R

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} X

þ

Aðr;sÞ

ðRði; jÞÞ2 2

ði;jÞ2R

X

Iðr þ i; s þ jÞ  Rði; jÞ

ð10:8Þ

ði;jÞ2R

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

B¼constant

Cðr;sÞ

If the term Aðr; sÞ is approximately constant, the remaining cross-correlation term is a measure of the similarity between the image and the reference image: Cðr; sÞ ¼

X

Iðr þ i; s þ jÞRði; jÞ

ð10:9Þ

ði;jÞ2R

The computation of this cross correlation can be performed either in original space or in Fourier space utilizing ðf  gÞðu; vÞ :¼

M1 X X N1

¼ f ðj; gÞgðju; gvÞ

ð10:10Þ

j¼0 g¼0

Thus, the cross-correlation coefficient can be calculated efficiently using the fast Fourier transform F by C ¼ F 1 ½F ðf ÞF  ðgÞ

ð10:11Þ

(F  indicates the complex conjugate of the Fourier transform). The peak intensity in the resulting cross-correlation image is located at a position described by the vector q that describes how features contained in one image shift, compared to another image that also contains those features. If the cross-correlation function is computed directly from two pictures, it is possible to determine the shift level only with

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resolution on whole pixels. If higher (subpixel) resolution is required, it is necessary to use interpolation [20]. There are several disadvantages that have to be considered if Eq. (10.9) is used for template matching: .

. .

If Aðr; sÞ varies with position, matching can fail if the correlation between the feature and an exactly matching region in the image is less than the correlation between the feature and a bright spot. The range of Cðr; sÞ depends on the size of the feature. Equation (10.9) is not invariant to changes in image amplitude such as those caused by changing lighting conditions across the image sequence.

The shift of pattern features relative to the pattern from the reference position is typically measured at numerous widely spaced subregions at exactly the same place in the CCD camera image. Figure 10.4 outlines the main steps in the image processing procedure used to determine the small displacements within the pattern. To avoid any leakage problems during the fast Fourier transform, the mean intensity of each subregion is subtracted from every pixel and a window function (i.e., Hanning, Gauss, or sin2) is applied to bring the edge values of each subregion smoothly to zero. The resulting image is then passed through a bandpass filter to sharpen the edges of the Kikuchi bands on the one hand and to remove noise on the other hand. In order to estimate the strain sensitivity of the “pattern shift” approach, different methodologies can be found. The first estimation of strain sensitivity has been given by Wilkinson who quoted a measurement repeatability of 2  104 in case of pattern shifts induced during beam shift and sample rotation experiments conducted on unstrained high-quality single crystal samples. These results have been in good accordance with measurements conducted on strained SiGe epilayers, where a standard deviation of 2.5  104 was found for each component of the gradient displacement tensor. One of the most comprehensive works regarding the attainable accuracy of strain measure-

Figure 10.4 (a) EBSD pattern from a silicon single crystal; (b) subregion centered on a certain pattern feature (i.e., zone axis); (c) edge values of each ROI are progressively brought to

zero using a weighting function; (d) Fourier transform of the subregion; (e) crosscorrelation with the ROI from the reference pattern.

10.5 Sampling Strategies: Sources of Errors

Figure 10.5 Dynamical electron diffraction simulations (a) and experimental narrow-angle EBSD pattern (b) of a Si(100) sample at 20 kV.

ment using EBSD has been presented by Villert et al. [21], who measured the difference between applied and measured values of the displacement gradient tensor in case of geometrically simulated diffraction patterns. There, he indicated an accuracy better than 104 for applied strains of 2  103 if a cross-correlation error of 0.05 pixel is guaranteed. More recently published paper [22, 23], which confirm previous results, demonstrated the potential of using dynamical, simulated electron backscatter diffraction patterns in order to estimate the possible strain sensitivity and possible influencing factors. Using dynamical simulated diffraction patterns almost any phase, orientation, EBSD geometry, and lattice distortion can be simulated, whereas extremely fine structures can be observed that are often blurred in experimental diffraction patterns (Figure 10.5). However, due to the potentially much higher image quality compared to real EBSPs, accuracy assessment based on such patterns can be seen as a measure of the lower limit of strain sensitivity.

10.5 Sampling Strategies: Sources of Errors

Modern EBSD systems generally offer two different computer-controlled sampling modes to collect diffraction patterns from individual points, linescans, or grids of points. Depending upon the type of analysis, EBSPs can be recorded using either control of the electron beam, in which the focused primary bean is moved across the stationary specimen surface, or by translating the specimen mechanically under the focused stationary electron beam, usually named as stage scan mode. The last mentioned scan method enables the accommodation of large measurement fields, only limited in size by the range of travel of the specimen stage, whereas the step size calibration is independent of the SEM magnification. In contrast, digital beam scanning offers an extremely high speed and precision in beam positioning. However, the diffraction geometry and thus pattern center position, specimen to screen distance as well as background intensity and focus settings will vary from point to point due to the tilted specimen surface and beam deflection.

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The sensitivity of the strain measurement results to calibration parameters (i.e., pattern center position and specimen-to-screen distance) has been examined by several authors [21, 22, 24, 25]. It has been shown that a precise knowledge of the source point position is not necessary to realize an accurate strain measurement if EBSPs from reference and strained material were obtained with the same projection parameters. This is of course the case only if EBSD scans are made using the stage scan mode on a perfectly flat specimen or if the change in projection parameters between individual points is well known. In the last case, the EBSD system has to be calibrated dynamically from spot to spot that is typically made by scanning the electron beam across a larger field on a single crystal and measuring the inherent shift of the EBSD pattern across the phosphor screen. Subsequently, the measured shifts of pattern features caused by strains and rotations can be separated from artificial shifts due to the projection parameter variations by simple subtraction. However, due to the small size of subsections that are used to measure the shift between two patterns, this procedure is applicable only up to a certain degree of beam deflection. Otherwise, unequal Fourier spectra will be analyzed during template matching and the cross correlation will fail. Additional errors could arise from defects in the phosphor screen. Also, pollution that cannot be removed from the screen would cause fixed features in exactly the same position for each pattern and could lead to artifacts in the analysis of the corresponding pattern shifts. In order to reduce the effect of these features, Wilkinson et al. [25] proposed the use of standard background correction and filtering in the frequency domain. Even if this approach is very useful to remove random noise and differences in background intensities from patterns, its effect on distinct defects and pollution is rather small. In fact, it is possible to consider these artifacts by capturing a background image and applying a dynamic background correction and stretching to it. The resulting image will be characterized by a widely homogeneous distribution of gray scale values over the entire image and small black features resulting from imperfections on the screen. This image can be analyzed through binarization by setting a threshold to a certain gray scale value and classifying all pixels with values above this threshold as white and all other pixels as black (Figure 10.6). By carefully adjusting this threshold and applying appropriate neighborhood criteria, imperfections of the screen can be detected. Hence, subregions containing imperfections of a certain size that could be defined by the operator could be excluded from the analysis also in automated processing procedures.

10.6 Resolution Considerations

One of the most attractive features of electron backscatter diffraction is its unique capability to perform rapid, automated diffraction analysis of crystalline materials with excellent spatial resolution. The attainable resolution in general is a function of the density and constitution of the material, the acceleration voltage of the primary electrons, the beam profile, the specimen tilt, and the software used for pattern

10.6 Resolution Considerations

Figure 10.6 (a) Filtered background image showing small black features resulting from imperfections on the screen and (b) corresponding binarized image.

analysis [26]. In particular, the highly tilted specimen geometry has important implications both for EBSD spatial resolution and for the surface sensitivity. In fact, due to the specimen tilt, the interaction volume for backscattered electrons is asymmetric along the beam direction. Consequently, the resolution needs to be defined with respect to three orthogonal directions: lateral resolution (within the specimen plane but normal to the beam direction), longitudinal resolution (within the specimen plane but parallel to the beam direction) that is typically three times the value of the lateral resolution and depth resolution (extent of depth information). To date, two different definitions of spatial resolution could be found in literature: the physical resolution that indicates how far away from a large-angle grain diffracted intensities from both crystals can be obtained and the effective resolution that is a measure of how accurate an orientation microscopy system may resolve a large-angle grain boundary using software algorithms to deconvolute overlapping patterns. Usually, the effective resolution is better than the physical. However, in case of strain determination using the “pattern shift” approach, the latter is the significant one. The lateral resolution is usually measured by carrying out small scan steps across a boundary standing perpendicular to the sample tilt axis and determining the distance over which patterns cannot be solved. In order to determine the lateral physical resolution for silicon, a string ribbon wafer surface with a large-angle grain boundary standing perpendicular to the surface has been analyzed (Figure 10.7). The beam was moved in steps of 10 nm toward the grain boundary and a diffraction pattern was taken at every position. The first weak Kikuchi bands of the second crystal became visible at (50  10) nm away from the boundary that is synonymic for the physical resolution. The depth of the surface layer that gives rise to an EBSP can be determined by applying layers of amorphous metal to the surface of a Si single crystal and measuring the EBSD pattern quality in dependence of coating thickness [27]. In the present

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Figure 10.7 SE image of a large-angle grain boundary used to determine the physical lateral resolution in case of silicon (measurements have been performed under conditions where the lateral resolution

measured in an SE image is on the order of the beam diameter). The resolution is defined as follows: x, lateral resolution; y, longitudinal resolution; and z, depth resolution.

study, the sensitivity of EBSD patterns to coating thickness has been explored by coating silicon with platinum thin films of known thickness ranging from 1 to 8 nm. Figure 10.8 outlines the pattern quality in dependence of coating thickness, expressed using a quality parameter proposed by Krieger Lassen et al. [28]. The results show an almost exponential decrease in pattern quality that is detectable for each acceleration voltage. As shown in Figure 10.8, reducing acceleration voltage 0.5

30kV 20kV 10kV

pattern quality

0.4

30keV no coating

10keV no coating

30keV-4nm

10keV-4nm

30keV-8nm

10keV-8nm

0.3

0.2

0.1 0

0

1

2 3 4 5 6 coating thickness [nm]

7

8

Figure 10.8 Pattern quality of EBSPs obtained from a silicon crystal that has been sputter-coated with differently thick layers of amorphous platinum in dependence of coating thickness.

10.7 Illustrative Application

leads to less beam penetration and thus blurred patterns from the underlying silicon compared to higher primary beam energies. The exact value of depth resolution depends on its definition. If we define the depth resolution as the coating thickness where 50% of the original intensity is left, we obtain a value of 4 nm for all three electron energies. This result seems to be inconsistent with the visual impression (Figure 10.8). However, one has to consider that pattern quality parameters are strongly affected by a variety of factors, so that the absolute values of pattern quality cannot be compared unambiguously. For instance, changes in experimental parameters such as acceleration voltage, beam current, and exposure time of the CCD camera will strongly influence the degradation of the patterns and thus the numerical value of the pattern quality. In the present study, the lower penetration depth in case of a 10 keV beam energy results in a premature saturation of the quality parameter that is the reason for the almost similar depth resolution compared to higher beam energies.

10.7 Illustrative Application

Several researchers successfully used electron backscatter diffraction patterns to investigate both elastic strain fields in multilayer compound semiconductor systems [14, 15] and lattice curvature in metallic materials [19, 25, 29, 30]. Indeed, most of the quantitative EBSD strain measurements have been realized on epitaxially grown Si1x Gex layers on silicon substrates [16, 17, 21]. Utilizing the cross correlation-based “pattern shift” methodology, the need of an unstrained reference position seems to be a clear limitation. According to this, most of the semiconductor systems investigated so far could be considered to be model systems that have been chemically patterned to provide zones where the silicon is accessible and thus reference EBSD patterns of unstrained material could be obtained. However, there is a multitude of possible applications far beyond academic research, in particular in the field of semiconductor technologies, where such a reference position could be found. In fact, the need for experimental methods that gain access to these positions could be seen to be the bigger challenge. In the following section, we will give an example application of how low-energy focused ion beams can be used in order to provide unstrained reference positions in strained Silicon-onInsulator (sSOI) layers and consequently how EBSD can be used to determine the elastic strain and rotation tensors of these nanosized single-crystal films. SOI technology offers CMOS performance enhancement with the use of an embedded oxide layer to isolate transistors from the substrate, which results in lower parasitic capacitance and reduced junction leakage. Combining SOI substrates with strained Si technology takes advantage of the performance enhancement by both SOI on the one hand and the increased carrier mobility of strained Si on the other hand. Typically, the method of choice to obtain Ge-free sSOI is to transfer strained Si grown on a relaxed Si1x Gex buffer layer onto an oxide layer using SMART CUTÔ technology [31] that employs a combination of hydrogen ion implantation and wafer

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bonding technology [32]. The built-in strain level can be modulated by selectively choosing the Ge doping content of Si1x Gex . The samples used in this study consist of a 60 nm-thick sSOI epilayer built on 145 nm-thick buried oxide above a 525 mm-thick Si(001) wafer (Figure 10.9a). The necessary unstrained reference position has been generated by milling a trench with nominal dimensions of 350 mm  350 mm  500 nm using a dual-source ion column sputter gun (thermal ionization Cesium source, three-lens ion column) at a beam energy of 2 keV. The trench is characterized by gently inclined sidewalls and a smooth bottom that is the result of the comparatively low incident angle of 45 and the absence of differential sputtering along this particular incoming direction. However, the most important characteristic is the extremely thin amorphization layer ( 3 nm) on top of the silicon substrate caused by the ion–solid interaction that allows highresolution diffraction patterns to be captured (Figure 10.10). The EBSD measurements were made using a Zeiss CrossbeamÒ 1540 EsB at beam energies of 10 keV, 20 keV, and 30 keV and a beam current of 8 nA. All EBSPs have been recorded at full resolution (1344  1024 pixel2, frame averaging over 10 frames) utilizing a NordlysI camera and a Channel5 software suite. Figure 10.9a illustrates the Cartesian axes system that has been used to describe the strain and rotation tensor: x1 along the  direction, and x3 along the wafer surface normal, ½110Si direction, x2 along the ½110 Si that is, ½001Si . Strains and rotations have been determined using the method described by Wilkinson et al. [18]. Linescans have been conducted on the surface of the sSOI layer ranging from the center of the milled trench to a position 700 mm away from the reference position. The quality of the individual measurements along the linescans has been measured using a pattern quality parameter and the normalized cross-correlation peak height proposed in Ref. [25] (Figure 10.10). As expected, both parameters evidence a significant reduction in data quality at the sidewalls of the trench where noncrystalline oxide is exposed to the sample surface. In order to avoid misinterpretation of strains and rotations at these positions, the data sets have been filtered using

Figure 10.9 (a) Reference axes used for the analysis of the strained Silicon-on-Insulator system and (b) white light interferometer image of the trench that has been milled in order to provide an unstrained reference position.

10.7 Illustrative Application

0.7

30kV 20kV 10kV

pattern quality

0.6 0.5 0.4 0.3 0.2 0.1

0

100

200

300 400 500 600 700 distance from reference point [μm]

mean peak height

1.4

800

30kV 20kV 10kV

1.2 1 0.8 0.6 0.4 0

100

200 300 400 500 600 distance from reference point [μm]

700

800

Figure 10.10 (a) Image quality and (b) mean cross-correlation peak height parameter.

threshold values for these two parameters. Furthermore, Figure 10.10 illustrates differences in pattern quality for the three beam energies as well as between the silicon substrate and the sSOI. In particular in case of 10 keV beam energy, the measured pattern quality of the unstrained reference area is significantly decreased. It is reasonable to assume a noticeable influence of the thin amorphization layer above the silicon substrate on the pattern quality, especially for EBSPs obtained from shallow penetration depths. In contrast, the distribution of pattern quality in case of 30 keV beam energy shows no significant difference between trench bottom and strained layer but a larger variation of pattern quality in the area of sSOI that might be caused by defects at higher depths or be influenced by the underlying oxide. Figure 10.11 shows the distribution of each of the strain components along the linescan. The experimental results confirm the expected biaxial strain state of the sSOI, which is evidenced by a tensile strain e11 and e22 , respectively. Furthermore, e33 takes compressive values in order to maintain the imposed s 33 equals zero condition. According to the sample symmetry, neither significant differences in shear strains nor rotations are observed. The averaged strain sensitivity measured by the standard deviation is 3  104 in case of experiments using 10 kV and 20 kV beam energy and 5  104 for 30 kV. This result is in good accordance with previous assessments

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normal strain ε11

0.02

30kV 20kV 10kV

0.015 0.01 0.005 0 -0.005 0

100

200

300

400

500

600

700

800

distance from reference point [μm]

normal strain ε22

0.02

30kV 20kV 10kV

0.015 0.01 0.005 0 -0.005

0

100

200 300 400 500 600 distance from reference point [μm]

800

30kV 20kV 10kV

0.004 0.002

normal strain ε33

700

0 -0.002 -0.004 -0.006 -0.008 -0.01

0

100

200 300 400 500 600 700 distance from reference point [μm]

800

Figure 10.11 Distribution of normal strains along the linescan.

of the sensitivity. Comparison of pattern quality and variations in measured strains demonstrates the direct correlation between both quantities that is also reflected using the mean cross-correlation peak height parameter. However, blurring of patterns obtained from the reference position using a 10 kV electron beam could not be measured using the cross-correlation peak height parameter. This is due to the invariance of the cross correlation to changes in image amplitude, such as those

10.8 Conclusions

caused by pattern blurring. For that reason, it seems to be useful to assess the quality of the produced data using both parameters.

10.8 Conclusions

The automated analysis of electron backscatter diffraction patterns has developed into a mature technique to characterize crystalline materials at the subgrain-size level. During the past decade, there have been major efforts to improve the angular resolution of the technique that have opened up an entire new field of applications: local strain measurements with a spatial resolution on the order of 50 nm and high sensitivity of 104. In particular, the cross-correlation-based pattern analysis, which is also commercially available2), enables elastic strain tensor measurements at the nanoscale level. Due to the need of an unstrained reference position and high-quality patterns, this methodology can be used for strain determination in silicon devices that are fundamental for a multitude of microelectronic devices and MEMS technology. In this case, the necessary reference position could be provided by either choosing a measurement position far away from the strained sample region or by using sophisticated target preparation techniques. Especially, the use of low-energy sputtering combined with sequential chemical analysis (i.e., AES, XPS, or TOFSIMS) seems to be a promising tool to locally remove different types of material in order to reveal the unstrained silicon substrate and to measure the displacement gradient tensor of strained silicon or SiGe. More recently, the use of simulated reference patterns has been discussed controversially within the EBSD community [24, 33, 34]. This approach relies on an iterative procedure that is used to minimize the distortion between experimental patterns and patterns simulated at the deformed state in order to reflect the strain state of the experimental EBSPs. Due to the inherent poor accuracy of Hough-based source position determination, significant errors could be expected that are unavoidable using standard calibration routines. Even the use of novel, more accurate methods [35] might not attain the precision required to use this methodology.Despitethis,acontinuousprogressinthedevelopmentofEBSDforstrain determination could be stated, ranging from new analysis routines and sampling strategies to avoidance of errors and novel fields of application.

Acknowledgments

The work at Fraunhofer Institute for Mechanics of Materials was conducted as part of the “nanoSTRESS” research project. The authors gratefully acknowledge financial support by the Fraunhofer Society. Additionally, the assistance of Dr. Aimo Winkelmann in producing the dynamical simulated patterns is gratefully acknowledged.

2) CrossCourt 3 software from BLG Productions, Bristol, UK (www.blgproductions.co.uk).

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and Kunze, K. (1993) Orientation imaging: the emergence of a new microscopy. Metall. Trans. A, 24A, 819–831. Bastos, A., Zaefferer, S., and Raabe, D. (2008) 3-Dimensional EBSD study on the relationship between triple junctions and columnar grains in electrodeposited Co–Ni films. J. Microsc., 230, 487–498. Sivakov, V.A., Broenstrup, G., Pecz, B., Berger, A., Radnoczi, G.Z., Krause, M., and Christiansen, S. (2010) Realization of vertical and zigzag single crystalline silicon nanowire architectures. J. Phys. Chem. C, 114, 3798–3803. Baba-Kishi, K.Z. and Dingley, D.J. (1989) Backscatter Kikuchi diffraction in the SEM for identification of crystallographic point groups. Scanning, 11, 305–312. Dingley, D.J. and Wright, S.I. (2009) Phase identification through symmetry determination in EBSD patterns, in Electron Backscatter Diffraction in Materials Science, Springer Science þ Business Media, LLC, pp. 97–107. Krause, M., Maerz, B., Bennemann, S., and Petzold, M. (2010) High resolution analysis of intermetallic compounds in microelectronic interconnects using electron backscatter diffraction and transmission electron microscopy. Proceedings IEEE 60th Electronic Components and Technology Conference (ECTC 2010), pp. 591–598. Winkelmann, A. (2008) Dynamical effects of anisotropic inelastic scattering in electron backscatter diffraction. Ultramicroscopy, 108, 1546–1550. Winkelmann, A. (2009) Dynamical simulation of electron backscatter diffraction patterns, in Electron Backscatter Diffraction in Materials Science, Springer Science þ Business Media, LLC, pp. 21–33. Winkelmann, A. and Nolze, G. (2010) Analysis of Kikuchi band contrast reversal in electron backscatter diffraction patterns of silicon. Ultramicroscopy, 110, 190–194.

10 Winkelmann, A., Trager-Cowan, C.,

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Sweeney, F., Day, A.P., and Parbrook, P. (2007) Many-beam dynamical simulation of electron backscatter diffraction patterns. Ultramicroscopy, 107, 414–421. Randle, V. and Engler, O. (2000) Texture Analysis: Macrotexture, Microtexture and Orientation Mapping, CRC Press. Wilkinson, A.J. (2000) Measuring strains using electron backscatter diffraction, in Electron Backscatter Diffraction in Materials Science, Kluwer Academic/Plenum Publishers, pp. 231–246. Maurice, C. and Fortunier, R. (2008) A 3d Hough transform for indexing EBSD and Kossel patterns. J. Microsc., 230, 520–529. Keller, R.R., Roshko, A., Geiss, R.H., Bertness, K.A., and Quinn, T.P. (2004) EBSD measurement of strains in GaAs due to oxidation of buried AlGaAs layers. Microelectron. Eng., 75, 96–102. Luo, J.F., Ji, Y., Zhong, T.X., Zhang, Y.Q., Wang, J.Z., Liu, J.P., Niu, N.H., Han, J., Guo, X., and Shen, J.D. (2006) EBSD measurements of elastic strain fields in a GaN/sapphire structure. Microelectron. Reliab., 46, 178–182. Troost, K.Z., van der Sluis, P., and Gravesteijn, D.J. (1993) Microscale elastic strain determination by backscatter Kikuchi diffraction in the scanning electron microscope. Appl. Phys. Lett., 62, 1110–1112. Wilkinson, A.J. (1996) Measurement of elastic strains and small lattice rotations using electron backscatter diffraction. Ultramicroscopy, 62, 237–247. Wilkinson, A.J., Meaden, G., and Dingley, D.J. (2006) High-resolution elastic strain measurement from electron backscatter diffraction patterns: new levels of sensitivity. Ultramicroscopy, 106, 307–313. Wilkinson, A.J., Meaden, G., and Dingley, D.J. (2006) High resolution mapping of strains and rotations using electron backscatter diffraction. Mater. Sci. Tech. Ser., 22, 1271–1278.

References 20 Dost, M., Kieselstein, E., and Erb, R.

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(2002) Displacement analysis by means of gray scale correlation at digitised images and image sequence evaluation for microand nanoscale applications. Micromater. Nanomater., 1, 30–35. Villert, S., Maurice, C., Wyon, C., and Fortunier, R. (2009) Accuracy assessment of elastic strain measurement by EBSD. J. Microscopy, 233, 290–301. Britton, T.B., Maurice, C., Fortunier, R., Driver, J.H., Day, A.P., Meaden, G., Dingley, D.J., Mingard, K., and Wilkinson, A.J. (2010) Factors affecting the accuracy of high resolution electron backscatter diffraction when using simulated patterns. Ultramicroscopy, 110, 1443–1453. Krause, M., Graff, A., and Altmann, F. (2010) Strain determination using electron backscatter diffraction. Proceedings 11th International Workshop on Stress-Induced Phenomena in Metallization. Kacher, J., Landon, C., Adams, B.L., and Fullwood, D. (2009) Bragg’s law diffraction simulations for electron backscatter diffraction analysis. Ultramicroscopy, 109, 1148–1156. Wilkinson, A.J., Dingley, D.J., and Meaden, G. (2009) Strain mapping using electron backscatter diffraction, in Electron Backscatter Diffraction in Materials Science, Springer Science þ Business Media, LLC, pp. 231–249. Humphreys, F.J., Huang, Y., Brough, I., and Harris, C. (1999) Electron backscatter diffraction of grain and subgrain structures: resolution considerations. J. Microscopy, 195, 212–216. Zaefferer, S. (2007) On the formation mechanisms, spatial resolution and intensity of backscatter Kikuchi patterns. Ultramicroscopy, 107, 254–266. Krieger Lassen, N.C., Juul Jensen, D., and Conradsen, K. (1994) Automatic

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recognition of deformed and recrystallized regions in partly recrystallized samples using electron back scattering patterns. Mater. Sci. Forum, 157–162, 149–158. Landon, C.D., Adams, B.L., and Kacher, J. (2008) High-resolution methods for characterizing mesoscale dislocation structures. J. Eng. Mater. Technol. ASME, 130, 021004. Miyamoto, G., Shibata, A., Maki, T., and Furuhara, T. (2007) Precise measurement of accommodation strain in austenite surrounding martensite by electron backscatter diffraction. Proceedings 1st International Symposium on Steel Science, The Iron & Steel Institute, Japan. Bruel, M., Aspar, B., and Auberton-Herve, A.J. (1997) Smart-cut: a new silicon on insulator material technology based on hydrogen implantation and wafer bonding. Jpn. J. Appl. Phys., 36, 1636–1641. Christiansen, S.H., Singh, R., Radu, I., Reiche, M., G€osele, U., Webb, D., Bukalo, S., and Dietrich, B. (2005) Strained silicon on insulator (SSOI) by wafer bonding. Mater. Sci. Semicond. Process., 8, 197–202. Kacher, J., Basinger, J., Adams, B.L., and Fullwood, D.T. (2010) Reply to comment by Maurice et al. in response to “Bragg’s law diffraction simulations for electron backscatter diffraction analysis.” Ultramicroscopy, 110, 760–762. Maurice, C., Fortunier, R., Driver, J., Day, A., Mingard, K., and Meaden, G. (2010) Comments on the paper “Bragg’s law diffraction simulations for electron backscatter diffraction analysis” by Josh Kacher, Colin Landon, Brent L. Adams & David Fullwood. Ultramicroscopy, 110, 758–759. Maurice, C., Dzieciol, K., and Fortunier, R. (2011) A method for accurate localisation of EBSD pattern centres. Ultramicroscopy, 111, 140–148.

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11 X-Ray Diffraction Analysis of Elastic Strains at the Nanoscale Olivier Thomas, Odile Robach, Stéphanie Escoubas, Jean-Sébastien Micha, Nicolas Vaxelaire, and Olivier Perroud

11.1 Introduction

It is well established that X-ray diffraction (XRD) is perfectly suited to analyze strains in crystals [1]. The sensitivity of XRD to the position of atoms together with the high resolution that may be achieved in diffraction space is the basis of structure solving. On the other hand, the size of conventional X-ray beams (100 mm–1 mm) prevented until recently any local strain determination relying on direct space resolution. During the past 10 years developments in X-ray optics [2] and in X-ray sources have paved the way for micro- or nanobeam diffraction [3] (expectations lie around 50 nm). With respect to electron beams, X-ray beams are clearly lagging behind when it comes to beam size and thus direct space resolution. But X-rays have a strong advantage: they are basically nondestructive and yield information on buried structures without any need for special sample preparation, which may drastically alter the original strain field in the sample. Moreover, as already mentioned, the limited real space resolution may be circumvented in some cases by reciprocal space resolution (a 5 nm resolution has been reported in Ref. [4]). With the development of nanoscience and nanotechnologies, the need for highresolution strain measurements at scales as small as possible has become an important issue. Because yield strengths are significantly higher in small dimensions [5], stresses play an important role in the unusual properties often encountered at small scales [6]. In technology and more specifically in microelectronics, the ability to measure strains in small features is critical. In small Cu interconnects, local stresses may trigger phenomena such as stress-induced voiding [7] that is one of the reasons for the failure of devices. In the very core of the transistors, the stress in the Si channel (32 nm wide in emerging technologies) is a tunable parameter to control the mobility of charge carriers [8]. This chapter is aimed at reviewing recent advances in local strain determination using XRD. A brief reminder of the basics of XRD within the framework of kinematic approximation is given in Section 11.2. Section 11.3 focuses on average strain

Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanb€ ucken, Pierre M€ uller, and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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measurements from Bragg peak shifts. Section 11.4 is devoted to submicron-sized Xray beams and their use for local strain measurements. In Section 11.5, we address the issue of strain determination from the distribution of intensity in reciprocal space. The specific issue of periodic systems is considered where the periodicity is used to enhance the diffracted signal from a single period and hence yield local information on the strain field. Coherent diffraction, a very promising technique for determining strains in single nanocrystals, is also described. Section 11.6 briefly addresses the issue of phase retrieval, which is a way to invert directly the diffracted signal and thus obtain the displacement field without any a priori model.

11.2 Strain Field from Intensity Maps around Bragg Peaks

We will first consider the simplest situation where an incident, fully monochromatic X-ray beam with wavelength l is scattered elastically by The interaction of X-rays with matter is very weak, and this allows the amplitude to be written as a simple Fourier transform [9] of the electron ð Að~ qÞ ¼ rð~ r Þei~q:~r d~ r

coherent a crystal. scattered density ð11:1Þ

where rð~ r Þ is electron density in real space and ~ q ¼~ k f ~ k i is the scattering vector defined as the difference between the incident wave vector and the scattered one. The modulus of ~ q is simply related to the scattering angle 2y q¼

4p siny l

ð11:2Þ

where l is the wavelength of the incoming radiation. Formula (11.1) remains an approximation and in the case of perfect crystals, it should be carefully justified. Generally, thin crystals with thickness smaller than the extinction length scatter waves in the kinematic regime [10]. For crystals larger than the extinction length, it is the crystal quality (mosaicity, density of defects, etc.) that will determine whether they scatter in the kinematic or dynamic regime. In the case of a crystal, considering the Bravais lattice ~ R m and the shape function sð~ r Þ, one gets " # X X X ~ ~mÞ ~ Að~ qÞ ¼ TF sð~ rÞ rc ð~ r Þ*dð~ r R m Þ ¼ Sð~ qÞ*Fð~ qÞ ei~q  R m / Sð~ qÞ*Fð~ qÞ dð~ qG m

m

m

ð11:3Þ

where rc ð~ r Þ is the unit cell electron density, Fð~ qÞ its Fourier transform, that is, the ~ m are reciprocal space structure factor, Sð~ qÞ is the Fourier transform of sð~ r Þ, and G vectors. This very well-known expression shows simply that the diffraction pattern of an unstrained crystal consists in well-defined Bragg peaks at positions given by the ~ m . Moreover, the Fourier transform Sð~ qÞ of the crystal shape is Laue condition:~ q¼G

11.2 Strain Field from Intensity Maps around Bragg Peaks

Figure 11.1 Coherent diffraction pattern from a square crystal calculated within the kinematic approximation. (a) At the origin of reciprocal space, the influence of crystal size is solely visible. (b) Around Bragg reflection, strain in the crystal distorts the diffraction pattern.

transferred by convolution on any reciprocal space node. Figure 11.1a shows the modulus jSð~ qÞj for a simple 2D square crystal. Periodic fringes are related to the size of the crystal, while streaks occur along the normal to the facets. Let us consider now the case of a strained crystal [11]. We will restrict ourselves in the first step to purely elastic strains. Moreover, considering that elasticity in crystals can only describe the distortion of the lattice [12] (and not of the basis) and that elastic strains are always small (even in nanocrystals they remain of the order of a few percent), we will assume that the structure factor Fð~ qÞ remains unchanged. In the strained crystal, the lattice is distorted 0 ~ R ¼~ R þ~ u

ð11:4Þ

where ~ u is the displacement field. It is worth noting that diffraction is sensitive to the displacement and not simply to the strain, that is, diffraction is also highly sensitive to lattice rotation. If i ¼ 1, 2, 3 are the principal orthogonal directions in space, the strain and rotation tensors [13] eij and vij are written as     1 @ui @uj 1 @ui @uj and vij ¼ ð11:5Þ eij ¼ þ  2 @xj 2 @xj @xi @xi within the framework of the small displacements approximation. The amplitude scattered from the strained crystal is written now as X ~ Að~ qÞ ¼ Sð~ qÞ*Fð~ qÞ ei~q  ðR m þ~u m Þ

ð11:6Þ

m

~ hkl , one may write ~ ~ þ~ In the vicinity of Bragg peak G q¼G g and neglect the term ~ g ~ u . Thus, X ~ ~ Að~ qÞ  Sð~ qÞ*Fð~ qÞ eiG hkl~u m  ei~q  R m ð11:7Þ m

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This shows that the amplitude scattered by the strained crystal may be described to a very good approximation [14] as the Fourier transform of a modified electron density: ~

Að~ qÞ  TF½rð~ r ÞeiG hkl ~u ð~r Þ 

ð11:8Þ

In consequence, a strained crystal behaves as if it had a complex electron density ~ hkl ~ with a phase factor G u ð~ r Þ. This phase strongly influences the intensity distribution as shown, for example, in Figure 11.1b where a displacement field has been introduced in the previous square crystal. Because the phase field considered here is not centrosymmetric, the diffraction pattern does not remain centrosymmetric, that is, Friedel’s law is not valid any more. The corresponding asymmetry in the diffraction pattern is a good sign for the presence of a strain field within the crystal. One should, however, remember that a centrosymmetric phase field would yield a centrosymmetric diffraction pattern. Hence, a strained crystal may still have a symmetric diffraction pattern. At variance with the size effect previously described, this strain effect is clearly Bragg peak dependent. When the strain is homogeneous throughout the crystal under investigation, the displacement field varies linearly with position and the amplitude remains unaffected. The strain is then deduced directly from the shift in peak position. This forms the basis of many methods to determine strains and stresses in materials. The examples illustrated in Figure 11.1 are typical of what is conventionally named coherent X-ray diffraction (CXD) where a crystal is illuminated by a fully coherent beam. The coherence length depends heavily on the source and optics used. The longitudinal coherence length is related to the energy spread in the beam. Typically, a Si(111) monochromator yields an energy spread of 104, hence a longitudinal coherence length of 1 mm at 8 keV. The transverse coherence length is inversely related to the source size. On third-generation synchrotron source, it may reach 100 mm; but on laboratory setups, 1 mm is an upper limit that calls for a very small divergence. When the beam size is larger than the coherence length, the intensity on the detector is an incoherent addition of intensities scattered by different parts of the sample under the beam. Because of the dispersion in size, strains, and so on in the beam footprint, one does not observe fringes any more but a diffraction peak whose width may be related to the average size distribution and strain distribution standard deviation.

11.3 Average Strains from Diffraction Peak Shift

Under Bragg conditions q ¼ Ghkl ¼ 2p=dhkl , where dhkl is the interplanar spacing. One can then derive the corresponding strain from the knowledge of the strain-free 0 spacing dhkl

11.3 Average Strains from Diffraction Peak Shift

Figure 11.2 Diffraction geometry and scattering vector.

e¼

0 dhkl dhkl 0 dhkl

ð11:9Þ

The change in the length of the scattering vector can be written as dq ¼ qq0 ¼ eq

ð11:10Þ

This shows that much better accuracy is obtained for large qs, that is, for large scattering angles. On a four-circle diffractometer, q-values may be measured with different orientations y and j with respect to the surface normal (Figure 11.2). The corresponding strain is thus measured in a reference frame that has been rotated with respect to the sample’s reference frame. Performing the proper tensor rotation (strain is a symmetric second-rank tensor) yields ewy ¼ eS11 cos2 j sin2 y þ eS12 sin 2j sin2 y þ eS22 sin2 j sin2 y þ eS33 cos2 y þ eS13 cos j sin 2y þ eS23 sin j sin 2y

ð11:11Þ

where eSij refers to the strain tensor components in the sample’s reference frame. Thus, the measurement of a large number of Bragg peak positions for different values of y and j provides a powerful and accurate means of determining the full strain tensor. In the simple case of an equal biaxial stress state, often encountered in thin films (here x3 is taken as the surface normal), there is no dependence on the in-plane azimuth j and the strain along y writes ey ¼ e== sin2 y þ e? cos2 y

ð11:12Þ

where es11 ¼ es22 ¼ e== and es33 ¼ e? . Equation (11.12) shows that ey ðsin2 yÞ is a straight line whose slope yields directly the in-plane stress s. This is the so-called sin2y technique [1]. Moreover, there is a particular direction y0 for which strain is zero and which depends solely on the

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5.658 GaAs (001)

5.656

ahkl (Å)

444

5.654

a0 224

5.652

115

5.65 004

5.648

0

0.2

0.4

0.6

0.8

1

Sin2 ψ Figure 11.3 Sin2y plot from an epitaxial GaAs layer on Ge(001).

material’s elastic constants. For a cubic material, measurement of lattice parameter at y0 yields directly the stress-free lattice parameter a0. A typical example of a sin2y plots can be found in Figure 11.3, where measurements taken [15] from an epitaxial GaAs layer grown on Ge(001) are shown. The slope indicates that the GaAs film is under tension. It is worth noting that what is actually measured with diffraction is the displacement field. The transition from displacement to strain requires the knowledge of strain-free lattice parameters [16], which may differ from bulk values because of the presence of impurities. X-ray diffraction and Eq. (11.11) may be used to investigate more complex stress states [17] as encountered, for example, in trenches where a triaxial stress state occurs. The transition from strain to stress is straightforward in the case of a single crystal since it requires the use of Hooke’s law with proper provision for anisotropic elasticity. In the case of polycrystals, it is more delicate since a proper account of the way X-ray diffraction averages over different grain populations together with a mechanical model for grain interaction [18] should be used. Let us consider a very simple example of a Cu thin film with two types of grains: {100} and {111}. Cu is highly anisotropic with an anisotropy factor A ¼ 3.2. It is thus expected that large grain-to-grain strain variations occur in such a film. Two extreme cases may be considered: (i) all grains are in the same state of stress (Reuss approximation) and (ii) all grains are in the same state of strain (Voigt approximation). The corresponding sin2y plots are shown in Figure 11.4. These extreme cases do not satisfy basic compatibility requirements for the fields. One should therefore look for more elaborate models (self-coherent approaches do respect compatibility requirements). It is, however, clear that mean field approaches, which do not take into account the exact environment of a given grain, are borne to miss the exact distribution of strains within the film.

11.3 Average Strains from Diffraction Peak Shift

Figure 11.4 Calculated Sin2y plots for (111) Cu grains (dashed line) and (001) grains (solid line) in the Reuss (a) or Voigt (b) approximation.

Using “conventional” X-ray beam sizes in the range 100–1000 mm, it is average quantities that are determined from X-ray diffraction. Since strain fields are most often highly inhomogeneous on short length scales, the diffracted intensity will result from an incoherent addition of intensities scattered from areas with different strain levels. This results in a broadening of Bragg peaks, and a considerable amount of work [19] has been devoted to what is called line profile analysis (LPA), which is mostly aimed at extracting the second moment of the strain distribution from the width of Bragg peaks. The major difficulty arises when different sources of strain inhomogeneity are present at the same time (dislocations, boundary conditions, elastic anisotropy, etc.). How can one determine the strain field at a scale as local as possible? The most obvious approach is to decrease the size of the X-ray beam in order to analyze as small an area as possible and monitor the average strain within the beam footprint via shifts in diffraction peaks. This is described in the next section.

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11.4 Local Strains Using Submicrometer Beams and Scanning XRD 11.4.1 Introduction

The comprehension of complex crystalline materials (e.g., polycrystalline and/or multiphased), and their inhomogeneous distributions of stress and plastic deformation, requires microprobe techniques. Two X-ray diffraction techniques will be addressed here, 3DXRD [20, 21] and Laue microdiffraction [22, 23]. The physical meaning, in real space, of the objects dealt with, and of the quantities measured, when using microprobe X-ray diffraction, first need to be reminded. Concerning space averages, diffraction is very sensitive to well-crystallized parts of the material, and very little sensitive to poorly crystallized parts: it acts as a filter of order. Any measured average of a quantity over the probed volume is therefore a “diffraction-wise” average, essentially a space average weighted by the degree of order. This needs to be taken into account when comparing “measured” strain with strain derived from a simulation of the displacement field. Here we define a grain as one orientation of the crystal unit cell, with a certain angular tolerance, inside the probe volume. A grain can contain several wellcrystallized domains separated by defective and/or differently oriented regions. The intragrain orientation distribution either may come from lattice curvature inside the well-crystallized domains or may come from rotations between the domains. When translating the material with respect to an X-ray microbeam, this angular tolerance is necessary to sort grains encountered at different positions of the probed volume according to their orientation, to draw the frontiers of the grains. The “local elastic stress” measurements consist in using the well-crystallized domains of a grain as “ideal” small local stress sensors, dispersed inside a possibly much less ideal material. An extreme example would be small diamond single crystals dispersed in a matrix of amorphous glass. The sensors react to stress with the elastic rigidity tensor of the macroscopic single crystal. This hypothesis allows converting the measured lattice parameters, and the derived strain tensor, into a local elastic stress tensor. In practice, the level of accuracy on strain (104) required for useful mechanical measurements often implies that the center-of-mass positions of the diffraction spots (which serve as input for deriving the lattice parameters) need to be determined with an uncertainty much smaller than the spot width. This accuracy is accessible for simple spot shapes, but it becomes much more problematic when spots are asymmetric or split, which is often the case for plastically deformed materials. In such cases, spot shape analysis and simulation in terms of microdisorientation and microstrain inside the probe volume are necessary to locate the average spots and derive the average lattice parameters with a reasonable accuracy. We will now describe how the two techniques proceed to map the positions, orientations, and elastic strain of the grains inside a polycrystalline material.

11.4 Local Strains Using Submicrometer Beams and Scanning XRD

11.4.2 High-Energy Monochromatic Beam: 3DXRD

The methodology for 3DXRD was developed by the group of H.F. Poulsen using two high-energy beamlines: ID11 at ESRF (Grenoble, France) and 1-ID at APS (Argonne, IL, USA). The basic version of 3DXRD allows to locate in 3D the center of mass (CM) of the grains and to determine their orientation, in the volume of a thick (a few millimeters to centimeters) polycrystal, with a spatial resolution around 5 mm. For successful measurements, the grains along the beam path should all be illuminated (sufficient transparency), in small numbers (