www.wiley-vch.de

Hanbücken · M

üllerW

ehrspohn (Eds.)M

echanical Stress on the N

anoscale

Bringing together experts from the various disciplines involved, this fi rst comprehensive overview of the current level of stress engineering on the nanoscale is unique in combining the theoretical fundamentals with simulation methods, model systems and characterization techniques. Essential reading for researchers in microelectronics, optoelectronics, sensing, and photonics.

From the contentsPart 1: Fundamentals of stress and strain on the nanoscale � Elastic strain relaxation: thermodynamics and kinetics � Fundamentals of stress and strain at the nanoscale level: Toward nanoelasticity � Onset of plasticity in crystalline nanomaterials � Relaxations on the nanoscale: an atomistic view by numerical simulationsPart 2: Model Systems with Stress-Engineered Properties � Accommodation of lattice misfi t in semiconductor heterostructure nanowires � Strained silicon nanodevices � Stress-driven nanopatterning in metallic systems � Semiconductor templates for the fabrication of nano-objectsPart 3: Characterization techniques of measuring stresses on the nanoscale � Strain analysis in transmission electron microscopy: How far can we go? � Determination of elastic strains using electron backscatter diffraction in the scanning electron microscope � X-ray diffraction analysis of elastic strains at the nanoscale � Diffuse X-ray scattering at low-dimensional structures in the system SiGe/Si � Direct measurement of elastic displacement modes by grazing incidence X-ray diffraction � Submicrometer-scale characterization of solar silicon by Raman spectroscopy � Strain-Induced Nonlinear Optics in Silicon

Margrit Hanbücken is Research Director in the French CNRS and director of the Competence Centre of Nanosciences and Nano-technologies of the Provence-Alpes-Côte d’Azur region. Her group at CINaM-CNRS in Marseille develops new strategies for the nano-fabrication and functionality of novel templates, sub-sequently used in different fi elds. Prof. Hanbücken has authored over 70 publications, patents and book chapters.

Pierre Müller is professor at the University Paul Cézanne and vice dean of the Science and Technology School of St Jérôme in Marseille, France. His research is dedicated to physics at the surface with a strong expertise in surface elasticity, surface thermodynamics and crystal growth mechanisms. Prof. Müller has authored more than 60 publications and has given 24 invited lectures.

Ralf Wehrspohn is Full Professor in Experimental Physics at the University of Halle-Wittenberg and Director of the Fraunhofer Institute for Mechanics of Materials in Halle, Germany. He has received the out-standing young inventor award of the German Science Foundation and is one of the TR100 nominated by the MIT Technology Review in 2003. Prof. Wehrspohn is author of more than 100 publications and co-inventor of nine patents.

Edited by M. Hanbücken,P. Müller, R. B. Wehrspohn

Mechanical Stress on the Nanoscale

Simulation, Material Systems and Characterization Techniques

57268File AttachmentCover.jpg

Edited by

Margrit Hanbücken,

Pierre Müller,

and Ralf B. Wehrspohn

Mechanical Stress

on the Nanoscale

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Edited byMargrit Hanbücken, Pierre Müller, and Ralf B. Wehrspohn

Mechanical Stress on the Nanoscale

Simulation, Material Systems and CharacterizationTechniques

The Editors

Dr. Margrit HanbückenCINaM-CNRSCampus LuminyMarseille, Frankreich

Dr. Pierre MüllerUniversité Paul CézanneCampus Saint-JérômeMarseille, Frankreich

Prof. Dr. Ralf B. WehrspohnFraunhofer Inst. fürWerkstoffmechanik HalleHalle, Germany

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Contents

Preface XVList of Contributors XVII

Part One Fundamentals of Stress and Strain on the Nanoscale 1

1 Elastic Strain Relaxation: Thermodynamics and Kinetics 3Frank Glas

1.1 Basics of Elastic Strain Relaxation 31.1.1 Introduction 31.1.2 Principles of Calculation 41.1.3 Methods of Calculation: A Brief Overview 61.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys 71.2.1 Spinodal Decomposition with No Elastic Effects 81.2.2 Elastic Strain Relaxation in an Alloy with Modulated Composition 91.2.3 Strain Stabilization and the Effect of Elastic Anisotropy 111.2.4 Elastic Relaxation in the Presence of a Free Surface 111.3 Diffusion 121.3.1 Diffusion without Elastic Effects 121.3.2 Diffusion under Stress in an Alloy 131.4 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers 141.4.1 Introduction 141.4.2 Elastic Strain Relaxation 151.4.3 Critical Thickness 161.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress 171.5.1 Introduction 171.5.2 Calculation of the Elastic Relaxation Fields 181.5.3 ATG Instability 191.5.4 Kinetics of the ATG Instability 211.5.5 Coupling between the Morphological and Compositional

Instabilities 211.6 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures 221.6.1 Quantum Dots 23

V

1.6.2 Nanowires 24References 24

2 Fundamentals of Stress and Strain at the Nanoscale Level:Toward Nanoelasticity 27Pierre Müller

2.1 Introduction 272.2 Theoretical Background 282.2.1 Bulk Elasticity: A Recall 282.2.1.1 Stress and Strain Definition 292.2.1.2 Equilibrium State 292.2.1.3 Elastic Energy 302.2.1.4 Elastic Constants 302.2.2 How to Describe Surfaces or Interfaces? 312.2.3 Surfaces and Interfaces Described from Excess Quantities 342.2.3.1 The Surface Elastic Energy as an Excess of the Bulk

Elastic Energy 342.2.3.2 The Surface Stress and Surface Strain Concepts 352.2.3.3 Surface Elastic Constants 372.2.3.4 Connecting Surface and Bulk Stresses 392.2.3.5 Surface Stress and Surface Tension 402.2.3.6 Surface Stress and Adsorption 412.2.3.7 The Case of Glissile Interfaces 422.2.4 Surfaces and Interfaces Described as a Foreign Material 422.2.4.1 The Surface as a Thin Bulk-Like Film 432.2.4.2 The Surface as an Elastic Membrane 432.3 Applications: Size Effects Due to the Surfaces 442.3.1 Lattice Contraction of Nanoparticles 442.3.2 Effective Modulus of Thin Freestanding Plane Films 462.3.3 Bending, Buckling, and Free Vibrations of Thin Films 482.3.3.1 General Equations 482.3.3.2 Discussion 502.3.4 Static Bending of Nanowires: An Analysis of the Recent

Literature 522.3.4.1 Young Modulus versus Size: Two-Phase Model 522.3.4.2 Young Modulus versus Size: Surface Stress Model 532.3.4.3 Prestress Bulk Due to Surface Stresses 532.3.5 A Short Overview of Experimental Difficulties 542.4 Conclusion 55

References 56

3 Onset of Plasticity in Crystalline Nanomaterials 61Laurent Pizzagalli, Sandrine Brochard, and Julien Godet

3.1 Introduction 613.2 The Role of Dislocations 63

VI Contents

3.3 Driving Forces for Dislocations 633.3.1 Stress 643.3.2 Thermal Activation 643.3.3 Combination of Stress and Thermal

Activation 643.4 Dislocation and Surfaces: Basic Concepts 653.4.1 Forces Related to Surface 653.4.2 Balance of Forces for Nucleation 663.4.3 Forces Due to Lattice Friction 663.4.4 Surface Modifications Due to Dislocations 683.5 Elastic Modeling 683.5.1 Elastic Model 683.5.2 Predicted Activation Parameters 703.5.3 What is Missing? 703.5.4 Peierls–Nabarro Approaches 723.6 Atomistic Modeling 723.6.1 Examples of Simulations 733.6.2 Determination of Activation Parameters 743.6.3 Comparison with Experiments 753.6.4 Influence of Surface Structure, Orientation, and

Chemistry 763.7 Extension to Different Geometries 783.8 Discussion 79

References 80

4 Relaxations on the Nanoscale: An AtomisticView by Numerical Simulations 83Christine Mottet

4.1 Introduction 844.2 Theoretical Models and Numerical Simulations 854.2.1 Energetic Models 854.2.2 Numerical Simulations 874.2.3 Definitions of Physical Quantities 894.3 Relaxations in Surfaces and Interfaces 914.3.1 Surface Reconstructions 924.3.2 Surface Alloys: a Simple Case of Heteroatomic

Adsorption 944.3.3 Heteroepitaxial Thin Films 964.4 Relaxations in Nanoclusters 984.4.1 Free Nanoclusters 994.4.2 Supported Nanoclusters 1004.4.3 Nanoalloys 1014.5 Conclusions 103

References 104

Contents VII

Part Two Model Systems with Stress-Engineered Properties 107

5 Accommodation of Lattice Misfit in SemiconductorHeterostructure Nanowires 109Volker Schmidt and Joerg V. Wittemann

5.1 Introduction 1095.2 Dislocations in Axial Heterostructure Nanowires 1115.3 Dislocations in Core–Shell Heterostructure

Nanowires 1135.4 Roughening of Core–Shell Heterostructure

Nanowires 1155.4.1 Zeroth-Order Stress and Strain 1175.4.2 First-Order Contribution to Stress and Strain 1205.4.3 Linear Stability Analysis 1225.4.4 Results and Discussion 1245.5 Conclusion 127

References 127

6 Strained Silicon Nanodevices 131Manfred Reiche, Oussama Moutanabbir, Jan Hoentschel, Angelika Hähnel,Stefan Flachowsky, Ulrich Gösele, and Manfred Horstmann

6.1 Introduction 1316.2 Impact of Strain on the Electronic Properties

of Silicon 1326.3 Methods to Generate Strain in Silicon Devices 1356.3.1 Substrates for Nanoscale CMOS Technologies 1356.3.2 Local Strain 1366.3.3 Global Strain 1396.3.3.1 Biaxially Strained Layers 1396.3.3.2 Uniaxially Strained Layers 1426.4 Strain Engineering for 22 nm CMOS Technologies

and Below 1426.5 Conclusions 146

References 146

7 Stress-Driven Nanopatterning in Metallic Systems 151Vincent Repain, Sylvie Rousset, and Shobhana Narasimhan

7.1 Introduction 1517.2 Surface Stress as a Driving Force for Patterning at Nanometer

Length Scales 1527.2.1 Surface Stress 1527.2.2 Surface Reconstruction and Misfit Dislocations 1537.2.2.1 Homoepitaxial Surfaces 1537.2.2.2 Heteroepitaxial Systems 1557.2.3 Stress Domains 156

VIII Contents

7.2.4 Vicinal Surfaces 1577.3 Nanopatterned Surfaces as Templates for the Ordered

Growth of Functionalized Nanostructures 1587.3.1 Metallic Ordered Growth on Nanopatterned

Surface 1587.3.1.1 Introduction 1587.3.1.2 Nucleation and Growth Concepts 1597.3.1.3 Heterogeneous Growth 1607.4 Stress Relaxation by the Formation of Surface-Confined

Alloys 1627.4.1 Two-Component Systems 1627.4.2 Three-Component Systems 1627.5 Conclusion 164

References 165

8 Semiconductor Templates for the Fabrication of Nano-Objects 169Joël Eymery, Laurence Masson, Houda Sahaf,and Margrit Hanbücken

8.1 Introduction 1698.2 Semiconductor Template Fabrication 1708.2.1 Artificially Prepatterned Substrates 1708.2.1.1 Morphological Patterning 1708.2.1.2 Silicon Etched Stripes: Example of the Use of Strain to

Control Nanostructure Formation and PhysicalProperties 171

8.2.1.3 Use of Buried Stressors 1718.2.2 Patterning through Vicinal Surfaces 1738.2.2.1 Generalities 1738.2.2.2 Vicinal Si(111) 1738.2.2.3 Vicinal Si(100) 1738.3 Ordered Growth of Nano-Objects 1758.3.1 Growth Modes and Self-Organization 1758.3.2 Quantum Dots and Nanoparticles Self-Organization with Control

in Size and Position 1768.3.2.1 Stranski–Krastanov Growth Mode 1768.3.2.2 Au/Si(111) System 1778.3.2.3 Ge/Si(001) System 1798.3.3 Wires: Catalytic and Catalyst-Free Growths with Control

in Size and Position 1798.3.3.1 Strain in Bottom-Up Wire Heterostructures: Longitudinal and

Radial Heterostructures 1818.3.3.2 Wires as a Position Controlled Template 1838.4 Conclusions 184

References 184

Contents IX

Part Three Characterization Techniques of Measuring Stresses onthe Nanoscale 189

9 Strain Analysis in Transmission Electron Microscopy:How Far Can We Go? 191Anne Ponchet, Christophe Gatel, Christian Roucau,and Marie-José Casanove

9.1 Introduction: How to Get Quantitative Information onStrain from TEM 192

9.1.1 Displacement, Strain, and Stress in Elasticity Theory 1929.1.2 Principles of TEM and Application to Strained

Nanosystems 1929.1.3 A Major Issue for Strained Nanostructure Analysis:

The Thin Foil Effect 1939.2 Bending Effects in Nanometric Strained Layers: A Tool

for Probing Stress 1949.2.1 Bending: A Relaxation Mechanism 1949.2.2 Relation between Curvature and Internal Stress 1959.2.3 Using the Bending as a Probe of the Epitaxial Stress:

The TEM Curvature Method 1969.2.4 Occurrence of Large Displacements in TEM Thinned

Samples 1979.2.5 Advantages and Limits of Bending as a Probe of Stress

in TEM 1999.3 Strain Analysis and Surface Relaxation in Electron

Diffraction 1999.3.1 CBED: Principle and Application to Determination of

Lattice Parameters 1999.3.2 Strain Determination in CBED 2019.3.3 Use and Limitations of CBED in Strain Determination 2029.3.4 Nanobeam Electron Diffraction 2039.4 Strain Analysis from HREM Image Analysis: Problematic

of Very Thin Foils 2039.4.1 Principle 2039.4.2 What Do We Really Measure in an HREM Image? 2059.4.2.1 Image Formation 2059.4.2.2 Reconstruction of the 3D Strain Field from a 2D Projection 2059.4.3 Modeling the Surface Relaxation in an HREM Experiment 2069.4.3.1 Full Relaxation (Uniaxial Stress) 2069.4.3.2 Intermediate Situations: Usefulness of Finite Element Modeling 2079.4.3.3 Thin Foil Effect: A Source of Incertitude in HREM 2079.4.4 Conclusion: HREM is a Powerful but Delicate Method

of Strain Analysis 2089.5 Conclusions 209

References 210

X Contents

10 Determination of Elastic Strains Using Electron BackscatterDiffraction in the Scanning Electron Microscope 213Michael Krause, Matthias Petzold, and Ralf B. Wehrspohn

10.1 Introduction 21310.2 Generation of Electron Backscatter Diffraction Patterns 21410.3 Strain Determination Through Lattice Parameter

Measurement 21510.4 Strain Determination Through Pattern Shift Measurement 21610.4.1 Linking Pattern Shifts to Strain 21610.4.2 Measurement of Pattern Shifts 21910.5 Sampling Strategies: Sources of Errors 22110.6 Resolution Considerations 22210.7 Illustrative Application 22510.8 Conclusions 229

References 230

11 X-Ray Diffraction Analysis of Elastic Strains at the Nanoscale 233Olivier Thomas, Odile Robach, Stéphanie Escoubas, Jean-Sébastien Micha,Nicolas Vaxelaire, and Olivier Perroud

11.1 Introduction 23311.2 Strain Field from Intensity Maps around Bragg Peaks 23411.3 Average Strains from Diffraction Peak Shift 23611.4 Local Strains Using Submicrometer Beams and Scanning XRD 24011.4.1 Introduction 24011.4.2 High-Energy Monochromatic Beam: 3DXRD 24111.4.3 White Beam: Laue Microdiffraction 24311.5 Local Strains Derived from the Intensity Distribution in

Reciprocal Space 24811.5.1 Periodic Assemblies of Identical Objects with Coherence

Length > Few Periods 24811.5.1.1 Introduction 24811.5.1.2 Reciprocal Space Mapping 24911.5.1.3 Applications 25111.5.2 Single-Object Coherent Diffraction 25211.6 Phase Retrieval from Strained Crystals 25411.7 Conclusions and Perspectives 255

References 256

12 Diffuse X-Ray Scattering at Low-Dimensional Structures in theSystem SiGe/Si 259Michael Hanke

12.1 Introduction 25912.2 Self-Organized Growth of Mesoscopic Structures 25912.2.1 The Stranski–Krastanow Process 26012.2.2 LPE-Grown Si1�xGex/Si(001) Islands 261

Contents XI

12.3 X-Ray Scattering Techniques 26212.3.1 High-Resolution X-Ray Diffraction 26212.3.2 Grazing Incidence Diffraction 26312.3.3 Grazing Incidence Small-Angle X-Ray Scattering 26412.4 Data Evaluation 26512.5 Results 26612.5.1 The Influence of Shape and Size on the GISAXS Signal 26612.5.2 HRXRD Measurement of Strain and Composition 26912.5.3 Positional Correlation Effects in HRXRD 27012.5.4 Iso-Strain Scattering 27112.6 Summary 273

References 274

13 Direct Measurement of Elastic Displacement Modes by GrazingIncidence X-Ray Diffraction 275Geoffroy Prévot

13.1 Introduction 27513.2 Elastic Displacement Modes: Analysis and GIXD Observation 27613.2.1 Fundamentals of Linear Elasticity in Direct Space 27613.2.1.1 Basic Equations 27613.2.1.2 Atomic Displacements and Elastic Interactions 27713.2.2 Greens Tensor in Reciprocal Space 27913.2.3 Grazing Incidence X-Ray Diffraction of Elastic Modes 28013.2.3.1 Diffraction by a Surface 28013.2.3.2 Contribution of the Elastic Modes 28013.2.3.3 Procedure for Analyzing the Systems 28113.3 Self-Organized Surfaces 28213.3.1 Force Distribution and Interaction Energy for Self-Organized

Surfaces 28213.3.2 A 1D Case: OCu(110) 28313.3.3 A 2D Case: NCu(001) 28613.4 Vicinal Surfaces 28913.4.1 Force Distribution and Interaction Energy for Steps 28913.4.2 Experimental Results for Vicinal Surfaces of Transition

Metals 29213.5 Conclusion 294

References 295

14 Submicrometer-Scale Characterization of Solar Siliconby Raman Spectroscopy 299Michael Becker, George Sarau, and Silke Christiansen

14.1 Introduction 29914.2 Crystal Orientation 30014.2.1 Qualitative Maps 30014.2.2 Quantitative Analysis 302

XII Contents

14.2.3 Comparison with Other Orientation MeasurementMethods 306

14.3 Analysis of Stress and Strain States 30714.3.1 General Theoretical Description 30714.3.2 Quantitative Strain/Stress Analysis in Polycrystalline

Silicon Wafers 30914.3.2.1 Assumptions 30914.3.2.2 Numerical Determination of Stress Components 31014.3.3 Experimental Procedure to Determine Phonon Frequency

Shifts 31114.3.4 Additional Influences on the Phonon Frequency Shifts 31114.3.4.1 Temperature 31114.3.4.2 Drift of the Spectrometer Grating 31314.3.5 Applications 31314.3.5.1 Mechanical Stresses at the Backside of Silicon Solar Cells 31314.3.5.2 Stress Fields at Microcracks in Polycrystalline Silicon Wafers 31514.3.5.3 Stress States at Grain Boundaries in Polycrystalline Silicon

Solar Cell Material and the Relation to the Grain BoundaryMicrostructure and Electrical Activity 316

14.3.6 Comparison with other Stress/Strain MeasurementMethods 318

14.4 Measurement of Free Carrier Concentrations 31814.4.1 Theoretical Description 31914.4.2 Experimental Details 32114.4.2.1 Small-Angle Beveling and Nomarski Differential Interference

Contrast Micrographs 32114.4.2.2 Evaluation of the Raman Data 32214.4.2.3 Calibration Measurements 32414.4.3 Experimental Results 32414.4.4 Comparison with other Dopant Measurement Methods 32814.5 Concluding Remarks 328

References 329

15 Strain-Induced Nonlinear Optics in Silicon 333Clemens Schriever, Christian Bohley, and Ralf B. Wehrspohn

15.1 Introduction 33315.2 Fundamentals of Second Harmonic Generation in Nonlinear

Optical Materials 33415.3 Second Harmonic Generation and Its Relation to Structural

Symmetry 33615.3.1 Sources of Second Harmonic Signals 33715.3.2 Bulk Contribution to Second Harmonic Generation 33815.3.3 Surface Contribution to Second Harmonic Generation 34115.4 Strain-Induced Modification of Second-Order Nonlinear

Susceptibility in Silicon 343

Contents XIII

15.5 Strained Silicon in Integrated Optics 34815.5.1 Strain-Induced Electro-Optical Effect 34815.5.2 Strain-Induced Photoelastic Effect 35015.6 Conclusions 352

References 353

Index 357

XIV Contents

Preface

The development of future integrated (‘‘smart’’) micro- and nanosystems is generallyfocusing on further improvements of functionality and performance, enhancementofminiaturization and integration density, and extension into new application fields.In addition to any of these technological developments, reliability, quality, andmanufacturing yield are key prerequisites for the development of any complexinnovative (‘‘smart’’) micro-/nanosystem application. Consequently, new methods,instruments, and tools adjusted to the specific boundary conditions of the miniatur-ization level down to the nanoscale have to be provided allowing the investigationand understanding of the microstructure, possible failure processes, and reliabilityrisks. In addition, methods and tools allowing the addressing and measurement oflocally affected material properties, such as residual stresses, in combination withthe microstructure are required. Such instruments and techniques are required tosupport a focused and rapid technological development and the time-efficient designof components and smart systems.

The particular results of microstructure and stress characterization do not onlyprovide the basis for technological process step improvement but are also requiredfor advanced simulation approaches and models that can be used to considerreliability properties already during the product development stage (‘‘design forreliability’’ concept). Such concepts gain increasing importance since they allow toreduce time-to-market and development cost.

Present local stress and strain measurements on the nanoscale are based onspecial transmission electron microscopy techniques such as CBED, HRTEM-GPA,or holographic dark field technology, special scanning electron microscopy techni-ques such as EBSD or adapted X-ray diffraction techniques such as coherent X-raydiffraction. This book brings together leading groups in these different disciplines toapply these techniques for local strain and stress measurement and its theoreticalbackground.

The book consists of three parts. Part One addresses the fundamentals of stressand strain on the nanoscale including an introduction to thermodynamics, kinetics,and models of elasticity, plasticity, and relaxation. Part Two addresses applicationswhere stress and strain on the nanoscale are relevant such as SiGe devices ornanowires. In Part Three, techniques for measuring stress and strain on the

XV

nanoscale are presented such as CBED-TEM, EBSD-REM, different ways to useX-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 increasingamount of the integration of various technologies, and various aspect ratios, it will becrucial to understand in detail processes and phenomena of nanostress.This work was stimulated by the cooperation of the Fraunhofer Society, the Max-

Planck-Society, the Carnot Association, and the CNRS via the CNano-PACA.This book is dedicated to Prof. Ulrich Gösele, who coinitiated this project.

February 28, 2011 Ralf WehrspohnHalle and Marseille Margrit Hanbücken

Pierre Müller

XVI Preface

List of Contributors

XVII

Michael BeckerMax Planck Institute ofMicrostructure PhysicsExperimental Department IIWeinberg 206120 HalleGermany

Sandrine BrochardInstitut PPRIME – CNRS UPR 3346Département de Physique et deMécanique des MatériauxEspace Phymat, BP 3017986962 Futuroscope Chasseneuil CedexFrance

Christian BohleyMartin-Luther-UniversityInstitute of PhysicsHeinrich-Damerow – Str. 406120 HalleGermany

and

Martin-Luther-UniversityCentre for Innovation CompetenceSiLi-nanoKarl-Freiherr-von-Fritsch-Str. 306120 Halle (Saale), Germany

Marie-José CasanoveCNRS-UPSCentre d’Elaboration de Matériauxet d’Etudes Structurales29, rue Jeanne Marvig, BP 9434731055 Toulouse Cedex 4France

Silke ChristiansenMax Planck Institute forthe Science of LightGuenther-Scharowsky – Str. 191058 ErlangenGermany

Stéphanie EscoubasAix-Marseille UniversitéIM2NP, Faculté des Scienceset TechniquesCampus de Saint-JérômeAvenue Escadrille NormandieNiemen, Case 14213397 Marseille CedexFrance

and

CNRS, IM2NP (UMR 6242)Faculté des Sciences et TechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

Joël EymeryCEA/CNRS/Université Joseph FourierCEA, INAC, SP2M17 rue des Martyrs38054 Grenoble Cedex 9France

Stefan FlachowskyGLOBALFOUNDRIES Fab 1Wilschdorfer Landstraße 10101109 DresdenGermany

Christophe GatelCNRS-UPSCentre d’Elaboration de Matériauxet d’Etudes Structurales29, rue Jeanne Marvig, BP 9434731055 Toulouse Cedex 4France

Frank GlasCNRSLaboratoire de Photonique et deNanostructuresRoute de Nozay91460 MarcoussisFrance

Julien GodetInstitut PPRIME – CNRS UPR 3346Département de Physique et deMécanique des MatériauxEspace Phymat, BP 3017986962 Futuroscope Chasseneuil CedexFrance

Ulrich Göseley

Max Planck Institute ofMicrostructure PhysicsWeinberg 206120 HalleGermany

Angelika HähnelMax Planck Institute ofMicrostructure PhysicsWeinberg 206120 HalleGermany

Margrit HanbückenCINaM-CNRSCampus de Luminy, Case 9133288 Marseille Cedex 9France

Michael HankePaul-Drude-Institute for Solid StateElectronicsHausvogteiplatz 5-710117 BerlinGermany

Jan HoentschelGLOBALFOUNDRIES Fab 1Wilschdorfer Landstraße 10101109 DresdenGermany

Manfred HorstmannGLOBALFOUNDRIES Fab 1Wilschdorfer Landstraße 10101109 DresdenGermany

Michael KrauseFraunhofer IWMWalter-Hülse – Str. 106120 HalleGermany

Laurence MassonCINaM-CNRSCampus de Luminy, Case 9133288 Marseille Cedex 9France

XVIII List of Contributors

Jean-Sébastien MichaINAC/SPrAMUMR 5819 (CEA-CNRS-UJF)CEA-Grenoble17 rue des Martyrs38054 Grenoble Cedex 9France

Christine MottetCINaM – CNRSCampus de Luminy, Case 91313288 Marseille Cedex 9France

Oussama MoutanabbirMax Planck Institute ofMicrostructure PhysicsWeinberg 206120 HalleGermany

Pierre MüllerAix Marseille UniversitéCenter Interdisciplinaire deNanoscience de MarseilleUPR CNRS 3118Campus de Luminy, Case 91313288 Marseille Cedex 9France

Shobhana NarasimhanJNCASRTheoretical Sciences UnitJakkur560 064 BangaloreIndia

Olivier PerroudAix-Marseille UniversitéIM2NP, Faculté des Scienceset TechniquesCampus de Saint-JérômeAvenue Escadrille NormandieNiemen, Case 14213397 Marseille CedexFrance

and

CNRS, IM2NP (UMR 6242)Faculté des Sciences et TechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

Laurent PizzagalliInstitut PPRIME – CNRS UPR 3346Département de Physique et deMécanique des MatériauxEspace Phymat, BP 3017986962 Futuroscope Chasseneuil CedexFrance

Anne PonchetCNRS-UPSCentre d’Elaboration de Matériauxet d’Etudes Structurales29, rue Jeanne Marvig, BP 9434731055 Toulouse Cedex 4France

Matthias PetzoldFraunhofer Institute for Mechanicsof Materials HalleWalter-Hülse-Str.106120 Halle

List of Contributors XIX

Geoffroy PrévotUniversité Pierre et Marie Curie-Paris 6UMR CNRS 7588, Institut desNanoSciences de ParisCampus Boucicaut, 140 rue de Lourmel75015 ParisFrance

Manfred ReicheMax Planck Institute ofMicrostructure PhysicsWeinberg 206120 HalleGermany

Vincent RepainCNRS et Université Paris DiderotMatériaux et Phénomènes QuantiquesBâtiment Condorcet – Case 702175205 ParisFrance

Odile RobachCEA-GrenobleINAC/SP2M/NRS17 rue des Martyrs38054 Grenoble Cedex 9France

Christian RoucauCNRS-UPSCentre d’Elaboration de Matériauxet d’Etudes Structurales29, rue Jeanne Marvig, BP 9434731055 Toulouse Cedex 4France

Sylvie RoussetCNRS et Université Paris DiderotMatériaux et Phénomènes QuantiquesBâtiment Condorcet – Case 702175205 ParisFrance

Houda SahafCINaM-CNRSCampus de Luminy, Case 9133288 Marseille Cedex 9France

George SarauMax Planck Institute of MicrostructurePhysicsExperimental Department IIWeinberg 206120 HalleGermany

and

Max Planck Institute forthe Science of LightGuenther-Scharowsky – Str. 191058 ErlangenGermany

Volker SchmidtMax Planck Institute ofMicrostructure PhysicsExperimental Department IIWeinberg 206120 HalleGermany

Clemens SchrieverMartin-Luther-UniversityInstitute of PhysicsHeinrich-Damerow – Str. 406120 HalleGermany

and

Martin-Luther-UniversityCentre for Innovation CompetenceSiLi-nanoKarl-Freiherr-von-Fritsch-Str. 306120 Halle (Saale), Germany

XX List of Contributors

Olivier ThomasAix-Marseille UniversitéIM2NP, Faculté des Sciences etTechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

and

CNRS, IM2NP (UMR 6242)Faculté des Sciences et TechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

Nicolas VaxelaireAix-Marseille UniversitéIM2NP, Faculté des Sciences etTechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

and

CNRS, IM2NP (UMR 6242)Faculté des Sciences et TechniquesCampus de Saint-JérômeAvenue Escadrille Normandie Niemen,Case 14213397 Marseille CedexFrance

Ralf B. WehrspohnMartin-Luther-UniversityInstitute of PhysicsHeinrich-Damerow – Str. 406120 HalleGermany

and

Fraunhofer Institute for Mechanicsof Materials HalleWalter-Hülse-Str. 106120 HalleGermany

Joerg V. WittemannMax Planck Institute ofMicrostructure PhysicsExperimental Department IIWeinberg 206120 HalleGermany

List of Contributors XXI

Part OneFundamentals 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.

j1

1Elastic Strain Relaxation: Thermodynamics and KineticsFrank Glas

1.1Basics of Elastic Strain Relaxation

1.1.1Introduction

Although frequently used, the phrase elastic strain relaxation is difficult to define. Itusually designates the modification of the strain fields induced in a solid by atransformation of part or whole of this solid. At variance with plastic relaxation, incrystals, 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 rearrange-ments.Usually the realization of the instability is conditioned by kinetic processes (inparticular, diffusion), which themselves depend on the stress state of the system.Elastic relaxation may also occur during the formation of part of a system, forinstance, by epitaxial growth. The state with respect to which the relaxation isassessed 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 sufficientlyactive for the system to reach its optimal configuration.

In the present introductory section, we give a general principle for the calculationof strain relaxation and briefly discuss some analytical andnumericalmethods. In thenext sections, we examine important cases where elastic strain relaxation plays acrucial part. Section 1.2 deals with strain relaxation in substitutional alloys withspatially varying compositions and with the thermodynamics and kinetics of theinstability of such alloys against composition modulations. Section 1.3 introduces akinetic process of major importance, namely, diffusion, and summarizes how it isaffected by elastic effects. Section 1.4 treats the case of a homogeneous mismatchedlayer of uniform thickness grownon a substrate. Section 1.5 showshowa systemwitha planar free surface submitted to a nonhydrostatic stress is unstable with respect to

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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.

the development of surface corrugations. Finally, Section 1.6 briefly recalls how thepresence of free surfaces in objects of nanometric lateral dimensions, such asquantum dots or nanowires (NWs), permits a much more efficient elastic strainrelaxation than in the case of uniformly thick layers.

1.1.2Principles of Calculation

At given temperature and pressure, any single crystal possesses a reference intrinsicmechanical state E0 in which the strains and stresses are zero, namely, the statedefined by the crystal lattice (and the unit cell) of this solid under bulk form. If thecrystal experiences a transformation (change of temperature, phase transformation,change of composition, etc.), this intrinsic mechanical state changes to E1, whereagain strains and stresses are zero (Figure 1.1a). The corresponding deformation isthe stress-free strain (or eigenstrain) e*ij with respect to state E0; for instance, for achange of temperature dT , e*ij ¼ dijadT , wherea is the thermal dilatation coefficient.If the crystal is mechanically isolated, it simply adopts its new intrinsic state E1; it isthen free of stresses. This is not the case if the transformation affects only part of thesystem. We then have two extreme cases. The transformation is incoherent if it doesnot preserve any continuity between the crystal lattices of the transformed part and ofits 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 andmatrix the untransformedpart of the system (indexed by exponents I and M). Coherency is obviouslyincompatible with the adoption by the inclusion of its stress-free state E1, the matrixremaining 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 thefollowing sense: if one imagines the inclusion having been transformed (forinstance, heated) but remaining in its original reference mechanical state E0 (whichrestores coherency, since the matrix has not been transformed from state E0), it issubjected to stresses, since forces must be applied at its boundary to bring it from itsnew intrinsic state E1 back to E0. 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 Eshelbysprocess.

4j 1 Elastic Strain Relaxation: Thermodynamics and Kinetics

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

1) One applies to the transformed inclusion (state E1) the strain�eI*ij , which bringsit back to state E0. This implies exerting on its external surface (whose externalnormal n has components nj) the forces�

P3j¼1 s

I�ij nj per unit area, where s

I�ij is

the stress associated1) with the stress-free strain eI�ij .

2) Having thus restored coherency between inclusion andmatrix, onemay reinsertthe former into the latter. The only change that then occurs is the change of thesurface density of forces applied at stage (1) into a body density fi, since thesurface of the inclusion becomes an internal interface.2)

3) The resulting state is not a mechanical equilibrium state, since forces fi must beapplied tomaintain it. One then lets the system relax by suppressing these forces,that is, by applying forces �fi , while at the same time maintaining coherencyeverywhere. One thus has to compute the strain field, in the inclusion (eIrij ) and inthematrix (eMrij ), solution of the elasticity equations for body forces�fi , under thecoherency constraint, which amounts to equal displacements uIr ¼ uMr at theinterface.

We may generalize this approach by not differentiating matrix and inclusion. Thewhole system experiences a transformation producing an inhomogeneous stress-free strain e�ijðrÞ (defined at any point r) with respect to initial uniform state E0 (perfectcrystal). One then applies the body forces producing strain �e�ij, namely,fi ¼

P3j¼1 @s

�ij=@xj, where s

�ijðrÞ is the stress associated with strain e�ij. Finally, one

calculates the relaxation field erij, the solution of the elastic problemwith forces�fiðrÞthat preserves coherency everywhere (displacements must be continuous). It isimportant to specify the reference state with respect to which one defines the finalstate of the system. It is often easier to visualize the relaxed state relative to theuniform state E0; strain is then simply erij. If, on the contrary, the elastic energy Wstored in the system is to be calculated, we must take as a reference state for eachvolume element its intrinsic state after transformation (E1), with respect to which thetotal strain is etij ¼ erij�e�ij. Hence,W ¼ ð1=2Þ

ÐV

P3i; j¼1 e

tijs

tijdV , where s

tij is the strain

associated with etij and where the integral is taken over the whole volume (in thereference state).

In case of an inclusion (Figure 1.1), one may easily show that

W ¼ � 12

ðI

X3i; j¼1

eI�ij sItij dv ð1:1Þ

This is a fundamental result obtained by Eshelby [1]. In particular, the total elasticenergy depends only on the stress in the inclusion.

An example of application to an infinite system with a continuously varyingtransformation will be given in Section 1.2. Eshelbys methodmay also be adapted to

1) Via the constitutive relations, for instance, Hookes law in linear elasticity.

2) In the present case (single inclusion), this density is nonzero only in the zero-thickness interfacelayer, so for a facet x ¼ x0, one has fi ¼ �sI�ixdðx�x0Þ.

1.1 Basics of Elastic Strain Relaxation j5

other problems. In particular, if the interface between matrix and inclusion does notentirely surround the latter (which happens if the inclusion has a free surface), it isnot necessary to apply strain�eI�ij to the inclusion at stage 1. It suffices to apply a strainthat restores the coherency in the interface, which may make the solution of theproblem simpler. An example is given in Section 1.4.

1.1.3Methods of Calculation: A Brief Overview

The problem thus consists in determining the fields relative to stage 3 of theprocess. One has to calculate the elastic relaxation of a medium subjected to a givendensity �fi of body forces. In addition to numerical methods, for instance, thosebased on finite elements, there exist several analytical methods for solving thisproblem, in particular, the Greens functions method [2] and the Fourier synthesismethod.

In elasticity, Greens function Gijðr; r0Þ is defined as the component along axis i ofdisplacement at point r caused by a unit body force along j applied at point r0. For asolid with homogeneous properties, it is a functionGijðr�r0Þ of the vector joining thetwo points. One easily shows that for an elastically linear solid (with elastic constantsCjklm), the displacement field at stage 3 is

uiðrÞ ¼ �X3

j;k;l;m¼1Cjklm

ðe�lmðr0Þ

@Gij@xk

ðr�r0Þd r0 ð1:2Þ

where the integral extends to all points r0 of the volume. Greens functions depend onthe elastic characteristics of themedium, but, once determined, any problem relativeto this medium is solved by a simple integration. However, if the Greens functionsfor an infinite and elastically isotropic solid have been known since 1882, only a fewcases have been solved exactly. If themedium is not infinite in three dimensions, theGreens functions also depend on its external boundary and on the conditions that areimposed to it. For epitaxy-related problems, the case of the half-space (semi-infinitesolid with planar surface) is particularly interesting. These functions have beencalculated for the elastically isotropic half-space with a free surface (no externaltractions) [3, 4]. Muras book gives further details [2]. The method also applies to therelaxation of two solids in contact via a planar interface; in this case, this surface isgenerally not traction-free and the boundary conditions may be on these tractions oron its displacements. Pan has given a general solution in the anisotropic case, validfor all boundary conditions [5].

In the Fourier synthesis method, one decomposes the stress-free strain distribu-tion into its Fourier components: e�ijðrÞ ¼

Ð~e�ijðkÞexpðikrÞd k, where k is the running

wave vector. In linear elasticity, the solution is simply the sum, weighted by theFourier coefficients ~e�ijðkÞ, of the solutions relative to each periodic wave of wavevector k, which are themselves periodic with the same wave vector. If the system isinfinite, the elementary solution is easily determined (see Section 1.2.2). The onlynontrivial point is then the integration. This method allows one to treat elegantly the

6j 1 Elastic Strain Relaxation: Thermodynamics and Kinetics