Preface ................................................ XV
List of Contributors ................................. XVII
Part One. Fundamentals of Stress and Strain on the
Nanoscale ....................................................... 2
1 Elastic Strain Relaxation: Thermodynamics and Kinetics ....... 3
Frank Clas
1.1 Basics of Elastic Strain Relaxation ..................... 3
1.1.1 Introduction ..................................... 3
1.1.2 Principles of Calculation ........................ 4
1.1.3 Methods of Calculation: A Brief Overview ......... 6
1.2 Elastic Strain Relaxation in Inhomogeneous
Substitutional Alloys ................................... 7
1.2.1 Spinodal Decomposition with No Elastic Effects ... 8
1.2.2 Elastic Strain Relaxation in an Alloy with
Modulated Composition ............................ 9
1.2.3 Strain Stabilization and the Effect of Elastic
Anisotropy ...................................... 11
1.2.4 Elastic Relaxation in the Presence of a Free
Surface ......................................... 11
1.3 Diffusion .............................................. 12
1.3.1 Diffusion without Elastic Effects ............... 12
1.3.2 Diffusion under Stress in an Alloy .............. 13
1.4 Strain Relaxation in Homogeneous Mismatched Epitaxial
Layers ................................................. 14
1.4.1 Introduction .................................... 14
1.4.2 Elastic Strain Relaxation ....................... 15
1.4.3 Critical Thickness .............................. 16
1.5 Morphological Relaxation of a Solid under
Nonhydrostatic Stress .................................. 17
1.5.1 Introduction .................................... 17
1.5.2 Calculation of the Elastic Relaxation Fields
................................................. 18
1.5.3 ATG Instability ................................. 19
1.5.4 Kinetics of the ATG Instability ................. 21
1.5.5 Coupling between the Morphological and
Compositional Instabilities ..................... 21
1.6 Elastic Relaxation of 0D and 1D Epitaxial
Nanostructures ......................................... 22
1.6.1 Quantum Dots .................................... 23
1.6.2 Nanowires ....................................... 24
References ...................................... 24
2 Fundamentals of Stress and Strain at the Nanoscale Level:
Toward Nanoelasticity ....................................... 27
Pierre Müller
2.1 Introduction ........................................... 27
2.2 Theoretical Background ................................. 28
2.2.1 Bulk Elasticity: A Recall ....................... 28
2.2.1.1 Stress and Strain Definition ........... 29
2.2.1.2 Equilibrium State ...................... 29
2.2.1.3 Elastic Energy ......................... 30
2.2.1.4 Elastic Constants ...................... 30
2.2.2 How to Describe Surfaces or Interfaces? ......... 31
2.2.3 Surfaces and Interfaces Described from Excess
Quantities ...................................... 34
2.2.3.1 The Surface Elastic Energy as an
Excess of the Bulk Elastic Energy ...... 34
2.2.3.2 The Surface Stress and Surface Strain
Concepts ............................... 35
2.2.3.3 Surface Elastic Constants .............. 37
2.2.3.4 Connecting Surface and Bulk Stresses ... 39
2.2.3.5 Surface Stress and Surface Tension ..... 40
2.2.3.6 Surface Stress and Adsorption .......... 41
2.2.3.7 The Case of Glissile Interfaces ........ 42
2.2.4 Surfaces and Interfaces Described as a Foreign
Material ........................................ 42
2.2.4.1 The Surface as a Thin Bulk-Like Film ... 43
2.2.4.2 The Surface as an Elastic Membrane ..... 43
2.3 Applications: Size Effects Due to the Surfaces ......... 44
2.3.1 Lattice Contraction of Nanoparticles ............ 44
2.3.2 Effective Modulus of Thin Freestanding Plane
Films ........................................... 46
2.3.3 Bending, Buckling, and Free Vibrations of Thin
Films ........................................... 48
2.3.3.1 General Equations ...................... 48
2.3.3.2 Discussion ............................. 50
2.3.4 Static Bending of Nanowires: An Analysis of
the Recent Literature ........................... 52
2.3.4.1 Young Modulus versus Size: Two-Phase
Model .................................. 52
2.3.4.2 Young Modulus versus Size: Surface
Stress Model ........................... 53
2.3.4.3 Prestress Bulk Due to Surface
Stresses ............................... 53
2.3.5 A Short Overview of Experimental Difficulties ... 54
2.4 Conclusion ............................................. 55
References ............................................. 56
3 Onset of Plasticity in Crystalline Nanomaterials ............ 62
Laurent Pizzagalli, Sandrine Brochard, and Julien Codet
3.1 Introduction ........................................... 62
3.2 The Role of Dislocations ............................... 63
3.3 Driving Forces for Dislocations ........................ 63
3.3.1 Stress .......................................... 64
3.3.2 Thermal Activation .............................. 64
3.3.3 Combination of Stress and Thermal Activation .... 64
3.4 Dislocation and Surfaces: Basic Concepts ............... 65
3.4.1 Forces Related to Surface ....................... 65
3.4.2 Balance of Forces for Nucleation ................ 66
3.4.3 Forces Due to Lattice Friction .................. 66
3.4.4 Surface Modifications Due to Dislocations ....... 68
3.5 Elastic Modeling ....................................... 68
3.5.1 Elastic Model ................................... 68
3.5.2 Predicted Activation Parameters ................. 70
3.5.3 What is Missing? ................................ 70
3.5.4 Peierls-Nabarro Approaches ...................... 72
3.6 Atomistic Modeling ..................................... 72
3.6.1 Examples of Simulations ......................... 73
3.6.2 Determination of Activation Parameters .......... 74
3.6.3 Comparison with Experiments ..................... 75
3.6.4 Influence of Surface Structure, Orientation,
and Chemistry ................................... 76
3.7 Extension to Different Geometries ...................... 78
3.8 Discussion ............................................. 79
References ............................................. 80
4 Relaxations on the Nanoscale: An Atomistic View by
Numerical Simulations ....................................... 83
Christine Mottet
4.1 Introduction ........................................... 84
4.2 Theoretical Models and Numerical Simulations ........... 85
4.2.1 Energetic Models ................................ 85
4.2.2 Numerical Simulations ........................... 87
4.2.3 Definitions of Physical Quantities .............. 89
4.3 Relaxations in Surfaces and Interfaces ................. 92
4.3.1 Surface Reconstructions ......................... 92
4.3.2 Surface Alloys: a Simple Case of Heteroatomic
Adsorption ...................................... 94
4.3.3 Heteroepitaxial Thin Films ...................... 96
4.4 Relaxations in Nanoclusters ............................ 98
4.4.1 Free Nanoclusters ............................... 99
4.4.2 Supported Nanoclusters ......................... 100
4.4.3 Nanoalloys ..................................... 101
4.5 Conclusions ........................................... 103
Reference ............................................. 204
Part Two. Model Systems with Stress-Engineered Properties ..... 107
5 Accommodation of Lattice Misfit in Semiconductor
Heterostructure Nanowires .................................. 109
Volker Schmidt and Joerg V. Wittemann
5.1 Introduction .......................................... 109
5.2 Dislocations in Axial Heterostructure Nanowires ....... 111
5.3 Dislocations in Core-Shell Heterostructure
Nanowires ............................................. 113
5.4 Roughening of Core-Shell Heterostructure Nanowires .... 115
5.4.1 Zeroth-Order Stress and Strain ................. 117
5.4.2 First-Order Contribution to Stress and
Strain ......................................... 120
5.4.3 Linear Stability Analysis ...................... 122
5.4.4 Results and Discussion ......................... 124
5.5 Conclusion ............................................ 127
Reference ............................................. 127
6 Strained Silicon Nanodevices ............................... 131
Manfred Reiche, Oussama Moutanabbir, Jan Hoentschel,
Angelika Hähnel, Stefan Flachowsky, Ulrich Cösele, and
Manfred Horstmann
6.1 Introduction .......................................... 131
6.2 Impact of Strain on the Electronic Properties of
Silicon ............................................... 132
6.3 Methods to Generate Strain in Silicon Devices ......... 135
6.3.1 Substrates for Nanoscale CMOS Technologies ..... 135
6.3.2 Local Strain ................................... 136
6.3.3 Global Strain .................................. 139
6.3.3.1 Biaxially Strained Layers ............. 139
6.3.3.2 Uniaxially Strained Layers ............ 142
6.4 Strain Engineering for 22 nm CMOS Technologies and
Below ................................................. 142
6.5 Conclusions ........................................... 146
Reference ............................................. 146
7 Stress-Driven Nanopatterning in Metallic Systems ........... 151
Vincent Repain, Sylvie Rousset, and Shobhana Narasimhan
7.1 Introduction .......................................... 151
7.2 Surface Stress as a Driving Force for Patterning at
Nanometer Length Scales ............................... 152
7.2.1 Surface Stress ................................. 152
7.2.2 Surface Reconstruction and Misfit
Dislocations ................................... 153
7.2.2.1 Homoepitaxial Surfaces ................ 153
7.2.2.2 Heteroepitaxial Systems ............... 155
7.2.3 Stress Domains ................................. 156
7.2.4 Vicinal Surfaces ............................... 257
7.3 Nanopatterned Surfaces as Templates for the Ordered
Growth of Functionalized Nanostructures ............... 158
7.3.1 Metallic Ordered Growth on Nanopatterned
Surface ........................................ 158
7.3.1.1 Introduction .......................... 158
7.3.1.2 Nucleation and Growth Concepts ........ 159
7.3.1.3 Heterogeneous Growth .................. 360
7.4 Stress Relaxation by the Formation of Surface-
Confined Alloys ....................................... 162
7.4.1 Two-Component Systems .......................... 162
7.4.2 Three-Component Systems ........................ 162
7.5 Conclusion ............................................ 164
Reference ............................................. 265
8 Semiconductor Templates for the Fabrication of Nano-
Objects .................................................... 269
Joël Eymery, Laurence Masson, Houda Sahaf, and Margrit
Hanbücken
8.1 Introduction .......................................... 269
8.2 Semiconductor Template Fabrication .................... 270
8.2.1 Artificially Prepatterned Substrates ........... 270
8.2.1.1 Morphological Patterning .............. 270
8.2.1.2 Silicon Etched Stripes: Example of
the Use of Strain to Control
Nanostructure Formation and Physical
Properties ............................ 172
8.2.1.3 Use of Buried Stressors ............... 272
8.2.2 Patterning through Vicinal Surfaces ............ 273
8.2.2.1 Generalities .......................... 173
8.2.2.2 Vicinal Si(111) ....................... 173
8.2.2.3 Vicinal Si(100) ....................... 173
8.3 Ordered Growth of Nano-Objects ........................ 175
8.3.1 Growth Modes and Self-Organization ............. 175
8.3.2 Quantum Dots and Nanoparticles Self-
Organization with Control in Size and
Position ....................................... 176
8.3.2.1 Stranski-Krastanov Growth Mode ........ 176
8.3.2.2 Au/Si(111) System ..................... 177
8.3.2.3 Ge/Si(001) System ..................... 179
8.3.3 Wires: Catalytic and Catalyst-Free Growths
with Control in Size and Position .............. 179
8.3.3.1 Strain in Bottom-Up Wire
Heterostructures: Longitudinal and
Radial Heterostructures ............... 181
8.3.3.2 Wires as a Position Controlled
Template .............................. 183
8.4 Conclusions ........................................... 184
Reference ............................................. 184
Part Three. Characterization Techniques of Measuring
Stresses on the Nanoscale ..................................... 189
9 Strain Analysis in Transmission Electron Microscopy:
How Far Can We Go? ......................................... 191
Anne Ponchet, Christophe Gatel, Christian Roucau, and
Marie-José Casanove
9.1 Introduction: How to Get Quantitative Information on
Strain from ÒÅÌ ....................................... 192
9.1.1 Displacement, Strain, and Stress in
Elasticity Theory .............................. 192
9.1.2 Principles of ÒÅÌ and Application to Strained
Nanosystems .................................... 192
9.1.3 A Major Issue for Strained Nanostructure
Analysis: The Thin Foil Effect ................. 193
9.2 Bending Effects in Nanometric Strained Layers:
A Tool for Probing Stress ............................. 194
9.2.1 Bending: A Relaxation Mechanism ................ 194
9.2.2 Relation between Curvature and Internal
Stress ......................................... 195
9.2.3 Using the Bending as a Probe of the Epitaxial
Stress: The ÒÅÌ Curvature Method ............... 296
9.2.4 Occurrence of Large Displacements in ÒÅÌ
Thinned Samples ................................ 197
9.2.5 Advantages and Limits of Bending as a Probe
of Stress in ÒÅÌ ............................... 199
9.3 Strain Analysis and Surface Relaxation in Electron
Diffraction ........................................... 199
9.3.1 CBED: Principle and Application to
Determination of Lattice Parameters ............ 199
9.3.2 Strain Determination in CBED ................... 201
9.3.3 Use and Limitations of CBED in Strain
Determination .................................. 202
9.3.4 Nanobeam Electron Diffraction .................. 203
9.4 Strain Analysis from HREM Image Analysis:
Problematic of Very Thin Foils ........................ 203
9.4.1 Principle ...................................... 203
9.4.2 What Do We Really Measure in an HREM Image? .... 205
9.4.2.1 Image Formation ....................... 205
9.4.2.2 Reconstruction of the 3D Strain
Field from a 2D Projection ............ 205
9.4.3 Modeling the Surface Relaxation in an HREM
Experiment ..................................... 206
9.4.3.1 Full Relaxation (Uniaxial Stress) ..... 206
9.4.3.2 Intermediate Situations: Usefulness
of Finite Element Modeling ............ 207
9.4.3.3 Thin Foil Effect: A Source of
Incertitude in HREM ................... 207
9.4.4 Conclusion: HREM is a Powerful but Delicate
Method of Strain Analysis ...................... 208
9.5 Conclusions ........................................... 209
Reference ............................................. 210
10 Determination of Elastic Strains Using Electron
Backscatter Diffraction in the Scanning Electron
Microscope ................................................. 213
Michael Krause, Matthias Petzold, and Ralf Â. Wehrspohn
10.1 Introduction .......................................... 213
10.2 Generation of Electron Backscatter Diffraction
Patterns .............................................. 214
10.3 Strain Determination Through Lattice Parameter
Measurement ........................................... 215
10.4 Strain Determination Through Pattern Shift
Measurement ........................................... 216
10.4.1 Linking Pattern Shifts to Strain ............... 226
10.4.2 Measurement of Pattern Shifts .................. 229
10.5 Sampling Strategies: Sources of Errors ................ 221
10.6 Resolution Considerations ............................. 222
10.7 Illustrative Application .............................. 225
10.8 Conclusions ........................................... 229
Reference ............................................. 230
11 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
11.1 Introduction .......................................... 233
11.2 Strain Field from Intensity Maps around Bragg Peaks ... 234
11.3 Average Strains from Diffraction Peak Shift ........... 236
11.4 Local Strains Using Submicrometer Beams and Scanning
XRD ................................................... 240
11.4.1 Introduction ................................... 240
11.4.2 High-Energy Monochromatic Beam: 3DXRD .......... 241
11.4.3 White Beam: Laue Microdiffraction .............. 243
11.5 Local Strains Derived from the Intensity
Distribution in Reciprocal Space ...................... 248
11.5.1 Periodic Assemblies of Identical Objects with
Coherence Length > Few Periods ................. 248
11.5.1.1 Introduction .......................... 248
11.5.1.2 Reciprocal Space Mapping .............. 249
11.5.1.3 Applications .......................... 251
11.5.2 Single-Object Coherent Diffraction ............. 252
11.6 Phase Retrieval from Strained Crystals ................ 254
11.7 Conclusions and Perspectives .......................... 255
Reference ............................................. 256
12 Diffuse X-Ray Scattering at Low-Dimensional Structures in
the System SiGe/Si ......................................... 259
Michael Hanke
12.1 Introduction .......................................... 259
12.2 Self-Organized Growth of Mesoscopic Structures ........ 259
12.2.1 The Stranski-Krastanow Process ................. 260
12.2.2 LPE-Grown Si1-xGex/Si(001) Islands ............. 261
12.3 X-Ray Scattering Techniques ........................... 262
12.3.1 High-Resolution X-Ray Diffraction .............. 262
12.3.2 Grazing Incidence Diffraction .................. 263
12.3.3 Grazing Incidence Small-Angle X-Ray
Scattering ..................................... 264
12.4 Data Evaluation ....................................... 265
12.5 Results ............................................... 266
12.5.1 The Influence of Shape and Size on the GISAXS
Signal ......................................... 266
12.5.2 HRXRD Measurement of Strain and Composition .... 269
12.5.3 Positional Correlation Effects in HRXRD ........ 270
12.5.4 Iso-Strain Scattering .......................... 272
12.6 Summary ............................................... 273
Reference ............................................. 274
13 Direct Measurement of Elastic Displacement Modes by
Crazing Incidence X-Ray Diffraction ......................... 275
Geoffrey Prévot
13.1 Introduction .......................................... 275
13.2 Elastic Displacement Modes: Analysis and GIXD
Observation ........................................... 276
13.2.1 Fundamentals of Linear Elasticity in Direct
Space .......................................... 276
13.2.1.1 Basic Equations ....................... 276
13.2.1.2 Atomic Displacements and Elastic
Interactions .......................... 277
13.2.2 Green's Tensor in Reciprocal Space ............. 279
13.2.3 Grazing Incidence X-Ray Diffraction of
Elastic Modes .................................. 280
13.2.3.1 Diffraction by a Surface .............. 280
13.2.3.2 Contribution of the Elastic Modes ..... 280
13.2.3.3 Procedure for Analyzing the Systems ... 282
13.3 Self-Organized Surfaces ............................... 282
13.3.1 Force Distribution and Interaction Energy for
Self-Organized Surfaces ........................ 282
13.3.2 A 1D Case: OCu(110) ............................ 283
13.3.3 A 2D Case: NCu(00l) ............................ 286
13.4 Vicinal Surfaces ...................................... 289
13.4.1 Force Distribution and Interaction Energy for
Steps .......................................... 289
13.4.2 Experimental Results for Vicinal Surfaces of
Transition Metals .............................. 292
13.5 Conclusion ............................................ 294
Reference ............................................. 295
14 Submicrometer-Scale Characterization of Solar Silicon
by Raman Spectroscopy ...................................... 299
Michael Becker, George Sarau, and Silke Christiansen
14.1 Introduction .......................................... 299
14.2 Crystal Orientation ................................... 300
14.2.1 Qualitative Maps ............................... 300
14.2.2 Quantitative Analysis .......................... 302
14.2.3 Comparison with Other Orientation Measurement
Methods ........................................ 306
14.3 Analysis of Stress and Strain States .................. 307
14.3.1 General Theoretical Description ................ 307
14.3.2 Quantitative Strain/Stress Analysis in
Polycrystalline Silicon Wafers ................. 309
14.3.2.1 Assumptions ........................... 309
14.3.2.2 Numerical Determination of Stress
Components ............................ 310
14.3.3 Experimental Procedure to Determine Phonon
Frequency Shifts ............................... 311
14.3.4 Additional Influences on the Phonon Frequency
Shifts ......................................... 311
14.3.4.1 Temperature ........................... 311
14.3.4.2 Drift of the Spectrometer Grating ..... 313
14.3.5 Applications ................................... 313
14.3.5.1 Mechanical Stresses at the Backside
of Silicon Solar Cells ................ 313
14.3.5.2 Stress Fields at Microcracks in
Polycrystalline Silicon Wafers ........ 315
14.3.5.3 Stress States at Grain Boundaries in
Polycrystalline Silicon Solar Cell
Material and the Relation to the
Grain Boundary Microstructure and
Electrical Activity ................... 316
14.3.6 Comparison with other Stress/Strain
Measurement Methods ............................ 318
14.4 Measurement of Free Carrier Concentrations ............ 318
14.4.1 Theoretical Description ........................ 319
14.4.2 Experimental Details ........................... 321
14.4.2.1 Small-Angle Beveling and Nomarski
Differential Interference Contrast
Micrographs ........................... 321
14.4.2.2 Evaluation of the Raman Data .......... 322
14.4.2.3 Calibration Measurements .............. 324
14.4.3 Experimental Results ........................... 324
14.4.4 Comparison with other Dopant Measurement
Methods ........................................ 328
14.5 Concluding Remarks .................................... 328
Reference ............................................. 329
15 Strain-Induced Nonlinear Optics in Silicon ................. 333
Clemens Schriever, Christian Bohley, and Ralf Â.
Wehrspohn
15.1 Introduction .......................................... 333
15.2 Fundamentals of Second Harmonic Generation in
Nonlinear Optical Materials ........................... 334
15.3 Second Harmonic Generation and Its Relation to
Structural Symmetry ................................... 336
15.3.1 Sources of Second Harmonic Signals ............. 337
15.3.2 Bulk Contribution to Second Harmonic
Generation ..................................... 338
15.3.3 Surface Contribution to Second Harmonic
Generation ..................................... 341
15.4 Strain-Induced Modification of Second-Order
Nonlinear Susceptibility in Silicon ................... 343
15.5 Strained Silicon in Integrated Optics ................. 348
15.5.1 Strain-Induced Electro-Optical Effect .......... 348
15.5.2 Strain-Induced Photoelastic Effect ............. 350
15.6 Conclusions ........................................... 352
Reference ............................................. 353
Index ................................................. 357
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