TU-Da

Kenneth A. Jackson
Kinetic Processes
Crystal Growth, Diffusion, and Phase Transitions in Materials
WILEYVCH
WILEY-VCH Verlag GmbH & Co. KGaA
Contents
Preface
XIII
1
Introduction
1.1
1.2
1.3
1.4
1.5
Arrhenius Plot 1
The Relationship between Kinetics and Thermodynamics
The Boltzmann Distribution 4
Kinetic Theory of Gases 6
Collisions 7
1
2
Diffusion in Fluids
2.1
2.2
2.2.1
2.2.2
2.2.3
Diffusion in a Gas 11
Diffusion in Liquids 12
Diffusion Distances 13
Molecular Dynamics Simulations of Diffusion in Liquids
Measurement of Diffusion Coefficients in Liquids 16
2
11
3
Diffusion in Amorphous Materials
3.1
3.2
3.2.1
3.2.2
3.3
3.4
3.5
3.6
3.7
Amorphous Materials 19
Network Glass Formers 20
Silica 20
Silicon and Germanium 20
The Glass Transition 22
The Free-Volume Model 23
Fictive Temperature 23
Diffusion in Polymers 24
The Stokes-Einstein Relationship
4
Diffusion in Crystals
4.1
4.2
4.2.1
4.2.2
Diffusion in a Crystal 27
Diffusion Mechanisms in Crystals
Vacancy Diffusion 28
Interstitial Diffusion 30
19
25
27
28
13
VI
Contents
4.4
4.4.1
4.4.2
4.5
4.6
4.7
4.8
4.9
4.10
4.11
A.12
4.13
Equilibrium Concentration of Vacancies 30
Thermodynamic Analysis 30
Kinetic Analysis 32
Simmons and Balluffi Experiment 34
Ionic and Covalent Crystals 34
Stoichiometry 36
Measurement of Diffusion Coefficients 36
Surface Diffusion 37
Diffusion in Grain Boundaries 38
Kirkendall Effect 38
Whisker Growth 39
Electromigration 40
5
Diffusion in Semiconductors
5.1
5.2.1
5.2.2
5.2.3
5.2.4
5.3
5.4
5.5
5.6
5.7
Introduction 45
Vacancy Diffusion in Silicon 47
Diffusion of Phosphorus in Silicon 49
Diffusion of Arsenic in Silicon 50
Diffusion of Boron in Silicon 50
Diffusion of Zinc in GaAs 51
Recombination-enhanced Diffusion 52
Doping of Semiconductors 53
Point-Defect Generation in Silicon during Crystal Growth 53
Migration of Interstitials (and Liquid Droplets)
in a Temperature Gradient 56
Oxygen in Silicon 57
Gettering 57
Solid-State Doping 58
5.8
5.9
5.10
45
6
Ion Implantation
6.1
6.2
6.3
6.4
6.5
6.6
Introduction 61
Ion Interactions 62
Implantation Damage 64
Rutherford Baekscattering 67
Channeling 69
Silicon-on-Insulator 71
61
7
Mathematics of Diffusion
7.1
7.2
7.3
7.3.1
7.3.2
7.3.3
7.3.4
Random Walk 75
The Diffusion Equation 77
Solutions to the Diffusion Equation 79
Gaussian Concentration Distribution 79
Error Function Concentration Distribution 82
p/n Junction Depth 84
Separation of Variables 85
75
Contents VII
7.4
7.A.I
7.4.2
7.4.3
7.4.4
7.5
Numerical Methods 90
Finite Difference Method for Diffusion 91
Initial Surface Concentration Boundary Conditions
Implanted Concentration Profile 92
Zero Flux Boundary Condition 93
Boltzmann-Matano Analysis 93
8
Stefan Problems
8.1
8.2
8.3
8.4
Steady-State Solutions to the Diffusion Equation 97
Deal-Grove Analysis 100
Diffusion-Controlled Growth of a Spherical Precipitate 102
Diffusion-Limited Growth in Cylindrical Coordinates 105
9
Phase Transformations
9.1
9.2
9.3
9.4
9.5
Transformation-Rate-Limited Growth
Diffusion-Limited Growth 111
Thermally Limited Growth 111
Casting of Metals 113
Operating Point 114
10
Crystal Growth Methods
10.1
10.1.1
10.1.2
10.1.3
10.1.4
10.1.5
10.2
10.2.1
10.2.2
10.2.3
10.3
10.4
Melt Growth 117
Czochralski Growth 117
Floating Zone 121
Bridgman Method 123
Chalmers Method 123
Horizontal Gradient Freeze 124
Solution Growth 124
Growth from Aqueous Solutions 125
Flux Growth 125
Hydrothermal Growth 125
Vapor-Phase Growth 126
Stoichiometry 126
11
Segregation
11.1
11.2
11.3
11.4
11.5
11.5.1
11.5.2
11.6
11.7
Segregation during a Phase Change 129
Lever Rule 130
Scheil Equation 131
Zone Refining 133
Diffusion at a Moving Interface 134
Steady-state Diffusion at a Moving Interface
Initial and Final Transients 137
Segregation in Three Dimensions 139
Burton, Primm and Schlicter Analysis 139
91
97
109
109
117
129
134
VIII
Contents
12
Interface Instabilities
12.1
12.2
12.3
Constitutional Supercooling 143
Mullins and Sekerka Linear Instability Analysis
Anisotropic Interface Kinetics 152
143
13
Chemical Reaction Rate Theory
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
The Equilibrium Constant 155
Reaction Rate Theory 155
Reaction Rate Constant 157
Transition State Theory 157
Experimental Determination of the Order of a Reaction 158
Net Rate of Reaction 160
Catalysis 162
Quasi-Equilibrium Model for the Rate of a First-Order phase Change
14
Phase Equilibria
14.1
14.2
14.3
First-Order Phase Changes 165
Second-Order Phase Changes 169
Critical Point between Liquid and Vapor
149
155
165
172
15
Nucleation
15.1
15.1.1
15.1.2
15.1.3
15.1.4
15.1.5
15.1.6
15.1.7
15.1.8
15.2
15.2.1
15.2.2
15.2.3
15.2.4
15.2.5
15.2.6
15.2.7
15.2.8
15.3
15.3.1
15.3.2
Homogeneous Nucleation 175
Volmer Analysis 176
Turnbull's Droplet Experiment 179
Surface Free Energy 181
Becker-Doring Analysis 183
Nucleation Rate 187
Limitations of the Becker-Doring Analysis 188
Assumptions in the Classical Nucleation Theory 188
Nucleation of a Precipitate Particle 189
Heterogeneous Nucleation 190
Heterogeneous Nucleation Theory 191
Nucleation Lore 192
Cavitation 193
Re-entrant Cavities 194
Cloud Seeding 195
Industrial Crystallization 195
Grain Refiners 196
Residues 196
Johnson-Mehl-Avrami Equation 197
Johnson-Mehl Equation 197
Johnson-Mehl-Avrami 199
175
16
Surface Layers 203
16.1.1
16.1.2
Langmuir Adsorption 203
CVD Growth by a Surface-Decomposition Reaction
204
163
Contents
16.1.3
16.2
16.2.1
16.2.2
16.2.3
16.3
16.3.1
16.3.2
16.4
16.5
16.6
16.7
16.8
16.9
16.9.1
16.9.2
16.9.3
Langmuir-Hinshelwood Reaction 205
Surface Nucleation 206
Nucleation on a Surface during Vapor Deposition
Cluster Formation 207
Rate Equations 209
Thin Films 210
Epitaxy 210
Deposited Surface Layers 212
Surface Reconstruction 213
Amorphous Deposits 214
Surface Modification 217
Fractal Deposits 217
Strain Energy and Misfit Dislocations 219
Strained-Layer Growth 223
Surface Modulation 223
Strained-Layer Superlartice 223
Graded-Strain Layers 224
17
Thin-Film Deposition
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
Liquid Phase Epitaxy 227
Growth Configurations for LPE 228
Chemical Vapor Deposition 229
Metal-Organic Chemical Vapor Deposition
Physical Vapor Deposition 234
Sputter Deposition 235
Metallization 235
Laser Ablation 236
Molecular Beam Epitaxy 237
Atomic Layer Epitaxy 238
227
18
Plasmas
18.1
18.2
18.3
18.4
18.5
18.6
18.7
Direct Current (DC) Plasmas 241
Radio-Frequency Plasmas 243
Plasma Etching 245
Plasma Reactors 247
Magnetron Sputtering 249
Electron Cyclotron Resonance 251
Ion Milling 252
241
19
Rapid Thermal Processing 253
19.1
19.2
19.3
19.4
19.5
19.6
Rapid Thermal Processing 253
Rapid Thermal Processing Equipment
Radiative Heating 254
Temperature Measurement 256
Thermal Stress 257
Laser Heating 258
254
233
206
IX
X
Contents
20
Kinetics of First-Order Phase Transformations
20.1
20.2
20.3
20.4
20.5
20.5.1
20.5.2
20.6
20.6.1
20.6.2
20.7
20.8
20.9
20.10
20.11
20.11.1
20.11.2
General Considerations 259
The Macroscopic Shape of Crystals 261
General Equation for the Growth Rate of Crystals 262
Kinetic Driving Force 263
Vapor-Phase Growth 264
Equilibrium 264
Kinetics of Vapor-Phase Growth 265
Melt Growth 267
Early Models for Melt Growth 268
Melt Growth Rates 269
Molecular Dynamics Studies of Melt Crystallization Kinetics
The Kossel-Stranski Model 272
Nucleation of Layers 273
Growth on Screw Dislocations 275
The Fluctuation Dissipation Theorem 277
Determination of the Kinetic Coefficient 277
Experimental Determination of Surface Tension 281
259
21
The Surface-Roughening Transition
21.1
21.2
21.3
21.4
21.5
21.5.1
21.5.2
21.5.3
21.5.4
21.5.5
21.6
21.6.1
21.6.2
21.7
21.8
21.9
Surface Roughness 283
The Ising Model 284
Cooperative Processes 285
Monte Carlo Simulations of Crystallization 287
Equilibrium Surface Structure 288
Thermodynamic Model for Surface Roughness 288
Application of Surface Roughening to Materials 291
Snow Flakes 294
Rate-Theory Analysis of Surface Roughness 295
Surface Roughness in the Ising Model 297
Computer Simulations 298
Determination of the Kinetic Coefficient 301
Simulations of Silicon Growth 302
Growth Morphologies 304
Kinetic Roughening 306
Polymer Crystallization 308
22
Alloys: Thermodynamics and Kinetics 311
22.1
22.2
22.3
22.4
22.5
22.6
Crystallization of Alloys 311
Phase Equilibria 313
Regular Solution Model 314
Near-Equilibrium Conditions 316
Phase Diagrams 317
The DLP Model 320
283
269
Contents
23
Phase Separation
23.1
23.2
23.3
23.4
Ordering versus Phase Separation 323
Phase Separation 324
Analytical Model for Spinodal Decomposition 326
Microstructure Resulting from Phase Separation 328
323
24
Non-Equilibrium Crystallization of Alloys
24.1
24.2
24.3
24.4
24.5
24.6
Non-Equilibrium Crystallization 331
Experiment 331
Computer Modeling 333
Analytical Model 335
Comparison with Experiment 336
Crystallization of Glasses 338
331
25
Coarsening, Ripening 341
25.1
25.2
25.3
ISA
25.5
25.6
25.7
25.8
Coarsening 341
Free Energy of a Small Particle 341
Coarsening in a Solution 342
Coarsening of Dendritic Structures 344
Sintering 345
Bubbles 346
Grain Boundaries 347
Scratch Smoothing 348
26
Dendrites
26.1
26.2
26.3
26.4
26.5
26.6
Dendritic Growth 351
Conditions for Dendritic Growth
Simple Dendrite Model 353
Phase Field Modeling 358
Faceted Growth 358
Distribution Coefficient 358
351
27
Eutectics
27.1
27.2
27.2.1
27.2.2
27.2.3
27.3
27.3.1
27.3.2
27.3.3
27'A
27.5
27.6
Eutectic Phase Diagram 361
Classes of Eutectic Microstrucrures 362
Class I Eutectics 362
Class II Eutectics 365
Class III Eutectics 367
Analysis of Lamellar Eutectics 367
Curvature of the Interface 368
Diffusion 370
Calculation of Eutectic Interface Shape 372
Off-Composition Eutectics 373
Coupled Growth 374
Third-Component Elements 378
351
361
XI
XII
Contents
28
Castings
28.1
28.2
Grain Structure of Castings 381
Dendrite Remelting 382
381
Subject Index by Page 387
Subject Index by Chapter Sections 403