Crystal Growth Processes Based on Capillarity: Czochralski, Floating Zone, Shaping and Crucible Techniques

556 Pages·2010·36.185 MB·English

Crystal Growth Processes
Based on Capillarity
Crystal Growth Processes Based on Capillarity: Czochralski, Floating Zone, Shaping and Crucible Techniques
Edited by Thierry Duffar
© 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-71244-3
Crystal Growth
Processes Based on
Capillarity
Czochralski, Floating Zone,
Shaping and Crucible Techniques
Edited by
THIERRY DUFFAR
Laboratoire SIMaP-EPM
INP Grenoble
Saint Martin d’Hères
France
A John Wiley & Sons, Ltd., Publication
This edition fi rst published 2010
© 2010 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data
Crystal growth processes based on capillarity : czochralski, fl oating zone, shaping and
crucible techniques / edited by Thierry Duffar.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-71244-3 (cloth)
1. Crystal growth. I. Duffar, Thierry.
QD921.C765 2010
548′.5—dc22
2009045996
A catalogue record for this book is available from the British Library.
ISBN HB: 9780470712443
Set in 10/12 Times by Toppan Best-set Premedia Limited
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is dedicated to my previous, present and future PhD students.
Each of them, with his/her personality, improved my perception of crystal growth and
inclined my scientifi c activity toward a human adventure.
Thank you!
Contents
Preface page xiii
Introduction xv
Acknowledgements xxv
Nomenclature xxvii
Contributors xxxi
1. Basic Principles of Capillarity in Relation to Crystal Growth 1
Nicolas Eustathopoulos, Béatrice Drevet, Simon Brandon and
Alexander Virozub
1.1 Defi nitions 1
1.1.1 Characteristic Energies of Surfaces and Interfaces 1
1.1.2 Capillary Pressure 3
1.1.3 Surface Energy versus Surface Tension 4
1.2 Contact Angles 4
1.2.1 Thermodynamics 5
1.2.2 Dynamics of Wetting 12
1.2.3 Measurements of Contact Angle and Surface Tension by
the Sessile Drop Technique 16
1.2.4 Selected Data for the Contact Angle for Systems of
Interest for Crystal Growth 17
1.3 Growth Angles 28
1.3.1 Theory 28
1.3.2 Measurements of Growth Angles: Methods and Values 35
1.3.3 Application of the Growth Angle Condition in
Simulations of Crystal Growth 38
1.3.4 Summary 45
Acknowledgements 45
References 46
2. The Possibility of Shape Stability in Capillary Crystal Growth
and Practical Realization of Shaped Crystals 51
Vitali A. Tatartchenko
2.1 Crucible-Free Crystal Growth – Capillary Shaping Techniques 52
2.2 Dynamic Stability of Crystallization – the Basis of Shaped Crystal
Growth by CST 54
2.2.1 Lyapunov Equations 57
2.2.2 Capillary Problem – Common Approach 59
2.2.3 Equation of Crystal Dimension Change Rate 62
viii Crystal Growth Processes Based on Capillarity
2.2.4 Equation of Crystallization Front Displacement Rate 63
2.2.5 Stability Analysis in a System with Two Degrees of Freedom 64
2.3 Stability Analysis and Growth of Shaped Crystals by
the Cz Technique 65
2.3.1 Capillary Problem 65
2.3.2 Temperature Distribution in the Crystal–Melt System 65
2.3.3 Stability Analysis and Shaped Crystal Growth 68
2.3.4 Dynamic Stability Problem for the Kyropoulos Technique 69
2.4 Stability Analysis and Growth of Shaped Crystals by the
Verneuil Technique 70
2.4.1 Principal Schemes of Growth 70
2.4.2 Theoretical Investigation 71
2.4.3 Practical Results of the Theoretical Analysis 76
2.4.4 Stability Analysis-Based Automation 78
2.5 Stability Analysis and Growth of Shaped Crystals by the
FZ Technique 80
2.6 TPS Techniques: Capillary Shaping and Impurity Distribution 81
2.6.1 Capillary Boundary Problem for TPS 81
2.6.2 Stability Analysis 92
2.6.3 Experimental Tests of the Capillary Shaping Theory
Statements 94
2.6.4 Impurity Distribution 100
2.6.5 Defi nition of TPS 104
2.6.6 Brief History of TPS 104
2.7 Shaped Growth of Ge, Sapphire, Si, and Metals: a Brief Presentation 108
2.7.1 Ge 108
2.7.2 Sapphire 109
2.7.3 Si 109
2.7.4 Metals and Alloys 110
2.8 TPS Peculiarities 110
References 111
3 Czochralski Process Dynamics and Control Design 115
Jan Winkler, Michael Neubert, Joachim Rudolph, Ning Duanmu and
Michael Gevelber
3.1 Introduction and Motivation 116
3.1.1 Overview of Cz Control Issues 116
3.1.2 Diameter Control 117
3.1.3 Growth Rate Control 117
3.1.4 Reconstruction of Quantities not Directly Measured 117
3.1.5 Specifi c Problems for Control in Cz Crystal Growth 118
3.1.6 PID Control vs. Model-Based Control 121
3.1.7 Components of a Control System 122
3.1.8 Modelling in Crystal Growth Analysis and Control 123
3.2 Cz Control Approaches 124
Contents ix
3.2.1 Proper Choice of Manipulated Variables 124
3.2.2 Feedforward Control 125
3.2.3 Model-Based Analysis of the Process 126
3.2.4 Stability 127
3.2.5 Model-Based Control 127
3.2.6 Identifi cation 130
3.2.7 Measurement Issues and State Estimation 130
3.3 Mathematical Model 132
3.3.1 Hydromechanical–Geometrical Model 133
3.3.2 Model of Thermal Behaviour 142
3.3.3 Linear System Model Analysis 148
3.4 Process Dynamics Analysis for Control 150
3.4.1 Operating Regime and Batch Implications 152
3.4.2 Actuator Performance Analysis 154
3.4.3 Curved Interface 157
3.4.4 Nonlinear Dynamics 157
3.5 Conventional Control Design 161
3.5.1 Control Based on Optical Diameter Estimation 161
3.5.2 Weight-Based Control 164
3.6 Geometry-Based Nonlinear Control Design 170
3.6.1 Basic Idea 170
3.6.2 Parametrization of the Hydromechanical–Geometrical Model
in Crystal Length 171
3.6.3 Flatness and Model-Based Feedback Control of the
Length-Parametrized Model 172
3.6.4 Control of Radius and Growth Rate 176
3.7 Advanced Techniques 181
3.7.1 Linear Observer Design 182
3.7.2 Nonlinear Observer Design 183
3.7.3 Control Structure Design for Batch Disturbance Rejection 194
References 199
4 Floating Zone Crystal Growth 203
Anke Lüdge, Helge Riemann, Michael Wünscher, Günter Behr,
Wolfgang Löser, Andris Muiznieks and Arne Cröll
4.1 FZ Processes with RF Heating 207
4.1.1 FZ Method for Si by RF Heating 207
4.1.2 FZ Growth for Metallic Melts 220
4.2 FZ Growth with Optical Heating 230
4.2.1 Introduction 230
4.2.2 Image Furnaces 230
4.2.3 Laser Heating 240
4.2.4 FZ Growth for Oxide Melts 242
4.3 Numerical Analysis of the Needle-Eye FZ Process 247
4.3.1 Literature Overview 248
x Crystal Growth Processes Based on Capillarity
4.3.2 Quasi-Stationary Axisymmetric Mathematical Model
of the Shape of the Molten Zone 249
4.3.3 Numerical Investigation of the Infl uence of Growth
Parameters on the Shape of the Molten Zone 256
4.3.4 Nonstationary Axisymmetric Mathematical Model for
Transient Crystal Growth Processes 258
Appendix: Code for Calculating the Free Surface During a FZ
Process in Python 267
References 270
5 Shaped Crystal Growth 277
Vladimir N. Kurlov, Sergei N. Rossolenko, Nikolai V. Abrosimov
and Kheirreddine Lebbou
5.1 Introduction 277
5.2 Shaped Si 279
5.2.1 EFG Method 281
5.2.2 Dendritic Web Growth 286
5.2.3 String Ribbon 288
5.2.4 Ribbon Growth on Substrate (RGS) 290
5.3 Sapphire Shaped Crystal Growth 292
5.3.1 EFG 293
5.3.2 Variable Shaping Technique (VST) 295
5.3.3 Noncapillary Shaping (NCS) 297
5.3.4 Growth from an Element of Shape (GES) 307
5.3.5 Modulation-Doped Shaped Crystal Growth Techniques 312
5.3.6 Automated Control of Shaped Crystal Growth 319
5.4 Shaped Crystals Grown by the Micro-Pulling
Down Technique (µ-PD) 333
5.4.1 Crucible–Melt Relation During Crystal Growth by the µ-PD
Technique 339
5.4.2 Examples of Crystals Grown by the µ-PD Technique 340
5.5 Conclusions 347
References 347
6 Vertical Bridgman Technique and Dewetting 355
Thierry Duffar and Lamine Sylla
6.1 Peculiarities and Drawbacks of the Bridgman Processes 356
6.1.1 Thermal Interface Curvature 356
6.1.2 Melt–Crystal–Crucible Contact Angle 358
6.1.3 Crystal–Crucible Adhesion and Thermomechanical Detachment 360
6.1.4 Spurious Nucleation on Crucible Walls 363
6.2 Full Encapsulation 366
6.2.1 Introduction 366
6.2.2 LiCl–KCl Encapsulant for Antimonides 368
6.2.3 BO Encapsulant 371
2 3
6.2.4 Conclusion 372
Contents xi
6.3 The Dewetting Process: a Modifi ed VB Technique 373
6.3.1 Introduction 373
6.3.2 Dewetting in Microgravity 374
6.3.3 Dewetting in Normal Gravity 378
6.3.4 Theoretical Models of Dewetting 394
6.3.5 Stability Analysis 404
6.4 Conclusion and Outlook 407
References 408
7 Marangoni Convection in Crystal Growth 413
Arne Cröll, Taketoshi Hibiya, Suguru Shiratori, Koichi Kakimoto
and Lijun Liu
7.1 Thermocapillary Convection in Float Zones 417
7.1.1 Model Materials 417
7.1.2 Semiconductors and Metals 417
7.1.3 Effect of Oxygen Partial Pressure on Thermocapillary
Flow in Si 419
7.1.4 Fluid Dynamics of Thermocapillary Flow in Half-Zones 422
7.1.5 Full Float Zones 435
7.1.6 The Critical Marangoni Number Ma 435
c2
7.1.7 Controlling Thermocapillary Convection in Float Zones 443
7.2 Thermocapillary Convection in Cz Crystal Growth of Si 448
7.2.1 Introduction 448
7.2.2 Surface Tension-Driven Flow in Cz Growth 448
7.2.3 Numerical Model 449
7.2.4 Calculation Results 452
7.2.5 Summary of Cz Results 456
7.3 Thermocapillary Convection in EFG Set-Ups 456
7.4 Thermocapillary Convection in Bridgman and Related Set-Ups 457
7.5 Solutocapillary Convection 457
References 460
8 Mathematical and Numerical Analysis of Capillarity Problems
and Processes 465
Liliana Braescu, Simona Epure and Thierry Duffar
8.1 Mathematical Formulation of the Capillary Problem 468
8.1.1 Boundary Value Problems for the Young–Laplace Equation 468
8.1.2 Initial and Boundary Conditions of the Meniscus Problem 472
8.1.3 Approximate Solutions of the Axisymmetric Meniscus
Problem 473
8.2 Analytical and Numerical Solutions for the Meniscus Equation in
the Cz Method 476
8.3 Analytical and Numerical Solutions for the Meniscus Equation in
the EFG Method 486
8.3.1 Sheets 486
8.3.2 Cylindrical Crystals 493
xii Crystal Growth Processes Based on Capillarity
8.4 Analytical and Numerical Solutions for the Meniscus Equation in
the Dewetted Bridgman Method 500
8.4.1 Zero Gravity 502
8.4.2 Normal Gravity 508
8.5 Conclusions 517
Appendix: Runge–Kutta Methods 518
A.1 Fourth-Order Runge–Kutta Method (RK4) 518
A.2 Rkfi xed and Rkadapt Routines for Solving IVP 520
References 523
Index 525