Crystal growth processes based on capillarity : czochralski, floating zone, shaping and crucible techniques

TLDR: In this paper, Eustathopoulos et al. presented a theoretical analysis of the shape stability in the Czochralski process with respect to the surface energy and surface tension.

Abstract: Preface. Introduction. Acknowledgements. Nomenclature. Contributors. 1. Basic Principles of Capillarity in Relation to Crystal Growth (Nicolas Eustathopoulos, Beatrice Drevet, Simon Brandon and Alexander Virozub). 1.1 Definitions. 1.1.1 Characteristic Energies of Surfaces and Interfaces. 1.1.2 Capillary Pressure. 1.1.3 Surface Energy versus Surface Tension. 1.2 Contact Angles. 1.2.1 Thermodynamics. 1.2.2 Dynamics of Wetting. 1.2.3 Measurements of Contact Angle and Surface Tension by the Sessile Drop Technique. 1.2.4 Selected Data for the Contact Angle for Systems of Interest for Crystal Growth. 1.3 Growth Angles. 1.3.1 Theory. 1.3.2 Measurements of Growth Angles: Methods and Values. 1.3.3 Application of the Growth Angle Condition in Simulations of Crystal Growth. 1.3.4 Summary. Acknowledgements. References. 2. The Possibility of Shape Stability in Capillary Crystal Growth and Practical Realization of Shaped Crystals (Vitali A. Tatartchenko). 2.1 Crucible-Free Crystal Growth Capillary Shaping Techniques. 2.2 Dynamic Stability of Crystallization the Basis of Shaped Crystal Growth by CST. 2.2.1 Lyapunov Equations. 2.2.2 Capillary Problem Common Approach. 2.2.3 Equation of Crystal Dimension Change Rate. 2.2.4 Equation of Crystallization Front Displacement Rate. 2.2.5 Stability Analysis in a System with Two Degrees of Freedom. 2.3 Stability Analysis and Growth of Shaped Crystals by the Cz Technique. 2.3.1 Capillary Problem. 2.3.2 Temperature Distribution in the Crystal Melt System. 2.3.3 Stability Analysis and Shaped Crystal Growth. 2.3.4 Dynamic Stability Problem for the Kyropoulos Technique. 2.4 Stability Analysis and Growth of Shaped Crystals by the Verneuil Technique. 2.4.1 Principal Schemes of Growth. 2.4.2 Theoretical Investigation. 2.4.3 Practical Results of the Theoretical Analysis. 2.4.4 Stability Analysis-Based Automation. 2.5 Stability Analysis and Growth of Shaped Crystals by the FZ Technique. 2.6 TPS Techniques: Capillary Shaping and Impurity Distribution. 2.6.1 Capillary Boundary Problem for TPS. 2.6.2 Stability Analysis. 2.6.3 Experimental Tests of the Capillary Shaping Theory Statements. 2.6.4 Impurity Distribution. 2.6.5 Definition of TPS. 2.6.6 Brief History of TPS. 2.7 Shaped Growth of Ge, Sapphire, Si, and Metals: a Brief Presentation. 2.7.1 Ge. 2.7.2 Sapphire. 2.7.3 Si. 2.7.4 Metals and Alloys. 2.8 TPS Peculiarities. References. 3 Czochralski Process Dynamics and Control Design (Jan Winkler, Michael Neubert, Joachim Rudolph, Ning Duanmu and Michael Gevelber). 3.1 Introduction and Motivation. 3.1.1 Overview of Cz Control Issues. 3.1.2 Diameter Control. 3.1.3 Growth Rate Control. 3.1.4 Reconstruction of Quantities not Directly Measured. 3.1.5 Specifi c Problems for Control in Cz Crystal Growth. 3.1.6 PID Control vs. Model-Based Control. 3.1.7 Components of a Control System. 3.1.8 Modelling in Crystal Growth Analysis and Control. 3.2 Cz Control Approaches. 3.2.1 Proper Choice of Manipulated Variables. 3.2.2 Feedforward Control. 3.2.3 Model-Based Analysis of the Process. 3.2.4 Stability. 3.2.5 Model-Based Control. 3.2.6 Identification. 3.2.7 Measurement Issues and State Estimation. 3.3 Mathematical Model. 3.3.1 Hydromechanical Geometrical Model. 3.3.2 Model of Thermal Behaviour. 3.3.3 Linear System Model Analysis. 3.4 Process Dynamics Analysis for Control. 3.4.1 Operating Regime and Batch Implications. 3.4.2 Actuator Performance Analysis. 3.4.3 Curved Interface. 3.4.4 Nonlinear Dynamics. 3.5 Conventional Control Design. 3.5.1 Control Based on Optical Diameter Estimation. 3.5.2 Weight-Based Control. 3.6 Geometry-Based Nonlinear Control Design. 3.6.1 Basic Idea. 3.6.2 Parametrization of the Hydromechanical Geometrical Model in Crystal Length. 3.6.3 Flatness and Model-Based Feedback Control of the Length-Parametrized Model. 3.6.4 Control of Radius and Growth Rate. 3.7 Advanced Techniques. 3.7.1 Linear Observer Design. 3.7.2 Nonlinear Observer Design. 3.7.3 Control Structure Design for Batch Disturbance Rejection. References. 4 Floating Zone Crystal Growth (Anke Ludge, Helge Riemann, Michael Wunscher, Gunter Behr, Wolfgang Loser, Andris Muiznieks and Arne Croll). 4.1 FZ Processes with RF Heating. 4.1.1 FZ Method for Si by RF Heating. 4.1.2 FZ Growth for Metallic Melts. 4.2 FZ Growth with Optical Heating. 4.2.1 Introduction. 4.2.2 Image Furnaces. 4.2.3 Laser Heating. 4.2.4 FZ Growth for Oxide Melts. 4.3 Numerical Analysis of the Needle-Eye FZ Process. 4.3.1 Literature Overview. 4.3.2 Quasi-Stationary Axisymmetric Mathematical Model of the Shape of the Molten Zone. 4.3.3 Numerical Investigation of the Influence of Growth Parameters on the Shape of the Molten Zone. 4.3.4 Nonstationary Axisymmetric Mathematical Model for Transient Crystal Growth Processes. Appendix: Code for Calculating the Free Surface During a FZ Process in Python. References. 5 Shaped Crystal Growth (Vladimir N. Kurlov, Sergei N. Rossolenko, Nikolai V. Abrosimov and Kheirreddine Lebbou). 5.1 Introduction. 5.2 Shaped Si. 5.2.1 EFG Method. 5.2.2 Dendritic Web Growth. 5.2.3 String Ribbon. 5.2.4 Ribbon Growth on Substrate (RGS). 5.3 Sapphire Shaped Crystal Growth. 5.3.1 EFG. 5.3.2 Variable Shaping Technique (VST). 5.3.3 Noncapillary Shaping (NCS). 5.3.4 Growth from an Element of Shape (GES). 5.3.5 Modulation-Doped Shaped Crystal Growth Techniques. 5.3.6 Automated Control of Shaped Crystal Growth. 5.4 Shaped Crystals Grown by the Micro-Pulling Down Technique ( -PD). 5.4.1 Crucible Melt Relation During Crystal Growth by the -PD Technique. 5.4.2 Examples of Crystals Grown by the -PD Technique. 5.5 Conclusions. References. 6 Vertical Bridgman Technique and Dewetting (Thierry Duffar and Lamine Sylla). 6.1 Peculiarities and Drawbacks of the Bridgman Processes. 6.1.1 Thermal Interface Curvature. 6.1.2 Melt Crystal Crucible Contact Angle. 6.1.3 Crystal Crucible Adhesion and Thermomechanical Detachment. 6.1.4 Spurious Nucleation on Crucible Walls. 6.2 Full Encapsulation. 6.2.1 Introduction. 6.2.2 LiCl KCl Encapsulant for Antimonides. 6.2.3 B2O3 Encapsulant. 6.2.4 Conclusion. 6.3 The Dewetting Process: a Modified VB Technique. 6.3.1 Introduction. 6.3.2 Dewetting in Microgravity. 6.3.3 Dewetting in Normal Gravity. 6.3.4 Theoretical Models of Dewetting. 6.3.5 Stability Analysis. 6.4 Conclusion and Outlook. References. 7 Marangoni Convection in Crystal Growth (Arne Croll, Taketoshi Hibiya, Suguru Shiratori, Koichi Kakimoto and Lijun Liu). 7.1 Thermocapillary Convection in Float Zones. 7.1.1 Model Materials. 7.1.2 Semiconductors and Metals. 7.1.3 Effect of Oxygen Partial Pressure on Thermocapillary Flow in Si. 7.1.4 Fluid Dynamics of Thermocapillary Flow in Half-Zones. 7.1.5 Full Float Zones. 7.1.6 The Critical Marangoni Number Mac2. 7.1.7 Controlling Thermocapillary Convection in Float Zones. 7.2 Thermocapillary Convection in Cz Crystal Growth of Si. 7.2.1 Introduction. 7.2.2 Surface Tension-Driven Flow in Cz Growth. 7.2.3 Numerical Model. 7.2.4 Calculation Results. 7.2.5 Summary of Cz Results. 7.3 Thermocapillary Convection in EFG Set-Ups. 7.4 Thermocapillary Convection in Bridgman and Related Set-Ups. 7.5 Solutocapillary Convection. References. 8 Mathematical and Numerical Analysis of Capillarity Problems and Processes (Liliana Braescu, Simona Epure and Thierry Duffar). 8.1 Mathematical Formulation of the Capillary Problem. 8.1.1 Boundary Value Problems for the Young Laplace Equation. 8.1.2 Initial and Boundary Conditions of the Meniscus Problem. 8.1.3 Approximate Solutions of the Axisymmetric Meniscus Problem. 8.2 Analytical and Numerical Solutions for the Meniscus Equation in the Cz Method. 8.3 Analytical and Numerical Solutions for the Meniscus Equation in the EFG Method. 8.3.1 Sheets. 8.3.2 Cylindrical Crystals. 8.4 Analytical and Numerical Solutions for the Meniscus Equation in the Dewetted Bridgman Method. 8.4.1 Zero Gravity. 8.4.2 Normal Gravity. 8.5 Conclusions. Appendix: Runge Kutta Methods. A.1 Fourth-Order Runge Kutta Method (RK4). A.2 Rkfixed and Rkadapt Routines for Solving IVP. References. Index.