Crucible-free Crystal Growth of Germanium - Experiments and Simulations [Elektronische Ressource] / Michael Wünscher. Betreuer: Eckehard Schöll

Diplom-Physiker Michael Wünscher

Sujets

• Physik

Informations

• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 46
• Langue: Deutsch
• Poids de l'ouvrage: 18 Mo

Extrait

Crucible-free Crystal Growth of Germanium –
Experiments and Simulations
von
Diplom-Physiker
Michael Wünscher
aus Berlin
der Fakultät II – Mathematik und Naturwissenschaften –
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. M. Kneissl
Berichter: Prof. Dr. E. Schöll, PhD
Berichter: Prof. Dr. R. Fornari
Tag der wissenschaftlichen Aussprache: 30. August 2011
Berlin 2011
D 83This thesis is dedicated to
my son Oskar and my wife Christin.Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii (in german) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1. Introduction 1
1.1. Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Growth Techniques from the Melt . . . . . . . . . . . . . . . . . . 4
1.3. Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4. Lateral Photo-Voltage Scanning and Photoluminescence Scanning Tech-
niques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. The Simulation Program FEMAG-FZ 15
2.1. Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2. Geometry and Boundary Conditions . . . . . . . . . . . . . . . . . 17
2.1.3. Model Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2. Test Cases - Cape Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3. Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1. First Results for Germanium . . . . . . . . . . . . . . . . . . . . . 29
2.3.2. Melting Interface Calculation . . . . . . . . . . . . . . . . . . . . . 29
2.3.3. Open Melting Front . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4. Test Case - Fast Pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1. Maximal Growth Velocity - A 1D Model after Billig . . . . . . . . 32
2.4.2. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3. Comparison with Experimental Data . . . . . . . . . . . . . . . . . 36
2.4.4. Growth Angle Influence on the Numerical Results . . . . . . . . . 37
3. Melt Surface During the Floating Zone Process 41
3.1. Numerical Model in an Axisymmetric Approach . . . . . . . . . . . . . . . 41
3.1.1. Model for the Free Surface of the Melt . . . . . . . . . . . . . . . . 41
3.1.2. Model for the Electromagnetic Pressure Calculation . . . . . . . . 43
3.1.3. Coupling of the Models . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.4. Optimization Scheme for Curve Fitting . . . . . . . . . . . . . . . 46
3.2. Comparison with the Melt Surface of Silicon . . . . . . . . . . . . . . . . . 47
3.3. with the Melt of Germanium . . . . . . . . . . . . . . 50
3.3.1. Influence of the Coil Position . . . . . . . . . . . . . . . . . . . . . 50
vContents
3.3.2. Melt Surface for a Sequence of Pictures . . . . . . . . . . . . . . . 53
3.4. Measurement of the Growth Angle from High-Resolution Photographs . . 54
3.4.1. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.2. Measurement for Silicon . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.3.nt for Germanium . . . . . . . . . . . . . . . . . . . . . 56
4. Floating Zone Crystal Growth of Germanium 59
4.1. The Differences to Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1. Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.2. Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2. Measures Against Bulges and Spirals . . . . . . . . . . . . . . . . . . . . . 62
4.2.1. Influence of the Hole Diameter on the Calculation . . . . . . . . . 62
4.2.2. Experiments with Different Hole Diameters and Induction Coil
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.3. Short-Circuit Plate under the Induction Coil . . . . . . . . . . . . 66
4.2.4. Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.5. Main Slit and Additional Side Slits . . . . . . . . . . . . . . . . . . 70
4.3. Cooling Effect of the Ambient Gas . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1. Estimation of the Cooling Effect . . . . . . . . . . . . . . . . . . . 74
4.3.2. Modeling the Gas Flow in the Chamber . . . . . . . . . . . . . . . 76
4.3.3. Experiments in Helium Atmosphere . . . . . . . . . . . . . . . . . 81
4.3.4. Measures Against Noses . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.5. Two Gas Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.6. Dash Method for Single Crystal Growth . . . . . . . . . . . . . . . 85
4.4. Numerical Calculations of the Melt Flow . . . . . . . . . . . . . . . . . . . 89
4.4.1. Influence of the Turbulence Model . . . . . . . . . . . . . . . . . . 90
4.4.2. Reduced Electromagnetic Force . . . . . . . . . . . . . . . . . . . . 91
4.4.3. Increasing Buoyancy Force . . . . . . . . . . . . . . . . . . . . . . 96
4.4.4. Rotation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.5. Comparison with Ansys . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.6. Silicon Parameters for the Molten Zone of Germanium . . . . . . . 101
5. Pedestal Crystal Growth of Germanium 105
5.1. Model Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.1. Determination of the Reference Case . . . . . . . . . . . . . . . . . 106
5.1.2. Angle of the Induction Coil . . . . . . . . . . . . . . . . . . . . . . 107
5.1.3. Position of the Upper Triple Point . . . . . . . . . . . . . . . . . . 108
5.1.4. Diameter of the Feed Rod . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.5. Crystal Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2. Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.1. Experimental Parameter - Setup . . . . . . . . . . . . . . . . . . . 110
5.2.2. Exptal Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.3. Measurement of Triple Points and Melt Surface . . . . . . . . . . . 112
5.2.4. Simulation of the Experimental Setup . . . . . . . . . . . . . . . . 115
viContents
5.2.5. Interfaces of the Molten Zone . . . . . . . . . . . . . . . . . . . . . 116
6. Summary 119
7. Outlook 121
A. Studies of the Open Melting Front During the Stay at FEMAGSoft 123
A.1. Model Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.2. Open Melting Front from the Process Picture . . . . . . . . . . . . . . . . 124
A.3. Adjusted Upper Triple Point . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.4. Silicon Results for 2 Inches . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.5. Comparison of Different Conductivity Models along the Open Melting Front132
B. Runge Kutta Implementation in Python 133
Bibliography 137
List of Figures 143
List of Tables 153
Acknowledgement 155
viiContents
Abstract
The thesis is focused on the crucible-free crystal growth of germanium where the
floating zone process and the pedestal method were investigated by experiment and
simulation. For both crystal growthds, experimental and numerical results
were compared in order to verify the simulations and to use the simulation forecast
to improve the experiments.
The classical setup for the floating zone process of silicon is not stable for germa-
nium, which here results in growing spiral-shaped crystals. This growth behavior
was stabilized by exchanging argon for helium as protective gas. The four times
higher heat transfer coefficient increases the cooling of the growing crystal and re-
duces the spiral growth. On the other hand this causes an irregular melting of the
feed rod. To conform with these two diverging requirements both gases were used
at the same time, helium around the crystal and argon around the feed rod. The
increased heat transfer to the gas could be numerically verified by using the program
Elmer for calculating the gas convection and comparing the results with the heat
loss by radiation.
The model for the floating zone process developed by Dupret et.al. was first
verified with former simulation results and experiments done with silicon. The
quasi-stationary calculation in axisymmetric approximation gave a good agreement.
mm mmIn a new experiment the pull speed was increased from 5.0 /min to 8.0 /min. The
mmagreement was good until 6.5 /min and for higher pull speeds deviations occurred
which could not yet be completely explained.
By adapting the floating zone model for germanium it became clear that the melt-
ing interface had to be included in the model, and with it the influence of parameters
of the induction coil on the temperature field were studied. The simulation of the
melt flow shows the dominance of the electromagnetic force over Bouancy as well as
Marangoni forces. The rotation rate had no reasonable influence on the mel