Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals
Abstract
:1. Introduction
2. Numerical Model
2.1. Overview of Modelling Scheme
2.2. Phase Boundaries
2.2.1. Electromagnetic Field
2.3. Species Transport in Melt
- Marangoni force and the EM force are applied on the free melt surface.
- Fixed velocity (crystal pulling speed and rotation)—on the crystallization interface.
- On the crystallization interface: where n is the normal coordinate, is the segregation coefficient, and is the the angle between the horizontal plane and the interface normal vector.
- On the melt free surface: due to the assumption of a pure gas atmosphere and lack of evaporation [24].
- On the melting interface: arb.u., i.e., the species concentration is normalized to the initial concentration in the feed rod, which is assumed to be homogeneous.
2.4. 2D Species Distribution in Crystal
- Importing the data about process dynamics (time-dependent , , and ) from transient phase boundary simulations with FZoneT.
- Creating an approximate description of the cone phase based only on the simplified crystal shape described as , where L is the crystal length:
- Due to the assumption of constant pulling velocity, .
- Cone surfaces are approximated as having constant slope, and thus .
- The free surface height above the external triple point is assumed to be constant even during the cone phases because it is impossible to predict its evolution for an arbitrary crystal shape (without experimental data); therefore, .
- The crystallized volume is proportional to the crystal cross-section: , where is the crystal pulling velocity, and is the crystal cross-section area. Therefore, .
- Due to silicon mass conservation, .
3. Results
3.1. Description of the Experiment
3.2. Phase Boundaries
3.2.1. Quasi-Stationary Simulations
3.2.2. Influence of Three-Dimensionality of the EM Field
3.2.3. Transient Simulations
3.3. Species Transport in Melt
3.3.1. Effective Segregation Coefficient
3.3.2. Increased Crystal Rotation Rate
3.4. Species Distribution in Crystal
4. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
References
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Parameter | Value |
---|---|
Crystal diameter | 102 mm (cylinder phase) |
Feed rod diameter | 90 mm |
Crystal pulling rate | 3.5 mm/min |
Feed rod push rate | 4.5 mm/min (cylinder phase) |
Crystal rotation rate | 6 rpm |
Feed rod rotation rate | −0.8 rpm |
Zone height | 27 mm (cylinder phase) |
Inductor frequency f | 3 MHz |
Parameter | Value |
---|---|
Silicon density | |
Silicon viscosity | |
Silicon heat conductivity | |
Silicon specific heat capacity | |
Silicon thermal expansion coefficient | |
Marangoni coefficient M | [26] |
Carbon diffusion coefficient D | [28] |
Carbon segregation coefficient | 0.07 [3] |
Boron diffusion coefficient | [29] |
Boron segregation coefficient | 0.8 [29] |
Total number of mesh elements | 614,000 |
Largest element size | 0.8–1.4 mm (inside the melt) |
Smallest element thickness | 0.02–0.03 mm (at the crystallization interface) |
Time step | 2 ms |
Total simulation time | 350–500 s |
Averaging interval for | 100 s |
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Surovovs, K.; Surovovs, M.; Sabanskis, A.; Virbulis, J.; Dadzis, K.; Menzel, R.; Abrosimov, N. Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals. Crystals 2022, 12, 1718. https://doi.org/10.3390/cryst12121718
Surovovs K, Surovovs M, Sabanskis A, Virbulis J, Dadzis K, Menzel R, Abrosimov N. Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals. Crystals. 2022; 12(12):1718. https://doi.org/10.3390/cryst12121718
Chicago/Turabian StyleSurovovs, Kirils, Maksims Surovovs, Andrejs Sabanskis, Jānis Virbulis, Kaspars Dadzis, Robert Menzel, and Nikolay Abrosimov. 2022. "Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals" Crystals 12, no. 12: 1718. https://doi.org/10.3390/cryst12121718