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Optimization of the Czochralski silicon growth process by means of configured magnetic fields

Optimization of the Czochralski silicon growth process by means of configured magnetic fields. F. Bioul, N. Van Goethem, L. Wu, B. Delsaute, R. Rolinsky, N. Van den Bogaert, V. Regnier, F. Dupret. Université catholique de Louvain. Bulk growth from the melt : basic techniques.

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Optimization of the Czochralski silicon growth process by means of configured magnetic fields

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  1. Optimization of the Czochralski silicon growth process by means of configured magnetic fields F. Bioul, N. Van Goethem, L. Wu, B. Delsaute, R. Rolinsky, N. Van den Bogaert, V. Regnier, F. Dupret Université catholique de Louvain

  2. Bulk growth from the melt : basic techniques Czochralski (Cz),Liquid Encapsulated Czochralski (LEC) Floating Zone (FZ) Vertical Bridgman

  3. Czochralski process

  4. Factors affecting crystal quality • Cylindrical shape (technological requirement) • Regularity of the lattice (reduction of defects : point defects, dislocations, twins…) • Impurities (oxygen in Si growth) • Crystal stoichiometry/dopant concentration (reduction of axial and radial segregation)

  5. Numerical modeling goals • Better understanding of the factors affecting crystal quality • Prediction of : • crystal and melt temperature evolution • solid-liquid interface shape • melt flow • residual stresses • dopant and impurity concentrations • defects and dislocations • Process design improvement • Process control and optimization

  6. Principal aspects of the problem • Coupled, global  interaction between heat transfer in crystal and melt, solidification front deformation and overall radiation transfer • Non-linear  physics of radiation, melt convection and solidification • Dynamic  critical growth stages: seeding, shouldering, tail- end, crystal detachment, post-growth • Inverse  natural output is prescribed (crystal shape), while natural input is calculated (heater power or pull rate)

  7. Melt convection = Significant heat transfer mechanism  defect and dislocation densities growth striations interface shape = Dominant mechanism for dopant and impurity transfer dopant and impurity (oxygen) distributions

  8. Typical flow pattern Ws Melt convection is due to • Buoyancy (1) • Forced convection - Coriolis (2) - Centrifugal pumping (3) • Marangoni effect (4) • Gas flow (5) crystal 5 4 3 1 2 melt crucible Wc

  9. Quasi-steady axisymmetric models • Objective Coupling with quasi-steady and dynamic global heat transfer models • Difficulties Structured temporal and azimuthal oscillations (3D unsteady effects) + superposed chaotic oscillations (turbulence)  average modeling required

  10. Melt flow model Hypotheses :Incompressible Newtonian fluid Boussinesq approximation Quasi-steady, turbulent or laminar flow Reynolds equations :  mA, kA : additional viscosity and conductivity

  11. General dynamic strategy Time-dependent simulation can provide quasi-steady source terms equivalent to transient terms Quasi-steady simulations with melt flow t0 t1 t2 t3 t4 t5 t6 t7 time Time-dependent simulation with interpolated flow effect Cone growth Body growth Tail-end stage

  12. Melt convection • How to modify the flow? Large electrical conductivity of semiconductor melts  Use of magnetic fields to control the flow • Available magnetic fields • DC or AC • Axisymmetric : vertical or configured • Transverse (horizontal) • Rotating • Difficulties • Horizontal fields (3D effects) • Numerical problems (Hartmann layers…) • 2D turbulence (?)

  13. Rigid magnetic fields Rigid magnetic field approximation : induced magnetic field is negligible Imposed steady axisymmetric magnetic field : Ohm’s law : Conservation of charge :

  14. Analytical solutions From Hjellming & Walker, 1993 • Existence of a free shear layer: plays an important role in oxygen and impurity transfer Hypotheses : High Hartmann number : Inertialess approximation (valid if B≥0.2T) :

  15. Crystal B Melt Crucible Analytical solution Case I:Case II : Crystal Free shear layer B Melt Crucible No magnetic field lines in contact with neither the crystal nor the crucible Magnetic field lines in contact with both the crystal and the crucible

  16. Quasi-steady numerical results FEMAG Software Material and geometrical parameters : Silicon crystal diameter : 100 mm Crucible diameter : 300 mm Molecular dynamic viscosity : 8.22e-4 kg/m.s Process parameters : Crystal rotational rate : - 20 rpm (- 2.09 rad/s) Crucible rotational rate : + 5 rpm (+ 0.523 rad/s) Pull rate : 1.8 cm/h (5.0e-6 m/s)

  17. Magnetic field generated by 2 coils with same radius (600 mm) Turbulence Model : Adapted Mixing Length Magnetic field lines Bmax=0.03T Bmax=0.7T B=0T Stokes stream function

  18. Magnetic field generated by 2 coils with different radii(600 mm and 75 mm) Turbulence model : Adapted Mixing Length Magnetic field lines Bmax=0.2T Bmax=0.9T B=0T Stokes stream function

  19. Inverse dynamic simulations of silicon growth FEMAG-2 software Run A Opposite crystal and crucible rotation senses Silicon Mixing length model m= 8.225 10-4 kg/m.s Wc= 0.52 s-1 Ws= -2.O9 s-1 Vpul = 5. 10-6 m/s Run B Same as A with a vertical magnetic field B = 0.32 Tesla

  20. Stream function for runs A and B A B

  21. Temperature field for runs A and B A B

  22. Off-line Control • Objective To determine the best evolution of the process parameters in order to optimize selected process variables characterizing crystal shape and quality Long-term time scales are considered (instead of short-term time scales for on-line control) • Methodology Dynamic simulations are performed under supervision of a controller

  23. Time-dependent simulator Start new time step with updated process parameters Do process variables satisfy the control objectives ? Off-line controller Off-line Control

  24. Conclusions • Accurate quasi-steady and dynamic simulation models are available using FEMAG-2 software • Simulations are in agreement with theoretical predictions • Turbulence modeling must be validated and improved if necessary • Numerical scheme should be able to control mesh refinement along boundary and internal layers • Off-line control is a promising technique for optimizing the magnetic field design

  25. k-l turbulence model Additional viscosity : Additional conductivity : • How to modify the flow? Turbulent kinetic energy equation : mean turbulent kinetic energy From Th. Wetzel where : parameters of the model : additional Prandtl number

  26. Dimensionless parameters crucible Reynolds number (related to Coriolis force) crystal rotation Reynolds number (related to centrifugal force) Grashoff number (related to natural convection) Prandtl number Hartmann number

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