Optimization of the Czochralski silicon growth process by means of configured magnetic fields
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Transcript of 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
Czochralski (Cz),Liquid Encapsulated
Czochralski (LEC)
Czochralski (Cz),Liquid Encapsulated
Czochralski (LEC)
Floating Zone (FZ)Floating Zone (FZ) Vertical BridgmanVertical Bridgman
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)
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
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)
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
Typical flow pattern
Melt convection is due to• Buoyancy (1)• Forced convection
- Coriolis (2)
- Centrifugal pumping (3)• Marangoni effect (4)• Gas flow (5)
12
34
5crystal
melt
s
ccrucible
Quasi-steady axisymmetric models
• Objective
Coupling with quasi-steady and dynamic global heat transfer models
• DifficultiesStructured temporal and azimuthal oscillations (3D unsteady effects) + superposed chaotic oscillations (turbulence)
average modeling required
Melt flow model
Reynolds equations :
A, kA : additional viscosity and conductivity
Reynolds equations :
A, kA : additional viscosity and conductivity
Hypotheses : Incompressible Newtonian fluid Boussinesq approximation Quasi-steady, turbulent or laminar flow
Hypotheses : Incompressible Newtonian fluid Boussinesq approximation Quasi-steady, turbulent or laminar flow
t0 t1 t2 t3 t4 t5 t6 timet7
Cone growth
Body growth
Tail-end stage
Quasi-steady simulationswith melt flow
Quasi-steady simulationswith melt flow
Time-dependent simulation with interpolated flow effect
Time-dependent simulation with interpolated flow effect
Time-dependent simulation can provide quasi-steady source terms equivalent to transient terms
Time-dependent simulation can provide quasi-steady source terms equivalent to transient terms
General dynamic strategy
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 (?)
Rigid magnetic fields
Ohm’s law :
Conservation of charge :
Ohm’s law :
Conservation of charge :
Rigid magnetic field approximation : induced magnetic field is negligible
Imposed steady axisymmetric magnetic field :
Rigid magnetic field approximation : induced magnetic field is negligible
Imposed steady axisymmetric magnetic field :
Analytical solutionsFrom 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) :
Hypotheses :
High Hartmann number :
Inertialess approximation (valid if B≥0.2T) :
Case I : Case II :
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
B
Crystal
Melt
Crucible
B
Crystal
Melt
Crucible
Free shear layer
Analytical solution
Quasi-steady numerical results
Material and geometrical parameters :Silicon crystal diameter : 100 mmCrucible diameter : 300 mmMolecular 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)
Material and geometrical parameters :Silicon crystal diameter : 100 mmCrucible diameter : 300 mmMolecular 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)
FEMAG SoftwareFEMAG Software
Magnetic field lines
Bmax=0.03T Bmax=0.7T
Magnetic field generated by 2 coils with same radius (600 mm)
Turbulence Model : Adapted Mixing Length
B=0T
Stokes stream function
Magnetic field lines
Magnetic field generated by 2 coils with different radii(600 mm and 75 mm)
Turbulence model : Adapted Mixing Length
Bmax=0.2T Bmax=0.9T
Stokes stream function
B=0T
Run A
Opposite crystal and crucible rotation senses
Silicon
Mixing length model
= 8.225 10-4 kg/m.sc= 0.52 s-1
s= -2.O9 s-1
Vpul = 5. 10-6 m/s
Run A
Opposite crystal and crucible rotation senses
Silicon
Mixing length model
= 8.225 10-4 kg/m.sc= 0.52 s-1
s= -2.O9 s-1
Vpul = 5. 10-6 m/s
Run B
Same as A with a vertical magnetic field
B = 0.32 Tesla
Run B
Same as A with a vertical magnetic field
B = 0.32 Tesla
Inverse dynamic simulations of silicon growth
FEMAG-2 softwareFEMAG-2 software
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)
• MethodologyDynamic simulations are performed under supervision of a controller
Off-line Control
Time-dependentsimulator
Time-dependentsimulator
Off-linecontrollerOff-line
controller
Doprocess variables
satisfy the controlobjectives ?
Startnew time step with updated process
parameters
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
k-l turbulence model• How to modify the flow?Additional viscosity :
Additional conductivity :
: mean turbulent kinetic energywhere
Turbulent kinetic energy equation
: parameters of the model
: additional Prandtl number
From Th. Wetzel
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
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