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

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

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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

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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)

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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

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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
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4
3
1
2
melt
crucible
Wc
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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

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Melt flow model
Hypotheses Incompressible Newtonian
fluid Boussinesq approximation
Quasi-steady, turbulent or laminar flow
Reynolds equations ? mA, kA
additional viscosity and conductivity
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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
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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 (?)

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Rigid magnetic fields
Rigid magnetic field approximation
induced magnetic field is
negligible Imposed steady axisymmetric magnetic
field
Ohms law Conservation of charge
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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 B0.2T)
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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

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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)
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Magnetic field generated by 2 coils with same
radius (600 mm) Turbulence Model Adapted
Mixing Length
Magnetic field lines
Bmax0.03T
Bmax0.7T
B0T
Stokes stream function
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Magnetic field generated by 2 coils with
different radii(600 mm and 75 mm) Turbulence
model Adapted Mixing Length
Magnetic field lines
Bmax0.2T
Bmax0.9T
B0T
Stokes stream function
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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
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Stream function for runs A and B
A
B
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Temperature field for runs A and B
A
B
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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

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Off-line Control
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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

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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
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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 Har
tmann number