Title: Optimization of the Czochralski silicon growth process by means of configured magnetic fields
1Optimization 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
2Bulk growth from the melt basic techniques
Czochralski (Cz),Liquid Encapsulated Czochralski
(LEC)
Floating Zone (FZ)
Vertical Bridgman
3Czochralski process
4Factors 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)
5Numerical 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
6Principal 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)
7Melt 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
8Typical 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
9Quasi-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
10Melt flow model
Hypotheses Incompressible Newtonian
fluid Boussinesq approximation
Quasi-steady, turbulent or laminar flow
Reynolds equations ? mA, kA
additional viscosity and conductivity
11General 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
12Melt 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 (?)
13Rigid magnetic fields
Rigid magnetic field approximation
induced magnetic field is
negligible Imposed steady axisymmetric magnetic
field
Ohms law Conservation of charge
14Analytical 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)
15Analytical solution
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
16Quasi-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)
17Magnetic 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
18Magnetic 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
19Inverse 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
20Stream function for runs A and B
A
B
21Temperature field for runs A and B
A
B
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25Off-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
26Off-line Control
27Conclusions
- 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
28k-l turbulence model
Additional viscosity Additional conductivity
Turbulent kinetic energy equation
mean turbulent kinetic energy
From Th. Wetzel
where
parameters of the model
additional Prandtl number
29Dimensionless 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