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

Czochralski process

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

BB

AA

Stream function for runs A and BStream function for runs A and B

Temperature field for runs A and BTemperature field for runs A and B

BB

AA

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