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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 affectingcrystal 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, globalinteraction between heat transfer in crystal and

melt, solidification front deformation and overallradiation transfer

Non-linear

physics of radiation, melt convection andsolidification

Dynamiccritical growth stages: seeding, shouldering, tail- end, crystal detachment, post-growth

Inversenatural output is prescribed (crystal shape), while

natural input is calculated (heater power or pull rate)

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Melt convection

= Significant heat transfer mechanismdefect and dislocation densities

growth striations

interface shape

= Dominant mechanism for dopant and

impurity transferdopant and impurity (oxygen) distributions

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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

Ws

Wccrucible

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Quasi-steady axisymmetric models

ObjectiveCoupling 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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t0 t1 t2 t3 t4 t5 t6 timet7

Cone

growth

Body

growth

Tail-end

stage

Quasi-steady simulations

with melt flow

Time-

dependent

simulation

with

interpolated

flow effect

Time-dependent

simulation can provide

quasi-steady source

terms equivalent to

transient terms

General dynamic strategy

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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

Ohms law :

Conservation of charge :

Rigid magnetic field approximation :induced magnetic field is negligible

Imposed steady axisymmetric magnetic field :

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Analytical solutionsFrom Hjellming & Walker, 1993

Existence of a free shear layer:plays an important role in oxygen andimpurity transfer

Hypotheses :

High Hartmann number :

Inertialess approximation (valid if B0.2T) :

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Case I: Case II :

No magnetic field lines incontact with neither thecrystal nor the crucible

Magnetic field lines in contactwith both the crystal and thecrucible

B

Crystal

Melt

Crucible

B

Crystal

Melt

Crucible

Free shear layer

Analytical solution

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Quasi-steady numerical results

Material and geometrical parameters :

Silicon crystal diameter : 100 mm

Crucible 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 Software

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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

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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

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Run A

Opposite crystal and crucible

rotation senses

Silicon

Mixing length model

m= 8.225 10-4kg/m.s

Wc= 0.52 s-1

Ws= -2.O9 s-1

Vpul= 5. 10-6m/s

Run B

Same as A with a vertical

magnetic field

B = 0.32 Tesla

Inverse dynamic simulations of silicon growth

FEMAG-2 software

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B

A

Stream function for runs A and B

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Temperature field for runs A and B

B

A

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Off-line Control

ObjectiveTo determine the best evolution of the processparameters 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

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Off-line Control

Time-dependent

simulator

Off-linecontroller

Doprocess variables

satisfy the control

objectives ?

Startnew time step with

updated process

parameters

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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 ifnecessary

Numerical scheme should be able to control mesh

refinement along boundary and internal layers

Off-line control is a promising technique for optimizingthe magnetic field design

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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

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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

Hartmann number