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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
CONTROLLING SEMICONDUCTOR GROWTH USING MAGNETIC FIELDS AND ROTATION
Baskar Ganapathysubramanian, Nicholas ZabarasMaterials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall
Cornell University Ithaca, NY 14853-3801
Email: zabaras@cornell.eduURL: http://mpdc.mae.cornell.edu
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
FUNDING SOURCES:
Air Force Research Laboratory
Air Force Office of Scientific Research
National Science Foundation (NSF)
ALCOA
Army Research Office
COMPUTING SUPPORT:
Cornell Theory Center (CTC)
ACKNOWLEDGEMENTSACKNOWLEDGEMENTS
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
OUTLINE OF THE PRESENTATION Introduction and motivation for the current study Numerical model of crystal growth under the influence of
magnetic fields and rotation Numerical examples Optimization problem in alloy solidification using time
varying magnetic fields and rotation Conclusions Current and Future Research
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Single crystals : semiconductors
Chips, laser heads, lithographic heads
Communications,
control …
SEMI-CONDUCTOR GROWTH
-Single crystal semiconductors the backbone of the electronics industry.
- Growth from the melt is the most commonly used method
- Process conditions completely determine the life of the component
- Look at non-invasive controls
- Electromagnetic control, thermal control and rotation
- Analysis of the process to control and the effect of the control variables
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
OBJECTIVES OF SOLIDIFICATION PROCESS DESIGNOBJECTIVES OF SOLIDIFICATION PROCESS DESIGN
MELTMELT
SOLIDSOLID
G ,VG ,V
qos
qol
g
B
Requirements for a
better crystal
• Flat growth interface with controlled growth velocity (V) and thermal gradient (G)
• Homogeneous distribution of solute
• Reduction in temperature and concentration striations during growth
• Minimize defects and dislocations
• Minimize residual stresses in the crystal
Controllable factors
• Interface motion
• Melt flow
• Thermal conditions
• Furnace design
DEVELOP INVERSE METHODS FOR:DEVELOP INVERSE METHODS FOR:
• Controlling the growth velocity V and the temperature gradient G
• Improving macroscopic and microscopic homogeneity of the final crystal
• Eliminating or reducing the effects of convection on the solidification morphology
• Delaying or eliminating morphological instability
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
DIFFERENT PHYSICAL PHENOMENA INVOLVED IN SINGLE CRYSTAL GROWTHDIFFERENT PHYSICAL PHENOMENA INVOLVED IN SINGLE CRYSTAL GROWTH
MELTMELT
CRYSTALCRYSTAL
INTERFACE
INTERFACE
InterfacialInterfacialThermodynamicsThermodynamics
CapillarityCapillarity
Buoyancy EffectsBuoyancy EffectsMarangoni Marangoni ConvectionConvection
MicrogravityMicrogravityEffectsEffects
DiffusionDiffusion
MorphologicalMorphologicalInstabilityInstability
ElectromagneticElectromagneticEffectsEffectsTurbulence EffectsTurbulence Effects
RotationalRotationalEffectsEffects Volume ChangeVolume Change
Induced FlowInduced Flow
Governing physics
SOLID
• Heat conduction
MELT
• Heat and solute transport• Incompressibility• Navier-Stokes equations with Lorentz, Kelvin & buoyancy force terms• Traction force on free surface due to surface tension variation (Marangoni convection)
SOLID-LIQUID INTERFACE
• Interfacial heat and solute balance• Thermodynamic equilibrium conditions
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
PHYSICAL MECHANISMS TO BE CONTROLLED DURING SOLIDIFICATIONPHYSICAL MECHANISMS TO BE CONTROLLED DURING SOLIDIFICATION
MEANS FOR DESIGN
• Control the boundary heat flux
• Multiple-zone controllable furnace design
• Rotation of the furnace
• Micro-gravity growth
• Electromagnetic fields
Heat flux design
• Sampath & Zabaras (2000, ..)
• Stable growth for given V
• Design for given V & G
• Required heat flux uneconomical
Electromagnetic fields
• Constant magnetic fields- damp convection, but large fields required
• Rotating magnetic fields, striations
• Combination of different magnetic fields?
MELTMELT
CRYSTALCRYSTAL
INTERFACE
INTERFACE
InterfacialInterfacialThermodynamicsThermodynamics
CapillarityCapillarity
Buoyancy EffectsBuoyancy EffectsMarangoni Marangoni ConvectionConvection
MicrogravityMicrogravityEffectsEffects
DiffusionDiffusion
MorphologicalMorphologicalInstabilityInstability
ElectromagneticElectromagneticEffectsEffectsTurbulence EffectsTurbulence Effects
RotationalRotationalEffectsEffects Volume ChangeVolume Change
Induced FlowInduced Flow
Micro-gravity growth
• Skylab experiments
• Suppression of convection
• Large, good quality crystals
• Very expensive
Furnace rotation
• Cz growth, floating zone method
• Forced convection via ‘Accelerated crucible rotation technique’ , etc.
• Material specific
Furnace design
• Time history and number of heating zones.
• Achieve growth for given V
• Furnace requirements impractical
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CCOORRNNEELLLL U N I V E R S I T Y
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GOVERNING EQUATIONSMomentum
Temperature
On all boundaries
Thermal gradient: g1 on melt side, g2 on solid side
Pulling velocity : vel_pulling
On the side wall
Electric potential
Interface Solid
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
• The solid part and the melt part modeled seperately
• Moving/deforming FEM to explicitly track the advancing solid-liquid interface
• Transport equations for momentum, energy and species transport in the solid and melt
• Individual phase boundaries are explicitly tracked.
• Interfacial dynamics modeled using the Stefan condition and solute rejection
• Different grids used for solid and melt part
FEATURES OF THE NUMERICAL MODEL
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
• The densities of both phases are assumed to be equal and constant except in the Boussinesq approximation term for thermosolutal buoyancy.
•The solid is assumed to be stress free.
• Constant thermo-physical and transport properties, including thermal and solute diffusivities viscosity, density, thermal conductivity and phase change latent heat.
• The melt flow is assumed to be laminar
• The radiative boundary conditions are linearized with respect to the melting temperature
• The melting temperature of the material remains constant throughout the process
IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS
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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS
• Phenomenological cross effects – galvomagnetic, thermoelectric and thermomagnetic – are neglected• The induced magnetic field is negligible, the applied field
• Magnetic field assumed to be quasistatic
• The current density is solenoidal,
• The external magnetic field is applied only in a single direction
• Spatial variations in the magnetic field negligible in the problem domains
• Charge density is negligible,
MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS
Electromagnetic force per
unit volume on fluid :
Current density :
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COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES
For 2D:•Stabilized finite element methods used for discretizing governing equations.
• For the thermal sub-problem, SUPG technique used for discretization
• The fluid flow sub-problem is discretized using the SUPG-PSPG technique
For 3D: •Stabilized finite element methods used for discretizing governing equations.
• Fractional time step method.
• For the thermal and solute sub-problems, SUPG technique used for discretization
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONSMATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS
solid-liquid interfacesolid-liquid interface
MELTMELTSOLIDSOLID
VV
gg
bs bl
ts tl
qos
os
ol
qolI
B(t)B(t)
GASGAS
hothotcoldcoldVelocity Velocity
profileprofile
FREE FREE SURFACESURFACE
Surface Tension GradientSurface Tension Gradient
DESIGN OBJECTIVESDESIGN OBJECTIVESFind the optimal magnetic field such that, in the presence Find the optimal magnetic field such that, in the presence
of coupled thermocapillary, buoyancy, and of coupled thermocapillary, buoyancy, and electromagnetic convection in the melt, a flat solid- liquid electromagnetic convection in the melt, a flat solid- liquid
interface with diffusion dominated growth is achievedinterface with diffusion dominated growth is achieved
Growth under diffusion dominated conditions ensures:
• Flat solid-liquid interface. This is crucial in crystal growth
• Uniform temperature gradients along the interface. This results in reduced stress in the cooling crystal. Found to be directly related to the life time of the component
• Uniform solute distribution. This leads to a homogeneous crystal. Further, this also results in reduced dislocations
• Suppression in temperature and solute fluctuations leading to reduced defects in the crystal
Micro-gravity based growth is purely diffusion based
Objective is to achieve some sort of reduced gravity growth
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Materials Process Design and Control Laboratory
INVERSE-DESIGN PROBLEMINVERSE-DESIGN PROBLEM
INVERSE PROBLEM INVERSE PROBLEM STATEMENTSTATEMENT
Find the magnetic field b(t) in Find the magnetic field b(t) in [0, t[0, tmaxmax] such that melt ] such that melt
convection is suppressedconvection is suppressed
solid-liquid interfacesolid-liquid interface
MELTMELTSOLIDSOLID
VV
gg
bs bl
ts tl
qos
os
ol
qol
I
GASGAS
hothotcoldcoldVelocity Velocity
profileprofile
FREE FREE SURFACESURFACE
Surface Tension GradientSurface Tension Gradient
With a guessed magnetic field, With a guessed magnetic field,
solve the following direct solve the following direct
problem for:problem for:
Melt regionMelt region::• Temperature field: Temperature field: TT((x, t; bx, t; b))• Concentration field: Concentration field: cc((x, t; bx, t; b))• Velocity field: Velocity field: vv((x, t; bx, t; b))• Electric potential: Electric potential: (x, t; (x, t; bb))
Solid regionSolid region::
• Temperature field: Temperature field: TTss((x, t; bx, t; b))
Measure of deviation from diffusion based growth
B(t)B(t)
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NONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEMNONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEM
Continuum Continuum sensitivity problemsensitivity problem
Define the inverse solidification problem as a unconstrained spatio- Define the inverse solidification problem as a unconstrained spatio- temporal optimization problemtemporal optimization problem
Solve the above unconstrained minimization problem using the Solve the above unconstrained minimization problem using the nonlinear Conjugate Gradient Method (CGM)nonlinear Conjugate Gradient Method (CGM)
Needs design gradient informationNeeds design gradient information Needs descent step sizeNeeds descent step size
Continuum Continuum adjoint problemadjoint problem
Find a quasi- solutionFind a quasi- solution: : boo LL22 ([0([0, , ttmaxmax]) ]) such thatsuch thatS S ((boo) ) S S ((b ) ) boo LL22 (( [0[0, , ttmaxmax]) ])
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CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control Laboratory
THE CONTINUUM ADJOINT PROBLEMTHE CONTINUUM ADJOINT PROBLEM
Adjoint equations
Gradient of the cost functional given in terms of the direct and the adjoint fields
Using integration by parts; Green’s theorem; Reynolds transport theorem and some vector algebra
1
2
3
4
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Materials Process Design and Control Laboratory
Design definition:Find the time history of the imposed magnetic field/gradient, such that diffusion- based growth is achieved in the presence of thermocapillary and buoyant forces
Material characteristics:Binary alloy/pure material,Non-conducting
Material specification
27% NaCl aqueous solution
Prandtl number: 0.007
Thermal Rayleigh number: 200000
Solutal Rayleigh number: 10000
Lewis number: 3000
Marangoni number: ~0
Stefan number: 0.12778
Ratio of thermal diffusivites: 1.25975
Setup specifications
Solidification in a rectangular cavity
Dimensions 2cm x 2cm
Fluid initially at 1 C
Left wall kept at -10 C
Driven by thermal and solutal buoyancy
Minimize the cost functional
DESIGNING TAILORED MAGNETIC FIELDSDESIGNING TAILORED MAGNETIC FIELDS
CCOORRNNEELLLL U N I V E R S I T Y
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CG iterations
Cos
tfun
ctio
nal
0 1 2 3 4
10-4
10-3
10-2
10-1
100
CG iterations
Gra
dien
tof
the
cost
func
tiona
l
0 1 2 3 410-2
10-1
100
Time
Gra
dien
t(T
2 /m)
0 0.025 0.05 0.075 0.131.95
31.96
31.97
31.98
31.99
32
32.01
32.02
Stopping tolerance ~ 5e-4. Initially quadratic convergence, superlinear later.
Optimal field : Reduces initially because of the increased solutal buoyancy due to the solute rejection into the melt at the interface. At later times, the concentration of the solute along the interface becomes uniform and hence solutal buoyancy decreases
Gradient of the cost functionalCost functional
Optimal field
DESIGNING TAILORED MAGNETIC FIELDSDESIGNING TAILORED MAGNETIC FIELDS
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DESIGNING TAILORED MAGNETIC FIELDSDESIGNING TAILORED MAGNETIC FIELDS
Comparison of the evolution of the velocity and temperature fields for the reference case (Left) and the optimal case (Right).
Velocity is damped out to a large extent. The maximum velocity for the optimal case is 0.76 compared to 36.0 for the reference case. There is some amount of vorticity near the interface due to the local gradients in temperature and concentration
Temperature evolution is primarily conduction based as can be seen by the motion of the isotherms.
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Reference case
Optimal magnetic field
There is significant reduction in vorticity (reduction by a factor of 200). Notice that in the reference growth, the larger flow near the walls causes a stratification of temperature as seen in the isotherms. This is avoided in the optimal growth. The temperature contours are more evenly distributed. The melt is almost quiescent. The concentration of the solute is more evenly distributed with the application of the magnetic gradient.
CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH –– RESULTS RESULTS
0.58
0.54
0.51
0.47
0.43
0.390.35
0.310.27
0.230.19
0.12
Solid
0 .04
0.04 0.04
0.04
0.04
0.040.04
0.04
2.52 2.52
2.52
2.52
2.52
2.52
5.015.01
5.01
5.01
5.01 7.49
7.49 7.49
7.49
7.49
7.49
9.97
9.979.97
9.97
9.97
12.46
12.46 12.46
12.46
14.9
4 14.9
4
Solid
0.8
75
0.7500.625
0.563
0.500
0.438
0.375
0.3130.250
0.063Solid
0.9
380
.87
50.8
13
0.7
50
0.6
88
0.625
0.563
0.5000.438
0.375
0.313
0.2
50
0.1
88
0.1
25
0.0
63Solid
1.00409
1.121281.30517
1.48508
1.8568
2.44984Solid
1.77892
1.77892
1.76946
1.79517
1.79517
1.77892
1.7
78
92
Solid
Streamline contours
Isotherms
Isochors
Comparison of streamline contours, isotherms and isochors during the growth
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Under normal growth conditions, are fluctuations in the temperature and concentration fields in the melt. This leads to striations or formation of banded solute layers in the solid. This has an implicit relation with the dislocation density, stress and defects in the crystal. The two figures show the concentration profiles at the interface during the time of growth.
CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH –– RESULTS RESULTS
A frequency domain representation of the concentration at the interface. (log(power) vs. frequency). The application of the magnetic gradient damps out fluctuations to a great extent. This has a direct effect on the quality of the crystal. 0 100 200 300 400 500 600 700 800 900 1000
0
1
2
3
4
5
6
7
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REFERENCE CASE: TAILORED MAGNETIC FIELDS AND ROTATION
Properties corresponding to GaAs
Non-dimensionalized
Prandtl number = 0.00717
Rayleigh number T= 50000
Rayleigh number C= 0
Direction of field : z axis
No gradient of field applied
Direction of rotation: y axis
Ratio of conductivities = 1
Stefan number = 0.12778
Pulling vel = 0.616; (5.6e-4 cm/s)
Melting temp = 0.0;
Biot num = 10.0;
Solute diffusivity = 0.0032;
Melting temp = 0.0;
Time_step = 0.002
Number of steps = 500
Computational details
Number of elements ~ 110,000
8 hours on 8 nodes of the Cornell Theory Centre
Finite time for the heater motion to reach the centre.
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Results in changes in the solute rejection pattern.
Previous work used gradient of magnetic field
Use other forms of body forces?
Rotation causes solid body rotation
Coupled rotation with magnetic field.
= 10
Solid body rotationDESIGN OBJECTIVES
- Remove variations in the growth velocity- Increase the growth velocity- Keep the imposed thermal gradient as less as possible
REFERENCE CASE: TAILORED MAGNETIC FIELDS AND ROTATION
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CCOORRNNEELLLL U N I V E R S I T Y
Time varying
magnetic fields with rotation
Spatial variations in the growth
velocity
Non-linear optimal control problem to determine time variation
Choosing a
polynomial basis Design parameter
set
DESIGN OBJECTIVESFind the optimal magnetic field B(t) in [0,tmax]determined by the set and the optimal rotation rate such that,
in the presence of coupled thermosolutal buoyancy, and electromagnetic forces in the melt, the crystal growth rate is
close to the pulling velocity
OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS
Cost Functional:
and
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OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS
Define the inverse solidification problem as an unconstrained spatio – temporal optimization problem
Find a quasi – solution : B ({b}k) such that
J(B{b}k) J(B{b}) {b}; an optimum
design variable set {b}k sought
Gradient of the cost functional:
Sensitivity of velocity field :
m sensitivity problems
to be solved
Gradient
information
Obtained from
sensitivity field
Direct ProblemContinuum
sensitivity equationsDesign differentiate
with respect to
Non – linear conjugate gradient method
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Momentum
Temperature
Electric potential
Interface
Solid
CONTINUUM SENSITIVITY EQUATIONS
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CCOORRNNEELLLL U N I V E R S I T Y
Run sensitivity problem with b; b
Run direct problem with field b
Run direct problem with field b+b
Find difference in all properties
Compare the propertiesCompare the properties
VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS
• Continuum sensitivity problems solved are linear in nature.• Each optimization iteration requires solution of the direct problem and m linear CSM problems.
In each CSM problem :• Thermal and solutal sub-problems solved in an iterative loop •The flow and potential sub - problem are solved only once.
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Direct problem run for the conditions specified in the reference case with an imposed magnetic field specified by bi=1, i=1,..,4 and rotation of Ω = 1
Direct problems run with imposed magnetic field specified by bi=1+0.05, i=1,..,4 and rotation of Ω = 1 + 0.05
Sensitivity problems run with Δ bi = 0.05
Temperature at x mid-plane
Error less than 0.05 %
X
Y
Z
4.85E-04 1.46E-03 2.43E-03 3.40E-03 4.37E-03 5.34E-03 6.31E-03 7.28E-03
Temperature iso-surfaces
VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS
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OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD
DETAILS OF THE CONJUGATE GRADIENT ALGORITHM
Make an initial guess of {b} and set k = 0
Solve the direct andsensitivity problems for all required fields
Set pk = -J’ ({b}0) if (k = 0)else pk = -J’({b}k) + γ pk-1
Set γ = 0, if k = 0; Otherwise
Calculate J({b}k) and J’({b}k) = J({b}k)
Check if
(J({b}k) ≤ εtol
γ
Calculate the optimal step size αk
αk =
Set {b}opt = {b}k
and stop
Update {b}k+1 = {b}k + α pk
Yes
No
{b}opt – final set of design parameters
Minimizes J({b}k) in the search direction pk
Sensitivity matrix M given by
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Properties corresponding to GaAs
Non-dimensionalized
Prandtl number = 0.00717
Rayleigh number T= 50000
Rayleigh number C= 0
Hartmann number = 60
Direction of field : z axis
Direction of rotation: y axis
Ratio of conductivities = 1
Stefan number = 0.12778
Pulling vel = 0.616; (5.6e-4 cm/s)
Melting temp = 0.0;
Biot num = 10.0;
Solute diffusivity = 0.0032;
Melting temp = 0.0;
Time_step = 0.002
Number of steps = 100
DESIGN PROBLEM: 1
Temp gradient length = 2
Pulling velocity = 0.616
Design definition:Find the time history of the imposed magnetic field and the steady rotation such that the growth velocity is close to 0.616
Optimize the reference case discussed earlier
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DESIGN PROBLEM: 1 Results
4 iterations of the Conjugate gradient method
Each iteration 6 hours on 20 nodes at Cornell theory center
Cost function reduced by two orders of magnitude
Optimal rotation 9.8
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Substantial reduction in curvature of interface.
Thermal gradients more uniform
Iteration 1
Iteration 4
DESIGN PROBLEM: 1 Results
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CCOORRNNEELLLL U N I V E R S I T Y
Properties corresponding to GaAs
Non-dimensionalized
Prandtl number = 0.00717
Rayleigh number T= 50000
Rayleigh number C= 0
Hartmann number = 60
Direction of field : z axis
Direction of rotation: y axis
Ratio of conductivities = 1
Stefan number = 0.12778
Pulling vel = 0.616; (5.6e-4 cm/s)
Melting temp = 0.0;
Biot num = 10.0;
Solute diffusivity = 0.0032;
Melting temp = 0.0;
Time_step = 0.002
Number of steps = 100
DESIGN PROBLEM: 2
Temp gradient length = 10
Pulling velocity = 0.616
Design definition:Find the time history of the imposed magnetic field and the steady rotation such that the growth velocity is close to 0.616
Reduce the imposed thermal gradient
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DESIGN PROBLEM: 2 Results
4 iterations of the Conjugate gradient method
Cost function reduced by two orders of magnitude
Optimal rotation 10.4
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DESIGN PROBLEM: 2 Results
Iteration 1
Iteration 4
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CONCLUSIONS
Developed a generic crystal growth control simulator
Flexible, modular and parallel.
Easy to include more physics.
Described the unconstrained optimization method towards control of crystal growth through the continuum sensitivity method.
Performed growth rate control for the initial growth period of Bridgmann growth.
Look at longer growth regimes
Reduce some of the assumptions stated.
B. Ganapathysubramanian and N. Zabaras, “Using magnetic field gradients to control the directional solidification of alloys and the growth of single crystals”, Journal of Crystal growth, Vol. 270/1-2, 255-272, 2004.
B. Ganapathysubramanian and N. Zabaras, “Control of solidification of non-conducting materials using tailored magnetic fields”, Journal of Crystal growth, Vol. 276/1-2, 299-316, 2005.
B. Ganapathysubramanian and N. Zabaras, “On the control of solidification of conducting materials using magnetic fields and magnetic field gradients”, International Journal of Heat and Mass Transfer, in press.
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