TAILORED MAGNETIC FIELDS FOR CONTROLLED SEMICONDUCTOR GROWTH

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TAILORED MAGNETIC FIELDS FOR CONTROLLED
SEMICONDUCTOR GROWTH
Baskar Ganapathysubramanian, Nicholas
Zabaras Materials 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_at_cornell.edu URL
http//mpdc.mae.cornell.edu
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ACKNOWLEDGEMENTS
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)
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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
• Numerical Examples
• Conclusions
• Current and Future Research

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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

Single crystals semiconductors
Chips, laser heads, lithographic heads
Communications, control
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GOVERNING EQUATIONS
Momentum
On all boundaries
Temperature
On the side wall
Electric potential
Thermal gradient g1 on melt side, g2 on solid
side Pulling velocity vel_pulling
Solid
Interface
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FEATURES OF THE NUMERICAL MODEL

• 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

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS

• 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

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IMPORTANT ASSUMPTIONS AND SIMPLIFICATIONS
MAGNETO-HYDRODYNAMIC (MHD) EQUATIONS

• 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,

Electromagnetic force per unit volume on fluid
Current density
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COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• 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 2D
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

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REFERENCE CASE
Properties corresponding to GaAs Non-dimensionaliz
ed 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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REFERENCE CASE
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.
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Solid body rotation

• DESIGN OBJECTIVES
• Remove variations in the growth velocity
• Increase the growth velocity
• Keep the imposed thermal gradient as less as
  possible

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OPTIMIZATION PROBLEM USING TAILORED MAGNETIC
FIELDS
Spatial variations in the growth velocity
Time varying magnetic fields with rotation
and
Choosing a polynomial basis
Non-linear optimal control problem to determine
time variation
Design parameter set
DESIGN OBJECTIVES Find the optimal magnetic field
B(t) in 0,tmaxdetermined 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
Cost Functional
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OPTIMIZATION PROBLEM USING TAILORED MAGNETIC
FIELDS
Find a quasi solution B (bk) such that
J(Bbk) ? J(Bb) ? b an optimum design
variable set bk sought
Define the inverse solidification problem as an
unconstrained spatio temporal optimization
problem
Non linear conjugate gradient method
Gradient of the cost functional
Sensitivity of velocity field
Gradient information
Obtained from sensitivity field
m sensitivity problems to be solved
Continuum sensitivity equations
Design differentiate with respect to
Direct Problem
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CONTINUUM SENSITIVITY EQUATIONS
Momentum
Temperature
Electric potential
Interface
Solid
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VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS
Run direct problem with field b
Find difference in all properties
Run direct problem with field b?b
Compare the properties
Run sensitivity problem with b ?b

• 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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VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS
Temperature at x mid-plane Error less than 0.05
Temperature iso-surfaces
Direct problem run for the conditions specified
in the reference case with an imposed magnetic
field specified by bi1, i1,..,4 and rotation of
O 1 Direct problems run with imposed magnetic
field specified by bi10.05, i1,..,4 and
rotation of O 1 0.05 Sensitivity problems run
with ? bi 0.05
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OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC
FIELD
DETAILS OF THE CONJUGATE GRADIENT ALGORITHM
bopt final set of design parameters
Make an initial guess of b and set k 0
Update bk1 bk a pk
Solve the direct and sensitivity problems for
all required fields
Calculate the optimal step size ak
Minimizes J(bk) in the search direction pk
ak
Calculate J(bk) and J(bk) J(bk)
Sensitivity matrix M given by
Set pk -J (b0) if (k 0) else pk
-J(bk) ? pk-1
Check if (J(bk) etol
No
Set ? 0, if k 0 Otherwise
?
Yes
Set bopt bk and stop
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DESIGN PROBLEM 1
Properties corresponding to GaAs Non-dimensionaliz
ed 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
Optimize the reference case discussed earlier
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
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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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DESIGN PROBLEM 1 Results
Iteration 4
Iteration 1
Substantial reduction in curvature of
interface. Thermal gradients more uniform
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DESIGN PROBLEM 2
Properties corresponding to GaAs Non-dimensionaliz
ed 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
Reduce the imposed thermal gradient
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
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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 4
Iteration 1
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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.