Development of Modelbased Control for Bridgman Crystal Growth

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Development of Model-based Control for Bridgman
Crystal Growth

• P. Sonda, A. Yeckel, P. Daoutidis, and J.J.
  Derby, Department of Chemical Engineering and
• Materials ScienceUniversity of Minnesota
• ACCGE-15
• Keystone, CO
• July 20-24, 2003

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The interplay between fluid dynamics and mass
transfer in a thermally destabilized Bridgman
system is studied

• Processing issue crystal striations
• Motivating Works
• Kim, Witt, and Gatos (1972) performed pioneering
  striation experiments in a thermally
  destabilizing furnace for Te-doped InSb
• Müller (1984) further characterized system
  behavior, as a function of Prandtl number and
  aspect ratio

Destabilized Bridgman crystal growth system
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Experimental results from Kim, Witt, and Gatos
show flow transitions as melt depth drops
Flow behavior inferred from thermocouple traces
Increasing Rayleigh Number
Growth direction
Oscillatory flow
Etched cross-section
Thermocouple traces
Steady flow
From Kim, Witt, and Gatos, J. Electrochem. Soc.
119 (1972) 1218.
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The vertical Bridgman model exhibits fascinating
nonlinear behavior
Direction of growth (decreasing melt height)
A
Spatially Averaged Melt Speed (cm/sec)
B
Transition to time-periodic flows
Rayleigh Number
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Model predicts Hopf bifurcation at critical melt
height (Ra) oscillatory to steady flow
Time-periodic flow
Hopf bifurcation
Speed (cm/sec)
Unstable steady flow
Stable steady flow
Rayleigh Number
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Vertical Bridgman crystal growth The control
problem

• Crystal growth time scale ?(hours)
• Computations are fast enough for real-time
  sensing and control
• Actuation Several practical options exist
• Thermal environment
• Magnetic field
• Vibration
• Our choice Crucible Rotation

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Two controllers are designed to control the
molten flows within the VB system

• Control Objective
• Oscillation attenuation/stabilization
• Output (E) Volume-averaged kinetic energy S2
• Manipulated Input Crucible rotation rate, ?

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Two controllers are designed to control the
molten flows within the VB system

• Control Objective
• Oscillation attenuation/stabilization
• Output (E) Volume-averaged kinetic energy S2
• Manipulated Input Crucible rotation rate, ?
• Proportional controller

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Two controllers are designed to control the
molten flows within the VB system

• Control Objective
• Oscillation attenuation/stabilization
• Output (E) Volume-averaged kinetic energy S2
• Manipulated Input Crucible rotation rate, ?
• Proportional controller
• Nonlinear control law via input-output
  linearization
• Enforces a first order closed-loop response

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Two controllers are designed to control the
molten flows within the VB system

• Control Objective
• Oscillation attenuation/stabilization
• Output (E) Volume-averaged kinetic energy S2
• Manipulated Input Crucible rotation rate, ?
• Proportional controller
• Nonlinear control law via input-output
  linearization
• Enforces a first order closed-loop response
• Controllers were applied to two vertical Bridgman
  systems
• Thermally stabilized system with time-varying
  furnace profile
• Thermally destabilized system (KWG)

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A thermally stabilized configuration is modeled
with a disturbance to test controller feasibility
At t 0, steady crucible rotation is applied
Time-varying disturbance added to furnace
temperature A 1 K b 0.045 / s
? 0
? 2 rpm
Speed (cm/sec)
Time (sec)
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The nonlinear controller is most effective in
stabilizing flow oscillations
At t 0, feedback control is applied
Thermally stabilized system
Kc 100
No control
Speed (cm/sec)
Kc 1000
Kc 2500
Controller setpoint
Nonlinear
Time (sec)
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The nonlinear controller is most effective in
stabilizing flow oscillations
At t 0, feedback control is applied
Thermally stabilized system
Kc 100
Proportional controllers exhibit offset, which is
dependent on the controller gain, Kc
No control
Speed (cm/sec)
Kc 1000
Kc 2500
Controller setpoint
Nonlinear
Time (sec)
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The nonlinear controller is most effective in
stabilizing flow oscillations
Thermally stabilized system
Nonlinear
Kc 2500
Rotation Rate (RPM)
Kc 1000
Kc 100
Time (sec)
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The nonlinear controller is most effective in
stabilizing flow oscillations
Thermally stabilized system
Nonlinear
Kc 2500
Increasing gain further causes system to go
unstable
Rotation Rate (RPM)
Kc 1000
Kc 100
Time (sec)
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Steady crucible rotation is unsuccessful in
stabilizing flow oscillations in thermally
destabilized system
Ra 3.6 x 105
Speed (cm/sec)
Critical Rayleigh Number
Time (sec)
Rotation Rate (RPM)
Steady crucible rotation encourages the onset of
time-periodic flows Rac decreases as ? is
increased
Steady crucible rotation applied to this
configuration exacerbates flow oscillations in
melt
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Proportional controller stabilizes inherent flow
oscillations in thermally destabilized system
Ra 3.6 x 105 Kc 0.75
Time to achieve suppression of oscillations
Speed (cm/sec)
Time (hr)
Proportional control is effective only for weakly
time-varying flows
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Rotation rate varies temporally to stabilize flow
oscillations in thermally destabilized system
The system rotates in step with the natural
oscillations to dampen them
Rotation Rate (RPM)
Time (hr)
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Rotation rate varies temporally to stabilize flow
oscillations in thermally destabilized system
The system rotates in step with the natural
oscillations to dampen them
Rotation Rate (RPM)
Time (hr)
Application of the nonlinear controller to this
configuration produces an unstable closed-loop
system
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Crucible rotation is shown to exacerbate flow
oscillations in thermally destabilized system
Ra 3.6 x 105 ? 2 rpm
A negative Rayleigh discriminant indicates a
possible flow instability
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Summary and future work

• Feedback controllers were designed to control
  flows within a Vertical Bridgman crystal growth
  system
• Controlled variable Crucible rotation rate
• Measurement variable Volume-averaged kinetic
  energy
• Two systems were studied
• Stabilized configuration with disturbance to
  thermal environment
• Crucible rotation is a feasible option for
  control
• Steady crucible rotation does not suppress
  oscillations
• Feedback control suppresses oscillations to
  varying degrees
• Nonlinear controller is optimal.
• Destabilized configuration with inherent flow
  oscillations
• Crucible rotation promotes the onset of
  time-varying flows
• Proportional control is effective but extremely
  slow for limited range of Ra
• Implementation of nonlinear controller produces
  an unstable system
• Investigate other actuation parameters

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Bridgman process model is solved via the finite
element method

• Mathematical model
• Based on KWG experiments
• 3D Axisymmetric
• Quasi-steady-state (QSS) or time-dependent
• Heat conduction in all materials
• Incompressible, Boussinesq fluid in melt
• Moving boundary at melting point for crystal-melt
  interface
• Simple radiation/convection boundary conditions
• Ampoule is assumed fixed, i.e. Va 0.
• Numerical method
• Galerkin FEM with isoparametric elements
• Adaptive mesh (front-tracking, algebraic
  generation)
• Trapezoid rule time integration
• Full Newton iteration method with direct matrix
  solver
• Computation time for N iteration 4.7
  seconds

Schematic