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TAILORED MAGNETIC FIELDS FOR CONTROLLED SEMICONDUCTOR GROWTH

TAILORED MAGNETIC FIELDS FOR CONTROLLED SEMICONDUCTOR GROWTH. Baskar Ganapathysubramanian, Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801

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TAILORED MAGNETIC FIELDS FOR CONTROLLED SEMICONDUCTOR GROWTH

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  1. 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@cornell.edu URL: http://mpdc.mae.cornell.edu

  2. 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)

  3. 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

  4. 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 …

  5. 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

  6. 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

  7. 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

  8. 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 :

  9. 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

  10. REFERENCE CASE 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.

  11. 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. = 10 Solid body rotation • DESIGN OBJECTIVES • Remove variations in the growth velocity • Increase the growth velocity • Keep the imposed thermal gradient as less as possible

  12. 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,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 Cost Functional:

  13. OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS Find a quasi – solution : B({b}k) such that J(B{b}k)  J(B{b})  {b}; an optimum design variable set{b}ksought 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

  14. CONTINUUM SENSITIVITY EQUATIONS Momentum Temperature Electric potential Interface Solid

  15. 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.

  16. 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 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

  17. OPTIMIZATION PROBLEM WITH A TIME VARYING MAGNETIC FIELD DETAILS OF THE CONJUGATE GRADIENT ALGORITHM {b}opt – final set of design parameters Make an initial guess of {b} and set k = 0 Update {b}k+1 = {b}k + αpk Solve the direct and sensitivity problems for all required fields Calculate the optimal step size αk MinimizesJ({b}k) in the search direction pk αk= Calculate J({b}k) and J’({b}k) = J({b}k) Sensitivity matrix M given by Set pk = -J’ ({b}0) if (k = 0) else pk = -J’({b}k) + γpk-1 Check if (J({b}k) ≤ εtol No Set γ= 0, if k = 0; Otherwise γ Yes Set {b}opt = {b}k and stop

  18. DESIGN PROBLEM: 1 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 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

  19. 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

  20. DESIGN PROBLEM: 1 Results Iteration 4 Iteration 1 Substantial reduction in curvature of interface. Thermal gradients more uniform

  21. DESIGN PROBLEM: 2 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 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

  22. DESIGN PROBLEM: 2 Results 4 iterations of the Conjugate gradient method Cost function reduced by two orders of magnitude Optimal rotation 10.4

  23. DESIGN PROBLEM: 2 Results Iteration 4 Iteration 1

  24. 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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