1. Materials Process Design and Control Laboratory 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 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu

2. Materials Process Design and Control Laboratory 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) ACKNOWLEDGEMENTS

3. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory 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

5. Materials Process Design and Control Laboratory GOVERNING EQUATIONS Momentum Temperature On all boundaries Thermal gradient: g 1 on melt side, g 2 on solid side Pulling velocity : vel_pulling On the side wall Electric potential Interface Solid

6. Materials Process Design and Control Laboratory 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

7. Materials Process Design and Control Laboratory 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

8. Materials Process Design and Control Laboratory 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 :

9. Materials Process Design and Control Laboratory 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

10. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory 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 OBJECTIVES Find the optimal magnetic field B(t) in [0,t max ]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

13. Materials Process Design and Control Laboratory 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 Problem Continuum sensitivity equations Design differentiate with respect to Non – linear conjugate gradient method

14. Materials Process Design and Control Laboratory Momentum Temperature Electric potential Interface Solid CONTINUUM SENSITIVITY EQUATIONS

15. Materials Process Design and Control Laboratory 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 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.

16. Materials Process Design and Control Laboratory Direct problem run for the conditions specified in the reference case with an imposed magnetic field specified by b i =1, i=1,..,4 and rotation of Ω = 1 Direct problems run with imposed magnetic field specified by b i =1+0.05, i=1,..,4 and rotation of Ω = 1 + 0.05 Sensitivity problems run with Δ b i = 0.05 Temperature at x mid- plane Error less than 0.05 % Temperature iso-surfaces VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS

17. Materials Process Design and Control Laboratory 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 and sensitivity problems for all required fields Set p k = -J’ ( {b} 0 ) if (k = 0) else p k = -J’ ( {b} k ) + γ p k-1 Set γ = 0, if k = 0; Otherwise Calculate J({b} k ) and J’({b} k ) = J({b} k ) Check if (J({b} k ) ≤ ε tol γ Calculate the optimal step size α k αk =αk = Set {b} opt = {b} k and stop Update {b} k+1 = {b} k + α p k Yes No {b} opt – final set of design parameters Minimizes J({b} k ) in the search direction p k Sensitivity matrix M given by

18. Materials Process Design and Control Laboratory 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

19. Materials Process Design and Control Laboratory 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. Materials Process Design and Control Laboratory Substantial reduction in curvature of interface. Thermal gradients more uniform Iteration 1 Iteration 4 DESIGN PROBLEM: 1 Results

21. Materials Process Design and Control Laboratory 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

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

23. Materials Process Design and Control Laboratory DESIGN PROBLEM: 2 Results Iteration 1 Iteration 4

24. Materials Process Design and Control Laboratory 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.