Materials Process Design and Control Laboratory CONTROLLING SEMICONDUCTOR GROWTH USING MAGNETIC FIELDS AND ROTATION 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 URL:

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

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 and rotation  Conclusions  Current and Future Research

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

Materials Process Design and Control Laboratory OBJECTIVES OF SOLIDIFICATION PROCESS DESIGN MELT SOLID G,V q os q ol 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: 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

Materials Process Design and Control Laboratory DIFFERENT PHYSICAL PHENOMENA INVOLVED IN SINGLE CRYSTAL GROWTH MELT CRYSTAL INTERFACE InterfacialThermodynamics Capillarity Buoyancy Effects MarangoniConvection MicrogravityEffects Diffusion MorphologicalInstability Electromagnetic ElectromagneticEffects Turbulence Effects Rotational RotationalEffects Volume Change Volume Change Induced 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

Materials Process Design and Control Laboratory PHYSICAL 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? MELT CRYSTAL INTERFACE InterfacialThermodynamics Capillarity Buoyancy Effects MarangoniConvection MicrogravityEffects Diffusion MorphologicalInstability Electromagnetic ElectromagneticEffects Turbulence Effects Rotational RotationalEffects Volume Change Volume Change Induced 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

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

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

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

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 :

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

Materials Process Design and Control Laboratory MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS solid-liquid interface MELT SOLID V g  bs  bl  ts  tl q os  os  ol q ol II B(t) GAS hot cold Velocityprofile FREESURFACE Surface Tension Gradient DESIGN OBJECTIVES Find the optimal magnetic field such that, in the presence of coupled thermocapillary, buoyancy, and electromagnetic convection in the melt, a flat solid- liquid interface 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

Materials Process Design and Control Laboratory INVERSE-DESIGN PROBLEM INVERSE-DESIGN PROBLEM INVERSE PROBLEM STATEMENT Find the magnetic field b(t) in [0, t max ] such that melt convection is suppressed solid-liquid interface MELT SOLID V g  bs  bl  ts  tl q os  os  ol q ol II GAS hot cold Velocityprofile FREESURFACE Surface Tension Gradient With a guessed magnetic field, solve the following direct problem for: Melt region: Temperature field: T(x, t; b) Temperature field: T(x, t; b) Concentration field: c(x, t; b) Concentration field: c(x, t; b) Velocity field: v(x, t; b) Velocity field: v(x, t; b) Electric potential:  (x, t; b) Electric potential:  (x, t; b) Solid region: Temperature field: T s (x, t; b) Temperature field: T s (x, t; b) Measure of deviation from diffusion based growth B(t)

Materials Process Design and Control Laboratory NONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEM Continuum sensitivity problem Define the inverse solidification problem as a unconstrained spatio- temporal optimization problem Solve the above unconstrained minimization problem using the nonlinear Conjugate Gradient Method (CGM) Needs design gradient information Needs descent step size Continuum adjoint problem Find a quasi- solution: o  L 2 ([0, t max ]) such that Find a quasi- solution: b o  L 2 ([0, t max ]) such that S ( o )  S ( )  o  L 2 ( [0, t max ]) S ( b o )  S ( b )  b o  L 2 ( [0, t max ])

Materials Process Design and Control Laboratory THE 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

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: Thermal Rayleigh number: Solutal Rayleigh number: Lewis number: 3000 Marangoni number: ~0 Stefan number: Ratio of thermal diffusivites: 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 FIELDS

Materials Process Design and Control Laboratory 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 functional Cost functional Optimal field DESIGNING TAILORED MAGNETIC FIELDS

Materials Process Design and Control Laboratory DESIGNING 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.

Materials Process Design and Control Laboratory Reference caseOptimal 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 – RESULTS CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH – RESULTS Streamline contours Isotherms Isochors Comparison of streamline contours, isotherms and isochors during the growth

Materials Process Design and Control Laboratory 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 – RESULTS CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH – 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.

Materials Process Design and Control Laboratory REFERENCE CASE: TAILORED MAGNETIC FIELDS AND ROTATION Properties corresponding to GaAs Non-dimensionalized Prandtl number = Rayleigh number T= 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 = Pulling vel = 0.616; (5.6e-4 cm/s) Melting temp = 0.0; Biot num = 10.0; Solute diffusivity = ; Melting temp = 0.0; Time_step = 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.

Materials Process Design and Control Laboratory 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 REFERENCE CASE: TAILORED MAGNETIC FIELDS AND ROTATION

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

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

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

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.

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 = 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

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

Materials Process Design and Control Laboratory Properties corresponding to GaAs Non-dimensionalized Prandtl number = Rayleigh number T= Rayleigh number C= 0 Hartmann number = 60 Direction of field : z axis Direction of rotation: y axis Ratio of conductivities = 1 Stefan number = Pulling vel = 0.616; (5.6e-4 cm/s) Melting temp = 0.0; Biot num = 10.0; Solute diffusivity = ; Melting temp = 0.0; Time_step = Number of steps = 100 DESIGN PROBLEM: 1 Temp gradient length = 2 Pulling velocity = Design definition: Find the time history of the imposed magnetic field and the steady rotation such that the growth velocity is close to Optimize the reference case discussed earlier

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

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

Materials Process Design and Control Laboratory Properties corresponding to GaAs Non-dimensionalized Prandtl number = Rayleigh number T= Rayleigh number C= 0 Hartmann number = 60 Direction of field : z axis Direction of rotation: y axis Ratio of conductivities = 1 Stefan number = Pulling vel = 0.616; (5.6e-4 cm/s) Melting temp = 0.0; Biot num = 10.0; Solute diffusivity = ; Melting temp = 0.0; Time_step = Number of steps = 100 DESIGN PROBLEM: 2 Temp gradient length = 10 Pulling velocity = Design definition: Find the time history of the imposed magnetic field and the steady rotation such that the growth velocity is close to Reduce the imposed thermal gradient

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

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

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, ,  B. Ganapathysubramanian and N. Zabaras, “Control of solidification of non- conducting materials using tailored magnetic fields”, Journal of Crystal growth, Vol. 276/1-2, ,  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.