Materials Process Design and Control Laboratory CONTROLLING SEMICONDUCTOR GROWTH USING MAGNETIC FIELDS AND ROTATION Baskar Ganapathysubramanian, Nicholas

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


CONTROLLING SEMICONDUCTOR GROWTH USING MAGNETIC FIELDS AND ROTATION

Baskar Ganapathysubramanian, Nicholas ZabarasMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall

Cornell University Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu




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)

ACKNOWLEDGEMENTSACKNOWLEDGEMENTS




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




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 DESIGNOBJECTIVES OF SOLIDIFICATION PROCESS DESIGN

MELTMELT

SOLIDSOLID

G ,VG ,V

qos

qol

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: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




DIFFERENT PHYSICAL PHENOMENA INVOLVED IN SINGLE CRYSTAL GROWTHDIFFERENT PHYSICAL PHENOMENA INVOLVED IN SINGLE CRYSTAL GROWTH

MELTMELT

CRYSTALCRYSTAL

INTERFACE

INTERFACE

InterfacialInterfacialThermodynamicsThermodynamics

CapillarityCapillarity

Buoyancy EffectsBuoyancy EffectsMarangoni Marangoni ConvectionConvection

MicrogravityMicrogravityEffectsEffects

DiffusionDiffusion

MorphologicalMorphologicalInstabilityInstability

ElectromagneticElectromagneticEffectsEffectsTurbulence EffectsTurbulence Effects

RotationalRotationalEffectsEffects Volume ChangeVolume Change

Induced FlowInduced 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




PHYSICAL MECHANISMS TO BE CONTROLLED DURING SOLIDIFICATIONPHYSICAL 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?

MELTMELT

CRYSTALCRYSTAL

INTERFACE

INTERFACE

InterfacialInterfacialThermodynamicsThermodynamics

CapillarityCapillarity

Buoyancy EffectsBuoyancy EffectsMarangoni Marangoni ConvectionConvection

MicrogravityMicrogravityEffectsEffects

DiffusionDiffusion

MorphologicalMorphologicalInstabilityInstability

ElectromagneticElectromagneticEffectsEffectsTurbulence EffectsTurbulence Effects

RotationalRotationalEffectsEffects Volume ChangeVolume Change

Induced FlowInduced 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




GOVERNING EQUATIONSMomentum

Temperature

On all boundaries

Thermal gradient: g1 on melt side, g2 on solid side

Pulling velocity : vel_pulling

On the side wall

Electric potential

Interface Solid




• 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




• 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




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 :




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




MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONSMATERIAL SOLIDIFICATION PROCESS DESIGN UNDER DIFFUSION CONDITIONS

solid-liquid interfacesolid-liquid interface

MELTMELTSOLIDSOLID

VV

gg

bs bl

ts tl

qos

os

ol

qolI

B(t)B(t)

GASGAS

hothotcoldcoldVelocity Velocity

profileprofile

FREE FREE SURFACESURFACE

Surface Tension GradientSurface Tension Gradient

DESIGN OBJECTIVESDESIGN OBJECTIVESFind the optimal magnetic field such that, in the presence Find the optimal magnetic field such that, in the presence

of coupled thermocapillary, buoyancy, and of coupled thermocapillary, buoyancy, and electromagnetic convection in the melt, a flat solid- liquid electromagnetic convection in the melt, a flat solid- liquid

interface with diffusion dominated growth is achievedinterface 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




INVERSE-DESIGN PROBLEMINVERSE-DESIGN PROBLEM

INVERSE PROBLEM INVERSE PROBLEM STATEMENTSTATEMENT

Find the magnetic field b(t) in Find the magnetic field b(t) in [0, t[0, tmaxmax] such that melt ] such that melt

convection is suppressedconvection is suppressed

solid-liquid interfacesolid-liquid interface

MELTMELTSOLIDSOLID

VV

gg

bs bl

ts tl

qos

os

ol

qol

I

GASGAS

hothotcoldcoldVelocity Velocity

profileprofile

FREE FREE SURFACESURFACE

Surface Tension GradientSurface Tension Gradient

With a guessed magnetic field, With a guessed magnetic field,

solve the following direct solve the following direct

problem for:problem for:

Melt regionMelt region::• Temperature field: Temperature field: TT((x, t; bx, t; b))• Concentration field: Concentration field: cc((x, t; bx, t; b))• Velocity field: Velocity field: vv((x, t; bx, t; b))• Electric potential: Electric potential: (x, t; (x, t; bb))

Solid regionSolid region::

• Temperature field: Temperature field: TTss((x, t; bx, t; b))

Measure of deviation from diffusion based growth

B(t)B(t)




NONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEMNONLINEAR OPTIMIZATION APPROACH TO THE INVERSE SOLIDIFICATION PROBLEM

Continuum Continuum sensitivity problemsensitivity problem

Define the inverse solidification problem as a unconstrained spatio- Define the inverse solidification problem as a unconstrained spatio- temporal optimization problemtemporal optimization problem

Solve the above unconstrained minimization problem using the Solve the above unconstrained minimization problem using the nonlinear Conjugate Gradient Method (CGM)nonlinear Conjugate Gradient Method (CGM)

Needs design gradient informationNeeds design gradient information Needs descent step sizeNeeds descent step size

Continuum Continuum adjoint problemadjoint problem

Find a quasi- solutionFind a quasi- solution: : boo LL22 ([0([0, , ttmaxmax]) ]) such thatsuch thatS S ((boo) ) S S ((b ) ) boo LL22 (( [0[0, , ttmaxmax]) ])




THE CONTINUUM ADJOINT PROBLEMTHE 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

1

2

3

4




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: 0.007

Thermal Rayleigh number: 200000

Solutal Rayleigh number: 10000

Lewis number: 3000

Marangoni number: ~0

Stefan number: 0.12778

Ratio of thermal diffusivites: 1.25975

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 FIELDSDESIGNING TAILORED MAGNETIC FIELDS




CG iterations

Cos

tfun

ctio

nal

0 1 2 3 4

10-4

10-3

10-2

10-1

100

CG iterations

Gra

dien

tof

the

cost

func

tiona

l

0 1 2 3 410-2

10-1

100

Time

Gra

dien

t(T

2 /m)

0 0.025 0.05 0.075 0.131.95

31.96

31.97

31.98

31.99

32

32.01

32.02

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 functionalCost functional

Optimal field






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.




Reference case

Optimal 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 CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH –– RESULTS RESULTS

0.58

0.54

0.51

0.47

0.43

0.390.35

0.310.27

0.230.19

0.12

Solid

0 .04

0.04 0.04

0.04

0.04

0.040.04

0.04

2.52 2.52

2.52

2.52

2.52

2.52

5.015.01

5.01

5.01

5.01 7.49

7.49 7.49

7.49

7.49

7.49

9.97

9.979.97

9.97

9.97

12.46

12.46 12.46

12.46

14.9

4 14.9

4

Solid

0.8

75

0.7500.625

0.563

0.500

0.438

0.375

0.3130.250

0.063Solid

0.9

380

.87

50.8

13

0.7

50

0.6

88

0.625

0.563

0.5000.438

0.375

0.313

0.2

50

0.1

88

0.1

25

0.0

63Solid

1.00409

1.121281.30517

1.48508

1.8568

2.44984Solid

1.77892

1.77892

1.76946

1.79517

1.79517

1.77892

1.7

78

92

Solid

Streamline contours

Isotherms

Isochors

Comparison of streamline contours, isotherms and isochors during the growth




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 CRYSTAL GROWTH – HORIZONTAL BRIDGEMAN GROWTH –– RESULTS 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. 0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

5

6

7




REFERENCE CASE: TAILORED MAGNETIC FIELDS AND ROTATION

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.




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 rotationDESIGN 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




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 OBJECTIVESFind 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

OPTIMIZATION PROBLEM USING TAILORED MAGNETIC FIELDS

Cost Functional:

and




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 ProblemContinuum

sensitivity equationsDesign differentiate

with respect to

Non – linear conjugate gradient method




Momentum

Temperature

Electric potential

Interface

Solid

CONTINUUM SENSITIVITY EQUATIONS




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 propertiesCompare 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.




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

Temperature at x mid-plane

Error less than 0.05 %

X

Y

Z

4.85E-04 1.46E-03 2.43E-03 3.40E-03 4.37E-03 5.34E-03 6.31E-03 7.28E-03

Temperature iso-surfaces

VALIDATION OF THE CONTINUUM SENSITIVITY EQUATIONS




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 andsensitivity problems for all required fields

Set pk = -J’ ({b}0) if (k = 0)else pk = -J’({b}k) + γ pk-1

Set γ = 0, if k = 0; Otherwise

Calculate J({b}k) and J’({b}k) = J({b}k)

Check if

(J({b}k) ≤ εtol

γ

Calculate the optimal step size αk

αk =

Set {b}opt = {b}k

and stop

Update {b}k+1 = {b}k + α pk

Yes

No

{b}opt – final set of design parameters

Minimizes J({b}k) in the search direction pk

Sensitivity matrix M given by





Non-dimensionalized




Hartmann number = 60






Melting temp = 0.0;

Biot num = 10.0;


Melting temp = 0.0;

Time_step = 0.002


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




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




Substantial reduction in curvature of interface.

Thermal gradients more uniform

Iteration 1

Iteration 4






Non-dimensionalized




Hartmann number = 60






Melting temp = 0.0;

Biot num = 10.0;


Melting temp = 0.0;

Time_step = 0.002


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





4 iterations of the Conjugate gradient method

Cost function reduced by two orders of magnitude

Optimal rotation 10.4





Iteration 1

Iteration 4




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.

Documents

Materials Process Design and Control Laboratory CONTROLLING SEMICONDUCTOR GROWTH USING MAGNETIC FIELDS AND ROTATION Baskar Ganapathysubramanian, Nicholas