control of multi-scale process systems panagiotis...

CONTROL OF MULTI-SCALE PROCESS SYSTEMS

Panagiotis D. Christofides

Department of Chemical and Biomolecular Engineering

Department of Electrical Engineering

University of California, Los Angeles

Nanomanufacturing Workshop

February 11, 2008

Funded by NSF

MODEL-BASED APPROACH TO CONTROLLER DESIGN

• Selection of inputs/outputs - Feedback control loop

ProcessControllerSet point Input Disturbances OutputController synthesis based on a process model

• Model construction: First-principles / System identification

¦ Possibility of improved closed-loop performance. Model accounts for inherent process characteristics (e.g., nonlinearity,

spatial variations, multiscale behavior)

¦ Characterization of limitations on achievable closed-loop stability,performance and robustness

LUMPED CHEMICAL PROCESSES

• Example: continuous stirred tank reactor / (CA, T : spatially-homogeneous)

CA, T

Cooling water

Tj

CA0 , T0 , F

CA , T , F

• Models: Systems of nonlinear ordinary differential equations (ODEs)

dx

dt= f(x) + g(x)u

y = h(x)

• Approaches for nonlinear controller design

¦ Geometric control

¦ Lyapunov-based control

¦ Model predictive control

THIN FILM GROWTH: A MULTISCALE PROCESS

Desorption

Gas phase

MigrationAdsorptionSurface

• Large disparity of time and length scales of phenomena occurring in gasphase and surface:¦ The assumption of continuum is not valid on the surface.

¦ Computationally impossible to model the whole system from amolecular point of view.

• Solution to bridge the macroscopic and microscopic scales:¦ Model the continuous gas phase by PDEs.

¦ Model the configuration of the surface by kinetic Monte-Carlo andstochastic PDEs.

¦ Incorporate the results of microscopic computations to PDEs viaboundary conditions.

CONTROL OF FILM SPATIAL UNIFORMITYRapid thermal CVD

Positioning ArmRetracting and Rotating

ReactantGasesLamp Bank AQuartz Process ChamberLamp Bank BLamp Bank CSteel Loading Chamber

Reduce film spatial non-uniformity

Plasma enhanced CVD

Substrate

Influent Gas Stream

Plasma

Showerhead

Effluent Gas Stream

Electrodes


• Energy balance to model wafer temperature ( nonlinear parabolic PDE ):

∂

∂t

(Cpw

(T )T)

= c01r

∂

∂r

(κ(T )r

∂T

∂r

)+ f(T, r)

CONTROL OF FILM SPATIAL UNIFORMITYRapid thermal CVD




Plasma enhanced CVD

Substrate

Influent Gas Stream

Plasma

Showerhead

Effluent Gas Stream

Electrodes


• Energy balance to model wafer temperature (nonlinear parabolic PDE):

∂

∂t

(Cpw

(T )T)

= c01r

∂

∂r

(κ(T )r

∂T

∂r

)+ f(T, r)

CONTROL OF THIN FILM ROUGHNESS

vapor phase

migration

adsorption

substrate

vapor phase molecule

surface molecule

bulk molecule

desorption

• Deposition processes: molecule adsorption, migration and desorption

• Control problem: regulate the thin film surface roughness to a desired level

• First-principles models of surface height evolution: kinetic latticeMonte-Carlo (discrete) / stochastic PDEs (continuous approximation):

∂h

∂t= c0(W,T ) + c1(W,T )(

∂2h

∂x2+

∂2h

∂y2) + ξ(x, y, t)

CONTROL OF THIN FILM COMPOSITION

• PECVD of ZrO2 in an electron cyclotron resonance (ECR) reactor

¦ Metalorganic precursors are used

• Real-time carbon content estimator based on optical emission spectroscopy(OES) measurements

• Feedback control of the carbon content of the ZrO2 film

FEEDBACK CONTROL OF WAFER TEMPERATURE PROFILEControl of a parabolic PDE



TemperatureMeasurements

• Energy balance to model wafer temperature (nonlinear parabolic PDE):∂

∂t

(Cpw(T )T

)= c0

1r

∂

∂r

(κ(T )r

∂T

∂r

)+ f(T, r, u)

• Feedback control of parabolic PDEs (Christofides, Birkhauser, 2001)

¦ Derivation of low-dimensional ODE models using singular perturbationsand approximate inertial manifolds

¦ Feedback controller design using methods for ODE systems

¦ Characterization of stability and performance of PDE system

RAPID THERMAL CHEMICAL VAPOR DEPOSITION

Closed-loop simulation results / (Baker and Christofides, IJC, 2000)

Spatiotemporal wafer temperature profile under nonlinear low-order control

0

0.5

1

0510152025303540

300500700900

11001300

rt (s)

T (K)

Final film thickness (t = 40 sec) under nonlinear low-order control

0.47

0.48

0.49

0.5

0.51

0.52

0 0.2 0.4 0.6 0.8 1

Dep

ositi

on (m

icro

met

ers)

r

CONTROL OF NONLINEAR DISTRIBUTED SYSTEMS(Christofides, Birkhauser, 2001; Kluwer Academic, 2002)

Control of NonlinearDPSTransport/Reaction ProcessesFluid Dynamic SystemsParticulate ProcessesUncertaintyTime-delaysConstraintsOptimal Actuator/ Sensor PlacementIntegro-differential Equations

Parabolic/Hyperbolic PDEsHigher-order PDEs/Navier-Stokes EquationsSYSTEMS APPLICATIONS

PRACTICAL CONTROL ISSUES .

MODELING OF SURFACE ROUGHNESS

vapor phase

migration

adsorption

substrate


surface molecule

bulk molecule

desorption

• Deposition processes: molecule adsorption, migration and desorption

• Surface micro-processes are assumed to be Poisson processes.

• Master equation: describes the time evolution of the probability that thesurface is in configuration α at time t

dP (α, t)dt

=∑

β

P (β, t)Wαβ −∑

β

P (α, t)Wβα

Computationally intractable approach to modeling


• Kinetic Monte-Carlo (kMC) simulation: provides unbiased realizations of astochastic process described by the Master equation

¦ Both the master equation and the Monte-Carlo algorithm can be derivedusing the same set of assumptions (Gillespie, J. Comput. Phys., 1976).

• Kinetic Monte-Carlo model for film growth (Vlachos, AIChE J., 1997):¦ First-nearest-neighbor interactions only.

¦ Solid-on-solid approximation of a simple cubic lattice.

¦ Periodic boundary conditions.

• Rates of adsorption, desorption, and migration:

ra =s0P

2a√

2πmkTCtot

rd(n) =ν0

2aexp(−nE

kT)

rm(n) =ν0A

2aexp(−nE

kT)

• Life time of an MC event is determined by a random number and the totalrate.


• Kinetic Monte-Carlo (kMC) simulation: provides unbiased realizations of astochastic process described by the Master equation

¦ Both the master equation and the Monte-Carlo algorithm can be derivedusing the same set of assumptions (Gillespie, J. Comput. Phys., 1976).

• Kinetic Monte-Carlo model for film growth (Vlachos, AIChE J., 1997):¦ First-nearest-neighbor interactions only.

¦ Solid-on-solid approximation of a simple cubic lattice.

¦ Periodic boundary conditions.

• Rates of adsorption, desorption, and migration:

ra =s0P

2a√

2πmkTCtot

rd(n) =ν0

2aexp(−nE

kT)

rm(n) =ν0A

2aexp(−nE

kT)

n =0

n =1 n =2 n =4

n =3

---- Bottom layer

---- Top layer

• Life time of an MC event is determined by a random number and the totalrate.

SIMULATION OF SURFACE ROUGHNESS

• Thin film microstructure and surface micro-processes.

¦ Adsorption events roughen the surface.

¦ Migration and desorption smoothen the surface.

• Effect of substrate temperature on surface roughness.

¦ High temperature reduces surface roughness by increasing the rates ofdesorption and migration.

¦ Left figure: configuration of film surface at T=550K.

¦ Right figure: configuration of film surface at T=700K.

CONTROL OF MICROSCOPIC PROCESSES(Christofides and co-workers)

• Control of microscopic properties - surface roughness.

¦ Control of surface roughness using kinetic Monte-Carlo models (AIChEJ., 2003; CES, 2003; CACE, 2004).

¦ Covariance control of surface roughness using stochastic PDEsconstructed directly from microscopic rules (AIChE J., 2005; CACE,2005).

¦ Application to processes described by the Edwards–Wilkinson andKuramoto–Sivashinsky equations.

• Construction of stochastic PDEs using kMC simulation results (ACC,2005; IECR, 2005).

¦ Predictive control of surface roughness subject to state and controlconstraints.

¦ Application to one- and two-dimensional deposition processes.

Application to control and model construction of other microscopic properties /processes.

PROCESS DESCRIPTION

vapor phase

migration

adsorption

substrate


surface molecule

bulk molecule

desorption

• Simple cubic lattice.

• Homogenous growth: indistinguishable molecules.

• Surface microprocesses: adsorption, migration and desorption.

• Surface molecule interaction: first nearest-neighbors only.

• Multiscale modeling approach can be used to couple the surfacemicrostructure model with the gas phase model.

SURFACE MICROSTRUCTURE MODEL (sPDE)(Ni & Christofides, IECR, 2005)

• Surface fluctuation model of the adsorption-migration-desorption process.

∂h

∂t= W

1− kw

W awe−kBT

Ew

+

kc

k2maxW ace

−kBT

Ec

52 h + ξ(x, y, t)

5h(0, y, t) = 5h(π, y, t), h(0, y, t) = h(π, y, t),

5h(x, 0, t) = 5h(x, π, t), h(x, 0, t) = h(x, π, t),

h(x, y, 0) = h0(x, y)

〈ξ(x, y, t)ξ∗(x′, y′, t′)〉 =π2

k2max

W

[1 +

e(at + ktW )T

eav + kvW

]δ(x− x′)δ(y − y′)δ(t− t′)

W : adsorption rate, T : substrate temperature,kw , kc , kt , kv , aw , ac , at , av , Ew and Ec : kinetic parametersdepend on the frequency constants and energy barriers associated with themicroscopic processes.

OPEN-LOOP SIMULATION RESULTS: kMC vs sPDE

• Surface snapshots generated by kMC (left) and sPDE (right) simulation ofa thin film deposited at T = 610 K and W = 0.5 1/s for 200 s .

• Surface snapshots generated by kMC (left) and sPDE (right) simulation ofa thin film deposited at T = 710 K and W = 0.5 1/s for 200 s .

PREDICTIVE CONTROL OF THIN FILM GROWTH• Thin film thickness (average surface height)/surface roughness

(root-mean-square of surface height)

h =

kmax∑

kx,ky=0

h(kxL, kyL)

k2max

r =

√√√√√√√

kmax∑

kx,ky=0

[h(kxL, kyL)− h]2

k2max

• Objective function

minW (tK+1),T (tK+1)

J(tK) = qh(hset − 〈hfinal(tK)〉)2 + qr(r2set − 〈rfinal(tK)〉2)2

• State and control constraints

¦ Substrate temperature constraint: Tmin ≤ T (t) ≤ Tmax

¦ Adsorption rate constraint: Wmin ≤ W (t) ≤ Wmax

¦ Nonnegative growth rate: h(tK+1) > h(tK)

• Predictive controller is design based on a low-order ODE model

CLOSED-LOOP SIMULATIONS

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

h closed-loop h open-loop

W (s

-1)

h (M

L)

t (s)

W

h

W closed-loop

0 20 40 60 80 100 120 140 160 180 2000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

500

520

540

560

580

600

620

640

660

T (K

)

r closed-loop r open-loop

r (M

L)

t (s)

T

r

T closed-loop

• Initial deposition conditions: W = 1.0 1/s and T = 610 K .

• Deposition length: tfinal = 200 s .

• Final thickness set-point: hset = 100 ML .

• Final surface roughness set-point: rset = 1.5 ML .

THIN FILM VARIANCE REDUCTION VIA FEEDBACKCONTROL

1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.700

2

4

6

8

10

12

open-loop3 =10%

coun

ts

roughness (ML)

1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.800

5

10

15

20

25

30 closed-loop3 =3%

coun

ts

• The variance of thin film properties is unavoidable due to the stochasticnature of the thin film growth process.

• Feedback control is able to compensate for the stochasticity of the thin filmgrowth process and reduce the variance among the thin films.

REAL-TIME CARBON CONTENT CONTROL IN PECVD(Ni et al., ACC, 2003; IEEE TSM, 2004)

• PECVD ZrO2 in an electron cyclotron resonance (ECR) reactor

¦ Metalorganic precursors are used

• Real-time carbon content estimator based on optical emission spectroscopy(OES) measurements

• Feedback control of the carbon content of the ZrO2 film

¦ Control the carbon content in the film by manipulating the mass flowrate of O2 / Controller utilizes real-time carbon content estimates

REAL-TIME CARBON CONTENT ESTIMATION USING OES

• Real-time measurements of optical emission intensity ratio of C2 and O

from OES

• Correlation of carbon content with optical emission intensity ratio of C2

and O obtained from XPS measurements (Cho et al., 2001)

0 2 4 6 8 10 120

20

40

60

80

100R=0.95

Carb

on co

nten

t in t

he fi

lm (%

)

IC

2

(516.52 nm)

/ IO (777.42 nm)

Ambient Contamination

• Real-time carbon content estimation model:

N(k) =4.69

k − k0γ(k) + N(k − 1)

k − k0 − 1k − k0

k > k0

EXPERIMENTAL RESULTS OF CLOSED-LOOP SYSTEM

0 200 400 600 800 1000 12000

1

2

3

4

5 Closed-loop Open-loop

XC (

%)

time (sec)1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5 Carbon Concentration

Actu

al C

arb

on

Co

nce

ntr

atio

n (

%)

Carbon Concentration Setpoint (%)

O / Z

r Ra

tio

O/Zr Ratio

• Film carbon content is controlled at the desired values

(verification via XPS) - Desired O/Zr ratio is achieved

• Film carbon content is significantly reduced under feedback control

OPTIMIZATION WITH MULTISCALE OBJECTIVES

Optimization Problem

Objective function:

min∫Ω

G(x, x, d)dz

Equality constraints:

0 = A(x) + f(x, d)xm(ti) = Π(xm(ti−1), δt)

• Discretize PDEs in space and time using finite differences (FD)

¦ Results in large NLPs, specialized algorithms required

• Microscopic model

¦ Unavailable in closed form, computationally expensive

Computationally-efficient solution (Armaou and co-workers)

• Spatial discretization of PDEs using Galerkins method

¦ Eigenfunctions computed using Proper Orthogonal Decomposition(POD)

¦ Considerably small NLP compared to FD

• In situ adaptive tabulation for the microscopic integrator

¦ Speeds-up the computation by efficient tabulation and interpolation

CONTROL OF MICROSTRUCTURAL DEFECTS

• Low pressure CVD reactor and thin film structure formation

αA B C

A B C

Pressure Sensor3-zone furnace Wafer PumpQuartz tubeGas inletLoad door

CONSTRUCTION OF STOCHASTIC PDEs

• One-dimensional nonlinear stochastic PDE describing theevolution of the thin film density

∂ρ

∂t= c0ρ + c1

∂ρ

∂x+ c2

∂2ρ

∂x2+ · · ·+ cw

∂wρ

∂xw+ c + f(ρ, x, t) + ξ(x, t)

• Periodic boundary conditions

∂jρ

∂xj(0, t) =

∂jρ

∂xj(π, t), j = 0, · · · , w − 1

• Initial conditions

ρ(x, 0) = ρ0(x)

• ξ(x, t) is a Gaussian noise with zero mean and covariance

〈ξ(x, t)ξ(x′, t′)〉 = ς2δ(x− x′)δ(t− t′)

CONTROL ALGORITHM FOR POROSITY REGULATIONUSING STOCHASTIC PDEs

• Model predictive control (MPC) formulations for stochasticPDE systems

minu(·)∈U

〈∫ t+T

tL(ρ(τ), u(τ))dτ + F (ρ(t + T ))〉

s.t.∂ρ

∂t= c0ρ + c1

∂ρ

∂x+ c2

∂2ρ

∂x2+ · · ·+ cw

∂wρ

∂xw+ f(ρ, x, t) + ξ(x, t)

P (χmin ≤ ρ(τ) ≤ χmax) ≥ pmin, τ ∈ [t, t + T ],

Model Predictive Controllerset

min <J(t, , u)> LPCVDProcess

ODE State Evaluation

(Fourier Transform )

u(t) (x, t)zn(t)

set

u

SUMMARY

• Control of macroscopic properties (film spatial uniformity) usingdistributed parameter system theory.

• Control and optimization of microscopic properties - surface roughness(New book with Birkhauser).

¦ Control of surface roughness using kinetic Monte-Carlo models.

¦ Covariance control of surface roughness using stochastic PDEsconstructed directly from microscopic rules.

¦ Application to processes described by the Edwards–Wilkinson andKuramoto–Sivashinsky equations.

¦ Construction of stochastic PDEs using kMC simulation results.

¦ Predictive control of surface roughness subject to state and controlconstraints.

¦ Application to one- and two-dimensional deposition processes.

¦ Optimization of thin film growth with multiscale control objectives.

• Applications to experimental system.

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