A convex optimization-based nonlinear filtering algorithm wi

224IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY2003 A Convex Optimization-Based Nonlinear Filtering

Algorithm With Applications to Real-Time

Sensing for Patterned Wafers

Ji-Woong Lee,Student Member,IEEE,and Pramod P.Khargonekar,Fellow,IEEE

Abstract—This paper is concerned with nonlinear filtering under unknown dynamics and high-complexity observations.We propose a convex optimization-based filtering algorithm,and show that the algorithm yields bounded error if the disturbances are small and bounded,and if the observations are redundant.An experimental result is presented to demonstrate that the algorithm is capable of accurate real-time estimation of patterned wafer parameters in a plasma etching process with optical observation. Index Terms—Nonlinear filtering,piecewise linearity,reactive ion etching,redundancy,spectroscopic reflectometry.

I.I NTRODUCTION

T HIS paper has been motivated by our work on in situ real-time metrology of patterned semiconductor wafers undergoing plasma etching.Optical metrology techniques such as spectral reflectometry and spectroscopic ellipsometry offer great promise in monitoring and control of the semiconductor manufacturing processes.In the case of unpatterned wafers, these techniques have been successfully applied for etch rate and depth estimation:for example,recursive etch rate estima-tion based on extended Kalman filtering with random-walk approximation of the dynamics[1].Applications of these methods on periodic patterned structures,however,are very limited[2],[3]due to the inherent electromagnetic diffraction problems that arise in the submicron regime.Recently,it has been shown that one can use vector diffraction theory to obtain a physical model of the optical phenomena leading to high-ac-curacy ex situ measurements of wafer state in the submicron regime[4],[5].Unfortunately,this physics based model turns out to have very high computational complexity and is not suitable for direct use in real-time in situ monitoring.(Indeed, computation of the output for a given wafer state requires running a computer simulation that takes up to a minute on a high-end workstation).

From a system theory perspective,the basic problem in these applications is that of state estimation from noisy ob-

Manuscript received May1,2001;revised February8,2002and May29, 2002.Recommended by Associate Editor G.DeNicolao.This work was sup-ported in part by the AFOSR/ARPA MURI Center under Grant F49620-95-1-0524.

J.-W.Lee is with the Department of Electrical Engineering and Computer Science,the University of Michigan,Ann Arbor,MI48109USA(e-mail:ji-woong@umich.edu).

P.P.Khargonekar is with the Department of Electrical and Computer Engineering,the University of Florida,Gainesville,FL32611USA(e-mail: ppk@ufl.edu).

Digital Object Identifier10.1109/TAC.2002.808467servations.Therefore,one might expect that techniques from nonlinear estimation and filtering such as extended Kalman filtering should be applicable[6]–[8].As is well known,these techniques require that the mathematical models for state dy-namics and output observation be known and be quite accurate. Moreover,for real-time applications,these models need to be of sufficiently low computational complexity so that all the required computations can be done at the desired sampling rate. However,in our problem of patterned wafer state estimation from optical measurements,a physics based dynamics model is not available,and more critically,the existing observation model is computationally very complex.These considerations led us to explore the possibility of developing new filtering schemes to overcome the challenges mentioned above.

In this paper,we propose an online recursive algorithm for state estimation that addresses the aforementioned problem.A particular version of it has been used in[9]to achieve the first re-ported real-time in situ estimation of submicron patterned wafer parameters during a plasma etching process.A key element in our approach is to use state-output pairs obtained via offline vector diffraction model simulation to represent a“sampled”version of the observation model,where explicit reconstruction or approximation of the physical model is not attempted.At each instant of time,the measurement update(or correction)step in-volves solving a convex optimization problem over an evolving constraint set in the observation space to compute the“best”approximation of the model output corresponding to a given measured output.Given a measured output,the best approxima-tion of the corresponding model output is expressed as a convex combination of the outputs in the model data,and then the state estimate is given by the same convex combination of the states in the model data.Systematic errors in the state estimate are par-tially compensated for by another constrained optimization over a compact set in the state–space.Finally,in the time update(or prediction)step,the constraint set in the state–space is appro-priately translated giving rise to the evolution of the constraint set in the observation space.

The proposed algorithm overcomes computational com-plexity of the observation model through its use of a sampled version of the observation model.The time update step for nonlinear filtering is done via a given subsidiary filter—see Section III-C—that approximates the unknown dynamics.A deterministic worst-case analysis shows that the algorithm yields bounded estimation error if the true dynamics is suf-ficiently nice,if the underlying uncertainties are small and bounded,and if the(high-dimensional)observations are redun-

0018-9286/03$17.00©2003IEEE

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM225

dant(see Section V for precise conditions).We include some experimental result to show that the algorithm is capable of accurate real-time estimation of the evolving feature geometry of patterned wafers during a plasma etching process.We also use it to illustrate the utility of analysis results in the design of the estimator.

Our algorithm is analogous in spirit to the standard bounded-error estimation techniques[10]–[16]in that no particular dis-turbance models are assumed except for boundedness.On the other hand,its form is reminiscent of the techniques in on-line optimization and model predictive control[17]–[20],where process performance is predicted and/or optimized using dy-namic constrained-process models.The unique feature of our algorithm is that an observation model with high computational complexity can be used when the model admits redundant ob-servations.What we call redundancy in observation defines the condition under which the high dimensionality of the measured output allows one to use sampled model data without explicit reconstruction of the physical model.

The remainder of this paper is organized as follows.In Section II,we describe the motivating application—the in situ sensing in plasma etching.The problem formulation is given in Section III,and the proposed nonlinear filtering algorithm is described in Section IV.Sections V and VI are devoted to analyzing the algorithm,and experimentally demonstrating its performance,respectively.

II.M OTIV ATING A PPLICATION

At the present time,many semiconductor manufacturing pro-cesses are operated without feedback control,and it has been recognized that the development of real-time feedback control technology holds great potential in the semiconductor manufac-turing industry.In particular,estimation and control of semi-conductor wafer topography is becoming increasingly crucial as pattern dimensions of modern integrated circuits continuously shrink.Our work is directed toward achieving real-time estima-tion of the evolving wafer topography with in situ optical obser-vation;more specifically,real-time extraction of the photoresist grating profile through the two-channel spectroscopic reflec-tometer(2CSR)measurements during the reactive ion etching (RIE)process.

A.In Situ Sensing in RIE Via2CSR

RIE is a critical technology for modern integrated circuit fab-rication at many steps of the manufacturing process.In RIE a plasma is generated to produce ions and other excited species that react with the exposed surfaces of the wafer to cause selec-tive etching of the patterned structure.The key issue is not only that physics based dynamic models for the RIE process are not currently available,but also that many of the crucial etch param-eters,i.e.,the wafer parameters,that define the feature geometry have not been measured in real time.

Since the process should not be disturbed,it is not possible to directly measure the wafer parameters.Spectral reflectometry (SR)or spectroscopic ellipsometry(SE)is one of the most favorable techniques for indirectly determining the

surface Fig.1.Two-channel spectroscopic reflectometer(2CSR).

properties of a sample being processed because of its nonde-structive nature and sensitivity.According to elementary optics, a linearly polarized single-wavelength light beam is elliptically polarized after reflection on a sample surface,so that the electric field of the reflected light has components polarized parallel and perpendicular to the plane of incidence.SR or SE measures this polarization of the reflected light over multiple wavelengths.For example,the2CSR[21],depicted in Fig.1, measures the

reflectances

in the spectral range of370–850nm,

where

.Successful ex situ extraction of wafer parameters in submicron regime has been reported using a vector diffraction theory based2CSR observation model[22]; namely,the rigorous coupled-wave analysis(RCWA)model.

B.RCWA Model for2CSR Measurements on Photoresist Grating

Fig.2shows the scanning electron microscope(SEM)photo of a typical submicron Photoresist grating.The grating profile is approximated to be trapezoidal,so given the grating period the feature geometry is defined by thickness,top width,and wall angle,as shown in Fig.3.An observation model for the2CSR measurements on the trapezoidal photoresist grating has been obtained[22]using vector diffraction theory through the RCW A as developed in[23].

The RCW A slices the groove of a given surface relief grating into thin layers so that the grating profile is approximated by a number of rectangular grating slabs:for each wavelength being considered,a set of space harmonics of the electromagnetic field generated by the incident wave is calculated on each layer,and the boundary conditions are applied at each interface.Suppose that measurements

for

, of the wafer state(thickness,top width,wall angle)to the cor-responding2CSR

measurements

226IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY

2003

Fig.2.SEM photo of a submicron photoresist

grating.

Fig.3.

Trapezoidal periodic grating.

is a function from a subset

of ,

where

and

.The number of layers and the number of terms in

the field space-harmonic expansion for each wavelength deter-mine the precision and computational complexity of the RCWA model.

plexity of RCWA Model

Because of the lack of sufficient knowledge,or because of intractability of the required analysis,in establishing a physics based dynamics model for the RIE process,one may assume that the wafer state evolution is “completely unknown.”That is,if the measured 2CSR output at a given time

is

of

is

sufficiently large,and if the effect of local minima is made small,then will be a good estimate

of

,given a wafer

state

,a

typical nonlinear least squares operation to obtain ,based on the

same

,

from .We have tried empirical approximation

of

is very high,often very lim-ited experimental data are available for satisfactory training.Our experience indicates that this approach is rather sensitive to the presence of systematic modeling errors

in

,and the high dimensionality

of the 2CSR output causes the difficulty in constructing an ap-proximation .On the other hand,the reported ex situ extrac-tion of wafer states has been made possible by using sufficiently large number of wavelengths;that is,the high dimensionality of the output acts on making it a plausible engineering issue to estimate wafer state through optical observation.These are the salient features of the RCW A model,and we need to overcome,or take advantage of,each of them to achieve real-time wafer state estimation.

III.P ROBLEM F ORMULATION

The photoresist grating RIE process with 2CSR observation is taken to be a discrete-time system.

Let

is assumed to

be governed by a state-space model of the following

form:

be a subset

of

,

is known,but of high computational complexity.The wafer

state

,

and

,even if they are independent

of

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM 227

unknown but within a class of sufficiently nice functions (see Section V-B for precise conditions

on

is the mapping representing the 2CSR obser-vation model.

For

and ,the computational complexity of the RCWA model prohibits one from evaluating

either

or in real time.However,one may obtain a sampled version

of

,and

compute

defines a sample

of

is fixed,and

that

.For example,

if

).

In (1),the

term ,

which we denote

by

is the measured output at

time

,

after obtaining a new

measurement

,and the other is the time

update (or prediction)step to

predict

,

where

-algebra generated

by

is a subsidiary filter that

approximates

on a

sample .

The filtering (and one-step prediction)problem based on (1)

can be stated as follows:

given

,produce an es-

timate

of .Any filtering technique tackling this problem

should be capable of dealing with unknown dynamics and com-putationally complex observation.

IV .P ROPOSED A LGORITHM

By taking the approach of nonlinear filtering,one may overcome the difficulties of direct approximation of non-linear least squares described in Section II-C.The filtering problem of Section III-C cannot be addressed with standard filtering approaches,so we begin with proposing a structure of state estimation algorithms.The proposed structure yields a nonlinear filtering scheme based in part on an approximate decomposition of the nonlinear least squares operation.A.Structure of State Estimation Algorithm

Let

.We propose that the filtering algorithm consist of the following steps at each

time

of

of

of

in Sec-tion II-C,where the second step is meant to be performed based

on

were affine and one-to-one,we could

write

(where we

interpret the points

of

.This can be viewed as a special case of the

first two proposed steps

with

given

by

is not affine,and if only a

sample

being

sufficiently nice)to choose a

suitable

that

has

228IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY 2003

B.Filtering Algorithm

Accepting the structure in Section IV-B,one may consider

two separate estimation problems:one is the estimation

of

and the state

prediction

.

These two are coupled through the choice and evaluation of

a

.

Let

,

denoted

by

.

If

f o

r ,a s e

t

F o r m a s a m p l

e

o

f

.

S t e p 1(O b s e r v a t i o n )G e t t h e m e a s u r e m e n

t a s a c o n v e x c o m b i n a t i o n o f t h e p o i n t s i

n

t

o .G o t o S t e p 1.F i g .4i l l u s t r a t e s t h e s t e p s i n v o l v e d i n A l g o r i t h m 1a t t i m e a n

d

,a n

d

,o r a v a i l a b l e a t t h e o u t s e t a s i n A l g o r i t h m 2.S t e p 2d e f i n e s

t h e e v o l u t i o n o f t h e c o n s t r a i n t s e t s o f t h e t w o o p t i m i z a t i o n s t e p s i n v o l v e d i n A l g o r i t h m 1—n a m e l y ,S t e p 3a n d S t e p 5—b y u p -

d a t i n g t h

e s u b s e t =

f d ;d

g .(b)^y

h (d h (d h (d h (^y d d

d =^f ),D ;d g

.,so

that

in the polyhedron [Fig.4(b)].Simi-

larly,Step 5performs a constrained optimization in the evolving

set

of ,which is the

set

,so that it is used to

update

covered

by [Fig.4(d)].

Algorithm 1looks similar to the recursive state-bounding al-

gorithms in the literature on bounded-error estimation.Since the

sets

are random elements deter-mined by the observation sequence,Algorithm 1allows a great

degree of freedom in choosing the set values it uses.Although

one can choose to keep track of correlations between the state

variables by utilizing structured set approximations that par-

allel those in bounded-error estimation techniques,actual error

bounds are not required to be used in the algorithm.On the other

hand,while most available results in bounded-error estimation

concern linear systems,our work focuses on a class of nonlinear

systems,and Algorithm 1exploits a special structure of the sys-

tems within the class (See Section V for details).

A particular example of Algorithm 1is the following,which is what we have implemented in the lab to produce the results in Section VI.Algorithm 2:Perform Algorithm 1

with for some box centered at the origin,

and

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM 229

A deeper insight to the properties of the proposed algorithm can be obtained by looking into Algorithm 2.Notice that the state

prediction

because for

all

when the system

dynamics

model

is affine,one-to-one,and given

by

for

each

is the output disturbance at

time

.The distributions

of

,

be a piecewise linear continuous

function given

by

and

has a linear left

inverse ,

where

,

let

be the closed intervals of the same

length

,where

denotes the interior

of

,

,

,

and

is

are linear functions whose

matrix representations

are

is contained

in

.

Define .

Let

,

,

and

in Algorithm 1be determined by the fol-

lowing:

and

are determined without refer-ence

to

.

From the existence of an affine left

inverse

and for

each

.

A numerical simulation result is shown in Fig.5.The true

state trajectory marked by x’s is generated

with

,,

each

,and

each -ball cen-tered at the origin

of

.The state estimates (a)

230IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY

2003

Fig.5.Simple example.(a)Algorithm 1with (2).(b)Algorithm 1with a

modification of (2)so that ^x

is the center of X )for each k .and (b)are obtained

with

.(Note that,as long

as

,

for some approximation

operation

of

is nonempty for

each

such

that

satisfying the following:

whenever

,

with

has full rank.

Then

,such

that

,is understood,instead of fixing a

particular

of the output space is sufficiently large relative to the

dimension

is redundant,or that we have redundant observations,with “re-dundancy

number”

has a (sufficiently large)

neighborhood

has an affine left inverse.

Lemma 1:Suppose that Assumption 1holds.

If

and a

vector

to

for

for any convex

combination

.

According to Lemma 1,

if

of

’s,there exists an affine left

inverse

to

with

,then we

have

,without explicit knowledge

about .

For example,consider the case

of

,depicted in Fig.6,which we hope will give the reader some intuition as to why Lemma 1holds true.

Let

;

let

,.

Then

.

Since

with ,we may

take

and

,it follows

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM231

that

,

are all bounded in the following sense.

Assumption2:

a)For

each is Lipschitz

on

,there exists a compact

set

such

that

are satisfied by all the realizations

of,

and

,and Assumption2(b)states that for

each

modulo a certain systematic

error.Recall

that.

Lemma2:Under Assumption2,Algorithm1yields the fol-

lowing.

a)

If,

then

,

then

for

to be close

to

such

that

),

and such

that

c)For

each

and

d)It holds

that

formed

by

for

some

of the

form

.If we

define

and for

any

,are small

and close to each other.Assumptions3(b)and3(c)say that for

each

;that

is,

:it is satisfied if for

example,

,the set of

vertices of the

box

are all boxes.

232IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY2003 We have the following result.

Theorem1:If Assumptions1–3hold,then we

have

(The

constants

form a regular grid on a compact

set

with

spacing,so that

the grid interval for

the

boxes

;

the

is

is a vertex

of

is defined

by

are Lipschitz

on

such

that and

all.

d)There exist compact

sets

such

that

;

th side-length of the

box,denoted

by

for

m grating period(Fig.2).The initial feature di-

mension was around

0.78-

wall angle.The grating in the photoresist layer was

on top of a thin

31.7-

flow,and

15-W RF power.

The grating profile was assumed to be trapezoidal,and

the2CSR measurements for the46

wavelengths

and.We

let the

set

deg),where the spacing of the grid

was

nm.The RCW A simulation

on

.

Fig.7.Real-time wafer state estimates produced by Algorithm2using boxes

W)=3l

;l.

A r e a l-t i m e

i n s i t u w a f e r s t a t e s e n s i n g s y s t e m w i t h A l g o r i t h m

2w a s i m p l e m e n t e d i

n,w h i c h w a s e x e c u t e d s i-

m u l t a n e o u s l y w i t

h

[22].

F i g.7s h o w s a t y p i c a l r e s u l t,w h i c h i s o b t a i n e d i n r e a l

u s i n g A l g o r i t h m2d u r i n g a n R I E p r o c e s s o f a p h o t o r e s i s t g

f o r a b o u t160s.F o r r e f e r e n c e,m a r k e d b y x’s a r e t h e o

n o n l i n e a r l e a s t s q u a r e s f i t s a t t i m e

s

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM

233Fig.8.Deviation of the real-time wafer state estimates produced by Algorithm

2from the offline nonlinear least squares fits.

VII.C ONCLUSION

We have proposed an innovative convex optimization based

filtering algorithm for a class of nonlinear systems without

physics based dynamics models and with a computationally

complex high-dimensional observation model.The algorithm

utilizes a sampled version of the observation model,which is

obtained via offline model simulation,to perform constrained

optimization steps in evolving constraint sets.It has been

shown that the estimation error is bounded if the underlying

disturbances and systematic errors are bounded and small,and

if the observation model leads to redundant measurements.

The algorithm has allowed us to achieve the initial success

in addressing the problem of real-time estimation of patterned

wafer state through optical metrology,which has been an open

research area.This is owing to the nature of the proposed al-

gorithm that does not suffer from the complexity of the optical

observation model.The generality of Algorithm 1leaves room

for further development of the algorithm:for example,in future

work,the random elements used in the algorithm can be made

to perform structured set approximations that resemble those in

bounded-error estimation techniques.Another possible applica-

tion of the algorithm is nonlinear filtering in power systems with

redundant meter readings.

A PPENDIX

A.Proof of Lemma 1

Without loss of generality,assume that the

set

consists

of

by

and ,

so

for is connected,there exists a finite

sequence such that the

sets

and are identical,and such

that it follows

that ,we deduce

that

Put ,

so Furthermore,given a convex

combination

,such

that ,so we

have

w h e r

e i s c o n v e x ,w e

h a v

e

f r o m w h i c h p a r t (a )f o l l o w s.P a r t (b )f o l l o w s f r o

m

234IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.2,FEBRUARY2003

and

there exists

a

,we

have

.

From Assumption3(b)and Lemma1,it follows that there is an

affine left

inverse

to

[where

such that(4)holds.Then

by Lemma2(c),we

have

,with As-

sumption3(c),we verify

that

with.

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Ji-Woong Lee(S’98)received the B.S.degree in

electronic engineering from Sogang University,

Seoul,Korea,the M.S.degree in electrical engi-

neering from the University of Maryland,College

Park,and the M.S.degree in mathematics and the

Ph.D.degree in electrical engineering from the

University of Michigan,Ann Arbor,in1990,1996,

and2002,respectively.

He is currently a Research Assistant at the Univer-

sity of Michigan,Ann Arbor.His research interests

are in nonlinear,stochastic,and decentralized sys-

tems,and statistical learning.

LEE AND KHARGONEKAR:CONVEX OPTIMIZATION-BASED NONLINEAR FILTERING ALGORITHM235 Pramod P.Khargonekar(S’81–M’81–SM’90–

F’93)received the B.Tech.degree in electrical

engineering from the Indian Institute of Technology,

Bombay,India,and the M.S.degree in mathematics

and the Ph.D.degree in electrical engineering from

the University of Florida,Gainesville,in1977,1980,

and1981,respectively.

After holding faculty positions at the University of

Florida and University of Minnesota,Minneapolis,

he joined the University of Michigan,Ann Arbor,in

1989where he last held the positions of Claude E.

Shannon Professor of Engineering Science and Chairman and Professor of Elec-

trical Engineering and Computer Science.In July2001,he rejoined the Univer-

sity of Florida as Dean of Engineering,Associate Vice President of the Engi-

neering and Industrial Experiment Station(EIES),and Eckis Professor of Elec-

trical and Computer Engineering.His current research interests are in learning

and intelligent systems,control of semiconductor manufacturing systems,logic

control of manufacturing systems,and control of color xeroxgraphy.He has

taught a wide range of courses in control systems.

Dr.Khargonekar was a recipient of the National Science Foundation Pres-

idential Young Investigator Award(1985),George Taylor Award for Research

from University of Minnesota(1987),the American Automatic Control

Council’s(AACC)Donald Eckman Award(1989),the George Axelby Best

Paper Award(1990),the IEEE W.R.G.Baker Prize Paper Award(1991),the

Japan Society for Promotion of Science Fellowship(1992),the Hugo Schuck

ACC Best Paper Award(1993),and a Distinguished Alumnus Award from the

Indian Institute of Technology,Bombay(1997).At the University of Michigan,

he received a Teaching Excellence Award from the Electrical Engineering and

Computer Science Department in1991,a Research Excellence Award from the

College of Engineering in1994,and the Arthur F.Thurnau Professorship from

1995to1998.He served as the Vice-Chair for Invited Sessions for the1992

American Automatic Control Conference.He was an Associate Editor of the

IEEE T RANSACTIONS ON A UTOMATIC C ONTROL,the SIAM Journal on Control

and Optimization,Systems and Control Letters,Mathematics of Control,

Signals,and Systems,and the International Journal of Robust and Nonlinear

Control.He is currently an Associate Editor of Mathematical Problems in

Engineering.

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