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A new algorithm for real-time thin film thickness estimation given in situ multiwavelength ellipsometry using an extended Kalman filter

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Ž .
Thin Solid Films 313
314 1998 156
160
A new algorithm for real-time thin ﬁlm thickness estimationgiven in situ multiwavelength ellipsometry using an extendedKalman ﬁlter
C.G. Galarza
a,
, P.P. Khargonekar
a
, N. Layadi
b
, T.L. Vincent
a
,E.A. Rietman
c
, J.T.C. Lee
c
a
The Uni
ersity of Michigan, EECS Department, 1201 Beal A
enue, Rm 4400, Ann Arbor, MI 48109-2122, USA
b
Bell Labs, Lucent Technologies, Orlando, FL 32819, USA
c
Bell Labs, Lucent Technologies, Murray Hill, NJ 07974-0636, USA
Abstract
We present a novel solution to the problem of thickness estimation from in situ ellipsometry measurements. A non-linear
Ž .
dynamic estimator is designed and implemented using the Extended Kalman Filter EKF theory. Major advantages of thisscheme include fast data processing, and robustness to measurement noise and errors in the initial ﬁlm thickness estimates.Since the number of function evaluations is much smaller than in the traditional non-linear least squares approach, EKF iscomputationally very efﬁcient and has great potential for real-time applications. This technique is demonstrated on both
Ž . Ž .
simulated and experimental data gathered during plasma etching of a polycrystalline silicon poly-Si on silicon dioxide SiO
2
Ž .
on silicon Si stack.
1998 Elsevier Science S.A.
Keywords:
In situ ellipsometry; Non-linear estimation; Poly-Si etching
1. Introduction
In situ spectroscopic ellipsometry is potentially a very useful technique for thin ﬁlm thickness estima-tion during a plasma etching. Thus, it is very impor-tant to develop a fast and robust algorithm to processellipsometry signals in real-time to obtain accuratethickness estimates.The traditional approach to this problem has been
Ž .
to solve a non-linear least squares problem NLLSQat each time step. Since the solution at one time stepis obtained independently from the previous estima-tions, the thickness estimates obtained with NLLSQ
Corresponding author. Tel.:
1 313 7636724; fax:
1 3137638041; e-mail: cgalarza@eecs.umich.edu
are very sensitive to measurement noise. In addition,the NLLSQ solution is computationally demandingbecause it requires several function evaluations pertime step. In this paper, we present an alternativesolution that does not suffer from these problems.Our scheme uses a dynamic non-linear system, known
Ž .
as the extended Kalman ﬁlter EKF , that implementsa non-linear estimation technique. This technique hasbeen successfully applied in different ﬁelds. However,it was just recently that these ideas were used to
Ž
.
process optical measurements of thin ﬁlms see 1,2 .
In Ref. 1 an EKF was designed to estimate etch ratefrom reﬂectometry measurements during plasma etch-ing. In this paper, we extend those results to ellip-sometry measurements. Since it is not our intentionto develop new optical models, we have based our
work on a well-established optical model 3 . Without
0040-6090
98
$19.00
1998 Elsevier Science S.A. All rights reserved
Ž .
PII
S0040-6090 97 00803-1
( )C.G. Galarza et al.
Thin Solid Films 313
314 1998 156
160
157
loss of generality, we derive an EKF for a two-layerstack of smooth thin ﬁlms. The results can be general-ized for any number of layers.
2. Thickness estimation using a non-linear ﬁlter
Consider a stack formed by three layers of smooththin ﬁlms illuminated by a monochromatic light beam.Let
be the incident wavelength,
the angle of
0
Ž .
incidence, and
N
N
,
N
,
N
the vector of refrac-
1 2 3
tive indices, where
N
is the refractive index of the
i
i
-th layer. Let
d
and
d
be the thicknesses of the ﬁrst
1 2
and second layer, respectively. We will assume thatthe third layer is semi-inﬁnite. Following the standardnotation, let
be the change in the polarization of
the reﬂected light with respect to the incident light.Then, the magnitudes
and
are deﬁned by the
relation
Ž .
j
Ž .
tan
e
1
The scalar
is a function of
d
,
d
,
and
N
.
1 2
Ž
.
Using a standard model to calculate
see 3 ,
and
can be expressed as functions of
d
,
d
,
,
1 2
and
N
:
Ž .
h d
,
d
,
,
N
1 2
Ž .
2
Ž .
h d
,
d
,
,
N
1 2
During an etching process,
d
is a function of time
1
that depends on the etch rate,
er
. Let
T
be the
s
Ž .
sampling time. We denote by
x k
the value of thesignal
x
evaluated at time
t
kT
, where
k
is the
s
sample number. Then,
Ž . Ž .
d k
1
d k
1 0
T
1 1
s
Ž . Ž .
d k
1
d k
0 1 0
2 2
0 0 1
Ž . Ž .
er k
1
er k
Ž .
w k
d
1
u
0
Ž .
u
0
Ž .
w k
3
d
2
Ž .
u k
er
Ž .
w k
er
which can be compactly written in the form
Ž . Ž . Ž . Ž . Ž .
x
k
1
Fx
k
u
k
w
k
4
t
Here
x
d d er
is the state of the system,
1 2
t
t
u
0 0
u
is the nominal input, and
w
w w w
er
d d er
1 2
is the process noise. The state
x
evolves from the
Ž . Ž .
initial state,
x
0 , to the actual state,
x
k
, describing atrajectory in the state space. While the input
u
de-termines the nominal trajectory, the noise
w
modelsthe small variations of the state trajectory around itsnominal trajectory. We introduce the input signal inthe model as a way to include our a priori knowledgeabout the dynamics of the system, i.e. the available apriori information about the etch rate.
Ž . Ž .
At each time
k
,
k
and
k
, are computed
Ž . Ž . Ž . Ž .
using Eq. 2 with
d k
and
d k
given by Eq. 3 .
1 2
Let
y
be the vector of measured signals. Then, if thereare
L
different incident wavelengths,
Ž .
k
1
1
Ž .
k
1
2
Ž . Ž .
y
k
5
Ž .
k
2
Ž
L
1
.
L
2
L
Ž .
k
L
where
is the measurement noise affecting the
i
-th
i
Ž .
component of
y
. Eq. 5 can be written as
Ž . Ž Ž .. Ž . Ž .
y
k
h x
k
v
k
6
Ž Ž ..
The function
h x
k
maps
d
,
d
,
...,
, and
1 2 1 L
N
, into
, ...,
, and
v
is the measurement noise
1
L
vector.
Ž . Ž . Ž .
The problem is to estimate
d k
,
d k
and
er k
1 2
Ž . Ž .
in an accurate manner given
y
1 ...
y k
1 . Theextended Kalman ﬁlter is an effective technique to
Ž . Ž .
solve this problem. Let
x
k
and
y
k
be the estimates
ˆ ˆ
Ž . Ž .
of
x
k
and
y
k
. Then, the EKF is implemented as
Ž
.
follows see Ref. 4
Ž . Ž . Ž Ž . Ž ..
x
k
1
Fx
k
L y
k
y
k
ˆ ˆ ˆ
k
Ž . Ž .
x
0
x
7
ˆ
0
Ž . Ž Ž ..
y
k
h x
k
ˆ ˆ
Here,
x
is the initial state estimation or initial
0
‘guess’ for
d
,
d
and
er
. The matrix
L
is a time
1 2 k
varying matrix that is recursively computed on-lineusing
Ž .
d
h x
Ž .
8
Ž .
x
x
k
ˆ
d
x
Note that the EKF is a discrete time system driven
Ž Ž . Ž ..
by the output estimation error
y
k
y
k
.
ˆ
3. Analysis of performance
Throughout this analysis, we have used a poly-Si
SiO
Si stack. The incident light has four differ-
2
˚ ˚ ˚
ent wavelength components, 3100 A, 3758 A, 4429 A,
˚
and 6200 A, and the angle of incidence is close to 70
.
( )C.G. Galarza et al.
Thin Solid Films 313
314 1998 156
160
158
˚
Ž .
Fig. 1. Results for
6200 A. Top, estimates dashdot line com-
ˆ ˆ
Ž . Ž . Ž .
pared to noisy measurements solid line : a
and
; b
and
. Bottom, errors between noise-free simulations and estimations:
ˆ
ˆ
Ž . Ž .
c
; d
.
For brevity, we will show only the results for
6200
˚ ˚ ˚
A and
3758 A. The estimates for 4429 A and 3100
˚ ˚
A are consistent with those obtained at 6200 A and
˚
3758 A.First we analyze the performance of the EKF whenthe data are noisy and the algorithm has initial valueerrors. For that purpose we have used simulated data
Ž
to avoid any uncertainty in the model e.g. uncertainty
.
in the refractive indices of the poly-Si layer . Firstly, we obtained the noise free simulations
and
, by
Ž . Ž .
using Eq. 3 and Eq. 5 with
T
1 s,
u
0 and
s
˚ ˚
Ž . Ž .
initial conditions
d
0
4000 A,
d
0
1000 A,
er
1 2
˚
1
65 As . To simulate noisy measurements, we thenadded white noise to
and
. Let
and
be thenoisy simulations. Fig. 1 and Fig. 2 show the results
ˆ ˚ ˆ
Ž . Ž .
when the EKF is initialized with
d
0
4500 A,
d
0
1 2
1
˚ ˚
Ž .
1100 A and
er
0
60 As .Fig. 1 shows the actual output estimation error, i.e.
Fig. 2. State estimates compared to actual states during a simu-
ˆ ˆ
lated etch. Solid line:
d
,
d
and
er
; dashed line:
d
,
d
and
er
.
1 2 1 2
ˆ
Ž . Ž . Ž .
Fig. 3. State estimation errors
d k
d k
using EKF solid line
1 1
Ž . Ž .
and NLLSQ dashdot line for two different noise levels: a set A
Ž . Ž .
noisier ; b set B.
Ž .
the error between the estimated outputs
,
and
Ž
.
the noise free signal
,
. We see that theestimates converge to the actual traces despite themeasurement noise in
and
. The initial outputestimation error shown in Fig. 1 is due to the mis-match in the initial conditions. This error is reducedduring the run while the thickness of the ﬁrst layer
Ž .
converges to the actual thickness see Fig. 2 .To complete this analysis we compare the solutions
Ž .
obtained with the traditional approach NLLSQ andthe EKF technique. In particular, we are interested inthe robustness of the estimation against measurementnoise. For this purpose, we simulate two sets of mea-surements, A and B. The noise level in A is higherthan in B. Fig. 3 shows the results for sets A and B
ˆ ˚
Ž . Ž . Ž .
when initial errors are
d
0
d
0
300 A,
d
0
1 1 2
ˆ ˚ ˆ
Ž .
d
0
100 A. The NLLSQ did not estimate
d
.
2 2
For each set, we have plotted on the same subﬁgure
ˆ
the errors
d
d
obtained with the EKF and with
1 1
the NLLSQ.It is obvious that the NLLSQ algorithm convergesfaster than the EKF. However, the former is verysensitive to measurement noise while the latter ex-hibits a smooth convergence regardless of the noiselevel. This slower convergence is often not a majordisadvantage since we are typically interested in thick-ness estimation towards the end of the etch process. As a measure of computational efﬁciency, we de-termine the average number of function calls per time
ˆ ˆ
Ž . Ž .
step required to obtain
d k
and
d k
. For the
1 2
previous example, NLLSQ made 24.3 function evalua-tions per time step while the EKF needed only 3function evaluations per time step. This makes EKFmuch more suitable for real-time applications.
4. Experimental results
The experimental data were obtained with a UV- visible spectroscopic phase-modulated ellipsometer
( )C.G. Galarza et al.
Thin Solid Films 313
314 1998 156
160
159
ˆ ˆ ˚ ˚
Ž . Ž . Ž . Ž .
Fig. 4. Estimates
and
dashdot line compared to experimental data
and
solid line : a
3758 A; b
6200 A.
made by ISA
Jobin Yvon. The measurements wereperformed during the plasma etching of a poly-Si
SiO
Si wafer. The details of the experiments can
2
be found in Ref. 5 . We initialized the estimation
ˆ ˚ ˆ ˚
Ž . Ž .
algorithm with
d
0
5000 A and
d
0
1000 A.
1 2
These values were rough estimates of the actualthicknesses. The nominal etch rate was considered
˚
1
constant and equal to 80 As . The sampling time was
T
0.63 s.
s
ˆ ˆ
Ž .
Fig. 4 shows the estimated outputs
and
and
Ž .
the measured outputs
and
as functions of time.In Fig. 5, we have plotted the corresponding outputestimation errors.Due to the high absorption coefﬁcient of the poly-Si
˚
for
3758 A,
and
and hence the output
ˆ ˆ
Ž . Ž .
estimation errors
and
are constant
ˆ ˆ
during the ﬁrst 50 s. The offsets in
and
are dueto errors in the optical model that we used to design
˚ ˚
Ž . Ž .
Fig. 5. Output estimation errors corresponding to Fig. 4: a
3758 A; b
6200 A.
( )C.G. Galarza et al.
Thin Solid Films 313
314 1998 156
160
160
the EKF. In particular, it is known that the refractiveindex of poly-Si depends on the processing conditions.Moreover, the surface of the poly-Si is usually roughand the optical model that we used does not accu-
rately model roughness 6 . During the etch, as
d
1
decreases,
d
becomes more readily observable from
2
ˆ
and
and the EKF is able to estimate
d
. Unfor-
2
tunately, if the initial estimated error in
d
is too
2
large, the EKF is not able to obtain a good estimateof
d
before the end of the run. As the ﬁrst layer
2
ˆ
becomes thinner, the estimation error
d
d
ap-
2 2
pears in the output estimation error. This causes the
ˆ ˆ
increase in
and
that we observe in Fig.5.
5. Conclusion
In this paper, we proposed the use of a non-linearestimator known as the extended Kalman ﬁlter tosolve the problem of thickness estimation during anetch process. The performance of the EKF for real-time thickness estimation was studied both throughsimulations and experimental data. It was shown thatthe EKF is able to obtain good thickness estimationin a very short time. This makes the EKF an excellentcandidate for the implementation of in situ real-timeprocessing of ellipsometry data. We have focused ourstudy on the etching of unpatterned wafers only. Weare now working on implementing this technique insitu in real-time on patterned wafers.
Acknowledgements
This work was supported in part by the AFOSR
ARPA MURI Center under grant no.F49620-95-1-0524. We thank Helen Maynard for allher help in this work.
References
1 T. Vincent, P. Khargonekar, F. Terry, Jr., MRS SymposiumProceedings, vol. 406, MRS, Pittsburgh, PA, 1996, p. 87.
2 W. Woo, S. Svoronos, H. Sankur, J. Bajaj, S. Irvine, AIChE J.
Ž .
42 1996 1319.
3 R. Azzam, N. Bashara, Ellipsometry and Polarized Light, NorthHolland, New York, 1989.
4 B. Anderson, J. Moore, Optimal Filtering, Prentice Hall, En-glewood Cliffs, NJ, 1979.
5 H. Maynard, N. Layadi, J.T.C. Lee, J. Vac. Sci. Technol. B 15
Ž .
1997 109.
6 T. Benson, Ph.D. dissertation, The University of Michigan,1996.

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