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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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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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