Nonlinear Dynamics Modeling of Correlated Functional ... - CiteSeerX

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IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 22, NO. 1, FEBRUARY 2009

Nonlinear Dynamics Modeling of Correlated Functional Process Variables for Condition Monitoring in Chemical–Mechanical Planarization Hui Wang, Xi Zhang, Ashok Kumar, and Qiang Huang

Abstract—This paper aims to investigate correlation mechanism among functional process variables (FPVs) for condition monitoring in chemical–mechanical planarization (CMP). During wafer polishing, critical process variables such as coefficient of friction and pad temperature vary with time and present in the shape of functional curves. Our previous work has demonstrated that correlation patterns among these FPVs could be related to polishing conditions. Since correlation is affected by both amplitude fluctuations and phase variability in FPVs, further study of timing correlation of FPVs measured in different units could bring more insight into the physical interactions and thereby enhance CMP condition monitoring. Existing research on FPVs in CMP mainly focuses on individual effects of FPVs and statistical correlations through experimental and theoretical analyses. In this paper, we intend to specifically reveal the timing correlation patterns in CMP. Using nonlinear dynamics, we first established a dynamic phase model to define the strength and patterns of FPV interaction. By monitoring the extracted patterns, we then developed a novel method of detecting CMP condition change and demonstrated the approach via a CMP experiment. The results show that the proposed method has a promising application in identifying the process changes that may not be easily detected otherwise. Index Terms—Correlation, multivariable systems, phase synchronization, process monitoring, signal processing.

I. INTRODUCTION

C

HEMICAL–MECHANICAL planarization (CMP) is widely used for wafer surface polishing in semiconductor manufacturing. When observing CMP process variables to monitor polishing quality online, variables such as coefficient of friction (COF) and polishing pad temperature could vary with time and these functional process variables (FPVs) contain rich information regarding process conditions. For instance, the COF between the wafer and the pad provides information regarding the tribological condition at the interface. An abrupt

Manuscript received January 06, 2008; revised September 25, 2008. Current version published February 04, 2009. This work was supported by the National Science Foundation under Grant CMMI-0600066. H. Wang is with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109 USA. X. Zhang and Q. Huang are with the Department of Industrial Management and Systems Engineering, University of South Florida, Tampa, FL 33620 USA (e-mail: [email protected]). A. Kumar is with the Department of Mechanical Engineering, University of South Florida, Tampa, FL 33620 USA. Digital Object Identifier 10.1109/TSM.2008.2011180

and large variation in COF could be a real-time indication of pad failure, large particles on the pads, or underlying barrier layer exposure on the wafer. Highly correlated to COF, pad temperature is another important FPV that depicts heat variations generated through friction and chemical reaction. Simultaneously analyzing these FPVs and their correlation patterns might bring additional insights into process condition changes and new opportunities for process improvement. Improved process performance could impact a large revenue stream for wafer fabrication [1]. In response to the functional nature of certain key CMP process variables, research works have been conducted to experimentally and theoretically investigate FPVs for understanding the material removal and failure mechanisms. For instance, Kim et al. [2] experimentally investigated the changes of COF and pad temperature over time. Experimental analyses were conducted to investigate the impact of heat generation on copper CMP defects such as dishing, erosion, and metal loss [3]. Chan et al. [4] employed a noncontact infrared camera to capture the temperature profile over time for a proper selection of non-Preston copper slurry in situ. By observing the patterns of time-varying temperature profile, Wang et al. [5] developed an end-point detection method for a tungsten CMP process using noncontact thermal sensors. Parallel to experimental studies, kinematic modeling provides an explanation of behaviors of FPVs. By studying the mechanism of energy flow, White et al. [6] established a dynamic model to predict the transient and steady-state heat transfer via pad and slurry. Using finite-element analysis, several studies [7]–[9] modeled thermal or mechanical behavior as functions of time or pad radius for process condition prediction. The temperature rise was also estimated through kinematic analysis in [10] to monitor pad life and predict removal uniformity. In addition, Hocheng et al. [11] established a statistical regression model of pad temperature rise to detect the end point in situ. However, process changes in CMP may not be easily detected without collectively studying these FPVs. For example, our previous study has shown that slurry condition change was indicated by the correlation pattern between temperature and COF [12]. But process change could not be identified by inspecting the two process variables separately. Therefore, process monitoring and improvement may require a thorough understanding of the interrelationship between mechanical and chemical variables in functional form. Previous research on the interrelationship involves statistical resemblance (correlation) [12], [13] and

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WANG et al.: NONLINEAR DYNAMICS MODELING OF CORRELATED FUNCTIONAL PROCESS VARIABLES

empirical interdependence or statistical interaction among multiple process variables on polishing quality [14]–[16]. Correlation was also employed to predict the uneasily measurable process variables by comparing its statistical correlation with FPVs that can be directly recorded through experiments [13]. However, most studies focused on general correlation analysis without differentiating amplitude and timing correlations. When two FPVs are measured in different physical units, it is a known fact that the statistical correlation should be carefully interpreted [17], [18]. It should also be pointed out that statistical correlation is a pure data-driven approach to analyze the interrelationship among multiple FPVs. There is an essential need to uncover the physical influence among multiple FPVs (as we call it physical interaction or interaction). Nevertheless, such interaction among dynamic FPVs has been studied far less, especially for improved quality control. This paper aims to address this issue by conducting nonlinear dynamics modeling of functional COF-temperature interaction and proposing a new method of detecting abnormal condition changes. Section II develops methods of nonlinear dynamics modeling of physical interaction. In Section III, a model-based process condition change detection/diagnosis method is then proposed by employing statistical process control tools. Section IV applies the methods to a CMP experimental data and discusses the results. Conclusions are given in Section V.

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Fig. 1. Strong cyclic pattern in process variables.

is the phase variable (a function of time ), is the where base angular frequency (frequency component with the highest is the white noise of the th oscillatory signals power), and . The phase variables can be obtained by constructing an analytical signal obtained from Hilbert transform as follows (for , the phase is a simple sine signal ): defined as

II. NONLINEAR DYNAMICS MODELING OF FPV TIMING CORRELATION Several approaches have been reported in the literature to model correlations among nonstationary continuous signals. The most commonly used method is the cross-correlogram which measures the cross-covariance of paired FPVs [19], [20]. This time-domain method is powerful, but if not used carefully can lead to spurious detection because of artifactual sharp peaks in the signals. Coherence and cross-spectrum methods aim to analyze the correlation of paired signals in the frequency domain and are most commonly used with continuous signals [21]. The correlation analysis based on these methods might be affected both by amplitude fluctuations and by phase variability in signals. Hence, the phase synchronization or phase-locking method has recently received increasing attention by studying the timing correlation in the phase domain while discarding the effect of the amplitude [22]. In CMP processes, phase synchronization modeling provides an effective tool to describe the timing correlation among critical FPVs such as COF and temperature since they show strong cyclic patterns (see Fig. 1 as an example). The remainder of this section will establish a model to identify the main effect and interaction effect between COF and temperature. The extracted interaction pattern can facilitate further monitoring and diagnosis of pattern change in timing correlation in Section III. Nonlinear dynamics theory suggests that the synchronization among oscillatory signals can be modeled by [23]

(1)

(2) is defined as a function describing interacThe term tion among these phase variable and is approximately pecan be approximated by Fourier expanriodic. Therefore, sion

(3) in the Fourier expansion are integers where the superscripts and cannot be zeros simultaneously. The values of coefficients and can be estimated through the ordinary least square (OLS) regression method. Prior to fitting the model, any signal trend must be removed from the data and the signal should fluctuate around zero (see example in Fig. 2). The trends in the middle graphs were extracted using cubic spline smoothing. The major challenge is to find interaction terms in the model with adequate orders. The model to be developed will not only

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be statistically adequate to avoid overfitting for robust prediction, but also physically interpretable for a better understanding of the synchronization mechanism. Grounded on these principles, a small Fourier expansion is preferred for a better physical interpretation. Initially, we can start with a small series and gradually increase the order of Fourier series through a model adequacy check. The Fourier expansion should be controlled in such a way that the base frequencies roughly go through the middle of the instantaneous frequencies. A model selection procedure, e.g., backward selection method, is adopted to screen the insignificant coefficients in the Fourier series. The statistical significance of model coefficients is determined by a small -value, e.g., 0.01 or less. To obtain a model with better interpretability, the interaction order and strength are defined as follows. Following the concept of statistical effects in the design of experiments [24], the main effect and interaction effect can be defined in a similar way. For example, the main effects include , , , , , and . The two-way interaction effects or the first-order interactions include , , etc. The three-way interaction effects or the second order , etc. With interactions contain this definition, basic principles like that of the hierarchical ordering principle are readily applied to the model selection procedure. For instance, the effect heredity principle suggests that in order for the interaction effect to be significant, normally at least one of its parent effects should be significant. Its parent effects are the main effects in the trigonometric identity . Due to the nature of Fourier expansion, the cosine and sine pairs can be represented in a complex form, e.g.,

The term represents

the

amplitude

in

signal processing and is related to the power of that frequency component. The strength of each frequency component in a main effect or an interaction effect can be thus defined using the concept of the power, e.g., . The strength of the main effects/interaction effects is defined as the summation of the power of every frequency component in all the main/interaction . The effects, e.g., proposed definition provides an opportunity to identify and analyze the important frequency components in each order of interactions. III. STATISTICAL PROCESS MONITORING BASED ON NONLINEAR DYNAMICS MODELING OF FPV TIMING CORRELATION

Statistical process control (SPC) is an effective tool to monitor process changes and reduce process variations. But standard SPC methods could not be directly applied to processes with FPVs [25]. Two main strategies have been deployed for process monitoring using functional data. The first strategy is to

extract features from curves, e.g., peak values, wavelet coefficients [26]–[29], or slope and intercept [30], [31]. Then, standard procedures developed in multivariate process control (e.g., control chart) can be applied to monitor those Hotelling’s features. The second strategy is nonparametric regression, i.e., to approximate curves with functions. Curves collected under different process conditions can then be discriminated into categories through baseline functions [32], [33]. Nevertheless, these approaches mainly focused on one single FPV except the modeling methods reviewed in Section II and a semi-parametric method [34] based on principal curve analysis [35]. In this section, we propose a new statistical method to detect change of timing correlation among FPVs for CMP processes. Prior to the detection procedures, data for FPVs should be collected in the following manner: sample data under condition 1 (normal condition), and sample data under condition 2 (abnormal condition). The data collected under the normal condition will be used as training data to establish the phase dynamics model proposed in (1). The changes of synchronization change pattern include systematic change or base frequency change. Systematic change in and interaction change or signal base frequency implies that significant process condition changes occur, which can be directly detected by visual inspection on the signal oscillatory pattern. Interaction change is related to moderate process condition change, which might not be identified by inspecting original signals. , and define patSince the model coefficients , terns that characterize process conditions, they will be used to detect interaction changes. Suppose synchronized signals are and as a modeled by (1). Denote stackup of coefficients in the interaction function . Given the data collected from the normal process condition, multichart) can be built variate control charts (e.g., Hotelling’s to monitor systematic pattern change and for , and interaction pattern change, respectively. Due to the irregularity, noise, and complex spatio-temporal patterns in real-time signals, the phase dynamics model may . The large model consist of a large set of coefficients dimension will adversely affect the performance of detection procedures. Principal component analysis (PCA) is an effective way of dealing with highly correlated parameter estimates. It is implemented along with the development of statistical detection procedures to reduce the model dimension for effective change detection. The basic idea is to monitor the first few prininstead of the coefficients themcipal components of statistic in terms of prinselves. For instance, Hotelling’s ciple components can be , where is the th principle component and is its corresponding sample variance. The number of principle components can be determined by the amount of total sample variance explained. Thus, the phase I control chart limit to monitor the principle compo, , where nents is is the number of samples and is the upper percentage point of beta distribution with parameters and . In this paper, is assumed to be 0.01. Before building a phase II control chart, it is necessary to remove the scores of out-of-control points in the phase I control chart and



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Fig. 2. Example of signal detrending.

Fig. 4. Example of recording temperature. Fig. 3. Experimental setup used in performing polishing experiments. TABLE I POLISHING PROCESS PARAMETERS

recompute the sample variance of the principle components. The phase II control chart to monitor the principle components can be then established by [36]

(4) where is the upper percentage point of distribu. The printion with degrees of freedom and comes from the future observation and ciple component comes from the phase I control chart. IV. CASE STUDIES A. Experiments To validate the proposed modeling and detection method, we designed validation experiments to generate abnormal process condition changes. The setup used in performing the polishing experiments is shown in Fig. 3. The polishing process was carried out on a bench-top CMP tester (model CP-4) manufactured by CETR, Inc. During polishing, lateral force and normal force of the contact interface were recorded at frequency 100 Hz and COF could

be recorded in situ by calculating the ratio of these two forces. Meanwhile, an FLIR infrared camera was shooting at the polishing area to monitor in situ the temperature distribution on the pad. The average temperature of the polishing zone (Fig. 4) on the pad was recorded at a frequency of 30 Hz. The online monitoring of the COF and temperature signals facilitates studying the interaction between chemical and mechanical process variables. The 6-in-diameter IC 1000 k grove polishing pad was attached on the rotating bottom platen in CETR and the 2-in wafer coupon was attached to the upper polishing head. The slider velocity was maintained at 3 mm/s during the whole experiment and the duration of each run was 3 minutes. Planerlite


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TABLE II EXPERIMENTAL CONDITIONS

Contaminated slurry was formulated with 350 mL slurry, 30 mL 30%-H

O

, and D.I. water contaminated with 20 mL faucet water.

Fig. 5. Example of signal recordings before and after slurry contamination.

7105 copper polishing slurry was mixed with 30% hydrogen peroxide in this experiment and was fed onto the center of the pad at the rate of 50 mL/min. The slurry temperature was maintained at 30 C using a controllable heater (manufactured by Corning, Inc.). The newly changed pad was conditioned for two 20-min runs with 1-min polishing of dummy samples in between. During the polishing process, the pad was conditioned ex situ after each run. Process parameters of the experimentation are summarized in Table I. In the experiment, we simulated the case when the slurry was contaminated by impurities and faucet water during the polishing process (Table II). Four samples (2 in) were polished with slurry without contamination for 3 minutes, followed by another 3 minutes of polishing using the contaminated slurry. Fig. 5 gives the temperature and COF recordings during polishing of one sample. The left panel shows the data when polishing the wafer with normal slurry (350:30:650 for Slurry: H O : D.I. water) while the right panel displays the signal profile when the slurry was contaminated during polishing. Apparently, visual inspection of these two variables is not easy to identify underlying pattern changes. B. Results and Analysis Instead of visual inspection, nonlinear dynamics modeling of phase synchronization assists statistical detection. This section demonstrates the method based on the data collected in Section IV-A. The nonlinear dynamics modeling [see (1)] results after statistical model selection are given in Fig. 6, where in the first row the dashed line represents instantaneous frequency extracted by

Fig. 6. Phase nonlinear dynamics modeling results.

Hilbert transform, and the solid line is for instantaneous frequency predicted by the model. The prediction residuals (as shown in the second row) exhibit random patterns and no systematic trend or pattern (e.g. cyclic fluctuation), which implies and are adequate. that the model orders In Fig. 7, the left and right panels compare the interaction strength (defined in Section II) before and after slurry contamination, respectively. The attached table shows the ratio between strengths of interaction and main effects. The interaction effect



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Fig. 7. Strength of main and interaction effects.

Fig. 8. Phase II control charts for main and interaction effects. (a) Base frequency monitoring. (b) interaction effect monitoring.

is significant because the ratio is far larger than 1 under two conditions. Compared to the normal condition (left panel), slurry contamination (right panel) significantly weakens the interac-

tion effect between temperature and COF. Such an interaction pattern could reflect process condition changes. For example, strong interaction between temperature and COF could be re-


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lated to the effective chemical reaction while the weakened interaction might indicate the chemical process is jeopardized or changed. Therefore, we propose a statistical detection method to identify the significant interaction pattern change. We built phase II T -Hotelling charts (Section III) to monitor, respectively, the change of interaction pattern after the slurry is the model term that shows the efcontamination, where fect of temperature on COF, and shows the effect of COF that usually correon temperature. The base frequency sponds to certain systematic process change is also monitored. We can see from Fig. 8 that all the sample points are under , control limit (dashed line) for monitoring main effects and are beyond the limits. This imwhereas all the samples of plies that the interaction effect of temperature on COF has significantly changed after slurry contamination. However, slurry contamination does not significantly affect the base frequency and the interaction of COF on temperature. Combining the rehas been sults of interaction strength, we can know that the significantly weakened after the slurry contamination. In this paper, such a type of interaction pattern change can be used as an indicator of less effective chemical reaction that could be related to slurry problems.

V. CONCLUSION This paper studied timing correlation of multiple FPVs in phase domain for CMP process condition monitoring and diagnosis. Considering the oscillatory pattern in the FPVs, we first established a nonlinear dynamics model to capture the main and interaction effect in the rhythm of cyclic components. It can be estimated through regression analysis followed by statistical screening procedures. Following the concept of statistical effects in design of experiments, we defined the order and strength of interaction, through which the directionality of interaction can be identified. Uncovering interaction directionality will assist us in understanding physical interaction among multiple FPVs. A statistical method of condition change detection was then developed by monitoring the interaction patterns using statistical process control tools. The extracted interaction patterns are especially helpful for detecting abnormal condition caused by hidden factors that may not be easily identified. The proposed method was applied to data analysis on CMP experiments, where we generated slurry contamination during CMP polishing. The modeling result showed strong interaction strength on both directions of COF-temperature interaction. Statistical control charts indicated that interaction effects of temperature on COF were significantly changed after slurry contamination, whereas the reversed interaction effect and base frequencies of both signals remained unchanged. Combining the results of interaction strength analysis, we can further conclude that the interaction effect of temperature on COF has been significantly weakened. In this paper, the weakened interaction pattern is an indicator of less effective chemical reaction due to slurry problems. These facts may lead to a new diagnosis method based on interaction modeling for abnormal process change caused by slurry problems.

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Hui Wang received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2001, the M.S.E. degree in mechanical engineering from the University of Michigan, Ann Arbor, in 2003, and the Ph.D. degree in industrial engineering from the University of South Florida, Tampa (advisor: Dr. Q. Huang), in 2007. He is currently a Research Fellow and an intermittent Lecturer in the Department of Mechanical Engineering, University of Michigan. His current research is focused on modeling and variation reduction for complex manufacturing processes.

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Xi Zhang received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, China, in 2006. He is currently working towards the Ph.D. degree at the University of South Florida, Tampa (advisor: Dr. Q. Huang). His dissertation topic focuses on process quality control on semiconductor manufacturing systems including modeling and diagnosis.

Ashok Kumar is a Professor in the Department of Mechanical Engineering, University of South Florida, Tampa. He is also affiliated faculty at Nanomaterials and Nanomanufacturing Research Center (NNRC) and the Clean Energy Research Center (CERC). His research is focused on development of surface engineered coatings for multifunctional applications. He has published two textbooks, edited six proceeding books, three invited review articles, five book chapters including 160 reviewed articles, and has presented approximately 190 papers in regional and national conferences. His other interests include K-12 educational outreach, gender and science education, and nanotechnology industrial outreach. Dr. Kumar has received numerous honors including the ASM Fellow, ASM-IIM Visiting Lecture Award (2007), Theodore and Venette Askounes Ashford Distinguished Scholar Award (2006), USF Outstanding Faculty Research Achievement Award (2004), USF President Faculty Excellence Award (2003), and NSF Faculty Early CAREER Development Award (2000). He has been an invited speaker and session organizer at many national and international meetings.

Qiang Huang is an Associate Professor in the Department of Industrial and Management Systems Engineering, University of South Florida, Tampa. His research interests are centered on modeling and analysis of advanced manufacturing processes such as micro/nanomanufacturing for quality and productivity improvement. Dr. Huang is a member of IIE, INFORMS, SME, ASME, and MRS.
