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Physical and statistical analysis of functional process variables for process control in semiconductor manufacturing
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Dissertation (Ph.D.)University of South Florida, 2009.
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ABSTRACT: The research aims at modeling and analyzing the interactions among functional process variables (FPVs) for process control in semiconductor manufacturing. Interaction is a universal phenomenon and different interaction patterns among system components might characterize the system conditions. To monitor and control the system, process variables are normally collected for observation which could vary with time and present in a functional form. These FPVs interact with each other and contain rich information regarding the process conditions. As an example in one of the semiconductor manufacturing processes, changes of interactions among FPVs like temperature and coefficient of friction (COF) might characterize different process conditions. This dissertation systematically developed a methodology to study interaction among FPVs through statistical and physical modeling.Three main topics are discussed in this dissertation: (1) Interaction patterns of FPVs under varying process conditions are studied both through experiments and statistical approaches. A method based on functional canonical correlation analysis (FCCA) is employed to extract the interaction patterns between FPVs and experiments of wafer polishing processes are conducted to verify the patterns of FPVs under varying process conditions. (2) Interaction among FPVs is further studied based on physics for process condition diagnosis. A mathematical model based on nonlinear dynamics is developed to study the strength of interaction and their directionalities, and advanced statistical control charts followed by this nonlinear dynamics model are established for process monitoring. (3) Complex interaction structures among multiple FPVs are analyzed based on nonlinear dynamics for a better understanding of process mechanism.An approach with extended nonlinear dynamics model is proposed to characterize process conditions, and combined engineering knowledge, complex interaction structure patterns are concluded accordingly for interpretation of process mechanism. The main contribution of this dissertation is to propose a novel methodology based on nonlinear dynamics, which could investigate interactions between components of systems and provide physical understanding of process mechanism for process monitoring and diagnosis. Through studies on interaction among FPVs in semiconductor manufacturing, this research provides guidance for improvement of manufacturing processes. Not limited to manufacturing, the developed methodology can be applied to other areas such as healthcare delivery.
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Advisor: Qiang Huang, Ph.D.
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Interaction patterns
Nonlinear dynamic model
Interaction structure
Statistical quality control
Interaction mechanism
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Physical and Statistical Analys is of Functional Process Vari ables for Process Control in Semiconductor Manufacturing by Xi Zhang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida Major Professor: Qiang Huang, Ph.D. Jos L. ZayasCastro, Ph.D. Michael Weng, Ph.D. Yuncheng You, Ph.D. Shekhar Bhansali, Ph.D. Date of Approval: July 16, 2009 Keywords: interaction patte rns, nonlinear dynamic model, interaction structure, statistical quality contro l, interaction mechanism Copyright 2009, Xi Zhang
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Dedication To my beloved parents
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Acknowledgements I would like to sincerely thank my adviso r Prof. Qiang Huang for offering me an opportunity to conduct research and pursue my doctorate degree under his guidance. Prof. Huangs serious attitude on research and in depth knowledge in applied statistics and engineering greatly inspire me, and his count less encouragement and critical mentoring in my research work present me with confidence and maturity both in research and my life. I would also like thank Prof. Qiang Huang fo r allowing me involving in NSF research project (NSF CMMI grant # 0600066) and pr oviding me an opportunity of parallel computation training program. Without him, I could not accomplish my dissertation. I would like to thank my di ssertation members Prof. Jo se Zayas, Prof. Michael Weng, Prof. Yuncheng You and Prof. Shekhar Bh ansali for their valuab le suggestions in my research. I would like to thank Prof. Ashok Kumar a nd his research group for experiment guidance. Additionally, I would e xpress my sincerely appreciation to Dr. Hui Wang for his assistance and enlightening me in my research. I woul d also like to thank Ms. Gloria Hanshaw, Ms. Jackie Stephe ns, Ms. Catherine Burton and Mr. Rafael. Moreover, I would like to thank my friends Ms. Yixin Wang, Mr. Shaoqiang Chen, Mr. Gang Liu, Mr. Yang Tan, Mr. Yu An, Mr. Ozan Ozcan and other workmates. Finally, I wish to express my purehearte d love to my beloved parents Jing Luo and Chunsheng Zhang. Without their constantly supports, continual encouragements and endless loves, I would have ne ver finished my dissertation.
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i Table of Contents List of Tables iv List of Figures v Abstract vii Chapter 1 Introduction 1 1.1 Studies in Semiconductor Manufacturing Processes 2 1.2 Related Work and the State of the Art 5 1.2.1 Related Works in Studying FPVs for Process Control 5 1.2.2 Related Works in Studying Interactions among FPVs 7 1.2.3 Summary of Literature Review 10 1.3 Dissertation Outline 11 Chapter 2 Analysis of Correlated Timevarying Process Variables for Condition Diagnosis in Semiconductor Manufacturing 13 2.1 Experimental Investigation of Slurry Thermal Effects and Correlation among Process Variables 13 2.1.1 CMP Experimental Setup and Design 15 2.1.1.1 Experimental Study of Slurry with Variation of Oxidizer Concentration 15
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ii 2.1.1.2 Experimental Study of Polishing Pad with Varying Diamond Sizes in Conditioner 18 2.1.2 Experimental Results and PostCMP Studies 18 2.2 Statistical Analysis of Corre lated Process Variables for Condition Diagnosis 26 2.2.1 Process Condition Charact erization through FCCA 26 2.2.2 Results and Analysis 28 2.3 Summary 32 Chapter 3 Nonlinear Dynamics Modeling of Correlated FPVs for Condition Monitoring in Semiconductor Manufacturing Processes 34 3.1 Nonlinea r Dynamics Modeling of FPV Timing Correlation 35 3.2 Statistical Process Monitori ng Based on Nonlinear Dynamics Modeling of FPV Timing Correlation 39 3.3 Case Studies 41 3.3.1 Experiments 41 3.3.2 Results and Analysis 44 3.4 Summary 48 Chapter 4 Analysis of Interaction Structure among Multiple FPVs for Process Control in Semiconductor Manufacturing 50 4.1 Review of Nonlinear Dynamics Model of FPVs 51 4.2 Analysis of Interaction Structure among Multiple FPVs 54 4.2.1 Extended Nonlinear Dyna mics Model for Interaction among FPVs 54
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iii 4.2.2 Interaction Structure Analysis 56 4.3 Application to Identification of Interaction Structure Patterns in Real CMP 63 4.4 Summary 69 Chapter 5 Conclusions and Future Work 71 5.1 Conclusions 71 5.2 Future Work 72 Cited References 75 About the Author End Page
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iv List of Tables Table 2.1 Polishing process parameters 16 Table 2.2 Experimental design 17 Table 2.3 Statistical summaries of withinsample uniformity from AFM (Unit: nm) 22 Table 2.4 Canonical correlation of slurry study 30 Table 2.5 Canonical correlation of conditioner study 32 Table 3.1 Wafer polishing process parameters 43 Table 3.2 Experimental conditions 44 Table 4.1 Interaction stru cture types and corresponding parameters 61
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v List of Figures Figure 1.1 The framework of dissertation 11 Figure 2.1 The thermal di stribution over polis hing pad under two polishing conditions 14 Figure 2.2 Experimental setup 15 Figure 2.3 Top view of one die on the wafer and sample cutting 17 Figure 2.4 Microscopic view of c onditioner with different abrasive size 18 Figure 2.5 An example of recording temperature 19 Figure 2.6 Observed functiona l process variables under 3 conditions 20 Figure 2.7 AFM 3D images for three levels of oxidizers 21 Figure 2.8 AFM measurements of wa fers under three levels of oxidizers 23 Figure 2.9 Optical images of wafers polished on pads conditioned with different abrasive sizes 24 Figure 2.10 Observed functional process variables under 3 different conditioners 25 Figure 2.11 An example of detrending (5% H2O2) 28 Figure 2.12 Canonical correlati on patterns under 3 levels of H2O2 29 Figure 2.13 Canonical co rrelation patterns under 3 leve ls of abrasive sizes 31 Figure 3.1 Strong cyclic patterns in process variables 35 Figure 3.2 An exam ple of signal detrending 37 Figure 3.3 Experimental setup us ed in performing polishing experiments 41 Figure 3.4 An example of recording temperature 42
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vi Figure 3.5 An example of signa l recordings before and after slurry contamination 44 Figure 3.6 Phase nonlinear dynamics modeling results 45 Figure 3.7 Strength of main and interaction effects 46 Figure 3.8 Phase II control char ts for main and interaction effects 47 Figure 4.1 Interaction structures represented as a network 54 Figure 4.2 Temporal patterns in FPVs 55 Figure 4.3 Procedure of analyzing in teraction structure 57 Figure 4.4 Selfoscillated variab les in system 57 Figure 4.5 Clockwise interacti ons among FPVs 58 Figure 4.6 Symmetric interactions among FPVs 58 Figure 4.7 Hidden FPVs interact th e network through Node 2 59 Figure 4.8 Four channel simulate d signals via van der Pol os cillators 60 Figure 4.9 Bar charts of intera ction strength under three interaction structures 62 Figure 4.10 FPVs with diamond particle size 8 m(L) and 100 m(R) 64 Figure 4.11 Sample data before and after detrending 65 Figure 4.12 One sample of model fitting result in a single window 66 Figure 4.13 Interactio n structure analysis: 100 m diamond particle size of conditioner 67 Figure 4.14 Interactio n structure analysis: 8 m diamond particle size of conditioner 69 Figure 5.1 Oscillatory in tegrated phrenic nerve signal and blood pressure 73
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vii Physical and Statistical Analys is of Functional Process Vari ables for Process Control in Semiconductor Manufacturing Xi Zhang ABSTRACT The research aims at modeling and anal yzing the interactio ns among functional process variables (FPVs) for process contro l in semiconductor manufacturing. Interaction is a universal phenomenon and different inte raction patterns among system components might characterize the system conditions. To monitor and control the system, process variables are normally collected for observati on which could vary with time and present in a functional form. These FPVs interact wi th each other and contain rich information regarding the process conditions. As an example in one of the semiconductor manufacturing processes, ch anges of interactions among FPVs like temperature and coefficient of friction (COF) might char acterize different process conditions. This dissertation systematically deve loped a methodology to study interaction among FPVs through statistical and physical m odeling. Three main topics are discussed in this dissertation: (1) Interaction patterns of FPVs under varying process conditions are studied both through experiments and sta tistical approaches. A method based on functional canonical correlation analysis (FCCA) is employed to extract th e interaction patterns between FPVs and experiments of wa fer polishing processes are conducted to verify the patterns of FPVs under varying process conditions. (2 ) Interaction among FPVs is further studied based on physics for process condition diagnos is. A mathematical
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viii model based on nonlinear dynamics is developed to study the strength of interaction and their directionalities, and advanced statisti cal control charts fo llowed by this nonlinear dynamics model are established for proces s monitoring. (3) Complex interaction structures among multiple FPVs are analyzed based on nonlinear dynamics for a better understanding of process mechanism. An approach with exte nded nonlinear dynamics model is proposed to characterize pro cess conditions, and combined engineering knowledge, complex interaction structure patterns are concluded accordingly for interpretation of process mechanism. The main contribution of this disser tation is to propose a novel methodology based on nonlinear dynamics, which could i nvestigate interactions between components of systems and provide physical understand ing of process mechanism for process monitoring and diagnosis. Through studies on interactio n among FPVs in semiconductor manufacturing, this research provides guidance for improvement of manufacturing processes. Not limited to manufacturing, th e developed methodology can be applied to other areas such as healthcare delivery.
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1 Chapter 1 Introduction Interaction among component s of a complex system is a universal phenomenon and usually can be defined as a kind of ac tion that occurs as two or more components having an effect on one another. Studying in teraction mechanism is important because interaction may characterize system. Theref ore, to monitor and control the system, interaction might be analy zed through collecting process variables. These process variables usually vary with time and interact with each other, containing rich information regarding the process conditions. As an example in one of the semiconductor manufacturing processes, changes of in teractions among process variables like temperature and coefficient of friction (C OF) might characterize different process conditions. Hence, to understand the intera ction mechanism may bring insights for process improvement. However, understanding the interaction m echanism is a quite challenging issue. There is still a lack of a science base to develop the interaction model upon which methods of detecting and diagnosing process conditions can be built. The difficulty of establishing such a sc ience base lies in the complex ity of the interaction phenomenon. The term of interaction, often vaguely de fined, has been used interchangeably with correlation, dependence and synchrony [1], which reflect s the different aspects or understanding of the interacti on phenomenon. For example in st atistics, term interaction
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2 means effects of various changes operate simultaneously, which is different from interaction in manufacturing pr ocess that dominant factors might trig interaction. It is therefore very challenging to mathematically define the interaction. In addition, FPV signals especially from semiconductor ma nufacturing have complex spatiotemporal patterns, irregularity and large noise. Moreove r, the interaction structure among three or more process variables can be extremel y complex. Thus, modeling, detection and diagnosis of the interacti on are very challenging. Therefore, an essential analysis of inte raction of FPVs for process improvement is required. The goal of this dissertation is to study the interaction of FPVs and generate knowledge of modeling, detection and diagno sis of the interaction to achieve an insightful understanding of interaction mechanism through combing nonlinear dynamics theory, the engineering knowledge and advanced statistical tools. 1.1 Studies in Semiconducto r Manufacturing Processes With the rapid development of techno logy in semiconductor manufacturing, more complex manufacturing processes are devel oped. For example according to Moores Law, the demand is the semiconductor industry with respect to the number of transistors per chip will be doubled every 2 year [2], and shrinking of device dimensions will bring complex manufacturing processes which coul d significantly impact the quality of products. Hence, advanced process control techniques are essentially needed for quality assurance in semiconductor manufacturing. To monitor and control the processes, FPVs are collected for observation. These FPVs usually interaction with each other and contain rich information and could characterize the varying process conditions. Hence, it is of great interest to understand the interaction mechanism for process improvement because
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3 further reduction of surface finish variations can affect 20% of wafer yield and impact a revenue stream of $2.8 billion in a single wafer fab [3]. In semiconductor manufacturing, usually ma ny process factors are involved to affect product quality. One of the important procedures in semiconductor manufacturing processes is chemicalmechanical planarizat ion (CMP) processes. CMP has been widely employed during semiconductor fabricatio n for planarizing the top surface of semiconductor wafers. In addition to abrasion du e to applied mechanic al pressure, slurry containing chemicals and particles is conti nuously fed onto the polishing pad to react with wafer materials on the interface for accelerated and improved planarization performance. CMP studies roughly follow three categor ies (1) physicsdriven modeling of polishing mechanism and predicting material removal rate (MRR) [4], (2) experimental investigation of process para meters effects on CMP performance, and (3) datadriven analysis of process variab les for condition monitoring and diagnosis. Since MRR is a commonly used criterion to evaluate CMP perf ormance, most research has been focusing on prediction of MRR based on a given set of process vari ables. Luo and Dornfeld [5] extended Prestons Equation and considered additional process parameters such as wafer hardness, pad roughness, abrasive size and ge ometry for better prediction. Considering the pad conditioning, Yi [6] first investigated the kinema tical relationship between wafer and polishing pad. He also employed the di stributed LuGre dynamic friction model to study the wafer/pad friction ch aracteristics. Sorooshian et al. [7] studied the effect of pad temperatures, and introduced a new energy parameter into new Prestons Equation by employing an Arrhenius argument. OsseoAsare [8] and Kaufman et al. [9] considered
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4 the chemical reaction between the slurry chem icals and wafer materials. They proposed a Tungsten CMP model by introducing oxidation reduction reaction which occurs in the passivation layer. In experimental investigation of CMP, process parameters such as slurry characteristics, pad temperature, polishing velocity, COF, and their effects on MRR have been studied. Mudhivarthi et al. [10] found that COF decreases wi th a rise of pressure and platen velocity, and MRR and COF incr ease when slurry temperature increases. Sorooshina et al. [11] investigated the effect of slurry temperature on COF, and they found that COF shows an increasing trend as polishing temper ature rises. Seal et al. [12] and Du et al. [13] concluded that the COF increases with increased peroxide concentration in the slurry and they interp reted this phenomenon as the cause of surface chemical decomposition of polyurethane material. Li et al. [14][15] studied the effect of slurry characteristics on friction mechanis m and they found slurry with different surfactants and abrasive sizes can significantly alter COF profile. They also found MRR decreases when slurry flow rate increases at a fixed relative rotating velocity. The least studied area is the datadriven analysis of sensing process variables for online process change detection and diagnosis. Hocheng et al. [16] investigated the distribution of the pad temper ature and established a regres sion model to detect the end point. Ganesan et al. [17][18] provided their waveletbased approaches based on sequential probability ration test to identify the delamination and end point online. Wang et al. [19] first studied the timing correlation be tween CMP process variables based on a new phase nonlinear dynamics model and used the model for proce ss change detection.
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5 It is known that factors such as applied force, pad property, and slurry flow rate would jointly (not independently) impact th e quality of polished wafers. The observed chemical and mechanical process variables are expected to be strong ly correlated during wafer polishing. Our hypothesis is that the co rrelation among sensing process variables could potentially be utilized to characterize process conditions for the purpose of process monitoring and diagnosis. Nevertheless, pro cess variables observed during manufacturing processes (e.g., the temperature distribution a nd coefficient of friction (COF) on a wafer), vary with time and present in a functi onal form. This significantly increases the complexity of analyzing correlation patterns and relating them with process conditions. Previous research focused on analysis of single process variable for online process change detection and diagnosis. However, process va riables like COF or temperature distribution alone cannot uni quely distinguish am ong process conditions. Jointly considering these correlated proce ss variables would assist to discover the hidden interaction mechanisms and a fundamental understanding the interaction mechanisms will assist to improve manufacturing processes. 1.2 Related Work and the State of the Art The study on interaction of FPVs is very limited. This section reviews the related research on process monitoring, diagnosis a nd control in manuf acturing processes. 1.2.1 Related Works in Studying FPVs for Process Control Due to the wide application of sens or technology in ma nufacturing, data processing has been significantly increased Some sensing data collected through multiple sensors simultaneously from manufacturing processes, often arise themselves in the functional form. Examples include tonna ge signals from forg ing processes [20],
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6 assembly force signals during seat and guide assembly in engine machining processes, and quadrupole mass spectrometry samples of rapid thermal chemical vapor deposition process in semiconductor manufacturing [21]. Comparing with observations of scalar or vector characteristics, func tional process variables (FPV s) contain richer process information which might potentially provi de additionally opportunities for process improvement and quality assurance. It is know n that statistical pro cess control (SPC) has been successfully applied to monitor vari ous manufacturing processes where process performance is measured by a scalar or vector characteristics. However, functional data imposes new dimensions and challenges for re altime process contro l because standard statistical procedures develope d in SPC are not directly app licable for continuous sample curves [22]. Two main strategies have been depl oyed in process control based on FPVs, depending on how the data is summarized. The first one is to extract features from functional data (e.g., wavelet coefficients [2325], or slope and inte rcept [26]) and apply standard procedures in mu ltivariate statistics (e.g., T2 control charts) to features for process monitoring [24][26]. The second stra tegy is nonparametric re gression, i.e., to approximate curves with functions nonpara metrically. Data collected under different process conditions can then be discriminated by estimated probability density functions [23] or baseline functions [27] Both of the two strategies assume that the collected functional data is wellsummarized by the extracted features, estimated probability density functions, or baseline. Changes in these summaries indicate process changes. The functional data studied in the SPC literature, however, is mainly univariate, if we treat a curve as one variable in some f unctional space. Multivariate curves occur from
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7 multisensor systems where location (in a gene ral sense) is an important source of variation. The correlation or interaction among these variables will be overlooked if we only investigate the mean curve or study each individual curves separately. Apparently, certain opportunity of variation reduction could be lost if multivariate curves are not modeled simultaneously in process control. A lthough it is rarely seen in SPC literature, analysis of nonstationary multivariate functional data has been investigated in statistics. The involved modeling techniques include nonp arametric regression [28][29], functional data analysis [30], spatial statistics [31], and principal curve [ 32]. These methodologies were mainly proposed for exploratory study an d are not directly applicable for process monitoring and control. Hence, this is a lack of methodology which could integrate both process control methods and modeling techniques for process improvement. 1.2.2 Related Works in Studying Interactions among FPVs Several approaches have been reported in the literature to model correlations among nonstationary continuous signals. The most commonly used method is the crosscorrelogram which measures the crosscovari ance of paired FPVs [ 33][34]. This timedomain 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 si gnals in the frequency domain and are most commonly used with continuous signals [35] 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 phaselocking method ha s recently received increasing attention by studying the timing correlation in the phase domain while discarding the effect of the amplitude [36] In CMP processes, phase synchronization
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8 modeling provides an effective tool to desc ribe the timing correlation among critical FPVs such as COF and temperature sin ce they show strong cyclic patterns. To detect the synchrony between these two signals, the phaselocking method and related test statistics focusing on the rhyt hm of signal pace should be applied while discarding the effect of amplitude. A m:n phase locking between a pair of signals is that the relative phase  m 1n 2 is bounded, i.e.,  m 1n 2
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9 Entropy Index [40]: This method also first empl oys the 1:1 phaselocking idea to compute j for each time point tk A histogram of relative phase j for j = kN, k the observation time window is then built So the entropy of th e series is then defined as (1.2) Where L is the number of bins and pj is the probability corresponding to the j th bin. For example, we could di vide the unit circle uniformly into 12 bins and then to calculate the pj and establish the histogram. Note that the time of observation is implicit in the fact that N prior phase values are used to generate the histogram from which hn is computed. In order to conv eniently evaluate the timingcorrelation in this method, the value s hould be normalized. For example, the maximum value can be obtaine d through uniform distribution. pj = 1/ L for all j and hN reaches its maximum value hn max = log L The normalized entropy is then hn* = ( hn max hn)/ hn max (1.3) and the value of hn* only between 0 and 1. hn measures the degree of clustering of the angular distribution, and it is therefor e different from the phase coherence in that it will achieve high values for mu ltimodal phase distributions as well as for the unimodal case. For example, the phase coherence of a di stribution that has two symmetrical opposing lobes in the circle will be zero, whereas the normalized entropy will yield a value closer to 1. Mutual Information Index [41]: Similarly to entropy index method, mutual information index is defined as following: 1log(1/)L Njj jhpp
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10 (1.4) where Pi, Pj is probability of corresponding bin from histogram of the individual phase series. Here Pij is probability from joint histogram of the pairs l 1, l 2. This statistic is computed over the histogram of N accumulated phase values at each point in time. N pairs of angular values l 1, l 2 from the phase trajectories of the individual channels are accumulated during an integration window T on a torus. The joint distribution can be obtained from a two dimensional histogram after dividing the torus in L X L bins. The mutual information can also be normalized to its maximum value, In = log L achieved when the series l 1, l 2 are identical. In* = ( In max In)/ In max (1.5) Those approaches mentioned above althrough could avoid effects from amplitude of singals or abrupt variations when analyzing interactions, limitation still exists because only paired correlated signals could be studied. Hence, a novel appoach to analyze complex interaction with mulitple variables is essentially required. 1.2.3 Summary of Literature Review Previous research has been focused on an alysis of the individual FPV or coupled FPVs. However, interaction mechanism could not be simply studied by only inspecting individual FPV or paired FPVs. Therefore, process m onitoring and improvement may require a thorough understandi ng of the interrelationship among those FPVs. Moreover, the interaction structure among three or more FPVs has not b een thoroughly investigated for the purpose of process control. There is a lack of physical model to describe the interaction structure of multiple FP Vs in manufacturing processes. , 11logLL ij Nij ij ijp Ip pp
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11 1.3 Dissertation Outline The insightful understanding the intera ction of FPVs for process improvement requires investigations in the following aspects: (1) experimentally and theoretically study interaction patterns of FPVs under varying process conditions. A method based on functional canonical correlation analysis (FCCA) is employed to extract th e interaction patterns between FPVs; (2) further study of interacti on among FPVs based on nonlinear dynamics for process condition diagnosis. A novel nonlinear dy namics model is developed to study the strengt h of interaction and their di rectionalities, and advanced statistical control charts are established for process monitoring; (3) analysis of complex interaction structures among multiple FPVs based on nonlinear dynamics for a better understanding of process mechanism. An approach with exte nded nonlinear dynamics model is proposed to characterize process co nditions, and complex interaction structure patterns are concluded accordingly for inte rpretation of process mechanism. The overall framework of this disse rtation is displayed in Fig 1.1. Figure 1.1 The framework of dissertation
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12 Chapter 1 mainly introduces the bac kground of monitoring and diagnosis in semiconductor manufacturing processes and re lated literature reviews for interaction analysis, process control and CMP studies. Chapter 2 presents an approach to im prove the predication of CMP performance based on the extracted correlati on patterns for online process c ontrol. The focus of this work is twofold: (1) experi mental investigation of th e correlation between process variables and the implication of correlati on pattern changes on process conditions, and (2) extraction and statistical analysis of co rrelation patterns from process variables in functional form. In Chapter 3, 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. In Chapter 4, we extended our previ ously developed nonlinear dynamics model by considering the autocorrelation in each FPV to uncover the interaction mechanism of multiple process variables for process condition identification. Chapter 5 makes the conclusion of this disse rtation. Perspectives of future work are also discussed.
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13 Chapter 2 Analysis of Correlated Timevarying Process Variables for Condition Diagnosis in Semiconductor Manufacturing This chapter mainly focuses on the two asp ects: (1) experimental investigation of the correlation between process variables and the implication of correlation pattern changes on process conditions, and (2) extracti on and statistical analysis of correlation patterns from process variables in functional form. The ultimate goal is to improve the predication of semiconductor manufacturin g performance based on the extracted correlation patterns for online process contro l. All the experiments and methods are validated through CMP processes. In this chapter, experimental investiga tion and statistical mo deling of correlation are presented in Section 2.1 and 2.2, respec tively, whereby two fa ilure modes (oxidizer and pad failures) are analyzed. A co nclusion is given in Section 2.3. 2.1 Experimental Investigation of Slurry Thermal Effects and Correlation among Process Variables Since the thermal effect of slurry with different percentages of oxidizers and the effect of conditionerdiamondsize induced pa d condition change have not been fully explored, we choose slurry and diamond abrasi ve size of conditioner in our experimental study as experimental factors. Different pro cess conditions will be created to investigate changes in correlation patterns (if there is any). The process variables to be observed
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14 online for correlation analysis include COF of the waferpad interface and the temperature distribution over the polishing area. (a) (b) Figure 2.1 The thermal distribution over polishing pad under two polishing conditions As a critical mechanical variable, th e timevarying COF reflects the realtime tribological property at the interface. Change s in COF indicate vari ations in abrasive performance due to pad failure, large partic les on the pads, or underlying barrier layer exposure on the wafer [42]. The thermal distribution over the polishing area is another indicator of process conditions and reflects heat generated through friction and chemical reactions. Figure 1 shows thermal imaging sn apshots of two process conditions using different slurries right after 15 seconds of polishing (wafer size 0.7.8 inch, slurry temperatures: 30oC). The temperature distribution is relatively uniform under the first process condition (Fig. 2.1a), while there is a bright ring around the polishing zone under the second polishing condition (Fig. 2.1b). Our i nvestigation suggests that the bright ring was due to the heat generated from both ch emical reactions and mechanical abrasion, while there was a lack of chemical r eaction under the first polishing condition. 21.8 30.7 22 24 26 28 30
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15 2.1.1 CMP Experimental Setup and Design The experiment was conducted to test th e hypothesis that the correlation among timevarying CMP process variab les could be utilized to ch aracterize process conditions. To be specific, the experiments aim to inve stigate (1) thermal eff ects of slurry with various percentages of oxidi zer, and (2) effects of pol ishing pads generated by conditioners with different diamond abrasive sizes. As shown in Fig. 2.2, the polishing was carried out on a bencht op CMP tester (model CP4) ma nufactured by CETR Inc. The realtime shear forces and normal forces at the contact interface were recorded at the frequency of 50Hz. The COF was calculated as the ratio of these two forces. Meanwhile, a FLIR infrared camera was employed to monito r the temperature distribution in situ on the pad with a sampling frequency of 50 Hz. Figure 2.2 Experimental setup 2.1.1.1 Experimental Study of Slurry with Variation of Oxidizer Concentration In this designed experiment, the 6inch diameter IC 1000A4 perforated polishing pad (manufactured by Rodel, Inc.) was attached on the rotating bottom platen of the CMP tester. The 2inch wafer coupon was attach ed to the upper polishing head. The upper polishing head and bottom platen rotated in counterclockwise direction in order to let the thermal camera capture the temperature on th e polishing zone. The upper slider swung
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16 from side to side to prevent pad from ex cessive local heat. The slider velocity was maintained at 3 mm/sec during the whole experiment and each run lasted 3 minutes. Cabot copper polishing slurry was mixed with hydrogen peroxide in three designated levels. The slurry was fed into the center of the pad at the rate of 50mL/min. The slurry temperature was maintained at 25oC for each single run by temperature controller (manufactured by Corning, Inc.). The pad wa s changed in every designated level and was conditioned for two 20min runs with 1min polishing of dummy samples in between. The process parameters employed for the expe rimentation are summarized in Table 2.1. Table 2.1 Polishing process parameters Description Value Wafer coupon LG Siltron silicon wafer 2 inches Polishing pad IC 1000A4 Slurry Cabot 5001 Copper slurry Oxidizer Hydrogen peroxide Slider movement Offset: mm, speed: 3mm/s Pad conditioning Pressure: 2psi, Revolution per minute (rpm): 150 Conditioner Diamond abrasive pad conditioner Slurry flow rate 50 mL/min Polishing head Rpm: 145 LG Siltron silicon wafers were employed during the CMP process to measure the nonuniformity, and the 50 m and 9 m copper lines were chosen. The wafer coupon size was designated 2 inches which contains patter ns from four identical dies on the wafer.
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17 Each sample was cut in such a way that the patterns from four di es could rotate on the same circular trajectory during the process, and thereby the patter ns could be polished under the same experimental condition (Fig. 2.3). Figure 2.3 Top view of one die on the wafer and sample cutting Three different levels (3 samples polished under each level) of slurry condition were designed (see Table 2.2) by adding hydrogen per oxide with different concentration to the Cobot 5001 slurry. The purpose of the design is to simulate the case when oxidizer fails during the polishing process a nd investigate how the oxidizer levels affect correlation patterns, which are to be discussed in Section III. Table 2.2 Experimental design Rotational velocity (Polishing head vs. pad) Pressure Polishing time Slurry Solution : H2O2 150 vs. 145 rpm 2 psi 2 minutes 900:150 900:75 900:0 9 _m line 50 _m line One die on the wafer A wafer with 9 dies Experiment sample
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18 2.1.1.2 Experimental Study of Polishing Pad with Varying Diamond Sizes in Conditioner To study CMP pad failure, the same proce ss parameters as 2.1.1.1 were applied except slurry and wafer samples. Here the Cobot 5001 copper slurry with 2.5% hydrogen peroxide and singlelayer copper wafer were used. Three conditioners with different diamond abrasive sizes (0.25m, 8 m and 10 0 m, Fig. 2.4) were used to generate different morphologies and roughness on polishing pads. In addition, different abrasive size can cause distinct failure patterns on the pad (e.g., scratches). Under each level of conditioners, experiments were replicated three times. (a) 0.25m (b) 8m (c) 100m Figure 2.4 Microscopic view of conditi oner with different abrasive size 2.1.2 Experimental Results and PostCMP Studies The image in Fig. 2.5 shows a snapshot of thermal distribution over the entire pad, while the data at the bottom panel is the timevarying temperature of a selected zone on the pad. The focused thermal zone was selected on the polishing pad adjacent to the waferpad interface to record the average te mperature. Since the temperature in this interface between wafer and polishing pad ca nnot be obtained directly, the zone we selected in this way might best indicate th e average process temperature 0. The thermal
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19 data were recorded on this se lected zone during the whole pr ocess, and the data could be observed instantaneously. Figure 2.5 An example of recording temperature In the first designed experi ments (Section 2.1.1.1), we collected thermal data and COF during the whole polishing process. For example, Figure 2.6 displays the COF and temperature vs. time under three conditions. Apparently, both signals have the similar patterns in their general trend, i.e., they in crease at the beginning of the cycle, decrease after the signals reach the peak values, and ev entually asymptotically approach a constant. The decreasing trend in the signals is cau sed by various factors such as improved lubrication of polishing after the process stabilize or improved interface between the waferpad friction pair after certain amount of copper has been removed. Zone of interest Thermal data
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20 Figure 2.6 Observed functional pr ocess variables under 3 conditions The post CMP surface characterization was carried out on an Atomic Force Microscope (AFM) to study the surface nonunif ormity, which is measured in terms of the vertical distance be tween the different locations on th e samples, i.e., the height of steps on the surface profile. Since the sample s (LG Siltron silicon wafers) we adopted have patterns under the copper layer, we need to evaluate and compare the nonuniformity on the same patterns. In this study, we chose two isolated lines (50 m and 9 m) on each die of the wafer (Fig. 2.7). 0 1000 2000 3000 4000 5000 6000 7000 0 0.2 0.4 0.6 0.8 Slurry with 5% OxidizerTime IndexCOF 0 1000 2000 3000 4000 5000 6000 7000 25 25.5 26 26.5 27 Time IndexTemperature 0 1000 2000 3000 4000 5000 6000 7000 0 0.2 0.4 0.6 0.8 Slurry with 2.5% OxidizerTime IndexCOF 0 1000 2000 3000 4000 5000 6000 7000 23 24 25 26 27 Time IndexTemperature 0 1000 2000 3000 4000 5000 6000 7000 0 0.2 0.4 0.6 0.8 Slurry with 0% OxidizerTime IndexCOF 0 1000 2000 3000 4000 5000 6000 7000 24.5 25 25.5 26 Time IndexTemperature
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t o l o l e ( t l i t h u t h i m The A o obtain the o cations. Fi g e vels of oxi d t he average i nes on four Table h erefore ha s sing the slu r h e variation m plies the c A FM cantile v profile ima g g ure 2.8 sh o d izer. Table 2 values vert i dies). Figure 2. 2.3 shows the lowest r ry with 2.5 % of step hei g c hemical re a v er and tip s g e, on whic h o ws the exa m 2 .3 gives th e i cal distanc e 7 AFM 3D that the s e nonunifor m % H2O2 dur i g ht over th e a ction spee d 21 s canned an a h we measur m ples of m e e statistical s e s h and s t ima g es for e cond level m ity. This i m i ng 2 minut e e same patt e d is differe n a rea of 30 m ed the step e asurement s ummaries o t andard dev i three level s has the s m m plies the p e s of polishi n e rns on all s n t over fou r m 30 m c o heights o n and AFM r e o f withinsa m i ation h ov e s of oxidize r m allest verti p olishing w a n g. Howeve r s amples is t h r dies on s a o vering eac h n sections at e adings for m ple unifor m e r 9 m or 5 r s cal distanc e a s more eff i r under this h e largest, w a mples. It i h line three three m ities 5 0 m e and i cient level, w hich s not
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22 surprising to observe that the third level has smaller variation in step height since the most copper remains intact due to a lack of chemical reaction. Table 2.3 Statistical summaries of within sample uniformity from AFM (Unit: nm) There is noticeable difference observed between the measurements on 9 m lines and 50 m lines. Surface over the 9 m line has lower nonuniformity than that over the 50 m line whereas the variation over the 50 m line is smaller. This fact implies that polishing on thin lines is more efficient th an polishing on thick lines in our experiment. In the second designed experiment (S ection 2.1.1.2), the data from both conditioning and polishing proc ess were recorded. Both COF and temperature have similar patterns and trends as in our first de signed experiment. The polished wafer surfaces were examined through Leitz Ergolux Optical Microscope. Figure 2.9 shows the optical images of wafer surfaces polished on pads conditioned with different abrasive sizes. It is concluded that th e pads conditioned with larger abrasives resulted in more number of scratches compared to the smaller abrasive size. 9 m line 50 m line Levels h h h h 5% H2O2 319.3933 44.29538390.723325.23637 2.5% H2O2 186.5692 51.28105231.017536.73086 0% H2O2 573.1422 19.80072472.257528.3114
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23 (a) 9 m line, 5% H2O2 (b) 50 m line, 5% H2O2 (c) 9 m line, 2.5% H2O2 (d) 50 m line, 2.5% H2O2 (e) 9 m line, 0% H2O2 (f) 50 m line, 0% H2O2 Figure 2.8 AFM measurements of waf ers under three levels of oxidizers
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24 (a) 0.25m (b) 8m (c) 100m Figure 2.9 Optical images of wafers polished on pads conditioned with different abrasive sizes We also collected thermal data and CO F during the whole process, and we ran three replicates in each condition. Fig. 2.10 showed one example of COF and temperature in each condition. It can be seen that the signal patterns under each condition are similar and difficult to distinguish with each other. From the first experiment (Section 2.1.1.1) the results have shown that the levels of oxidizer play a significant role affecting the polishing quality and efficiency. Process with slurry using 2.5% H2O2 gives more efficient polishing than 5% H2O2. This can be explained by the nonlinear relationshi p between the removal rate and H2O2 concentration, i.e., adding H2O2 will significantly increase the polishing rate whereas further increases in H2O2 will lower the polishing rate 0. The decr ease in polishing rate may be caused by more copper oxides generated by high level of oxidizer, which can reduce or prevent copper layer from further chemical reaction. The second experiment has shown that surface defects (scratches) on polished wafers could be attributed to the pad failure (rough pad) generated by a larger abrasive size of conditioners.
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25 Figure 2.10 Observed functional process var iables under 3 different conditioners It is essential to detect the process ch ange caused by large va riations in oxidizer level or pad failure. However, inspecting crit ical process variables such as COF and pad temperature separately is incapable of dis tinguishing different conditions. Nevertheless, we could observe that the os cillatory patterns in COF an d temperature bear certain similarities under each condition. This fact may lead to a new method to characterize and detect the process condition by uncovering the latent correlation patterns between the critical process variables. 0 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 Time IndexCoFConditioner with 0.25 um diamonds 0 1000 2000 3000 4000 5000 6000 7000 24 25 26 Time IndexTemperature 0 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 Time IndexCoFConditioner with 8 um diamond 0 1000 2000 3000 4000 5000 6000 7000 22 24 26 28 Time IndexTemperature 0 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 Time indexCoFConditioner with 100um diamond 0 1000 2000 3000 4000 5000 6000 7000 25 25.5 26 26.5 Time IndexTemperature
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26 2.2 Statistical Analysis of Correlated Pr ocess Variables for Condition Diagnosis Results from Section 2.1 show that insp ection of COF or temperature distribution alone cannot uniquely distinguish all process co nditions. In this Section, we will jointly consider these correlated process variables by analyzing their correlation. The objective is to discover the correlation patterns under different process conditions and thereby to detect the condition changes. The modeling approach is based on functional canonical correlation analysis (FCCA), which is to assess the correlation through measuring the statistical similarities among f unctional data. The statistica l correlation patterns under different polishing condi tions can be captured. 2.2.1 Process Condition Characterization through FCCA In applied statistics, canonical correl ation analysis (CCA) maximizes the correlation between a linear co mbination of two random vector s. FCCA is motivated to find functional canonical variates that maxi mize the covariance function of two process variables (or stochastic proce sses). To be specific, suppose x ( t ) and y ( t ) are two process variables with zero mean and covariance functions [ x ( t ) x ( s )], [ y ( t ) y ( s )], and [ x ( t ) y ( s )]. Also assume there are N observed pairs of data curves ( xi( t ) and yi( t )) for the two variables. The variance and covariance can then be estimated using these N observations, e.g., E[ x ( t ) y ( s )]= N1ixi( t ) yi( s ). The FCCA is to find weight functions u ( t ) and v ( t ) to maximize the squared correlation (denoted as ccorsq( u v )) of u ( t ) xi( t )d t and v ( t ) yi( t )d t i.e. [0], v u ,maxccorsq( u v )= u ( t ) [ x ( t ) y ( s )] v ( s ) dsdt (2.1) s t u ( t ) [ x ( t ) x ( s )] u ( s ) dsdt +  D2u ( t )2=1 and v ( t ) [ y ( t ) y ( s )] v ( s ) dsdt +  D2v ( t )2=1,
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27 where is a positive smoothing parameter, D is a differential operator d / dt .2 is an operator that computes (.)2dt and  D2( )2 gives a penalty term that considers the roughness of the functions u ( t ) and v ( t ). Appropriately select ed smoothing parameter can yield fairly smooth weight functions and a correlation that is no t unreasonably low. It can be chosen subjectively or according to crossvalidation procedures. Functional curves u ( t ) and v ( t ) can be estimated by seeking expansions in terms of a fixed number of basis functions i( t ), e.g., Fourier basis for periodical signals and Bspline basis for nonperiodical signals. Then functional curves are estimated as ibu ii( t ) for u ( t ) and ibv ii( t ) for v ( t ), where bu i and bv i are coefficients. It should be noted that the process va riables need to be detrended and zerocentered prior to implementing FCCA. The reason is that both COF and temperature have similar general trends and therefore their trends definitely are highly correlated and possess similar correlation patterns. For th e purpose of monitoring process condition using correlation analysis, we s hould look into the details of variations by removing the effect of general trend. Therefore, zerocente ring the data is a requirement of FCCA. In this study, cubic spline smoothing is employed to separate the data trend from the process variables. Figure 2.11 gives an example of th e detrending results for one replicate of polished samples (5% H2O2).
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28 Figure 2.11 An example of detrending (5% H2O2) Solutions to Eq. (2.1) (denoted as uI and vI) give a pair of leading canonical variates that maximize ccorsq( u v ). Thus, uI( t ) and vI( t ) reflect a latent pattern of correlation. Other latent pa tterns can be identified by the second and higher order canonical variates, denoted as ( uII, vII), ( uIII, vIII), and so on. For example, the second pair ( uII, vII) is the functions u and v that maximize the same correlation ccorsq( u v ) subject to the constraint that they are to be uncorrelated with the lead ing pair of canonical variates. The number of potential canonical variates is eq ual to the number of basis functions in fitting the functional curves. 2.2.2 Results and Analysis As pointed out in Section 2.2.1, the weight functions u ( t ) and v ( t ) determine the correlation patterns, which are ab le to characterize the process conditions. In 2.1.1.1, we study the correlation patterns unde r three levels of oxidizer (H2O2) by plotting (see Fig. 2.12) the weight functions for the first two pairs of canonical variat es, where solid curve gives the weight function of COF and dash ed line shows the weight function of temperature. Since the detrended signals oscillate, Fourier basis is chosen to fit the data to 0 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 COFOriginal Sigal 0 1000 2000 3000 4000 5000 6000 7000 0.5 Trend 0 1000 2000 3000 4000 5000 6000 7000 0.1 0 0.1 Time IndexDetrended signal 0 1000 2000 3000 4000 5000 6000 700 0 25 26 27 Pad TemperatureTemperature 0 1000 2000 3000 4000 5000 6000 700 0 25.5 26 26.5 27 Trend 0 1000 2000 3000 4000 5000 6000 700 0 0.5 0 0.5 Time IndexDetrended signal
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29 functional curve. Table 2.4 shows the values of corresponding correlations for the first two pairs of canonical variates. Leading pa ir of canonical variat es has very large correlation values and reflects major correlation patterns. Figure 2.12 Canonical correlation patterns under 3 levels of H2O2 Comparing the weight functions for slurri es with different oxidizer percentage, we observe that the change in oxidizer wi ll significantly alter the shape of weight function. To interpret such pattern changes, recall that the signals have been detrended and the functional data show a characteristic of periodical oscillation. Corresponding to the slurry with 2.5% H2O2 the solid curve and dashed curve of the first pair of canonical 5% H2O2 2.5% H2O2 0% H2O2
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30 variates have almost constant weight func tions. This indicates the original process variables themselves would achieve the maxi mum correlation (not tr ue in the other two conditions). Since the process with 2.5% H2O2 yields the best polishing results, the associated correlation pattern can serve as a ba seline pattern for future process diagnosis. A large fluctuation in weight function of te mperature is observed under the first level (5% H2O2) while the two curves significantly devi ate from each other under the third level (0 .0% H2O2). Table 2.4 Canonical correlation of slurry study Canonical variates 1s t 2n d Slurry w. 5% H2O2 0.928 0.259 Slurry w. 2.5%H2O2 0.989 0.283 Slurry w/o. H2O2 0.996 0.128 The second pair of canonical variates has a relatively small correlation; however, it still provides a reference to distinguish between different levels, especially for the levels of 2.5% H2O2 and 0% H2O2 since the correlation patterns are very different. We also analyzed the correlation patterns under three levels of abrasive size of conditioners and the weight functions of canoni cal variates are given in Fig. 2.13. Table 2.5 gives the correlation for the first two pa irs of canonical variates. Prior to the FCCA, the trend is removed and the Fourier basis wa s chosen in the same way as the case in 2.2.1.1. From Fig. 2.13, we again observe d different weighting function patterns under different conditions. When the abrasive size 0.25 m is chosen, the weighting function shapes of both COF and temperature appear al most the same, and this indicated original
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31 two process variables achieved maximum corre lation (not true in other two conditions). When 100 m abrasive size is used, in addition to fluctuation in dashed curve, two curves significantly depart from each other. Figure 2.13 Canonical correlation patterns under 3 levels of abrasive sizes From results of both cases, we may observe that the best pr ocess condition seems to be related to the correlation pattern that original process variables achieve the maximum correlation. Further experimental studies are needed to confirm this observation. 0.00.20.40.60.81.0 1.00.00.51.01.5 Can. Fn. 1Normalized timeWeight function 0.00.20.40.60.81.0 1.00.00.51.01.5 Can. Fn. 2Normalized timeWeight function 0.00.20.40.60.81.0 1.00.01.0 Can. Fn. 1Normalized timeWeight function 0.00.20.40.60.81.0 1.00.01.0 Can. Fn. 2Normalized timeWeight function 0.00.20.40.60.81.0 1.00.00.51.0 Can. Fn. 1Normalized timeWeight function 0.00.20.40.60.81.0 1.00.00.51.0 Can. Fn. 2Normalized timeWeight function 0.25m abrasive size 8m abrasive size 100m abrasive size
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32 Table 2.5 Canonical correlation of conditioner study Canonical variates 1s t 2n d 0.25 m abrasive size 0.992 0.142 8 m abrasive size 0.985 0.466 100 m abrasive size 0.840 0.253 2.3 Summary This chapter conducted an experimental and statistical anal ysis of correlation among timevarying process variables for CM P process condition monitoring and process change detection. We proposed that correlati on patterns can help to characterize process conditions. Therefore, two e xperiments were designed to study the correlation between process variables. In the first experiment, we polished silicon wafer using three levels of concentration of oxidizer (H2O2) in slurry to simulate oxi dizer failure and study its impact on polishing quality and COFtemperature correlation pattern. COF was recorded by the embedded pressure sensors in a CMP machine and temperature on polishing pad (including spatial and temporal variations) was captured by a thermal camera. PostCMP analysis conducted on an AFM has shown that the level of oxidizer has a huge impact on polishing quality and effici ency. Slurry with 2.5% H2O2 yielded the lower nonuniformity than other levels. In a similar fashion, th e second experiment investigated pad failures and their impact on COFtemper ature correlation. Three t ypes of abrasive sizes of conditioners were used to condition the pad that could cause scratches on the polished wafer. Experimental results show that it is difficult to distinguish among process conditions by investigating sensing variables COF or temperature alone.
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33 The statistical analysis intends to explore the implication of correlation analysis on process condition change det ection and monitoring. The f unctional data (timevarying function) were first expanded under certain fu nctional basis (e.g., Fourier basis). Based on the expansions, the FCCA then measures statistical similarities among timevarying process variables and identif ied the weight functions th at maximize the correlation among variables. The weight functions captu re the correlation pa tterns corresponding to different process conditions. F CCA analysis showed that the slurry problem and pad failure can significantly change the shape of COFtemperature canonical variates. The correlation between the leading pair of canoni cal variates reveals the major correlation patterns. From both experiments, we found that original COF and te mperature signals are more likely to achieve maximum correlation u nder the baseline conditions and less likely under faulty conditions. This fact leads to a new diagnostic method for abnormal process change caused by certain latent factors (e.g., slurry contamination or pad failure) that cannot be easily detected. Future research in next chapters invol ves interaction patt erns among multiple process variables and modeling improvement for a better interpretation of correlation patterns.
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34 Chapter 3 Nonlinear Dynamics Modeling of Correlated FPVs for Condition Monitoring in Semiconductor Manufacturing Processes This chapter aims to investigate interaction between functional process variables (FPVs) for condition monitoring in chemical mechanical planarization (CMP). During wafer polishing, critical proce ss variables such as coefficient of friction (COF) and pad temperature vary with time and present in the shape of functional curves. In previous chapter, we have demonstrated that corr elation patterns among these FPVs could be related to polishing conditions. Since co rrelation is affected by both amplitude fluctuations and phase variab ility in FPVs, further study of timing correlation of FPVs measured in different units could bring mo re insights into physic al interactions and thereby enhance CMP condition monitoring. Existing research on FPVs in CMP mainly focuses on individual effects of FPVs and st atistical 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 patter ns of FPV interaction. By monitoring the extracted patterns, we then developed a novel method of de tecting CMP condition change and demonstrated the approach via a CMP experiment. The results showed that the proposed method has a promising application in identifying the proc ess changes that may not be easily detected otherwise.
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35 This chapter is organized as follows. S ection 3.1 develops methods of nonlinear dynamics modeling of physical interaction. In Section 3.2, a model based process condition change detection/di agnosis method is then propos ed by employing statistical process control tools. Section 3.3 applies th e methods to a CMP experimental data and discusses the results. Conclusi ons are given in Section 3.4. 3.1 Nonlinear Dynamics Modeling of FPV Timing Correlation 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 patte rns (see Fig. 3.1 as an exam ple). The remaining of this section will establish a model to identify the main effect and intera ction effect between COF and temperature. The extracted interact ion pattern can facilitate further monitoring and diagnosis of pattern change in timing correlation in Section 3.2. Figure 3.1 Strong cyclic patt erns in process variables Nonlinear dynamics theory suggests that the synchronization among p oscillatory signals can be modeled by [46] 12()/[(),(),...,()],1,2,...,,kkkpkdtdtQtttkp (3.1) 0 1000 2000 3000 4000 5000 6000 7000 0 0.2 0.4 0.6 0.8 Time IndexCOF 0 1000 2000 3000 4000 5000 6000 7000 24.5 25 25.5 26 Time IndexTemperature
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36 where k( t ) is the phase variable (a function of time t ), k is the base angular frequency (frequency component with the highest power), and k is the white noise of the k th oscillatory signals xk( t ). The phase variables k( t )s can be obtained by constr ucting an analytical signal obtained from Hilbert transform as follows (For a simple sine signal x ( t )= A sin( 0t + 0), the phase ( t ) is defined as ( t ) = 0t + 0), ()arg{()[1/()/()]},1.kkktxtixtdi (3.2) The term Qk(.) is defined as a function describing interaction among these phase variable k( t ) and is approximately periodic. Therefore, Qk(.) can be approximated by Fourier expansion: 1212 12,,...,,,..., 12 ,,...,11(,,...,)[cos()sin()]pp ppp mmmmmm kpjjjkjj mmmjjQambm k =1,2, p (3.3) where the superscripts mjs in the Fourier expansion ar e integers and cannot be zeros simultaneously. The values of coefficients 12,,...,pmmm ja and 12,,...,pmmm kb can be estimated through the Ordinary Least Square (OLS) regr ession method. Prior to fitting the model, any signal trend must be removed from the data and the signal should fluctuate around zero (see an example in Fig 3.2). The trends in the middle graphs were extracted using cubic spline smoothing.
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37 Figure 3.2 An example of signal detrending The major challenge is to find interaction terms in the model with adequate orders. The model to be developed will not only be st atistically adequate to avoid overfitting for robust prediction, but also physically inte rpretable for a bette r understanding of the synchronization mechanism. Grounded on these pr inciples, a small Fourier expansion is preferred for a better physical in terpretation. Initially we can start with a small series and gradually increase the order of Fourier se ries through a model adequacy check. The Fourier expansion should be controlled in su ch a way that the base frequencies roughly go through the middle of the instantaneous freque ncies. A model selection procedure, e.g., backward selection method, is adopted to sc reen the insignificant coefficients in the Fourier series. The statistical significance of model coefficients is determined by a small p value, e.g., 0.01 or less. To obtain a model w ith better interpretability, the interaction order and strength are defined as follows. Following the concept of statistical effect s in the design of experiments [47], the main effect and interaction effect can be defi ned in a similar way. For example, the main 0 200 400 600 800 1000 1200 1400 1600 1800 0 0.5 1 Signal Detrending (COF)COF 0 200 400 600 800 1000 1200 1400 1600 1800 0.2 0.4 0.6 Trend of COF 0 200 400 600 800 1000 1200 1400 1600 1800 0.5 0 0.5 Detrended COFTime Index 0 200 400 600 800 1000 1200 1400 1600 1800 15 20 25 30 Signal Detrending (Temperature)COF 0 200 400 600 800 1000 1200 1400 1600 1800 15 20 25 30 Trend of COF 0 200 400 600 800 1000 1200 1400 1600 1800 2 0 2 Detrended COFTime Index
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38 effects include cos [ 1( t )], cos [ 2( t )], cos [ 3( t )], sin [ 1( t )], sin [ 2( t )], and sin [ 3( t )]. The twoway interaction effects or the first order interactions include cos [ 1( t )+ 2( t )], cos [ 2 1( t )+ 3( t )], etc. The threeway interaction eff ects or the second order interactions contain cos [ 1( t )+ 2( t )+ 3( t )], etc. With 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 sugges ts that in order for the interaction effect cos [ 1( t )+ 2( t )] to be significant, normally at leas t one of its parent effects should be significant. Its parent effects are the main effects in the trigonometric identity cos [ 1( t )+ 2( t )] = cos [ 1( t )] cos [ 2( t )]sin [ 1( t )] sin [ 2( t )]. Due to the nature of Fourier expansi on, the cosine and sine pairs can be represented in a complex form, e.g., 123123,,,, 22()()mmmmmm kkab exp{ im11( t )+ im22( t )+ im33( t )}. The term 2 , 2 ,) ( + ) (w r s k w r s kb a represents the amplitude in signal processing and [( a123,, mmm k)2+( b123,, mmm k)2] is related to the pow er 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., [( a123,, mmm k)2+( b123,, mmm k)2]. The strength of the main effe cts/interaction effects is defined as the summation of the power of every frequency component in all the main/interaction effects, e.g., 123,, mmm [( a123,, mmm k)2+( b123,, mmm k)2]. The proposed definition provides an opportunity to identify and analyze the important frequency component s in each order of interactions.
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39 3.2 Statistical Process Monitoring Based on Nonlinear Dynamics Modeling of FPV Timing Correlation Statistical process control (SPC) is an e ffective tool to monitor process changes and reduce process variations. Bu t standard SPC methods could not be directly applied to processes with FPVs [48]. Tw o main strategies have been deployed for process monitoring using functional data. The first strate gy is to extract features from curves, e.g., peak values, wavelet coefficients [4954], or slope and intercept [53][54]. Then standard procedures developed in multivariate process control (e.g., Hotellings T2 control chart) can be applied to monitor those features. Th e second strategy is nonparametric regression, i.e., to approximate curves with functions Curves collected under different process conditions can then be discriminated into ca tegories through baseline functions [55][56]. Nevertheless, these approaches mainly focused on one single FPV except the modeling methods reviewed in Section II and a semiparametric method based on principal curve analysis [58]. In this sec tion, we propose a new statistical method to detect change of timing correla tion among FPVs for CMP processes. Prior to the detection procedures, data for FPVs should be collected in the following manner: sample data under conditi on 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 Eq. (3.1). The changes of synchronization pattern include systematic change or base frequency k change and interaction change or Qk() change. Systematic change in signal base frequency implies that significant process cond ition changes occur, which can be directly detected by visual inspection on the signal oscillatory pattern. Interaction change is
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40 related to moderate process condition ch ange, which might not be identified by inspecting original signals. Since the model coefficients k, aw r s k ,s, and bw r s k ,s, define patterns that characterize process conditions, they will be used to detect interaction changes. Suppose p synchronized signals are modeled by Eq. (3.1). Denote = [ 1 2,, p]T and [ak bk]T as a stackup of coefficients in the interaction function Qk(). Given the data collected from the normal process condition, multivar iate control charts (e.g., Hotellings T2 chart) can be built for and [ak bk]T to monitor systematic pattern change and interaction pattern change, respectively. Due to the irregularity, noise, and comple x spatiotemporal patterns in realtime signals, the phase dynamics model may cons ist of a large set of coefficients [ak bk]T. The large model 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 principal components of [ak bk]T instead of the coefficients themselves. For instance, Hotellings T2 statistic in terms of principal components can be 222 1/A aa aTts where ta is the a th principal component and s2 a is its corresponding sample variance. The number of principal components A can be determined by the amount of total samp le variance explained. Thus, the phase I control chart limit to monitor the principal components is UCL=( m 1)2/ m A ( m A 1)/2, LCL=0, where m is the number of samples and A ( m A 1)/2 is the upper percentage point of beta distribu tion with parameters A and ( m A 1)/2. In this paper, is assumed to
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41 be 0.01. Before building a phase II control char t, it is necessary to remove the scores of outofcontrol points in the phase I contro l chart and recompute the sample variance s2 a of the principal components. The phase II control chart to monitor the principal components can be then established by [58], 22 ,, 1()/(1)/()A new aaAmA atsAmmAF (3.4) where F A m A is the upper percentage point of F distribution with degrees of freedom A and ( m A 1)/2. The principal component new at comes from the future observation and s2 a comes from the phase I control chart. 3.3 Case Studies In this part, the real CMP experiments we re conducted to validate our approach. 3.3.1 Experiments To validate the proposed modeling and det ection method, we designed validation experiments to generate abnorma l process condition changes. Figure 3.3 Experimental setup used in performing polishing experiments Setup used in performing the polishing e xperiments is shown in Fig. 3.3. The polishing process was carried out on a benc htop CMP tester (model CP4) manufactured by CETR Inc. During polishing, lateral forc e and normal force of the contact interface
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42 were recorded at frequency 100Hz and COF coul d be recorded in situ by calculating the ratio of these two forces. Meanwhile, a FLIR infrared camera was shooting at the polishing area to monitor in situ the temp erature distribution on the pad. The average temperature of the polishing zone (Fig. 3.4) on the pad was recorded at a frequency of 30 Hz. The online monitoring of the COF and te mperature signals facilitates studying the interaction between chemical and mechanical process variables. Figure 3.4 An example of recording temperature The 6inch diameter IC 1000k grove polishi ng pad was attached on the rotating bottom platen in CETR and 2inch wafer coupon was attached to the upper polishing head. The slider velocity was maintained at 3 mm/sec during the whole experiment and the duration of each run was 3 minutes. Pl anerlite 7105 copper polishing slurry was mixed with 30% hydrogen peroxide in this e xperiment and was fed onto the center of the pad at the rate of 50mL/min. The slu rry temperature was maintained at 30oC using a controllable heater (manufact ured by Corning, Inc.). The newly changed pad was conditioned for two 20min runs with 1min polishing of dummy samples in between. Average Temperature Average TemperatureTime (s) Temperature (oC)
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43 During polishing process, the pad was conditioned ex situ after each run. Process parameters of the experimentati on are summarized in Table 3.1. Table 3.1 Wafer polishing process parameters Description Value Wafer coupon, 2in Polishing pad IC1000k groove Slurry Planerlite 7105 copper slurry Oxidizer 30% hydrogen peroxide Slider movement Offset: 5mm, speed: 3mm/s Pad conditioning Pressure: 2psi, rpm: 150 Conditioner Diamond pad conditioner Slurry flow rate 50mL/min In the experiment, we simulated the cas e when the slurry was contaminated by impurities and faucet water during the polishing process (Table 3.2). Four samples (2 in.) were polished with slurry without contamin ation for 3 minutes, followed by another 3 minutes of polishing using the contaminated slurry. Figure 3.5 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: H2O2: D.I. water) while the right panel displays the signal profile when the slurry was contaminated during polishing. Apparently, vi sual inspection of these two variables is not easy to identify underlying pattern changes.
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44 Table 3.2 Experimental conditions Rotational velocity (Polishing head vs. pad) Pressure Polishing time Slurry : H2O2 : D.I. water 150 vs. 145 rpm 3 psi 3 min 350:30:650 (mL) 3 min Contaminated slurry * Contaminated slurry was formulated with 350 mL slurry, 30 mL 30%H2O2, and D.I. water contaminated with 20 mL faucet water. Figure 3.5 An example of signal rec ordings before and after slurry contamination 3.3.2 Results and Analysis Instead of visual inspection, non linear dynamics modeling of phase synchronization assists statisti cal detection. This section demonstrates the method based on the data collected in Section 2.2.1. 0 500 1000 1500 2000 2500 3000 0 0.2 0.4 0.6 0.8 Recorded SignalsCOF 0 500 1000 1500 2000 2500 3000 15 20 25 30 35 Pad TemperatureTime Index 0 200 400 600 800 1000 1200 1400 1600 1800 0 0.2 0.4 0.6 0.8 Recorded SignalsCO F 0 200 400 600 800 1000 1200 1400 1600 1800 15 20 25 30 35 Pad TemperatureTime Index Normal Slurry Contaminated Slurry
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45 The nonlinear dynamics modeling (Eq. (3 .1)) results after statistical model selection are given in Fig. 3.6, where in the first row the dash line represents instantaneous frequency extracted by Hilbert transform, and the solid line is for instantaneous frequency predicted by the mode l. The prediction residuals (as shown in the second row) exhibit random patterns and no systematic trend or pattern (e.g. cyclic fluctuation), which implies that the model orders p =14 and q = 14 are adequate. Figure 3.6 Phase nonlinear dynamics modeling results In Fig. 3.7, the left and right panels co mpare the interaction strength (defined in Section II) before and after sl urry contamination, respectivel y. The attached table shows the ratio between strengths of interaction and main effects. The in teraction effect 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 interaction effect between te mperature and COF. Such interaction pattern could reflect process condition changes. For example, strong interaction between temperature and 0 500 1000 0.4 0.6 0.8 1 1.2 1.4 Data Points (Time)Fitted vs Extracted IF 0 500 1000 0.3 0.2 0.1 0 0.1 0.2 Data Points (Time)Residuals 0 500 1000 0 0.5 1 1.5 2 Data Points (Time)Fitted vs Extracted IF 0 500 1000 0.5 0 0.5 Data Points (Time)Residuals COF Pad Temperature IF model prediction IF extracted from data IF model prediction IF extracted from data
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46 COF could be related to effective chemical reaction while the weaken ed interaction might indicate the chemical process is jeopardi zed or changed. Therefore, we propose a statistical detection method to identify the sign ificant interaction pattern change. Interaction/Main Effect Normal Contamination Effect of Temperature on COF 3.1222 2.6731 Effect of COF on Temperature 3.0458 2.6561 Figure 3.7 Strength of main and interaction effects We built phase II T2Hotelling charts (Section III) to monitor respectively the change of interaction pattern afte r the slurry contamination, where Q1 is the model term that shows the effect of temperature on COF and Q2 shows the effect of COF on temperature. The base frequency ( 1, 2) that usually corresponds to certain systematic process change is also monitored. 0 1 2 3 4 Strength of Main Effect and Interaction (Normal)Strength of COF 0 2 4 6 8 10 12 Strength of Pad TemperatureMain Effect Main Effect Interaction Effect Interaction Effect 0 1 2 3 4 Strength of Main Effect and Interaction (After Conatmination)Strength of COF 0 2 4 6 8 10 12 Strength of Pad TemperatureMain Effect Main EffectInteraction Effect Interaction Effect Effect of COF on Temperature Effect of Temperature on COF
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47 (a) Base frequency monitoring (b) Interaction effect monitoring Figure 3.8 Phase II control charts fo r main and interaction effects We can see from Fig. 3.8 that all the sa mple points are under control limit (dash line) for monitoring main effects and Q2 whereas all the samples of Q1 are beyond the limits. This implies that the interaction effect of temperature on COF has significantly changed after slurry contamination. Ho wever, slurry contamination does not significantly affect the base frequency a nd the interaction of COF on temperature. Combining the results of interacti on strength, we can know that the Q1 has been 1 2 3 4 0 20 40 60 80 100 120 140 Sam p le No.T2 Phase II Control Chart for ( 1 2)Base Frequencies 1 2 3 4 100 200 300 400 500 600 700 800 900 1000 Sample No.T2Phase II Control Chart for Q1 Effect of Temperature on COF 1 2 3 4 0 50 100 150 200 250 300 350 400 Sample No.T2Phase II Control Chart for Q2 Effect of COF on Temperature
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48 significantly weakened after the slurry contamination. In this study, such type of interaction pattern change can be used as an indicator of less effective chemical reaction that could be related to slurry problems. 3.4 Summary This chapter studied timing correlation of multiple functional process variables (FPVs) in phase domain for CMP pro cess condition monitoring and diagnosis. Considering the oscillatory pattern in the FPV s, we first establishe d a nonlinear dynamics model to capture the main and interaction effect in the rhyt hm of cyclic components. It can be estimated through regr ession analysis followed by sta tistical 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 directi onality will assist to understand 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 interacti on patterns are especially helpful for detecting abnormal condition caused by hidden factors th at 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 bot h directions of COFtemperat ure interaction. Statistical control charts indicated that interaction eff ects of temperature on COF were significantly changed after slurry contamination, whereas the reversed interaction effect and base frequencies of both signals remained unchang ed. Combining the results of interaction strength analysis, we can furt her conclude that the interac tion effect of temperature on
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49 COF has been significantly weakened. In this study, the weakened interaction pattern is an indicator of less effective chemical reac tion 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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50 Chapter 4 Analysis of Interaction Structure amon g Multiple FPVs for Process Control in Semiconductor Manufacturing From previous chapters, we have show n that the complex interaction patterns among functional process vari ables (FPVs) in semiconduc tor manufacturing processes could indicate process condition changes. We developed a nonlinear dynamics model to describe interactions among FPVs, which was further used to monitor process condition changes. However, the interaction structur e among three or more FPVs has not been thoroughly investigated for the pur pose of process control. In this work, we first extend our previously developed nonlinear dynamics m odel by considering the autocorrelation in each FPV. A generalized least square (GLS) me thod is applied to estimate the extended model. The interaction structure among FPVs is represented as a complex network in which the directionality and strength of interaction are discovered from the extended nonlinear dynamics model. To validate the proposed method, we first conduct simulation study using van der Pol oscillators. Then tw o sets of real experimental data from chemical mechanical planariz ation process are used to investigate the interaction structure change over a polishing cycle. The re sults show that the extracted patterns of interaction structure among FPVs could aid to uncover the polishing mechanisms and provide more insights for c ondition monitoring and diagnosis.
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51 The chapter is organized as follows. Sec tion 4.1 briefly reviews the work done in previous chapters. In Section 4.2 we firs t extend our previously developed nonlinear dynamics model by consideri ng the autocorrelati on in each FPV. A generalized least square (GLS) method is used to estimate th e extended model. The interaction structure among FPVs is represented as a complex netw ork in which the directionality and strength of interaction are discovered from th e extended nonlinear dynamics model. We demonstrate the interaction analysis appr oach through a simulation of van der Pol oscillators. In Section 4.3, two sets of real experimental data from chemical mechanical planarization process are used to investigate the interaction structure change over a polishing cycle. Conclusion is given in Section 4.4. 4.1 Review of Nonlinear Dynamics Model of FPVs In semiconductor manufacturing many pro cess factors are involved to affect product quality. As an example in chemical mechanical planarizat ion (CMP), factors such as applied force, pad property, and slurry flow rate would join tly (not independently) impact the quality of polished wafers. The in teractions among process factors can be very complex. To understand the ways that process factors affect product quality, process variables such as coefficient of fricti on (COF) and polishing pad temperature are collected online to predict th e realtime process conditions If process variables are continuously observed and vary with time, th ese functional process variables (FPVs) may contain rich process information. For in stance, COF between the wafer and the pad provides information regarding the tribologica l condition at the interface. An abrupt and large variation in COF could be a realtime indication of pad failure, large particles on the pads, or underlying barrier layer exposure on the wafer. Highly correlated to COF, pad
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52 temperature is another important FPV that depicts heat variati ons generated through friction and chemical reaction. Previous studies have investigated FPVs both experimentally and analytically [60][61]. Our experience suggested that certain process changes may not be easily detected w ithout collectively studying these FPVs. Simultaneously analyzing these FPVs and their interaction patterns could bring additional insights into process conditi on changes and new opportunitie s for process improvement [62][63]. In [62][63] we have shown that the complex interaction patterns among FPVs in semiconductor manufacturing processes could indicate process condition changes. We developed a nonlinear dynamics model to desc ribe interactions am ong FPVs, which was further used to monitor pro cess condition changes. However, the interaction structure among three or more FPVs has not been th oroughly investigated for the purpose of process control. Below we br iefly review related studies. The common approaches of analyzing co rrelation or interaction are crosscorrelogram and crossspectrum methods [6466]. They might be easily affected by artifacts and lead to imprope r detections, especially fo r the nonstationary signals collected in CMP process. 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 [65][39]. Phase synchronization was de veloped to detect timingcorrelation in phase domain while discarding the effect of the amplitude of signals [39][40]. Similar to the correlation coefficient in the time domain, coherence, entropy, or mutual information indices in phase domain ha ve been proposed to detect synchrony in
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53 paired signals [7072]. A nonlinear dynamics model wa s further developed in [73] to study phase synchronization: ) ( ) ( ) ( / ) ( ) ( ) ( /2 1 2 2 2 2 2 2 1 2 1 1 1 1 1 1t f q dt d t f q dt d (4.1) where 1,2 are the phases of c oupling variables, 1,2 are the base angula r frequencies or natural frequencies, and 1,2 are the noisy perturbations. 1,2 are defined as functions which involve interactions be tween coupling variables, and q1,2 are selfprovoked functions. This model can be easily extended to the case with more than two coupling variables. The major challenge, however, is to find interaction functions 1,2 in Eq. (4.1) with adequate orders after Fourier transf ormation. To improve the mode, we defined main effects and interaction effects of FPVs in [62] and further demonstrated it in CMP process monitoring. However, the temporal patterns, especia lly the often strong autocorrelation in FPVs were not considered th erein. This could lead to inadequate phase dynamics models and provide an incomplete picture of the complex spatiotemporal patterns in FPVs. Furthermore, the complex interaction structure among three or more FPVs has not been thoroughly investigated. As shown in Fig. 4.1, the general correlation analysis could not rev eal the directionality and strengt h of interaction, and network structure among multiple FPVs. As shown in our case study, analyzing interaction structures could assist to understand more insights of the polishing mechanisms.
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54 Figure 4.1 Interaction structures represented as a network Rosenblum et al. [74] first investigated interacti on structure based on mutual prediction. In their study the ca nonical structure (three oscillato rs in a ring) was identified through pairwise analysis of coupled osc illators. More complex structures were investigated using partial directed coherence method [75]. All these methods examine the directionality of twoway interactions to id entify interaction structures. Threeway or high order interactions are not considered. Mo reover, variables in th e interacting network are assumed to be known, and hence hidden influential variables could be missed. Therefore, a new method that could distinguis h different orders of interaction would be preferable so as to identify more complex interacting mechanisms. 4.2 Analysis of Interaction Structure among Multiple FPVs 4.2.1 Extended Nonlinear Dynamics Model for Interaction among FPVs In [62] we characterized the inte raction among multiple FPVs by 12()/[(),(),...,()],1,2,...,,kkkpkdtdtQtttkp (4.2) 200 250 300 350 400 450 0.15 0.2 0.25 0.3 0.35 0.4 0.45 time 21.8 30.7 22 24 26 28 30 Pad temperature Acoustic Emission Coefficient of Friction
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55 where phase variables k( t )s are obtained through Hilb ert transform of FPVs, and Qk(.) describes interaction am ong these phase variable k( t ). Qk(.) is approximately periodic which can be approximated by Fourier expansion: 1212 12,,...,,,..., 12 ,,...,11(,,...,)[cos()sin()]pp ppp mmmmmm kpjjjkjj mmmjjQambm k =1,2, p (4.3) where the superscripts mjs in the Fourier expansion ar e integers and cannot be zeros simultaneously. The values of coefficients 12,,...,pmmm ja and 12,,...,pmmm kb were estimated through the Ordinary Least Squa re (OLS) regression method. When sampling frequency is high, model (4.2) might overlook the strong autocorrelation or temporal patterns in FPVs. Figure 4.2 shows the fitted results for one segment of COF. The order of the Fourier expansion is m =4. As can be seen, although the fitted model (dash line) c ould capture the main trend of the original instantaneous frequency (bold line), the residuals show a st rong autocorrelation. Without considering the potential autocorrelation could leads to approximating Qk(.) with many sine and cosine terms, which may be hard to interpre t physically. Therefore, we extend the model (4.2) by imposing a structure on residuals, Figure 4.2 Temporal patterns in FPVs 0 70 140 210 280 350 420 490 560 1.8 1.9 2 2.1 2.2 2.3 0.02 0.015 0.01 0.005 0 0.005 0.01 0.015 0.02 0.02 0.015 0.01 0.005 0 0.005 time index Instantaneous frequency Fitted Values for COF using OLS residual tresidual t1
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56 Cov[ k( t ), k( t +1)]= 2 kl, time lag l =0,1,2, (4.4) Compared with the previous mode l based on the assumption of noise independence, the noise term in the new model permits an autocorrelated structure, i.e., error terms at time t and t + l have correlation coefficients l. For example, the error 1 of the COF may follow a firstorder auto regressive model (AR(1) model), 1( t )= 1( t 1)+ 1( t ), where 1( t ) is the white noise, 1( t )~ N (0, 2 ). Then the coefficient l= l and Cov[ 1( t ), k( t + l )]= 2 1 = 2 /(12). The first derivatives of phase k( t )s will serve as response variables while cosine and sine functions in the F ourier expansion of Qk(.)s will be predictors. The model coefficients i, 12,,..., p mmm ja, 12,,..., p mmm jb, and l can be estimated using the Generalized Leas t Squares (GLS) method [76]. 4.2.2 Interaction Structure Analysis The proposed interaction struct ure analysis approach is outlined in Fig. 3. Since the complexity of interaction analysis incr eases exponentially with the number of FPVs, we use a network involving three nodes as an example to demonstrate the procedure. With the extended phase dynamics model we will first identify the strength of interactions. As defined in 62, the main effects include cos [ 1( t )], cos [ 2( t )], cos [ 3( t )], sin [ 1( t )], sin [ 2( t )], and sin [ 3( t )]. The twoway interaction effects or the first order interactions include cos [ 1( t )+ 2( t )], cos [ 2 1( t )+ 3( t )], etc. The threeway interaction effects or the second order interactions contain cos [ 1( t )+ 2( t )+ 3( t )], etc. The term 2 , 2 ,) ( + ) (w r s k w r s kb a represents the amplitude in signal processing and [( a123,, mmm k)2+( b123,, mmm k)2] is related to the power of that fr equency component. The strength of each frequency component in a main effect or an interaction effect can be thus defined using
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57 the concept of the power, e.g., [( a123,, mmm k)2+( b123,, mmm k)2]. The strength of the main effects/interaction effects is defined as the summati on of the power of every frequency component in all the main/in teraction effects, e.g., 123,, mmm [( a123,, mmm k)2+( b123,, mmm k)2]. Then we could construct a bar chart for each FPV which shows the strength of main effect and interaction effects (see an example in Fig. 4.9). Figure 4.3 Procedure of analyz ing interaction structure From the bar charts we will investigate interaction patterns. We analyze four important cases of interaction structures: Selfoscillated FPV As illustrated in Fig. 4.4, if main effect of a FPV is dominant and all of the twoway and high order interaction effects are insignificant, the FPV does not interact with others. Figure 4.4 Selfoscillated variables in system
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58 Clockwise interaction pattern In this case, each FPV is affected by the other in clockwise manner (Fig. 4.5). From the ba r chart, both interaction effects are significant, but one is stronger than the other. Meanwhile, the main effect and threeway or high order intera ction are relatively weak. Figure 4.5 Clockwise interactions among FPVs Symmetric interactions among FPVs If strengths of all tw oway interactions are similar and higher order interactions are insignificant, we may encounter symmetric interaction structure (Fig. 4.6). Figure 4.6 Symmetric int eractions among FPVs Hidden FPVs in a network If only one threeway interaction effect is significant, we may suspect that at least one hidden variable exists in the network. As shown in Fig. 4.7, the bar chart displays interaction strength for node 2. If
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59 threeway interaction effects are weak in all the other FPVs, a hidden FPV might interact with the network through node 2. Similar idea for identifying hidden variables could be found in [77]. Figure 4.7 Hidden FPVs interact with the network through Node 2 The last issue is to determine the directionality of interaction. If the strength of twoway intera ction in node i is stronger than that of the corresponding twoway interaction in node j then node j has stronger influence on node i To validate the proposed approach for an alyzing interaction structures, fourchannel van der Pol oscillator syst em is simulated using Matlab [78] as follows: 2 22 11 1111221133114411 2 2 22 22 2222332244221122 2 2 22 33 3333443311332233 2 2(1)()()() (1)()()() (1)()()() dxdx uxxxxxxxx dtdt dxdx uxxxxxxxx dtdt dxdx uxxxxxxxx dtdt d 22 44 4444114422443344 2(1)()()() xdx uxxxxxxxx dtdt (4.5) where x1,2,3,4s are four FPVs u is the nonlinear selfweight parameter, 1,2,3,4 s are linear selfweight parameter; j i s are weights from c oupled variables, and 1,2,3,4 are white noises with normal distributions.
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60 Figure 4.8 Four channel simulated si gnals via van der Pol oscillators Three canonical types of interaction stru ctures are generated by varying parameter values (Table. 4.1). The figures on the right column depict interaction structures under variant dynamical systems. Nodes 1, 2, 3 and 4 are represented simulated FPVs, and arrows between nodes represent directed inte ractions determined by those parameters. Since some important process variable s might be missed in a real dynamical system or process, we choose node 4 as hidde n variable to be disc overed by the proposed approach. The nonlinear parameter u is fixed to 3 for high no nlinear feature in dynamical systems. To avoid one FPV being modulated by another, small values were given toj i and i. 5% Gaussian white noise of original sign al in decibel was added to each channel. 200 300 400 500 600 700 800 900 1000 4 2 0 2 4 200 300 400 500 600 700 800 900 1000 4 2 0 2 4 200 300 400 500 600 700 800 900 1000 4 2 0 2 4 200 300 400 500 600 700 800 900 1000 4 2 0 2 4 node 1 node 2 node 3 node 4time index time index time index time index
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61 Table 4.1 Interaction structure types and corresponding parameters Type A 1 = 0.5 2 = 0.5 3 = 0.5 4 = 0.5 1 2= 0.5 1 3= 0.5 1 4= 0 2 3= 0.5 2 4= 0 2 1= 0 3 4= 0 3 1= 0.5 3 2= 0 4 1= 0 4 2= 0 4 3= 0 Type B 1 = 0.5 2 = 0.5 3 = 0.5 4 = 0.5 1 2= 0.5 1 3= 0 1 4= 0 2 3= 0.5 2 4= 0 2 1= 0.7 3 4= 0 3 1= 0.5 3 2= 0.7 4 1= 0 4 2= 0 4 3= 0 Type C 1 = 0.5 2 = 0.5 3 = 0.5 4 = 0.5 1 2= 0.5 1 3= 0 1 4= 0 2 3= 0.5 2 4= 1 2 1= 0 3 4= 0 3 1= 0.5 3 2= 0 4 1= 0 4 2= 0 4 3= 0 1500 data points were sampled for each FPV The initial 200 points of data were cut off from original data se ries to avoid instability caused by initial values. Figure 4.8 displays the simulated signa ls from each node in type A structure. Since the interaction relationship could not be di stinguished via visual study, the proposed nonlinear dynamics models are required to unveil the interaction mechanism.
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62 According to the principles for modeling fitting discussed in Section 4.2.1, low order of Fourier expansion was preferred. Stepwise regression was applied to screen out those insignificant term s in the phase dynamics model. Te rms with very low strengths of main effects or interactions effect were also dropped off. The orde r of Fourier expansion was finally set to 2 with p value being 0.05 in model se lection. By computing the strength of main effects and interaction eff ects for each node, we obtain bar charts shown in Figure 4.9. Figure 4.9 Bar charts of interaction strength und er three interaction structures It can be seen clearly in the first row of Fig. 4.9 that both in teraction effects are significant, but one is stronge r than the other in each node. For instance, in node 2, twoway interaction 23 and 2 1 are both significant, but 2 3 is more dominant. The pattern repeats at each node which suggests cl ockwise interaction st ructure (nodes in a Main 2way_2, 2way_3 3way 0 0.1 0.2 0.3 0.4 0.5 Strength@Type A Main 2way_3, 2way_1 3way 0 0.1 0.2 0.3 0.4 0.5 Main 2way_1, 2way_2 3way 0 0.1 0.2 0.3 0.4 0.5 Main 2way_2, 2way_3 3way 0 0.05 0.1 0.15 0.2 0.25 0.3 Strength@Type B Main 2way_3, 2way_1 3way 0 0.1 0.2 0.3 0.4 Main 2way_1, 2way_2 3way 0 0.1 0.2 0.3 0.4 Main 2way_2, 2way_3 3way 0 0.1 0.2 0.3 0.4 0.5 0.6 Strength@Type C Main 2way_3, 2way_1 3way 0 0.1 0.2 0.3 0.4 0.5 0.6 Main 2way_1, 2way_2 3way 0 0.1 0.2 0.3 0.4 0.5 Node2 Node3 Node1
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63 ring pattern) in type A nonlinear system. Twoway interaction 2 1 in node 2 actually represents an indirect infl uence between nodes 1 and 2. In second row of the charts twoway interactions 1 2 (node 1), 23 and 2 1 (node 2), and 3 1, 32 (node 3) are significant. This indicates the structure of type B in Table 4.1. The third row is similar to the first row except that the threeway interaction 2 (1, 3) in node 2 appears significant. Since all the twoway interact ions suggest a ring pattern, the unexpected threeway interaction 2 (1, 3) may be due to a hidden variable (node 4). Node 4 influences the network through node 2. As clearly shown in the simulatio n study, our method determines the directionality of interaction via the definition of strength. Furthermore, we consider threeway or higher order interac tions which assists to understand complex structure. The mutual prediction algorithm [74] constructs an index with value from 1 to 1 using two oscillators each time. Depending on the sign of the index, the directionalities of each oscillator are able to be obtai ned for structures like Type A. Since high order interactions are not modeled, complex structur es with hidden factors are ha rd to be determined by that method. 4.3 Application to Identification of Interaction St ructure Patterns in Real CMP Understanding the interaction structur es has important implication in semiconductor manufacturing process control. Specifically the disc overed interaction patterns among process variables over time could assist to unders tanding the underlying physical mechanisms. In this section we will demonstrate this point using CMP process.
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64 Two CMP process conditions are investigated : diamond conditioners with abrasive size 8 m and 100 m chosen to condition the polis hing pads, respectively. The whole polishing proce ss was conducted on a benc htop CMP tester (model CP4) manufactured by CETR Inc.. The 6inch diameter IC 1000A4 perforated polishing pad (manufactured by Rodel, Inc.) was attached on the rotating bottom platen of the CMP tester. The 2inch copper wafer coupon was attached to the upper polishing head. Cabot 5003 copper polishing slurry was mixed with 2.5% hydrogen peroxide. The slurry was fed into the center of the pad at the rate of 50mL/min. The pressure were set to 2psi both on conditioning and polishing pr ocess, and polishing head ro tating speed was 150 rpm. Figure 4.10 FPVs with diamond particle size 8 m(L) and 100 m(R). We made three replicates under each pad condition and each run lasted 3 minutes. During each polishing process, three FPVs we re collected simultaneously at sampling frequency 20Hz. The coefficient of fricti on could be obtained by gathering shear and normal forces with sensors installed on th e polishing head, and acoustic emission (AE) could be collected through AE sensor on th e back of wafer holde r. A FLIR infrared 0 500 1000 1500 2000 2500 3000 3500 4000 0.5 0 0.5 1 1.5 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 0.2 0.4 0.6 0.8 0 500 1000 1500 2000 2500 3000 3500 4000 296 296.2 296.4 296.6 296.8 297 time index time index time indexAE COF Temperature 0 500 1000 1500 2000 2500 3000 3500 4000 0.5 0 0.5 1 1 5 0 500 1000 1500 2000 2500 3000 3500 4000 0.2 0.3 0.4 0.5 0.6 0.7 0 500 1000 1500 2000 2500 3000 3500 4000 296 296.5 297 297.5 time index time index time indexAE COF Temperature
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65 thermal camera was used to collect thermal data. The focused thermal zone was selected on the polishing pad adjacent to the waferpad interface to reco rd the average temperature. Since the temperature in this interface be tween wafer and polishing pad cannot be obtained directly, the zone we selected in this way migh t best indicate the average process temperature [79]. It should be addressed here that these three process variables could properly represent the chemical mechanical polishing mechanism [80][0]. Figure 10 shows one sample of the original signa ls collected under each process condition. Directly observing original signal s provides limited information. Since the main trend of each FPV w ould not be dramatically altered by interactions, we first remove the trend from collected data through cubicspline smoothing method. Figure 4.11 disp lays the one sample of te mperature data before and after detrending. Figure 4.11 Sample data before and after detrending To understand the dynamics of inter action mechanisms during the whole polishing process, we monitor the proce ss via sliding windows. When determining 0 500 1000 1500 2000 2500 300 0 293 294 295 296 297 0 500 1000 1500 2000 2500 300 0 294.5 295 295.5 296 296.5 297 0 500 1000 1500 2000 2500 300 0 0.2 0.1 0 0.1 0.2 time index time index time inde x original data trend data after detrending t empera t ure
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66 window size and length of ove rlapping period, we intend to (1) maintain a smooth transition between adjacent windows based on the belief that there is no sharp change of polishing mechanism in a short period; a nd (2) avoid large wi ndow size for potential missdetection of changes. We first start wi th large window size to test the polishing condition, and reduce the window size until difference is appare nt between two windows. Finally, our sliding window si ze is determined to be 25 seconds (500 data points). We update the sliding window every 5 seconds (100 data points) for good computing efficiency without losing any information of condition change. The overlapping period is 400 points to have smooth transition between adjacent windows. In each window, the dynamics model was fitted by GLS, and the model adequacy was checked. Figure 4.12 shows one sample of the instantaneous freque ncy extracted from original data versus fitted one in single window, and the residuals are also plotted to check the adequacy of autoregressive terms. It is found that model order n = 2 and autoregressive order l = 5 would have good prediction. Figure 4.12 One sample of model fitting result in a single window Strength of main effects, twoway and th reeway interaction effects is computed in each window. The interaction structures a nd their changes are able to be deduced by 0 70 140 210 280 350 420 470 560 3.8 4 4.2 4.4 4.6 4 3 2 1 0 1 2 3 4 x 103 6 4 2 0 2 4 x 103 time index Instantaneous frequency residual t1residual tFitted Values for COF using GLS
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67 analyzing the bar charts. Fi gure 4.13 shows structure ch anges under polishing condition using 100 m diamond particle size of conditioner. As can be seen, the twoway interactions AEtemperature and COF temperature are signifi cant at the beginning, and they decrease gradually over time. Mean while, the strength of twoway interactions AECOF and temperature COF increase. Other inter action effects seem not significant enough to affect interaction structures. Figure 4.13 Interaction structure analysis: 100 m diamond particle size of conditioner The dynamics of interaction change are represented by network structure change in Fig. 4.13. The solid line with arrow repr esents significant effects, and dash line with 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 COF AE T emperature time index time index time index Main effect 2way Interaction btw COF&AE 2way Interaction btw COF&T 3 way Interaction 3 way Interaction 2way Interaction btw AE&T 2way Interaction btw AE&COF Main effect 3 way Interaction 2way Interaction btw T&AE 2way Interaction btw T&COF Main effect
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68 arrow represents weak effects in term of strength. From the network, we could see temperature had greater impact on AE and COF initially, and COF influenced AE and temperature more afterwards. It is known th at temperature on pa d reflects the heat generated by friction and chemical reaction, AE is related to the material deformation, and COF is determined by the interface properties [80][0]. We therefore interpret the network structure change as follows. The chemical reaction was dominant at the beginning and temperature on was mainly aff ected heat released from wafer surface softening and weakening by slur ry. Afterwards, the mechanical friction played a bigger role because of the nonuniformity on the wafer surface in our polishing process. Figure 4.14 shows the analysis of polishing condition using 8 m diamond particle size of conditioner. Compared with Fig. 13, strength of interaction effects is relatively mild. Initially only the twoway interaction COF temperature is significant. This interaction also fades as the polishing process goes. The twoway interaction AE COF appears relative strong compared to other interacti on effects during the later period of the polishing process. This pattern might be inte rpreted as that the sm aller diamond particle size will generate smoother the polishing pad. This will lead to a weaker mechanical reaction relative to it s chemical reaction.
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69 Figure 4.14 Interaction structure analysis: 8 m diamond particle size of conditioner 4.4 Summary This chapter developed a methodology to analyze the dynamic interaction structures among multiple functional process va riables (FPVs). The analysis started with a phase dynamics model extended from our pr evious work by considering the temporal patterns in FPVs. By analyzing twoway and hi gher order interactions and their strength, directionality and interaction structures can be deduced from bar charts. The ability to 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 COF AE temperature Main effect 2way Interaction btw AE&COF 2way Interaction btw AE&T 3 way Interaction 3 way Interaction 2way Interaction btw COF&T 2way Interaction btw COF&AE Main effect 3 way Interaction 2way Interaction btw AE&T 2way Interaction btw AE&COF Main effect time index time index time index
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70 model high order interaction am ong three or more variables is one unique feature of the proposed method, which enable us to analyze complex structures. The method was validated using fourc hannel van der Pol oscillators. Four important cases of interaction structures were analyzed. The analysis can be extended to network with five more FPVs. To de monstrate the importa nce of understanding interaction structures in semiconductor ma nufacturing, we investigated two polishing conditions. The results show that uncoveri ng the interaction patterns among process variables over time brings new insights about the underlying physical mechanisms. These results also provide a new perspectiv e for process monitoring and diagnosis.
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71 Chapter 5 Conclusions and Future Work 5.1 Conclusions Correlated FPVs usually contain very rich information, and investigation of the interaction among these FPVs could unveil the physical phe nomenon through integrating engineering knowledge and statis tical approaches. In this research, in teractions of FPVs are studied to improve the manufacturing processes. Th e proposed methodology brings more insights for understanding the interacti on mechanism during processes. The major contributions of this dissertation are summarized as follows: Experimentally and theo retically study interaction patterns of FPVs. A method based on FCCA was borrowed to extract the interaction patterns between FPVs under different process condi tions and experiments were conducted to verify the relationship between the in teraction patterns and va rying process conditions. Establish nonlinear dynamics model fo r complex interactions among multiple FPVs. A novel nonlinear dynamics model was developed to describe the interaction by considering not only the complex spatial patterns among multiple FPVs signals, but also the temporal pa tterns within individual signals. The proposed model formulation provides better insight into inte raction mechanism, such as interaction order and strength. This improved understanding will establish
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72 a solid base for effective detection and diagnosis of failure modes during processes. Analysis of complex interacti on structures among multiple FPVs An approach with extended nonlinear dyna mics model is proposed to characterize process conditions and complex interaction struct ure patterns are co ncluded accordingly for interpretation of process mechanism. This approach could bring physical understanding of processes, and prov ide guidance for process diagnosis. 5.2 Future Work The purpose of this dissertation is to develop a methodology to study interaction of FPVs for process improveme nt. It is known that the interaction is a common phenomenon existing in many fields. Besides the semiconductor manufacturing processes, healthcare is another applicable area. For inst ance of Parkinsons disease, the interaction of many neurons results in th e tremor activity. Yet diagnosi s of medical problems with complex physiologic interactions often relies on either a trialanderror approach or expensive medical procedures not widely availa ble. Thus, it is highly desirable to develop generic detection and diagnosis techniques for the improved understa nding of physiologic interactions and reduction of medical errors in the treatmen t of a wide range of medical problems. Following case show certain pr omise as an application of proposed methodology to diagnose disease. To validate the approach, in vivo experi ments have been conducted on adult cats to study interaction of physiologic signals under a variet y of perturbations that alter breathing and cardiovascular parameters. In th is experiment, process variables phrenic nerve signal and blood pressure are collected simultaneously. Figur e 5.1 shows snapshots
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73 of oscillatory phrenic nerve signal and blood pressu re of an adult cat before and after introducing intermittent hypoxia, a severe exte rnal intervention (sampling frequency: 200Hz). It could not be judge d that whether hypoxia occurs on this adult cat only based on these original signals. Figure 5.1 Oscillatory integr ated phrenic nerve signal and blood pressure The previously proposed nonlinear dynamics model with inco rporation of GLS might be employed to detect the change s before and after hypoxia. Similarly as semiconductor manufacturing processes, thro ugh model selection, determination of interaction order and interaction strength de finition, we may draw some conclusion to discover the root causes of hypoxi a, and provide better guidan ce for doctors to reduce the medical errors. 0 50 100 2000 1800 1600 1400 1200 1000 1 2 3 1000 500 0 500 0 50 100 2500 2000 1500 1000 500 1 2 3 2000 1500 1000 500 0 Blood Pressure Blood Pressure Phrenic Nerve Signal Phrenic Nerve Signal Time (sec) Time (sec) Time (sec) Time (sec) Phrenicnerve signal Phrenicnerve signal Blood pressure Blood pressureBefore Hypoxia After Hypoxia 0 50 100 2000 1800 1600 1400 1200 1000 1 2 3 1000 500 0 500 0 50 100 2500 2000 1500 1000 500 1 2 3 2000 1500 1000 500 0 Blood Pressure Blood Pressure Phrenic Nerve Signal Phrenic Nerve Signal Time (sec) Time (sec) Time (sec) Time (sec) Phrenicnerve signal Phrenicnerve signal Blood pressure Blood pressure 0 50 100 2000 1800 1600 1400 1200 1000 1 2 3 1000 500 0 500 0 50 100 2500 2000 1500 1000 500 1 2 3 2000 1500 1000 500 0 Blood Pressure Blood Pressure Phrenic Nerve Signal Phrenic Nerve Signal Time (sec) Time (sec) Time (sec) Time (sec) Phrenicnerve signal Phrenicnerve signal Blood pressure Blood pressureBefore Hypoxia After Hypoxia
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74 In the future, more medical problems are expected to be diagnosed through interaction studies and our approaches. For example, we are attempting to identify obstructive sleeping apnea (OSA) from patients. Hence, stud ies of interaction based on nonlinear dynamics have shown promise to understand the interac tion mechanisms for future complex disease diagnosis. Besides, pr ocess control in many other fields may also be attempted through our proposed methodology.
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About the Author Xi Zhang received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, China, in 2006. He is curren tly working towards the Ph.D. degree at the University of South Florida, Tampa. While in the Ph.D. program at the University of South Florida, Mr. Zhang focused on the res earch of modeling and diagnosis for process quality control on semiconductor manufacturing sy stems. He also has two publications in IEEE transactions on Semiconductor Manufactur ing and one journal paper submitted. He also made several paper presentations at a nnual meetings of INFORMS. He is a member of INFORMS, IIE.
