IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY—PART C, VOL. 20, NO. 4, OCTOBER 1997
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`295
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`Equipment Fault Detection Using Spatial Signatures
`
`Martha M. Gardner, Jye-Chyi Lu, Ronald S. Gyurcsik, Member, IEEE, Jimmie J. Wortman, Senior Member, IEEE,
`Brian E. Hornung, Member, IEEE, Holger H. Heinisch, Student Member, IEEE, Eric A. Rying, Student Member, IEEE,
`Suraj Rao, Joseph C. Davis, Member, IEEE, and Purnendu K. Mozumder, Senior Member, IEEE
`
`Abstract—This paper describes a new methodology for equip-
`ment fault detection. The key features of this methodology are
`that it allows for the incorporation of spatial information and
`that it can be used to detect and diagnose equipment faults si-
`multaneously. This methodology consists of constructing a virtual
`wafer surface from spatial data and using physically based spatial
`signature metrics to compare the virtual wafer surface to an
`established baseline process surface in order to detect equipment
`faults. Statistical distributional studies of the spatial signature
`metrics provide the justification of determining the significance
`of the spatial signature. Data collected from a rapid thermal
`chemical vapor deposition (RTCVD) process and from a plasma
`enhanced chemical vapor deposition (PECVD) process are used to
`illustrate the procedures. This method detected equipment faults
`for all 11 wafers that were subjected to induced equipment faults
`in the RTCVD process, and even diagnosed the type of equipment
`fault for 10 of these wafers. This method also detected 42 of 44
`induced equipment faults in the PECVD process.
`
`Index Terms— Equipment fault diagnosis, process improve-
`ment, simulation, statistical metrology.
`
`I. INTRODUCTION
`
`The use of site-specific models has been shown to have
`better sensitivity, with respect to spatially dependent process
`variations, than mean-based models [1]. However, detection
`of equipment faults identified from models based on data
`from different sites can have inconsistent results; i.e., some
`site models may detect a certain type of equipment fault,
`while other site models do not [1]. Saxena, et al. [2] have
`used a monitor wafer controller (MWC) to fix this to some
`degree. It has also been shown that
`the use of a virtual
`wafer surface, rather than specific sites on a wafer, captures
`even more information about the spatial signatures generated
`from different equipment conditions [3]. Kibarian and Strojwas
`[4] have also developed models which account for spatial
`dependencies and shown how the models can be used to
`separate spatial dependencies from other causes.
`The detection and diagnosis of equipment faults in semi-
`conductor processes is usually a two step procedure. Detection
`refers to the identification of the occurrence of an equipment
`fault, whereas diagnosis refers to the classification of equip-
`ment faults. Faults are detected using one method. Then faults
`are classified using another method. Current research in the
`literature has concentrated on equipment fault diagnosis, rather
`than the detection of the existence of equipment faults. For
`example, pattern recognition techniques including statistical
`discriminant analysis techniques [1], fuzzy logic techniques
`[5], and neural networks [6] have been used for diagnosis
`purposes. Hu et al. [7], Butler and Stefani [8], and Bombay
`and Spanos [9] have applied empirical (or semi-empirical)
`polynomial modeling techniques to relate process outputs
`to process settings, and May and Spanos [10] have used
`evidential reasoning to integrate in-line, off-line, and main-
`tenance data for fault diagnosis. However, methods such as
`statistical discriminant analysis do not make use of the spatial
`information and the physical knowledge of equipment faults.
`The equipment fault detection methodology described in
`this paper is unique not only in that it incorporates the use
`of integrated spatial information in a virtual wafer surface, but
`also in that it can be used to detect and classify equipment
`faults at the same time. The main focus of this paper is using
`the spatial signatures of the differences between observed and
`expected virtual wafer surfaces to construct physically based
`metrics which can be used to detect and diagnose various types
`of equipment faults. When establishing an equipment fault sig-
`nature library, it would be ideal to have experiments conducted
`to model wafer spatial measurements at process conditions
`without faults and with certain known faults; however, his-
`torical data on existing faults may also be used. Using the
`1083–4400/97$10.00 ª
`
`EQUIPMENT faults are often the cause of major variations
`
`in semiconductor manufacturing processes. Considering
`the expense of processing, these variations can cause dramatic
`yield losses [1]. Traditionally, the mean or signal-to-noise ratio
`of the wafer surface data is modeled, and the resulting model is
`used to detect equipment faults according to statistical process
`control (SPC) techniques; however, as wafer sizes increase
`and film thicknesses are reduced, the use of integrated spatial
`information will have a greater impact on detecting equipment
`faults.
`
`Manuscript received March 24, 1997; revised September 29, 1997. This
`work was supported in part by Texas Instruments, Inc. and the NSF Engineer-
`ing Research Centers Program through the Center for Advanced Electronics
`Materials Processing, Grant CDR 8721505,
`the Semiconductor Research
`Corporation, SRC Contract 94-MP-132, and the SRC SCOE program at
`NCSU, SRC Contract 94-MC-509.
`M. M. Gardner was with the Department of Statistics, North Carolina
`State University, Raleigh, NC 27695 USA and is now with General Electric,
`Nishayuna, NY 12309 USA
`J.-C. Lu are with the Department of Statistics, North Carolina State
`University, Raleigh, NC 27695 USA.
`R. S. Gyurcsik is with the Semiconductor Research Corporation, Research
`Triangle Park, NC 27709 USA.
`J. J. Wortman, H. H. Heinisch, and E. A. Rying are with the Department
`of Electrical and Computer Engineering, North Carolina State University,
`Raleigh, NC 27695 USA.
`B. E. Hornung is with Motorola, Austin, TX 78721 USA.
`S. Rao, J. C. Davis, and P. K. Mozumder are with Texas Instruments, Inc.,
`Dallas, TX 75265 USA.
`Publisher Item Identifier S 1083-4400(97)09163-8.
`
`1997 IEEE
`
`Applied Materials, Inc. Ex. 1012
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`(a)
`
`(b)
`
`Fig. 1. Equipment fault detection and diagnosis chart.
`
`experimental data, one can construct physically based signa-
`ture metrics to detect and identify equipment faults. When the
`basic faults are understood, new faults can be added into the
`study. Fig. 1 shows a flow chart of all the steps in the process.
`If a certain type of fault is known to have a specific shape,
`then classification of faults can also be verified by comparing
`a newly fitted surface to the known fault surface. In this case,
`by treating the fault surface as the “target,” the methodology of
`spatial signature metrics can be used to statistically compare
`the newly fitted surface to this “target” to determine if the
`newly fitted surface belongs in this fault class.
`Section II describes the equipment fault detection method-
`ology using spatial signatures in detail. Sections III and IV
`provide illustrating examples from experiments conducted at
`North Carolina State University (NCSU) and Texas Instru-
`ments, Inc. (TI), respectively. Section V draws conclusions
`from this study and points to potential future work.
`
`II. FAULT DETECTION METHODOLOGY
`Fig. 2(a) and (b) show how equipment faults can be manifest
`in the spatial response of the process. Fig. 2(a) shows the gate
`oxide thickness surface of a wafer that was processed under
`fault-free conditions. Fig. 2(b) shows the gate oxide thickness
`surface of a wafer processed under known equipment faults.
`and
`represent the
`and
`distances from the center of
`the wafer. Not only is there an apparent decrease in thickness
`between the two surfaces, but also a change in spatial pattern.
`The next five subsections present the new methodology of
`using spatial signatures to detect equipment faults:
`1) modeling wafer surface data using thin-plate splines;
`2) estimation of the baseline or “fault-free” surface;
`3) construction of physically based signature metrics for
`comparing wafer surfaces;
`4) estimation of the statistical distribution of metrics;
`5) use of spatial metrics for equipment fault detection.
`
`Fig. 2. Fitted wafer surfaces from wafers processed (a) with no equipment
`faults and (b) with known equipment faults.
`
`A. Modeling Wafer Surface Data Using Thin-Plate Splines
`While recognizing that other modeling methods are avail-
`able, this study uses thin-plate splines to model the virtual
`wafer surface. A virtual wafer surface model of spatial process
`behavior is less sensitive to the position of measurement sites,
`measurement error, and angular orientation than techniques
`focusing on individual data points [11]. The thin-plate spline
`can be viewed as a multi-dimensional extension of the cubic
`smoothing spline. Although splines, in general, are constrained
`to pass through the knots of the function [e.g., gate oxide
`thickness measurements at ( ,
`) distances from the center of a
`wafer], the thin-plate spline attempts to produce the smoothest
`curve possible between the knots and, therefore, does not
`have the requirement that the surface actually pass through the
`knots. The estimator of the thin-plate spline
`is the minimizer
`of the following penalized sums of squares [12]:
`
`(1)
`
`is the
`where the first term represents the lack of fit,
`roughness penalty function, and
`is the spline smoothing
`parameter. For this study, thin-plate spline fittings were formed
`by using a collection of routines called FUNFITS written for
`use in the S-plus statistical software [13]. A thin-plate spline
`can then be used to predict the response at any location on the
`wafer and thus can be used to predict the entire wafer surface.
`In this study,
`is set to be very small (0.001) which gives
`more of an interpolating surface as recommended by Davis
`et al. [3].
`
`B. Estimation of the Target Surface
`A target surface needs to be specified for evaluating equip-
`ment performance. In many cases, the target surface may be
`known; e.g., a non uniformity study where the target thickness
`is 70 ˚A at all locations. However, due to process effects as
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`Individual wafer surface from wafer processed under fault-free
`Fig. 4.
`conditions in a PECVD experiment.
`
`(a)
`
`(b)
`
`Fig. 3. Replicate wafer surfaces from wafers processed under fault-free
`conditions in a RTCVD experiment.
`
`Fig. 5. Two replicates from fault-free conditions averaged to form target
`surface.
`
`a result of equipment design, the wafer surface may not be
`flat even though there is no equipment fault. Assuming a flat
`target surface in this situation may lead to incorrect equipment
`fault detection. Thus, if the experimental wafer surfaces under
`equipment fault-free conditions are not flat, then these surfaces
`should be used as the target surface rather than a constant. For
`example, the two surfaces shown in Fig. 3(a) and (b) are the
`surfaces from two replicates at the fault-free condition in a
`RTCVD experiment conducted at NCSU and have a nonlinear
`pattern. Fig. 4 shows a surface from the fault-free condition
`in a PECVD silicon nitride experiment conducted at Texas
`Instruments, and this surface has a linear pattern. However,
`none of the surfaces shown in Fig. 3(a) and (b) or Fig. 4 reflect
`a constant baseline process surface.
`In addition, the target surface should be validated after
`preventive maintenance or any other procedure which alters
`the tool. The proposed methodology can be used to determine
`if any significant changes in the tool have occurred. If no
`significant changes have occurred, then the new data can be
`used in conjunction with the historical data to update the target
`surface. The following method allows for a target surface to
`be estimated from data collected from fault-free runs.
`Data is collected from wafers under the equipment fault-
`free condition to obtain a good estimate of the target surface. If
`there is slow drift, and this slow drift is considered to be typical
`phenomenon, then the wafers are still considered fault-free.
`Statistical outlier diagnosis can be used to screen the data. Af-
`ter fitting the spline surface to each set of wafer measurements,
`
`the target surface is obtained by averaging location specific
`parameter estimates from the individual spline equations. As
`an example, by averaging the two surfaces in Fig. 3(a) and (b)
`from the RTCVD experiment at NCSU, we obtain the target
`surface as shown in Fig. 5. A randomization procedure for use
`with wafer surfaces processed under the fault-free condition is
`currently being studied to better incorporate wafer-to-wafer
`variation in the proposed methodology, but this randomization
`procedure is beyond the scope of this paper.
`A typical method for deriving the target surface is to first
`average the data collected from specific sites on “replicated”
`wafers at the fault-free condition and then fit a spline surface
`to the averaged data to create the target surface. However, this
`approach averages the data at
`sites, where
`is the number
`of data collected on a wafer, and requires that data be collected
`at the same sites on all wafers, as well as does not take into
`account any wafer-to-wafer variation. The approach described
`in the previous paragraph averages the spline estimates, which,
`in the intuitive sense, averages the spline surface at all possible
`sites. In the spatial signature metrics developed in next section,
`a grid of close to 700 sites is used for prediction.
`If the purpose of the statistical test is to compare an average
`of wafer surfaces to a target, then averaging the wafer data
`first would be appropriate. However, our concentration in this
`work is to compare the spatial surface of a single wafer to
`the target, subject to random variation, and the variance is
`underestimated if the data from the target wafers are averaged
`before the spline is fit. For example, if the spline fits from three
`replicate wafers have individual variances
`,
`, and
`, then
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`
`averaging the variance after the splines are fit yields a variance
`of
`. Now if we let
`represent the vector
`of averaged data, then since the variance of a mean
`is
`where
`is the variance of
`, by averaging the data first, an
`extra factor of 1/3 will be introduced into the variation before
`the spline is ever fit. Thus, the variance of the spline fit when
`averaging the wafer data first will be (
`) times smaller than
`the variance of the spline fit to the individual wafer data, where
`is the number of replicated wafers.
`Another alternative is to treat data from all wafers pro-
`cessed under the fault-free condition as coming from a single
`wafer, then construct a spline to estimate the target surface.
`Although this method can capture variation at a particular site
`without deflating it, this method also loses individual wafer
`characteristics.
`
`C. Construction of Physically Based Spatial Signature Metrics
`Different equipment faults may produce distinct spatial
`signatures. For instance, an equipment fault may affect only
`a specific region of the wafer surface rather than the entire
`wafer surface. In this case, a certain performance evaluation
`metric may better detect this particular type of equipment fault.
`It is also possible that several metrics may have to be used
`simultaneously to detect certain types of faults. Understanding
`the physical processes that create faults, and their resulting
`signatures, also greatly aids in constructing and deciding what
`types of evaluation metrics to use. Four metrics are presented
`below as examples of how different metrics may be needed for
`detecting certain fault signatures. The metrics are extensions
`of the uniformity metrics presented by Davis et al. [3] with
`an expected surface used as the target surface. All metrics dis-
`cussed here are based on loss functions. For all these metrics,
`denotes a newly fitted thin-plate spline surface,
`denotes
`the target surface, and
`denotes the wafer surface region.
`The quadratic and absolute loss functions are commonly
`used in many fields to quantify the penalty from departing
`from the target. The first
`two metrics used in this work
`are a squared deviate from target metric and an absolute
`value deviate from target metric. Both statistics are general
`metrics used to quantify the surface difference (
`) and are
`nonlinear functions of the error volume between two spline
`surfaces. The metrics are calculated as
`
`(2)
`
`The squared metric,
`, penalizes much more than the
`absolute metric,
`, with respect to larger departures from
`the target. Both metrics cover the entire wafer surface and
`place equal weights on information at all wafer sites. In
`addition, both metrics yield the same equipment fault detection
`results in this study.
`The following metric is an example of a metric that can be
`used to detect an equipment fault that leads to a thicker wafer
`surface. This metric is calculated as
`
`(3)
`
`where
`) could be any functions. For example,
`(
`could be the squared error loss function and
`set to 0. This
`example would only penalize surfaces thicker than the target.
`Another example is to place different weights on the penalties
`for
`than for
`. Again, understanding the reasons for
`getting equipment faults and their resulting spatial signatures
`plays an important role in the selection of the
`functions.
`Another type of metric that should be considered is one
`which allows for different regions of the wafer surface to
`be weighted differently. For instance, error in the center of
`the wafer may be of more importance than error toward the
`edge of the wafer. Also, certain equipment faults may cause
`defects, such as a thinner surface, in specific regions of the
`wafer surface rather than the entire wafer surface. An example
`of a metric that weights wafer surface regions differently is
`calculated as
`
`(4)
`
`where
`denotes the number of nonoverlapping regions, and
`denote the weight and penalty functions for the th
`and
`region, respectively. This metric has the potential to be very
`useful, particularly in the stage of equipment fault diagnosis,
`since it is more general than the other suggested metrics. In
`fact, the previous metrics may be considered as special cases
`of this metric.
`
`D. Estimation of Signature Metrics
`The calculation of the metrics discussed in Section II-C
`involves integration of the difference of two spline surfaces
`over certain regions. This integration can be done by using a
`numerical integration technique where the fitted surface
`is
`evaluated on an
`grid (with points outside the radius of
`the wafer removed) and the evaluated results, e.g.,
`, are
`then summed for each of the
`grid points and multiplied
`by the area of one of the grid elements. The metrics may be
`approximated as [3]
`
`metrics
`
`(5)
`
`where
`is the loss function incorporated into the metric,
`vector of the target,
`is an
`matrix
`is an
`of thin-plate spline coefficients for the measurements,
`is
`an
`vector of the measurements,
`is the number of
`measurements taken on the wafer, and
`is the area of one
`of the grid elements. This approximation was shown to have
`good results for
`30 [3]. Thus,
`30 is used in the
`experimental examples presented in this paper.
`
`E. Use of the Signature Metrics in Equipment Fault Detection
`If the metrics indicate that the surface of a newly processed
`wafer is statistically significantly different from the target
`wafer surface, then the conclusion is that an equipment fault
`has occurred. Therefore, the null distributions of the metrics
`should be studied in order to set up the “cut-off value” for
`determining if there is a significant difference between a newly
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`
`TABLE I
`WAFER LABELS FOR WAFERS PROCESSED UNDER
`EACH COMBINATION OF EXPERIMENTAL CONDITIONS
`
`fitted surface and the target wafer surface. In other words, the
`distributions of the metrics under the “fault-free” condition are
`needed to determine the “cut-off values” in the tail(s) of the
`distributions for a specified level of significance.
`An analytical approach based on standard statistical asymp-
`totic normal approximation theory was first considered. Ap-
`proximation theory cannot be applied in this study since
`traditional distributional spline results are for the independent
`identically distributed case; however, as
`(the number of
`spatial measurements) goes to a very large number, many
`devices are being sampled on a fixed size wafer, and the data
`become more dependent because of spatial correlations. This
`resulting increasing dependence is called infill-asymptotics
`[14]. An alternate Bayesian (simulation) approach can be taken
`to determine the null distribution of the metrics using the
`following steps.
`1) According to a procedure given in Green and Silverman
`[12], assuming a Gaussian prior distribution, the poste-
`rior distribution of the spline surface
`has the following
`multivariate normal distribution [12]:
`
`MVN
`
`(6)
`
`where
`is calcu-
`is the vector of fitted values,
`lated as: (the residual sums of squares about the fitted
`curve)/equivalent error degrees of freedom, and
`is the projection matrix which maps the vector of ob-
`served values to their predicted values. Since there were
`multiple wafers processed independently at the baseline
`conditions in this study, the averages of the
`’s and
`’s from all baseline wafers were used in (6).
`2) The following parametric bootstrapping approach can
`be used to simulate independent observations from the
`null posterior distribution of the metric. First, 5000 sets
`of
`observations are simulated from the multivariate
`normal model (6). A spline surface is fitted to each
`set of observations, and the spatial signature metrics
`are calculated using (2)–(5). As a result, 5000 indepen-
`dent observations are obtained from the null posterior
`distribution of each signature metric.
`3) With a pre-specified significance level,
`0.01 (or
`0.05), a “cut-off” value is defined as the 99th (or 95th)
`percentile from the null distribution of a specific metric.
`If the calculated metrics (2), (3), or (4) for a newly fitted
`surface are larger than their respective “cut-off values,” it
`is statistically significant at level
`that the wafer was not
`processed under fault-free conditions. The decision rule
`is still applicable even if the sampling scheme changes
`since the target surface remains the same. The decision
`rule is selected to balance the Type I error (false positive)
`
`and the Type II error (false negative) as follows: Fix
`Prob(Type I error) below a chosen level, e.g., 0.01, and
`select the test that minimizes
`Prob(Type II error).
`The choice of
`is up to the discretion of the practitioner.
`In this study, more conservative “cut-off” values were
`desired, so
`0.01 was selected.
`
`III. EXPERIMENTALVERIFICATION WITH
`NCSU LABORATORY EQUIPMENT
`
`A. Design of Experiment and Data Collection
`To test the proposed methodology, a laboratory experiment
`was conducted at NCSU where the following two types of
`equipment faults were induced in a prototype RTP (rapid ther-
`mal processing) single wafer system: lamps burning out and
`a miscalibrated SiH /Ar mass flow controller. The response
`measured was SiO thickness. There were 15 wafers available
`for the experiment. Preliminary experiments, in which one
`lamp at a time was removed, were performed to decide on how
`many lamps should be disengaged during an experiment. The
`removal of a single lamp caused a marked decrease in oxide
`thickness. For example, when a side lamp was disengaged,
`a 13.5 ˚A decrease in average oxide thickness was observed.
`When a bottom center lamp was disengaged, the average oxide
`thickness decreased by 10.22 ˚A.
`The final experimental design used to induce equipment
`faults was an unbalanced 3 design. While the N O flow was
`held constant at 500 sccm, three 10% SiH /Ar flow rates of
`20, 25, and 30 sccm were used for low, medium, and high flow
`rates, respectively. Table I shows all possible experimental
`conditions along with the wafer labels of each wafer processed
`under each combination of conditions. The baseline “fault-
`free” state was the condition with all lamps working and
`with medium flow rate. Three replicates were allocated for
`estimating the target surface and for constructing the posterior
`distributions of the spatial signature metrics. For each wafer,
`the oxide thickness was estimated at 17 points. The sampling
`scheme was designed to cover the wafer regions evenly as
`shown in Fig. 6.
`The oxide thickness was estimated by measuring the
`–
`characteristics of capacitors located at the measurement points
`using a Keithley 595 Quasistatic Meter and a Keithley 590
`– Analyzer. The gate oxide thickness was then extracted
`–
`data while accounting for polysilicon depletion
`from the
`and quantum effects using a program written by Hauser [15].
`Unfortunately, data could not be collected from wafer 5 (low
`flow rate and a side lamp out) since the wafer was damaged
`during processing. However, since the goal of this study is not
`an experimental design analysis, this was not a major concern.
`
`B. Application of the Proposed Methodology
`to the Experimental Data
`The laboratory experiment had three runs processed under
`the baseline condition. One of the wafers appeared to be
`consistently thicker than the other two wafers at all sampled
`sites. The data from this wafer was excluded from the esti-
`mation of the target surface, but was set aside for verification
`
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`Metrics 1–3 are all general metrics used to determine whether
`or not an equipment fault has occurred. Metric 3 incorporates
`specification limits on the target surface to increase robustness
`to a given noise level in the data. It was noted that the
`experimental runs with high flow rate and all lamps working
`yielded gate oxide thicknesses much thicker than the target
`surface. Metric 4 was designed to specifically detect this type
`of equipment fault. It was also noted that whenever a lamp was
`disengaged, the resulting surface was much thinner. Metric 5
`was designed to specifically detect this type of equipment fault.
`The null posterior distributions of each of these metrics were
`simulated using the average predicted surface of wafers 1 and
`11 as the target surface. 5000 observations were simulated for
`each of the five metrics using parametric bootstrapping from
`a MVN
`is the average of the predicted
`values at the actual 17 measurement locations using the thin-
`plate spline fittings for wafers 1 and 11, and
`is
`the average of the covariance matrices from the thin-plate
`spline fittings for wafers 1 and 11. A set of 5000 simulated
`observations was then used to calculate each metric resulting
`in a series of 5000 independent observations from the null
`distribution of each metric. The resulting null distributions for
`metrics 1–5 are shown in Fig. 7.
`Empirical
`“cut-off values” for each metric were
`then determined from the null distributions of the metrics, i.e.,
`the point from the null distribution where 1% of the simulated
`observations fall above that point. Each metric was also calcu-
`lated for each wafer (excluding the two wafers used to estimate
`the target surface). These calculated values are compared to
`the “cut-off values” in order to determine whether or not a
`fault is detected. Recall that the experimental combination
`of all lamps working and a medium flow rate represent the
`fault-free condition, with all other experimental combinations
`representing fault conditions. The general metrics (Metrics
`1–3) detected equipment faults for all wafers which were
`subjected to induced equipment faults (wafers 2–4, 6–10, and
`13–15). Metric 4, which was designed to detect equipment
`faults that caused a thicker than target surface, only detected
`equipment faults for the 2 wafers with a high flow rate, but all
`the lamps working (wafers 3 and 8) and, therefore, effectively
`detected a specific type of equipment fault: high flow rate
`only. Metric 5, which was designed to detect equipment faults
`that caused a thinner than target surface, detected equipment
`faults for all wafers subjected to a disengaged lamp, no matter
`what the flow rate was. It was also observed that the metric 5
`calculations (25.991, 31.827, and 21.46) for a disengaged side
`lamp (wafers 6, 9, and 13) were much higher than the metric
`5 calculations (2.346, 10.781, 10.766, 2.655, and 1.052) for
`the wafers with the disengaged bottom lamp (wafers 2, 4,
`10, 14, and 15). Thus, metric 5 also effectively detected a
`specific type of equipment fault: side lamp or bottom lamp
`burning out. No faults were detected for wafer 12 by any
`of the metrics, which was expected since this wafer was
`the replicate from the fault-free condition withheld from the
`estimation of the target surface for verification purposes. Table
`II shows the numerical results for each metric along with the
`0.01 level of significance
`corresponding “cut-off value” at
`for each metric. One-tailed tests are used here because of the
`
`Fig. 6. Sampling scheme.
`
`purposes in later wafer surface comparisons. The remaining
`two wafers had thin-plate splines fitted to their gate oxide
`thickness measurements, and the surfaces were predicted on a
`30
`30 grid of equally spaced points which was then trimmed
`to a circle with radius equal to 2 in. The predicted points from
`the two wafers were then averaged across the remaining 692
`locations to construct the target surface.
`is
`As stated in the introduction, experimental data that
`collected at fault conditions will help identify physically based
`spatial signature metrics for equipment fault detection. General
`metrics are used to detect the presence of an equipment fault,
`and specific metrics are used to capture distinct shapes of
`spatial surfaces to detect specific fault patterns. Thus, the
`following general and specific metrics were designed:
`
`1) M
`
`2) M
`
`3) M
`
`4) M
`
`5) M
`
`d
`
`d
`
`d
`
`(squared metric)
`
`(absolute value metric)
`
`˚A
`
`otherwise (spec limits metric)
`
`d
`
`d
`
`if
`
`(square above—absolute value below metric)
`
`d
`
`if
`
`d
`
`if
`
`(square below—absolute value above metric).
`
`Applied Materials, Inc. Ex. 1012
`Applied v. Ocean, IPR Patent No. 6,836,691
`Page 6 of 10
`
`

`

`GARDNER et al.: EQUIPMENT FAULT DETECTION USING SPATIAL SIGNATURES
`
`301
`
`TABLE II
`RTCVD CALCULATED METRICS AND “CUT-OFF” VALUES
`
`(a)
`
`(b)
`
`(c)
`
`(d)
`
`(e)
`
`Fig. 7. Null distributions of metrics for RTCVD experiment: (a) Squared
`deviate from target metric, (b) absolute deviate from target metric, (c)
`spec limits metric, (d) square above—absolute below metric, and (e) square
`below—absolute above metric.
`
`physically based design of the metrics, particularly metrics
`4 and 5 which are used to detect thicker than target surfaces
`and thinner than target surfaces, respectively. The shaded areas
`represent the wafers for which equipment faults were detected,
`i.e., wafers for which the metric values are greater than the
`“cut-off values.”
`
`IV. EXPERIMENTAL VERIFICATION
`WITH PECVD SILICON NITRIDE DATA
`In the previous section an example was given where metrics
`were designed to detect certain types of equipment faults
`when the faults were known to have a certain type of spatial
`signature. However,
`in practice,
`there may be too many
`potential faults, and the relationships between the faults and
`the spatial signatures may not be clear. Even in this case,
`the general metrics can still be used to detect the presence of
`
`equipment faults. To illustrate this, the proposed equipment
`fault detection methodology was also applied to experimental
`data collected at Texas Instruments, Inc. An experiment was
`conducted for the purpose of developing equipment models for
`use in a diagnosis system prototype [1]. The data was collected
`from a six factor central composite design experiment with
`45 combinations of equipment conditions (without replicates)
`plus an additional five runs replicated at the center point.
`The six factors were pressure,
`flow, sum of silane and
`ammonia flows (SiH
`NH ), ratio of silane and ammonia
`flows (SiH /NH ), ratio of RF power to electrode g