METHOD OF DETERMINING ETCH ENDPOINT USING PRINCIPAL COMPONENTS ANALYSIS OF OPTICAL EMISSION SPECTRA

Abstract: A method is provided for determining an etch endpoint (855). The method includes collecting intensity data (820) representative of optical emission spectral wavelengths during a plasma etch process (810). The method further includes calculating Scores (845) from at least a portion of the collected intensity data (820) using at most first, second, third and fourth Principal Components derived from a model (835). The method also includes determining the etch endpoint (855) using Scores (845) corresponding to at least one of the first, second, third and fourth Principal Components as an indicator (835) for the etch endpoint (855).

Description: METHOD OF DETERMINING ETCH ENDPOINT USING PRINCIPAL COMPONENTS ANALYSIS OF OPTICAL EMISSION SPECTRA SPECIFIC REFERENCE TO PROVISIONAL APPLICATION 5 The present application claims priority to provisional application Serial No 60/152 897 (Attornev Docket No 2000 010190) Filed September 8 1999 and to provisional application Serial No 60/163 868 (Attornev Docket No 2000 015390), filed November s 1999 the entire texts and figures oi w hich are incoφorated herein by reference without disclaimer TECHNICAL FIELD 10 This invention relates generally to semiconductor fabrication technologv, and more particularly to monitoring etching processes during semiconductor fabrication using optical emission spectroscopy BACKGROUND ART Typicallv during semiconductor processing, an etching process such as a reactive ion etch ( RI E) process is empioved for etelling tine line patterns in a silicon wafer RIE involves positioning a masked wafer in a chamber l i that contains a plasma The plasma contains etchant gases that are dissociated m a radio trequencv ( RF) field so that reactive ions contained in the etchant gases are vertically accelerated toward the wafer surface I he accelerated reactive ions combine chemicallv with unmasked material on the water surface As a result volatile etch products are produced During such etching single or multiple lavers ot material or films mav be removed Such material includes tor example, silicon dioxide ( SIOT ). polysilicon ( poly) and silicon nitride (Sι-N,) fndpoint 20 determination or detection refers to control of an etch step and is useful in etching processes in general and in RIE processes in particular As a layer of unmasked material is etched, the volatile etch products are incorporated into the plasma As the RIE process approaches the interlace or end ot the layer being etched the amount of volatile etch product found in the plasma decreases since the amount ot unmasked material being etched is reduced due to the etching 2 The amount of volatile etch product in the plasma mav be tracked to determine the endpoint ot the RIE process In other words the depletion or reduction in the amount ot volatile etch product in the plasma durιn« the RIL process tvpicaliv can be used as an indicatoi tor the end of the etching process It is also possible to track a reactive species such as one of the etchant or input gases used to etch a laver of material As the laver is etched the reactive species will be depleted and relatively low concentrations ot the 30 reactive species will be found in the plasma However, as more and more of the laver is consumed the reactive species will be found in the plasma in increasingly higher concentrations A time trace of the optical em issions from such a reactive species w ill show an increase in intensity as the laver is etched awa\ Tracking the intensity ot a wavelength for a particular species using optical emission spectroscopv (OES) mav also be used for endpoint determination or control of an etch process such as an RIE process 35 Conventionally OES has been used to track the amount of either volatile etch products or reactive species as a function of film thickness These techniques examine emissions from either the volatile etch products or reactive species in the plasma As the film interface is reached duπng etching the emission species related to the etch of the film will either decrease in the case ot volatile etch products or increase, in the case of reactive species More specifically, during an RIE process, plasma discharge materials, such as etchant neutral and 40 reactive ions in the plasma are continuously excited by electrons and collisions giving off emissions ranging from ultraviolet to infrared radiation \n optical emission spectrometer diffracts this light into its component wavelengths Since each species emits light at a wavelength characteristic only ot that species, it is possible to associate a certain wavelength with a particular species, and to use this information to detect an etch endpoint As an example, in etching SIOT with CHF3, carbon combines with oxygen from the water to form carbon monoxide (CO) as an etch product It is known that CO emits light at a w avelength of 451 nm, and that this wavelength can be monitored for detecting the endpoint for such an etch When the oxide is completely etched there is no longer a source of oxygen and the CO peak at 451 nm decreases, thus signaling an etch endpoint In the above example, it is known that light emitted from CO at a wavelength ot 45 1 nm would be used for etch endpoint determination or detection However, such specific wavelength information is generally unavailable, and it has been a formidable task to determine or select an appropriate wavelength to use for accurate etch endpoint deter ination or control This difficulty exists because ot the numerous possibilities for emissions In other words, any molecule may emit light at a multitude of different wavelengths due to the many transition states available for Therefore, given the process, the gases utilized and the material being etched, it is tvpicallv not readily known which wavelength in the spectrum to monitor for etch endpoint determination or control In this regard the OES spectrum tor a typical RIE etch, tor example ma\ be composed ot hundreds, or even thousands, of wavelengths in the visible and ultraviolet bands Additionally there is a trend towards using high-densitv plasma sources to replace RIE One example is in the use ot a high-densitv inductivelv-coupled plasma ( ICP) Another example is in the use ot electron cyclotron resonance ( ECR), which differs from RIE in plasma tormation Generally, CCR operates at a lower pressure than a conventional RIE system, and is, therefore able to etch finer line trenches anisotropicallv Comparison studies ot the emissions from high-densitv ICP, ECR and RIE plasmas show emphasis on different species and different wavelengths for similar input gas compositions The excitation mechanisms and interactions ot the particles at higher densities and or lower pressures are believed to account tor many ot these differences Consequently, the experience and knowledge accumulated from RIE emissions may not carrv over to high-densitv ICP emissions and ECR emissions In other words, it mav not be possible to monitor the same species or wavelengths for etch endpoint determination or detection in high-densitv ICP or ECR as were monitored for RIE. even if similar materials are being etched using similar input gas compositions Conventional techniques for determining an endpoint in an etching process using OES spectra are described for example, in U S Patent No 5 288,367 to Angell et al . entitled ' End-point Detection and in U S Patent No 5,658,423, to Angell et al , entitled "Monitoring and Controlling Plasma Processes via Optical Emission Using Principal Component Analysis ' These conventional techniques ty pically still entail singling out one wavelength to be used for signaling an etch endpoint, however A conventional technique tor effecting process control bv statistical analysis of the optical spectrum ot a product produced in a chemical process is described, for example, in U S Patent No 5,862,060 to Murray, Jr , entitled "Maintenance ot process control by statistical analysis of product optical spectrum" (the '060 patent) The '060 patent describes measuring the optical spectrum of each member of a calibration sample set of selected products, determining bv Principal Component Analysis (PCA) (or Partial Least Squares PLS) not more than four Principal Components to be used in the calibration sample set. determining the differences in Scores of the Principal Components between a standard ' target" product and a test product, and using the differences to control at least one process variable so as to minimize the differences However, one drawback associated with conventional techniques for determining an endpoint in an etching process using PCA applied to OES spectra is the uncertainty of how many Principal Components to use in the PCA analysis This general question in conventional PCA applications is described, tor example, in A User s Guide to Pi mcipal Components, by J Edward Jackson (Wiley Series in Probability and Mathematical Statistics, New York. 1991 ), particularly at pages 41 -58 Typically, the more Principal Components that are used, the better the PCA approximates the system being analyzed, but the longer it takes to perform the PCA For example, if all the Principal Components are used, the PCA exactly reproduces the system being analyzed, but the full-Principal Component PCA takes the longest time to perform However, determining the optimal number of PCA Principal Components to retain is also costly in terms ot time and resources involved The '060 patent describes, tor example, that a very small number ot Principal Components, usually no more than 4. suffice to define accurately that sample spectrum space for the purpose ot process control and that in some cases onlv 2 or 3 Principal Components need to be used However, that still leaves an undesirable amount of uncertainty in whether to use 2. 3 or 4 Principal Components furthermore, this uncertainty can lead these conventional techniques to be cumbersome and slow and difficult to implement on the fly" duπng real-time etching processes, for example Moreover, modern state-ot-the art OES systems are capable ot collecting thousands of frequencies or wavelengths ot optical emission spectra emanating trom the glow discharge ot gases in a plasma etch chamber t hese w avelengths mav be associated with the specific chemical species generated trom entering reactant gases and their products These products may result from gas phase reactions as well as reactions on the water and chamber wall surfaces As the surface composition ot the water shifts trom a steadv-state etch ot exposed surfaces to the complete removal of the etched material, the wavelengths and frequencies ot the optical emission spectra also sh itt Detection ot this shift may allow for etch endpoint determination, indicating the completion of the required etch Detection ot this shift also mav allow tor termination of the etch process before deleterious effects associated with an over-etch can occur However, the sheer number of OES Irequencies or wavelengths available to monitor to determine an etch endpoint makes the problem of selecting the appropriate OES frequencies or wavelengths to monitor even more severe Λn additional set of problems is posed bv the sheer number ot OES trequencies or wavelengths available to monitor I he monitoring typically generates a large amount of data 1 or example a data file for each water monitored may be as large as 2-3 megabytes ( MB), and each etcher can tv picallv process about 500-700 waters per dav C onventional storage methods would require over a gigabytes (GB) of storage space per etcher per day and over 6^ GB per etcher per year Further, the raw OES data generated in such monitoring is typically ' noisy and unenhanced I he present invention is directed to overcoming, or at least reducing the ettects of, one or more of the problems set forth above DISCLOSURE OF INVENTION In one aspect of the present invention a method is provided tor determining an etch endpoint The method includes collecting intensity data representative of optical emission spectral wavelengths during a plasma etch process I he method further includes calculating Scores from at least a portion ot the collected intensity data using at most tirst, second, third and fourth Principal Components derived from a model The method also includes determining the etch endpoint using Scores corresponding to at least one ot the first second, third and fourth Principal C mponents as an indicator for the etch endpoint In another aspect of the present invention, a computer-readable, program storage device is provided, encoded w ith instructions that when executed bv a computer, perform a method the method including collecting intensity data representative of optical emission spectral wavelengths duπng a plasma etch process The method further includes calculating Scores from at least a portion of the collected intensity data using at most first, second, third and fourth Principal Components derived from a model The method also includes determining the etch endpoint using Scores corresponding to at least one of the first, second, third and fourth Principal Components as an indicator for the etch endpoint In yet another aspect of the present invention, a computer programmed to perform a method is provided, the method including collecting intensity data representative of optical emission spectral wavelengths during a plasma etch process The method further includes calculating Scores from at least a portion of the collected intensity data using at most first, second, third and fourth Principal Components deπved from a model The method also includes determining the etch endpoint using Scores corresponding to at least one of the first, second, third and fourth Principal Components as an indicator for the etch endpoint BRIEF DESCRIPTION OF THE DRAWINGS The invention may be understood by reference to the following description taken in conjunction with the accompanying drawings, in which the leftmost significant dιgιt(s) in the reference numerals denote(s) the first figure in which the respective reference numerals appear, and in which Figures 1-7 schematically illustrate a flow diagram for various embodiments of a method according to the present invention. Figures 8- 14 schematically illustrate a flow diagram tor various alternative embodiments of a method according to the present invention, Figures 15-21 schematically illustrate a flow diagram for yet other various embodiments of a method according to the present invention, Figures 22 and 23 schematically illustrate first and second Principal Components tor respective data sets. Figure 24 schematically illustrates OES spectrometer counts plotted against wavelengths. Figure 25 schematically illustrates a time trace of OES spectrometer counts at a particular wavelength, Figure 26 schematically illustrates representative mean-scaled spectrometer counts tor OES traces of a contact hole etch plotted against wavelengths and time. Figure 27 schematically illustrates a time trace of Scores for the second Principal Component used to determine an etch endpoint, Figures 28 and 29 schematically illustrate geometrically Principal Components Analysis for respective data sets. Figure 30 schematically illustrates a time trace of OES spectrometer counts at a particular wavelength and a reconstructed time trace of the OES spectrometer counts at the particular wavelength. Figure 3 1 schematically illustrates a method for fabricating a semiconductor device practiced in accordance with the present invention, and Figure 32 schematically illustrates workpieces being processed using a high-density plasma (HDP) etch processing tool, using a plurality of control input signals, in accordance with the present invention While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and are herein described in detail It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims MODE(S) FOR CARRYING OUT THE INVENTION Illustrative embodiments of the invention are described below In the interest of clarity, not all features of an actual implementation are described in this specification It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the developers ' specific goals, such as compliance with system-related and business-related constraints, which will vary from one implementation to another Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure Using optical emission spectroscopy (OES) as an analytical tool in process control (such as in etch endpoint determination) affords the opportunity tor on-line measurements in real time A calibration set of OES spectra bounding the acceptable process space within which a particular property (such as the etch endpoint) is to be controlled may be obtained by conventional means Applying a multivaπant statistical method such as Principal Components Analysis (PCA) to the calibration set ot OES spectra affords a method of identifying the most important characteristics (Principal Components and respective Loadings and corresponding Scores) of the set of OES spectra that govern the controlled property, and are inherentiv related to the process Control then is effected by using only up to tour ot such characteristics ( Principal Components and respective Loadings and corresponding Scores), which can be determined quicklv and simply trom the measured spectra, as the control criteria to be applied to the process as a whole The result is a very effective way ot controlling a complex process using at most four criteria (the first through fourth Principal Components and respective Loadings and corresponding Scores) objectively determined from a calibration set. w hich can be applied in real time and virtually continuously, resulting in a well-controlled process that is (ideally) invariant In particular, we have found that the second Principal Component contains a very robust, high signal-to-noise indicator for etch endpoint determination We have also found that the first four Principal Components similarly may be useful as indicators tor etch endpoint determination as well as for data compression of OES data In various illustrative embodiments PCA mav be applied to the OES data, either the whole spectrum or at least a portion ot the whole spectrum If the engineer and/or controller knows that onlv a portion of the OES data contains useful information. PCA mav be applied onlv to that portion, for example In one illustrative embodiment of a method according to the present invention, as shown in Figures 1 -7, described in more detail below, archived data sets of OES w avelengths (or frequencies), trom wafers that had previously been plasma etched, mav be processed and Loadings for the first through fourth Principal Components determined from the archived OES data sets mav be used as model Loadings to calculate approximate Scores corresponding to newlv acquired OES data These approximate Scores, along with the mean values for each wavelength, may then be stored as compressed OES data In another illustrative embodiment of a method according to the present invention, as shown in Figures 8- 14. described in more detail below, archived data sets of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, may be processed and Loadings for the first through fourth Principal Components determined from the archived OES data sets may be used as model Loadings to calculate approximate Scores corresponding to newly acquired OES data 1 hese approximate Scores may be used as an etch endpoint indicator to determine an endpoint for an etch process In yet another illustrative embodiment ot a method according to the present invention, as shown in Figures 15-21. described in more detail below archived data sets of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, may be processed and Loadings tor the first through fourth Principal Components, determined from the archived OES data sets, may be used as model Loadings to calculate approximate Scores corresponding to newly acquired OES data These approximate Scores, along with model mean values also determined trom the archived OES data sets, may then be stored as more compressed OES data These approximate Scores may also be used as an etch endpoint indicator to determine an endpoint for an etch process Any ot these three embodiments may be applied in real-time etch processing Alternatively, either ot the last two illustrative embodiments may be used as an identification technique when using batch etch processing, with archived data being applied statistically, to determine an etch endpoint tor the batch Various embodiments of the present invention are applicable to any plasma etching process affording a characteristic data set whose quality may be said to define the ' success ' of the plasma etching process and whose identity may be monitored by suitable spectroscopic techniques such as OES The nature ot the plasma etching process itself is not critical, nor is the specific identity ot the workpieces (such as semiconducting silicon waters) whose spectra are being obtained However, an 'ideal or target characteristic data set should be able to be defined, having a known and/or determinable OES spectrum, and v ariations in the plasma etching process awav trom the target characteristic data set should be able to be controlled (; c , reduced) using identifiable independent process variables, e g , etch endpoints, plasma energy and/or temperature, plasma chamber pressure etchant concentrations, flow rates, and the like The ultimate goal of process control is to maintain a properly operating plasma etching process at status quo When a plasma etching process has been deemed to have reached a proper operating condition, process control should be able to maintain it bv proper adjustment ot plasma etching process parameters such as etch endpoints, temperature, pressure, flow rate, residence time in the plasma chamber, and the like At proper operating conditions, the plasma etching process affords the "ideal" or ' target ' characteristics One feature of illustrative embodiments ot our control process is that the spectrum ot the stream hereafter referred to as the test stream mav be obtained continuously (or in a near-continuous manner) on-line, and compared to the spectrum ot the target ' stream, hereafter referred to as the standard stream The ditterence in the two spectra may then be used to ad|ust one or more ot the process variables so as to produce a test stream more nearlv identical with the standard stream However, as envisioned in practice, the complex spectral data will be reduced and/or compressed to no more than 4 numerical values that define the coordinates ot the spectrum in the Principal Component or T actor space of the process sub|ect to control Typically small, incremental ad|ustments will be made so that the test stream approaches the standard stream while minimizing oscillation particularly oscillations that will tend to lose process control rather than exercise control Adjustments may be made according to one or more suitable algorithms based on process modeling, process experience, and/or artificial intelligence feedback, for example In principle, more than one process variable may be sub|ect to control, although it is apparent that as the number ot process variables under control increases so does the complexity of the control process Similarly, more than one test stream and more than one standard stream may be sampled, either simultaneously or concurrently, again with an increase in complexity In its simplest form, where there is one test stream and one process variable under control, one mav analogize the foregoing to the use of a thermocouple in a reaction chamber to generate a heating voltage based on the sensed temperature, and to use the difference between the generated heating voltage and a set point heating voltage to send power to the reaction chamber in proportion to the difference between the actual temperature and the desired temperature which, presumably, has been predetermined to be the optimum temperature Since the result of a given change in a process variable can be determined quickly, this new approach opens up the possibility of controlling the process by an automated "trial and error" feedback system, since unfavorable changes can be detected quickly Illustrative embodiments of the present invention may operate as null-detectors with feedback from the set point of operation, where the feedback signal represents the deviation of the total composition of the stream from a target composition In one illustrative embodiment, OES spectra are taken of characteristic data sets of plasma etching processes of various grades and quality, spanning the maximum range of values typical of the particular plasma etching processes Such spectra then are representative of the entire range of plasma etching processes and are often referred to as calibration samples Note that because the characteristic data sets are representative of those formed in the plasma etching processes, the characteristic data sets constitute a subset of those that define the boundaries of representative processes It will be recognized that there is no subset that is unique, that many different subsets may be used to define the boundaries, and that the specific samples selected are not critical Subsequently, the spectra of the calibration samples are subjected to the well-known statistical technique of Principal Component Analysis (PCA) to afford a small number of Principal Components (or Factors) that largely determine the spectrum of any sample The Principal Components which represent the major contributions to the spectral changes, are obtained from the calibration samples by PCA or Partial Least Squares (PLS) Thereafter, any new sample may be assigned various contributions ot these Principal Components that would reproduce the spectrum of the new sample The amount of each Principal Component required is called a Score, and time traces of these Scores, tracking how various of the Scores are changing with time, are used to detect deviations from the "target spectrum In mathematical terms, a set of m time samples of an OES for a workpiece (such as a semiconductor wafer having various process layers formed thereon) taken at n channels or wavelengths or frequencies may be arranged as a rectangular nxm matrix X In other words, the rectangular nxm matrix X may be comprised of 1 to n rows (each row corresponding to a separate OES channel or wavelength or frequency time sample) and I to m columns (each column corresponding to a separate OES spectrum time sample) The values of the rectangular nxm matrix X may be counts representing the intensity of the OES spectrum, or ratios of spectral intensities (normalized to a reference intensity), or logarithms of such ratios, for example The rectangular nxm matrix X may have rank r, where r<mιn{m,n} is the maximum number of independent variables in the matrix X The use of PCA, for example, generates a set of Principal Components P (whose "Loadings," or components, represent the contributions of the various spectral components) as an eigenmatrix (a matrix whose columns are eigenvectors) of the equation ((X-M)(X-M) : )P=VP. where M is a rectangular nxm matrix of the mean values of the columns of X (the m columns of M are each the column mean vector j nx | of X, lxm ), A" is an nxn diagonal matrix of the squares of the eigenvalues λ„ ι=l 2, ,r of the mean-scaled matrix X-M, and a Scores matrix, T, with X-M=PT T and (X-M) T =(PT T ) T =(T T ) T P T =TP r so that ((X-M)(X-M) T )P=((PT T )(TP T ))P and ((PT T XTP T ))P=(P(T T T)P T )P-P(T'T)=Λ : P The rectangular nxm matrix X, also denoted X nxm , may have elements x, j , where ι=l,2, ,n, and j= 1.2, ,m, and the rectangular mxn matrix X τ , the transpose of the rectangular nxm matrix X, also denoted (X r ) m „, may have elements X j „ where ι=l ,2, ,n, and j=l,2, ,m The nxn matrix X-M)(X-M) is (m- 1 ) times the covaπance matrix S ny „, having elements s , where ι ~ l 2 n and j= defined so that S = corresponding to the rectangular nxm matrix X n*m For the puψoses of the process control envisioned in this application we have found that at most 4 Principal Components are needed to accommodate the data tor a large range of plasma etching processes trom a v ariety ot plasma etching chambers The spectrum of the standard sample is then expressed in terms of time traces ot the Scores of the 4 Principal Components used the spectrum ot the test sample is similarly expressed and the differences in the time traces of the Scores are used to control the process variables Thus no direct correlations between the sample spectrum and plasma etch endpoints need be known In fact the nature of the sample itself need not be known as long as there is a standard the OES spectrum of the standard is known a set of at most 4 Principal Components is identified tor the class ot test stream samples and one can establish how to use the 4 Principal Components to control the process variables (as discussed more tullv below) T he spectrum ot any sample mav be expressed as a 2-dιmensιonal representation of the intensitv of mission at a particular w avelength vs the wavelength That is one axis represents intensity the other wavelength The toregoing characterization ot a spectrum is intended to incorporate various transformations that are mathematically covaπant c <ξ instead ot emission one might use absorption and/or transmission, and either mav be expressed as a percentage or logarithmically Whatever the details each spectrum may be viewed as a vector The group of spectra arising trom a group ot samples similarly corresponds to a group ot vectors If the number ot samples is N there are at most N distinct spectra If in fact none ot the spectra can be expressed as a linear combination of the other spectra then the set of spectra define an /V-dιmensιonal spectrum space However, in the cases of interest here w here a particular stream in an invariant plasma etching process is being sampled w e have observed that in fact anv particular spectrum mav be accurately represented as a linear combination of a small number Λ/ - 4 of other spectra — their Principal Components that w e refer to as working spectra These working spectra mav be viewed as the new basis set / c hnearlv independent vectors that define the 4-dιmensιonal spectrum space in w hich the samples reside The spectrum ot anv other sample is then a linear combination of the w orking spectra ( or is at least proiectable onto the 4-dιmensιonal spectrum space spanned bv the working spectra) Our experience demonstrates that the samples tvpicallv reside in at most a 4-dιmensιonal spectrum space and that this 4-dιmensιonal model suffices as a practical matter Statistical methods are available to determine the set of working spectra appropriate for anv sample set and the method of PCA is the one most tavored in the practice of the present invention although other methods e g partial least squares non-linear partial least squares, (or with less confidence multiple linear regression) also may be utilized The working spectra or the hnearlv independent vectors defining the sample spectrum space are called Principal Components or Factors Thus the spectrum of anv sample is a linear combination of the Factors The fractional contribution ot anv Factor is called the Score tor the respective Factor Hence the spectrum of anv sample completely defines a set of Scores that greatly reduces the apparent complexity of comparing different spectra In fact it has been found that tor many processes of interest in semiconductor processing a very small number of Factors no more than 4. suffice to define accurately the sample spectrum space for the puφose of process control This means that the process of characterizing the difference between a test sample and the standard sample comes down to the difference between onlv 4 numbers — the Scores of the respective 4 Factors for the sample and target It is significant to note that the small number ot Scores embodies a great deal of information about the samples and the process, and that only 4 numbers are adequate to control the process within quite close tolerances. By using the null approach of illustrative embodiments of the present invention, the use of Scores is simplified to teaching small shifts (and restoring them to zero) rather than drawing conclusions and or correlations from the absolute values of the Scores Although other methods may exist, four methods for computing Principal Components are as follows 1 eigenanalysis (EIG), 2 singular value decomposition (SVD), 3 nonlinear iterative partial least squares (NIPALS), and 4 power method Each of the first two methods, EIG and SVD, simultaneously calculates all possible Principal Components, whereas the NIPALS method allows for calculation of one Principal Component at a time However, the power method, described more fully below, is an iterative approach to finding eigenvalues and eigenvectors, and also allows for calculation of one Principal Component at a time There are as many Principal Components as there are channels (or wavelengths or frequencies). The power method may efficiently use computing time For example, consider the 3x2 matrix A, its transpose, the 2x3 matrix A 1 , their 2x2 matrix product A T A, and their 3x3 matrix product AA T (\ 1 1 ^ A ' = Λ 0 - EIG reveals that the eigenvalues λ of the matrix product A T A are 3 and 2 The eigenvectors of the matrix product A r A are solutions t of the equation (A T A)t = λt, and may be seen by inspection to be t, τ = ( l ,0) and t = (0, 1 ), belonging to the eigenvalues λ, = 3 and λ 2 = 2, respectively The power method, for example, may be used to determine the eigenvalues λ and eigenvectors p of the matrix product AA , where the eigenvalues λ and the eigenvectors g are solutions p of the equation (AA T )rj = λp A trial eigenvector rj = ( 1 , 1 , 1 ) may be used This indicates that the trial eigenvector p τ = (1,1 ,1) happened to correspond to the eigenvector 2ι Γ = ( 1 , 1 , 1 ) belonging to the eigenvalue λ, = 3 The power method then proceeds by subtracting the outer product matrix _ι _ι T from the matrix product AA r to form a residual matrix R, Another trial eigenvector β T = ( 1 ,0,- 1 ) may be used (AA' - p n T )p = R, p λ 2 / This indicates that the trial eigenvector p τ = ( 1,0,-1 ) happened to correspond to the eigenvector p_ ' = ( 1 ,0,- 1 ) belonging to the eigenvalue λ 2 = 2 The power method then proceeds by subtracting the outer product matrix p_-2- T f rorn tne residual matrix R, to form a second residual matrix R The fact that the second residual matrix R, vanishes indicates that the eigenvalue λτ = 0 and that the eigenvector £ 3 is completely arbitrary The eigenvector 3 may be conveniently chosen to be orthogonal to the eigenvectors 2ι = (1, 1.1 ) and 22 = ( ' -0,- 1 ), so that the eigenvector pj T = ( 1 ,-2, 1 ) Indeed, one may readily verify that Similarly, SVD of A shows that A = PT T where P is the Principal Component matrix and T is the Scores matrix SVD confirms that the singular values ot A are 3 and V2, the positive square roots of the eigenvalues λ, - 3 and λ 2 = 2 of the matrix product A T A Note that the columns of the Principal Component matrix P are the orthonormalized eigenvectors of the matrix product AA T Likewise, SVD of A τ shows that A τ = TP T SVD confirms that the (non-zero) singular values of A τ are 3 and V2, the positive square roots of the eigenvalues λ., = 3 and λ 2 = 2 of the matrix product AA T Note that the columns of the Principal Component matrix P (the rows of the Principal Component matrix P τ ) are the orthonormalized eigenvectors of the matrix product AA r Also note that the non-zero elements of the Scores matrix T are the positive square roots v/3 and 2 ot the (non-zero) eigenvalues λ| = 3 and λi = 2 ot both of the matrix products A T A and AA r Taking another example, consider the 4x3 matrix B, its transpose, the 3x4 matrix B τ , their 3x3 matrix product B r B, and their 4x4 matrix product BB 1 EIG reveals that the eigenvalues of the matrix product B T B are 4, 2 and 2 The eigenvectors of the matrix product B T B are solutions t of the equation (B τ B)t = λt, and may be seen by inspection to be t, τ = (1,0,0), , 1 = (0, 1,0), and t-, τ = (0,0,1), belonging to the eigenvalues λ, = 4, λ 2 = 2, and λ 3 = 2, respectively The power method, for example, may be used to determine the eigenvalues λ and eigenvectors of the matrix product BB T , where the eigenvalues λ and the eigenvectors are solutions of the equation (BB T )2 = λ A trial eigenvector 2 T = ( 1 , 1 , 1 , 1 ) may be used This indicates that the trial eigenvector 2 T = ( U M ) happened to correspond to the eigenvector 2ι r = ( 1 , 1.1 , 1 ) belonging to the eigenvalue λ t = 4 The power method then proceeds by subtracting the outer product matrix 2ιEι T f rom tne matrix product BB T to form a residual matrix R, Another trial eigenvector p = ( 1 ,0,0,- 1 ) may be used This indicates that the trial eigenvector p r = ( Α0,- l ) happened to correspond to the eigenvector p = ( 1 ,0,0,- 1 ) belonging to the eigenvalue λ = 2 The power method then proceeds by subtracting the outer product matrix 2 from the residual matrix R, to form a second residual matrix R 2 Another trial eigenvector 2 r' = (0.1 ,- 1.0) may be used This indicates that the trial eigenvector 2 ? = (0,1,-1,0) happened to correspond to the eigenvector 2 3 T = (0, 1 ,- 1 ,0) belonging to the eigenvalue λ 3 = 2 The power method then proceeds by subtracting the outer product matrix p^ . -/ from the second residual matrix R 2 to form a third residual matrix R 3 The fact that the third residual matrix R3 vanishes indicates that the eigenvalue λ 4 = 0 and that the eigenvector 1S completely arbitrary. The eigenvector p^ may be conveniently chosen to be orthogonal to the eigenvectors p./ = ( 1 1 , 1.1 ), 2/ = ( 1 ,0,0,- 1 ), and rjj 1 = (0, 1 ,- 1 ,0), so that the eigenvector j r = ( 1,- 1 ,-1 ,1) Indeed, one may readily verify that In this case, since the eigenvalues λ 2 = 2 and λ 3 = 2 are equal, and, hence, degenerate, the eigenvectors p_ 2 T = ( 1 ,0,0,- 1 ) and p = (0, 1,-1 ,0) belonging to the degenerate eigenvalues λ 2 = 2 = λ 3 may be conveniently chosen to be orthonormal A Gram-Schmidt orthonormalization procedure may be used, for example Similarly, SVD of B shows that B = PT T , where P is the Principal Component matrix and T is the Scores matrix SVD confirms that the singular values of B are 2. 2 and 2, the positive square roots of the eigenvalues λ| = 4, λ 2 = 2 and λ - 2 of the matrix product B T B Likewise, SVD of B r shows that SVD confirms that the (non-zero) singular values of B τ are 2, V2, and 2, the positive square roots of the eigenvalues λi = 4, λ-. - 2 and λ-, = 2 of the matrix product AA T Note that the columns of the Principal Component matrix P (the rows ot the Principal Component matrix P τ ) are the orthonormalized eigenvectors of the matrix product BB r Also note that the non-zero elements of the Scores matrix T are the positive square roots 2 x/2 and Λ/2 of the (non-zero) eigenvalues λ, = 4 λ7 = 2 and λ - 2 of both ot the matrix products B r B and BB T The matrices A and B discussed above have been used for the sake of simplifying the presentation of PCA and the power method and are much smaller than the data matrices encountered in illustrative embodiments of the present invention For example in one illustrative embodiment for each water 8 scans of OES data over 495 wavelengths may be taken during an etching step with about a 13 second interval between scans In this illustrative embodiment 18 wafers may be run and corresponding OES data collected The data mav be organized as a set of 1 8x495 matrices X, = (X,,) s where s= l 2, 8, for each of the different scans and X, j is the intensity ot the lth water run at the jth wavelength Putting all 8 of the 18x495 matrices λ - (X,,) s for s- 1 2 8 next to each other 10 produces the overall OES data matrix X, an 18x3960 matrix X = [X , X X,] = [(X ,), (X„ ,(X,,) 8 ] Each row in X represents the OES data from 8 scans over 495 wavelengths tor a run Brute force modeling using all 8 scans and all 495 wavelengths would entail using 3960 input variables to predict the etching behavior ot 18 sample wafers an ill-conditioned regression problem Techniques such as PCA and/or partial least squares (PLS also known as pro)ectιon to latent structures) reduce the complexity in such cases bv revealing the hierarchical ordering l i ot the data based on levels of decreasing variability In PC \ this involves finding successive Principal Components In PLS techniques such as NIPALS this involves finding successive latent vectors As shown in Figure 22 a scatteφlot 2200 ot data points 2210 mav be plotted in an n-dimensional variable space (n-3 in T ιgure 22) The mean vector 2220 may he at the center ot a p-dimensional Principal Component ellipsoid 2230 (p=2 in Figure 22) The mean vector 2220 mav be determined bv taking the average of the columns 20 of the overall OES data matrix X The Principal Component ellipsoid 2230 mav have a first Principal Component 2240 (maior axis in Figure 22) with a length equal to the largest eigenvalue of the mean-scaled OES data matrix X-M and a second Principal Component 2250 (minor axis in Figure 22) with a length equal to the next largest eigenvalue of the mean-scaled OES data matrix X-M For example the 3x4 matrix B r given above mav be taken as the overall OES data matrix X (again for the sake of simplicity) corresponding to 4 scans taken at 3 wavelengths \s shown in Figure 23 a scatteφlot 2300 of data points 23 10 mav be plotted in a 3-dιmensιonal variable space 1 h mean vector 2320 u mav lie at the center of a 2-dιmensιonal Principal Component ellipsoid 2330 (reallv a circle a degenerate ellipsoid) The mean vector 2320 u mav be determined bv taking the average ot the columns ot the overall OES 3x4 data matrix B r The Principal Component ellipsoid 2330 mav have a first Principal Component 2340 ( maior axis in Figure 23) and a second Principal Component 2350 ( minor axis in Figure 23) Here the eigenvalues ot the mean-scaled OES data matrix B τ -M are equal and degenerate so the lengths of the major and m inor axes in Figure 23 are equal As shown in Figure 23 the mean vector 2320 y . is given bv and the matrix M has the mean vector 2320 μ_ for all 4 columns 35 In another illustrative embodiment 5500 samples ot each wafer mav be taken on wavelengths between about 240-1 100 nm at a high sample rate of about one per second For example, 555 1 sampling points/spectrum/second (corresponding to 1 scan per wafer per second taken at 5551 wavelengths or 7 scans per water per second taken at 793 wavelengths or 13 scans per water per second taken at 427 wavelengths or 61 scans per wafer per second taken at 91 wavelengths) may be collected in real time, during etching ot a contact hole using an Applied Materials AMAT 5300 Centura etching chamber, to produce high resolution and broad band OES spectra As shown in Figure 24, a representative OES spectrum 2400 of a contact hole etch is illustrated • > Wavelengths, measured in nanometers (nm) are plotted along the horizontal axis against spectrometer counts plotted along the vertical axis As shown in Figure 25, a representative OES trace 2500 ot a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against spectrometer counts plotted along the vertical axis As shown in Figure 25, by about 40 seconds into the etching process, as indicated by dashed line 2510, the 10 OES trace 2500 of spectrometer counts "settles down" to a range ot values less than or about 300. for example As shown in Figure 26 representative OES traces 2600 ot a contact hole etch are illustrated Wavelengths, measured in nanometers (nm) are plotted along a first axis, time, measured in seconds (sec) is plotted along a second axis, and mean-scaled OES spectrometer counts, for example are plotted along a third (vertical) axis As shown in Figure 26, over the course ot about 150 seconds of etching three clusters of wavelengths 2610, 2620 and ! -> 2630 respectively, show variations in the respective mean-scaled OES spectrometer counts In one illustrative embodiment, any one of the three clusters of wavelengths 2610 2620 and 2630 mav be used, either taken alone or taken in any combination with any one (or both) ot the others, as an indicator variable signaling an etch endpoint In an alternative illustrative embodiment, only the two clusters of wavelengths 2620 and 2630 having absolute values of mean-scaled OES spectrometer counts that exceed a preselected threshold absolute mean-scaled OES 20 spectrometer count value (for example, about 200, as shown in Figure 26) may be used, either taken alone or taken together, as an indicator variable signaling an etch endpoint In vet another alternative illustrative embodiment, only one cluster ot wavelengths 2630 having an absolute value ot mean-scaled OES spectrometer counts that exceeds a preselected threshold absolute mean-scaled OES spectrometer count value (for example, about 300. as shown in Figure 26) mav be used as an indicator variable signaling an etch endpoint ^ As shown in Figure 27, a representative Scores time trace 2700 corresponding to the second Principal C omponent during a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against Scores (in arbitrary units) plotted along the vertical axis As shown in Figure 27, the Scores time trace 2700 corresponding to the second Principal Component during a contact hole etch mav start at a relatively high value initially, decrease with time, pass through a minimum value, and then begin increasing before leveling >0 off We have found that the inflection point (indicated by dashed line 2710 and approximately where the second derivative ot the Scores time trace 2700 with respect to time vanishes) is a robust indicator for the etch endpoint Principal Components Analysis (PCA) may be illustrated geometrically For example, the 3x2 matrix C (similar to the 3x2 matrix A given above) C = 0 may be taken as the overall OES data matrix X (again tor the sake of simplicity ), corresponding to 2 scans taken at 3 wavelengths As shown in Figure 28, a scatteφlot 2800 of OES data points 2810 and 2820. with coordinates ( 1 1.1) and (- 1,0,1 ). respectively, may be plotted in a 3-dιmensιonal variable space where the variables are respective spectrometer counts for each of the 3 wavelengths The mean vector 2830 u. mav he at the center of a 1 -dimensional Principal Component ellipsoid 2840 (really a line, a very degenerate ellipsoid) The mean vector 2830 u may be determined by taking the average of the columns of the overall OES 3x2 matrix C The Principal Component ellipsoid 2840 mav have a first Principal Component 2850 (the "major" axis in Figure 28. with length V5, lying along a first Principal Component axis 2860) and no second or third Principal Component lying along second or third Principal Component axes 2870 and 2880, respectively Here, two of the eigenvalues of the mean-scaled OES data matrix C-M are equal to zero, so the lengths of the "minor" axes in Figure 28 are both equal to zero. As shown in Figure 28, the mean vector 2830 u_ is given by and the matrix M has the mean vector 2830 μ. for both columns As shown in Figure 28. PCA is nothing more than a. principal axis rotation of the original variable axes (here, the OES spectrometer counts for 3 wavelengths) about the endpoint of the mean vector 2830 μ_, with coordinates (0,1/2, 1 ) with respect to the original coordinate axes and coordinates [0,0,0] with respect to the new Principal Component axes 2860, 2870 and 2880 The Loadings are merely the direction cosines of the new Principal Component axes 2860, 2870 and 2880 with respect to the original variable axes The Scores are simply the coordinates of the OES data points 2810 and 2820, [5 θ 3 /2 0.0] and [-5 0 : 72,0,0], respectively, referred to the new Principal Component axes 2860, 2870 and 2880 The mean-scaled 3x2 OES data matrix C-M, its transpose, the 2x3 matrix (C-M) 1 , their 2x2 matrix product (C-M) T (C-M), and their 3x3 matrix product (C-M) (C-M) τ are given by 1 1 / 2 0 ' {C - M)' = 1 - 1 / 2 0 J The 3x3 matrix (C-M)(C-M) T is the covaπance matrix S J 3 , having elements s, j , where 1=1 ,2,3, and 2 ∑ C ,^,A - ∑ c ,„ ∑ c A j= 1 ,2,3, defined so that = *=ι *=ι , corresponding to the rectangular 3x2 matrix C 1x2 2(2 - 1) EIG reveals that the eigenvalues λ of the matrix product (C-M) T (C-M) are 5/2 and 0, for example b finding solutions to the secular equation = 0 The eigenvectors of the matrix product (C-M) T (C-M) are solutions t of the equation (C-M) T (C-M)t = λt, which may be rewritten as ((C-M) τ (C-M)-λ)t = 0 For the eigenvalue λ, = 5 2, the eigenvector t, may be seen by = 0 to be t = (1,-1) For the eigenvalue λi = 0, the eigenvector t> may be seen by t = 0 to bet/ = (1.1) The power method, for example, may be used to determine the eigenvalues λ and eigenvectors 2 of the matrix product (C-M)(C-M) T , where the eigenvalues λ and the eigenvectors p are solutions of the equation ((C-M)(C-M) T )2 = λ A trial eigenvector 2 T = (1,1,1) may be used ((C-M)(C-M) r )p = = 3q ((C-M)(C-M)' = λ l , This illustrates that the trial eigenvector 2 T = ( , , 1 > ') g ets replaced by the improved trial eigenvector / = (1,1/2,0) that happened to correspond to the eigenvector p = (1,1/2,0) belonging to the eigenvalue λ, = 5/2 The power method then proceeds by subtracting the outer product matrix 2ιEι T fr° m the matrix product (C-M)(C-M) r to form a residual matrix R, Another trial eigenvector p = (- 1.2.0), orthogonal to the eigenvector ι (1,1/2,0) may be used This indicates that the trial eigenvector 2 T = (-U2,0) happened to correspond to the eigenvector p_, τ = (- 1 ,2,0) belonging to the eigenvalue κ 2 = 0 The power method then proceeds by subtracting the outer product matrix p_>2 2 T from the residual matrix R, to form a second residual matrix Ri Another trial eigenvector 2 T = (0,0,1 ), orthogonal to the eigenvectors 2 ι T = ( 1.1/2.0) and p/ = (-1,2,0) may be used ((C - M)(C - M) r - p v[ - p, p' )p ((C - M)(C - M)' - p t p] - P 2 P 2 ' )P = R 2 P = = λ i P This indicates that the trial eigenvector 2 ? = (0,0, 1 ) happened to correspond to the eigenvector p = (0,0, 1 ) belonging to the eigenvalue λ, = 0 Indeed, one may readily verify that ((C - M)(C - M)' )p 3 = λ, . Similarly SVD of C-M shows that C-M = PT r , where P is the Principal Component matrix (whose columns are orthonormalized eigenvectors proportional to pi, £ 2 an d 2 3 , an ^ whose elements are the Loadings, the direction cosines of the new Principal Component axes 2860, 2870 and 2880 related to the original vaπable axes) and T is the Scores matrix (whose rows are the coordinates of the OES data points 2810 and 2820, referred to the new Principal Component axes 2860, 2870 and 2880) The transpose of the Scores matrix (T τ ) is given by the product of the matrix ot eigenvalues of C-M with a matrix whose rows are orthonormalized eigenvectors proportional to t, and f> As shown in Figure 28, the direction cosine (Loading) of the first Principal Component axis 2860 with respect to the wavelength I counts axis is given by COS©, , and the direction cosine (Loading) of the first Principal Component axis 2860 with respect to the wavelength 2 counts axis is given by COS©.,, = / r- Similarly, the direction cosine (Loading) of the first Principal Component axis 2860 with respect to the wavelength 3 counts axis is given by CθsΘ 3 | = 0 Similarly, the direction cosine (Loading) of the second Principal Component axis 2870 with respect to the wavelength 1 counts axis is given by Cθs Θ p = ~ γ \— , the direction cosine (Loading) of the second Principal Component axis 2870 with respect to the wavelength 2 counts axis is given by COs Θ n = 7 r • and the direction cosine (Loading) of the second Principal Component axis 2870 with respect to the wavelength 3 counts axis is given by cos Θ, 2 = 0 Lastly, the direction cosine ( Loading) of the third Principal Component axis 2880 with respect to the wavelength 1 counts axis is given by CθsΘ π = COsAM = 0 , the direction cosine (Loading) of the third Principal Component axis 2880 with respect third Principal Component axis 2880 with respect to the wavelength 3 counts axis is given by cosΘ„ = cos(θ) = 1 SVD confirms that the singular values ot C-M are 5/ 2 and 0. the non-negative square roots of the eigenvalues M 5 2 and λ 2 = 0 of the matrix product (C-M) T (C-M) Note that the columns ot the Principal Component matrix P are the orthonormalized eigenvectors of the matrix product (C-M)(C-M) r Taking another example, a 3 x4 matrix D (identical to the 3 4 matrix B r given above) may be taken as the overall OES data matrix X (again tor the sake of simplicity), corresponding to 4 scans taken at 3 wavelengths As shown in Figure 29, a scatteφlot 2900 of OES data points with coordinates ( 1 , 1 ,0), ( 1 ,0, 1 ), ( 1 ,0.- 1 ) and ( 1 ,- 1 ,0), respectively, may be plotted in a 3-dιmensιonal variable space where the variables are respective spectrometer counts for each of the 3 wavelengths The mean vector 2920 u mav e at the center of a 2-dιmensιonal Principal Component ellipsoid 2930 (really a circle, a somewhat degenerate ellipsoid) The mean vector 2920 u may be determined by taking the average of the columns of the overall OES 3 χ4 matrix D The Principal Component ellipsoid 2930 may have a first Principal Component 2940 ( the ma|or " axis in Figure 29 with length 2, lying along a first Principal Component axis 2950), a second Principal Component 2960 ( the minor axis in Figure 29, also with length 2 lving along a second Principal Component axis 2970). and no third Principal Component lying along a third Principal Component axis 2980 Flere. two of the eigenvalues ot the mean-scaled OES data matrix D-M are equal, so the lengths of the "major " and m inor axes of the Principal Component ellipsoid 2930 in Figure 29 are both equal, and the remaining eigenvalue is equal to zero, so the length of the other "minor" axis of the Principal Component ellipsoid 2930 in F igure 29 is equal to zero As shown in Figure 29. the mean vector 2920 u is given by and the matrix M has the mean vector 2920 μ_ for all 4 columns As shown in Figure 29, PCA is nothing more than a principal axis rotation of the original vaπable axes (here, the OES spectrometer counts for 3 wavelengths) about the endpoint of the mean vector 2920 u_, with coordinates ( 1 ,0 0) w ith respect to the original coordinate axes and coordinates [0,0,0] with respect to the new Principal Component axes 2950, 2970 and 2980 The Loadings are merely the direction cosines of the new Principal Component axes 2950. 2970 and 2980 with respect to the original vaπable axes The Scores are simply the coordinates of the OES data points, [1,0,0], [0,1,0], [0,-1,0] and [-1,0.0], respectively, referred to the new Principal Component axes 2950, 2970 and 2980 The 3x3 matrix product (D-M)(D-M) T is given by The 3x3 matrix (D-M)(D-M) T is 3 times the covaπance matrix S 3v3 , having elements s, j , where ι= 1,2,3, and corresponding to the rectangular 3x4 matrix EIG reveals that the eigenvalues of the matrix product (D-M)(D-M) T are 0, 2 and 2 The eigenvectors of the matrix product (D-M)(D-M) T are solutions p of the equation ((D-M)(D-M) T )2 = λβ. and may be seen by inspection to be pi =(01,0), p_ 2 =(0,0,1), and p = ( 1,0,0) belonging to the eigenvalues , =2, λ 2 = 2, and λ, = 0 respectively (following the convention of placing the largest eigenvalue first) As mav be seen in Figure 29, the direction cosine (Loading) of the first Principal Component axis 2950 with respect to the wavelength 1 counts axis is given by COS©,, = cos )= 0, the direction cosine (Loading) of the first Principal Component axis 2970 with respect to the wavelength 2 counts axis is given by cosθ^ =cos(θ)=l, and the direction cosine (Loading) of the first Principal Component axis 2860 with respect to the wavelength 3 counts axis is given by COS© , = cos /~l=0 Similarly, the direction cosine (Loading) of the second Principal Component axis 2970 with respect to the wavelength 1 counts axis is given by CθsΘ l2 =cos / } )=0 the direction cosine (Loading) ot the second Principal Component axis 2970 with respect to the wavelength 2 counts axis is given by cosθ,, = COs A 1= 0 , and the direction cosine (Loading) of the second Principal Component axis 2970 with respect to the wavelength 3 counts axis is given by = cos(θ)= 1 Lastly, the direction cosine (Loading) of the third Principal Component axis 2980 with respect to the wavelength 1 counts axis is given by CθsΘ H = cos(θ)= 1 , the direction cosine (Loading) of the third Principal Component axis 2980 with respect to the wavelength 2 counts axis is given by COS0, 3 an d tn e direction cosine (Loading) of the third Principal Component axis 2980 with respect to the wavelength 3 counts axis is given by COS Θ ,, = COsA ) = 0 The transpose of the Scores matrix T τ may be obtained simply by multiplying the mean-scaled OES data matrix D-M on the left by the transpose of the Principal Component matrix P, whose columns are pi, 2 2 , E 3> the orthonormalized eigenvectors of the matrix product (D-M)(D-M) T T' = P' (D - M) The columns of the transpose of the Scores matrix T τ (or. equivalently, the rows of the Scores matrix T) are. indeed, the coordinates of the OES data points. [ 1 ,0,0], [0.1 0], [0,- 1 ,0] and [- 1 ,0,0], respectively, referred to the new Principal Component axes 2950, 2970 and 2980 We have found that the second Principal Component contains a very robust, high signal-to-noise indicator for etch endpoint determination The overall mean-scaled OES rectangular nxm data matrix X llm -M nm may be decomposed into a portion corresponding to the first and second Principal Components and respective Loadings and Scores, and a residual portion (t' λ x - M = A τ/, ( + p n = ( Pi p 2 ) ■ + p n ^ V- J where P Pr is an nχ2 matrix whose columns X - M = Pl tl + p + P n ' i = \ rι + - are the first and second Principal Components. r P( is an mχ2 Scores matrix tor the first and second Principal Components, T PC T is a 2 χ m Scores matrix transpose, P PΓ I , ' = X P( is an nxm matrix, P a is an nx(m-2) matrix whose columns are the residual Principal Components, T as is an mχ(m-2) Scores matrix tor the residual Principal Components, T rι5 r is an (m-2) χ m Scores matrix transpose, and P, LS 1 r / = X res is an nxm matrix The kth column ot X-M, X k , k= l ,2, ,m. an n χ l matrix, similarly decomposes into x^ = (xpc) k +(V s ) k , where (x PC ) k = P r Pp( r X k is the projection of t into the Principal Component subspace ( PCS) spanned by the first and second Principal Components, (x rcs ) k = ( I n « n -P cPpc r ) k ls the projection ot x into the residual subspace orthogonal to the PCS spanned by the first and second Principal Components, ( pc) k ' ~~ is the pro|ectιon of x k ' into the Principal Component subspace (PCS) spanned by the first and second Principal Components, ( s ) = H U n χ n - pc ) is the projection of x into the residual subspace orthogonal to the PCS spanned by the first and second Principal Components, and l n „ is the nxn identity matrix Using the projection of x into the PCS spanned bv the first and second Principal Components and the projection of into the residual subspace orthogonal to the PCS spanned by the first and second Principal Components, there are two tests of the amount of variance in the data that are accounted for by the first and second Principal Components One test is the Q-test, also known as the Squared Prediction Error (SPE) test where SPE = Ul l „ n - P n P,', ( )x A = il Another test is the Hotell g T 2 test, a multivaπate generalization of the well-known Student s r-test, where T 1 = x A Λ "2 P/ ( Λ A = x[ P n P n ' Λ "2 A P n ' x, = (χ n )[ A '2 (χ p( ) k ≤ χ α 2 , where A 2 is a 2x2 diagonal matrix of the squares ot the eigenvalues λ„ ι= l ,2, belonging to the first and second Principal Components of the overall mean-scaled OES rectangular nxm data matrix X nm -M nm Both the SPE test and the Hotellmg T test can be used to monitor the etching process for example We have also found that the first through fourth Principal Components are similarly useful for containing high signal-to-noise indicators tor etch endpoint determination as well as being useful for OES data compression The overall mean-scaled OES rectangular nxm data matrix X„ m -M nm may be decomposed into a portion corresponding to the first through fourth Principal Components and respective Loadings and Scores, and a residual portion- X - M which expands out to X — M — p t ] + p t 2 + pf + p t + P r T n ,^ = X n + X n ,, , where P PC is an nχ4 matrix whose columns are the first through fourth Principal Components. T PC - is an mχ4 Scores matrix for the first through fourth Principal Components, Tp is a 4 χ m Scores matrix transpose, Pp r T PC T = Xpc is an nxm matrix, P res is an nχ(m-4) matrix whose columns are the residual Principal Components, T rcs is an mχ(m-4) Scores matrix for the residual Principal Components, T rL is a (m-4) χ m Scores matrix transpose, and P res T rCi r = X res ' s an nxm matrix The kth column of X-M, x , k ~ 1.2, ,m, an nx l matrix, similarly decomposes into k = (xpck + (V s ) k , w here ( '( ) k = Ppc PpA k >s the projection ot x k into the Principal Component subspace (PCS) spanned by the first through fourth Principal Components, (x r( . s ) k = U„, n -Ppc Pιτ H is the projection ot x k into the residual subspace orthogonal to the PCS spanned by the first through fourth Principal Components. (Xpc ) k ' = H Ac >s the projection ot x k ' into the Principal Component subspace (PCS) spanned by the first through fourth Principal Components. (λ es ) k T = X k r (I π π - pcP c r ) ' s the pro|ectιon of x into the residual subspace orthogonal to the PCS spanned by the first through fourth Principal Components, and !„„„ is the nxn identity matrix. Using the projection ot x k into the PCS spanned by the first through fourth Principal Components and the projection of X k into the residual subspace orthogonal to the PCS spanned by the first through fourth Principal Components, there are again two tests of the amount of variance in the data that are accounted for by the first through fourth Principal Components One test is the Q-test, also known as the Squared Prediction Error (SPE) test. where SPE = |( A, AAA ^ || AΛ ≤ δ r , Another test is the Hotellmg T " test, a multivaπate generalization of the well-known Student ' s /-test, where T 2 = x k ' P n A 2 P/, ( x k = x[ P r( / > A /> P, x k = (x r( )[ A 2 (x r( ) A < χ j , where Λ : ,s a 4x4 diagonal matrix of the squares of the eigenvalues λ„ ι= 1 ,2, 3,4 belonging to the first through fourth Principal Components of the overall mean-scaled OES rectangular nxm data matrix X nm -M nm More generally, the overall mean-scaled OES rectangular nxm data matrix X nm -M πm of rank r w here r≤mιn{m,n} , may be decomposed into a portion corresponding to the first through rth Principal Components and respective Scores, and a residual portion X - M = P c T l i ( + + P n , T n , which expands out to X - M = pf[ + p 2 t 2 + ■ ■ ■ + p r d r ' , + p r t r r + P rc T r ' c = X n + X n , , where P PC is an nxr matrix whose columns are the first through rth Principal Components. T Γ is an mxr Scores matrix for the first through rth Principal Components. T PC is an rxm Scores matrix transpose, P PC T PC T = X PC is an nxm matrix. P res is an nχ( -r) matrix whose columns are the residual Principal Components (if m = r, P res = 0). T res is an χ(m-r) Scores matrix for the residual Principal Components (if m = r, T res = 0), T rcs τ is an (m-r)χm Scores matrix transpose (if m = r, T rcs τ = 0). and T res P res = X res is an nxm matrix (if m = r. X rcs = 0) The kth column of X-M. x k . k= l ,2,. ,m, an nx l matrix, similarly decomposes into x k = (xpc) k +(X rcs ) k , where (x Pr ) k = PpcPp k is the projection of x k into the Principal Component subspace (PCS) spanned by the first through rth Principal Components, (V es ) k = (I π χ n -PpcPpc T )H is the projection of x k into the residual subspace orthogonal to the PCS spanned by the first through rth Principal Components, λ A = k ? Ppc pc T |S the projection of x k τ into the Principal Component subspace (PCS) spanned by the first through rth Principal Components. ( res ) k r = H T (I n*n - cPpc T ) >s the projection of X k ' into the residual subspace orthogonal to the PCS spanned by the first through rth Principal Components, and l„ y „ is the nxn identity matrix Using the projection ot x k into the PCS spanned by the first through rth Principal Components and the projection of x k into the residual subspace orthogonal to the PCS spanned by the first through rth Principal Components, there are likewise two tests ot the amount ot variance in the data that are accounted for by the first through rth Principal Components One test is the Q-test. also know n as the Squared Prediction Error ( SPE) test where SPE - (A, "A-c 'c ) χ HI A JI ' ≤ <\ 2 Another test is the Hotellmg 1 " test, a multivaπate generalization of the well-known Student s /-test. where T 1 = x i P r< ^ '2 p π Λ" A = l P n P,', ( A 2 P r( P n ' χ k = (χ n )[ A '2 (x n ) k ≤ / , where \ 2 is an rxr diagonal matrix of the squares of the eigenvalues λ,. ι= 1.2, ,r belonging to the first through rth Principal Components of the overall mean-scaled OES rectangular nxm data matrix X πm -M nm ot rank r, where r<mιn { m.n } In one illustrative embodiment of a method according to the present invention, as shown in Figures 1 -7, archived data sets (Y nxm ) °f OES wavelengths (or frequencies), trom waters that had previously been plasma etched, may be processed and the weighted linear combination ot the intensity data, representative of the archived OCS w avelengths (or frequencies ) collected over time during the plasma etch defined by the first through pth Principal Components, mav be used to compress newly acquired OES data The rectangular nxm matrix Y (Y„,, m ) may have rank r, where r<mιn { m,n } is the maximum number of independent variables in the matrix Y Here p<r, in various illustrative embodiments, p is m a range of 1 -4, in various alternative illustrative embodiments, p = 2 T he first through pth Principal Components may be determined trom the archived data sets (Y n χ m ) of OES wavelengths (or frequencies), for example, by any of the techniques discussed above For example, archived data sets (Y„« m ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, may be processed and Loadings (Q n* ) tor the first through fourth Principal Components determined trom the archived OES data sets (Y n IT ,) may be used as model Loadings (Q n , ) to calculate approximate Scores (T ln <4 ) corresponding to newly acquired OES data (X n*m ) These approximate Scores (T mx ), along with the mean values tor each wavelength (M nxm ), effectively the column mean vector (u_. X | ) of the newly acquired OES data (X n*m ), mav then be stored as compressed OES data As shown in Figure I . a workpiece 100, such as a semiconducting substrate or wafer, having one or more process layers and/or semiconductor devices such as an MOS transistor disposed thereon, for example, is delivered to an etching preprocessing step j 105 where j may have any value from j = 1 to j = N- l The total number N ot processing steps, such as masking, etching, depositing material and the like used to form the finished workpiece 100, may range from N = 1 to about anv finite value As shown in Figure 2. the workpiece 100 is sent from the etching preprocessing step j 105 to an etching step j+ 1 1 10 In the etching step j+ 1 1 10. the workpiece 100 is etched to remove selected portions from one or more process layers formed in any of the previous processing steps (such as etching preprocessing step j 105, where j may have any value from j = 1 to j = N- 1 ) As shown in Figure 2. if there is further processing to do on the workpiece 100 (if j < N- l ), then the workpiece 100 may be sent trom the etching step j+ 1 1 10 and delivered to a postetchmg processing step j+2 1 15 for further postetch processing, and then sent on from the postetching processing step j +2 1 15 Alternatively, the etching step j+ 1 1 10 may be the final step in the processing of the workpiece 100 In the etching step j+ 1 1 10, OES spectra are measured in situ bv an OES spectrometer (not shown), producing raw OES data 120 (X nxm ) indicative of the state of the workpiece 100 during the etching In one illustrative embodiment, about 5500 samples ot each wafer mav be taken on wavelengths between about 240- 1 100 nm at a high sample rate ot about one per second For example, 5551 sampling points/spectrum/second (corresponding to 1 scan per water per second taken at 555 1 wavelengths) may be collected in real time, during etching ot a contact hole using an Applied Materials ΛMAT 5300 Centura etching chamber to produce high resolution and broad band OES spectra As shown in Figure 3, the raw OES data 120 (X„χ m ) is sent trom the etching step j+ 1 1 10 and delivered to a mean-scaling step 125, producing a means matrix (M n m ), whose m columns are each the column mean vector (u_ nx ι ) of the raw OES data 120 (X n x m ), and mean-scaled OES data (X-^- - ,„) In the mean-scaling step 125, in various illustrative embodiments, the mean values are treated as part of a model built from the archived data sets (Y nxm ) of OCS wavelengths (or frequencies), from wafers that had previouslv been plasma etched In other words, a means matrix (N nxm ) previously determined from the archived data sets (Y„ m ) of OES wavelengths (or frequencies), trom waters that had previously been plasma etched, is used to generate alternative mean-scaled OES data ( X n .„, N n ,„) In various alternative illustrative embodiments, the mean values tor each wafer and/or mean value tor each wavelength, for example, are determined as discussed above and are used to generate the mean-scaled OES data ( n m -M n x m ) As shown in Figure 4, the means matrix (M nxm ) and the mean-scaled OES data (X n χ m -M πxrn ) 130 are sent from the mean scaling step 125 to a Scores calculating step 135, producing approximate Scores (T mxp ) In the Scores calculating step 135 in various illustrative embodiments, the mean-scaled OES data (X„χ m -M π m ) are multiplied on the left by the transpose of the Principal Component (Loadings) matrix Q„„ p , with columns rj , o^, c| p , that are the first p orthonormalized eigenvectors of the matrix product (Y-N)(Y-N) r ~ -A » χ m , producing the transpose of the Scores matrix T 1 of the approximate Scores ( mxp ) The columns of the transpose of the Scores matrix T , or, equivalents, the rows of the approximate Scores matrix ( T m , p ), are the coordinates of the OES data points referred to new approximate Principal Component axes In the Scores calculating step 135, in various illustrative embodiments the approximate Scores (T mxp ) are calculated using the Loadings (Q nxp ) derived trom the model built from the archived mean-scaled data sets (Y nxm -X. m ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched In other words the Loadings (Q nxp ), previously determined from the archived mean-scaled data sets (Y nxm -N π χ m ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, are used to generate the approximate Scores (T mxp ) corresponding to the mean-scaled OES data (X„ xm -M n m ) deπved from the raw OES data 120 (X nxm ) The Loadings (Q„ xp ) are defined by the first through pth Principal Components Here p≤r, in various illustrative embodiments, p is in a range of 1-4, in various alternative illustrative embodiments, p = 2 The first through pth Principal Components may be determmed off-line from the archived data sets (Y„ xm ) of OES wavelengths (or frequencies), for example, by any of the techniques discussed above The values of the rectangular nxm matrix Y (Y nxm ) f° r the archived data sets may be counts representing the intensity of the archived OES spectrum or ratios of spectral intensities (normalized to a reference intensity), or logarithms of such ratios, for example The rectangular nxm matrix Y (Y nxm ) for the archived data sets may have rank r where r<mιn{m,n} is the maximum number of independent variables in the matrix Y The use of PCA tor example, generates a set of Principal Component Loadings (Q nxp ), representing contributing spectral components, an eigenmatrix (a matrix whose columns are eigenvectors) of the equation ((Y-N)(Y-N) T )Q=A 2 P, where N is a rectangular nxm matrix of the mean values of the columns of Y (the m columns of N are each the column mean vector u_ nx ] of Y nxm ), A 2 is an nxn diagonal matrix of the squares of the eigenvalues λ„ ι= l ,2, .r, of the mean-scaled matrix Y-N, and a Scores matrix U with Y-N=QU T and and ((QU r )(UQ I ))Q=(Q(lI τ U)Q τ )Q=Q(U r U)=Λ , Q The rectangular nxm matrix Y also denoted Y nxrn , may have elements y,, where ι= 1 2, ,n, and j=l ,2, ,m, and the rectangular mxn matrix Y τ the transpose of the rectangular nxm matrix Y, also denoted (Y τ ) mx n, may have elements y,,, where ι=l ,2, ,n, and j= 1.2, ,m The nxn matrix (Y-N)(Y-N) r is (m- 1 ) times the covaπance matrix S nxn , having elements s, j , where ι= 1.2, ,n, and j= l ,2, ,n, W ∑^ - ∑^A ∑A defined so that Λ = k=\ k — \ k = \ , corresponding to the rectangular nxm matrix Y„χ m m(tn - \) As shown in Figure 5, a feedback control signal 140 may be sent from the Scores calculating step 135 to the etching step j+1 1 10 to adjust the processing performed in the etching step j+ 1 1 10 For example, based on the determination of the approximate Scores (T mxp ) calculated using the Loadings (O n P ) derived from the model built from the archived mean-scaled data sets ( n χ π rN nx m) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, the feedback control signal 140 may be used to signal the etch endpoint As shown in Figure 6, the means matrix (M πxm ) and the approximate Scores (T, nxp ) 145 are sent from the Scores calculating step 135 and delivered to a save compressed PCA data step 150 In the save compressed PCA data step 150, the means matrix (M nxm ) and the approximate Scores (T mxp ) 145 are saved and/or stored to be used in reconstructing X , κm , the decompressed approximation to the raw OES data 120 (X„ xm ) The decompressed approximation X mm to the raw OES data 120 (X n x m ) may be reconstructed from the means matrix (M nxm ) and the approximate Scores (T mxp ) 145 as follows X , κm = Q nxp ψ' )„„,„ + M mm In one illustrative embodiment, n = 5551, m = 100, and p = 4, so that the raw OES data 120 (X 5551X100 ) requires a storage volume of 5551 x 100, and generates a means matrix (M 555 1oo ) that only requires a storage volume of 5551 x 1, since all the 100 columns of the means matrix (M 555lx ι 0o ) are identical (each of the 5551 rows of each column being the mean value for that wavelength or channel over the 100 scans) The Loadings (Q 5551 X 4 ) are determined off-line from archived data sets of OES wavelengths (or frequencies), for example, by any of the techniques discussed above, and need not be separately stored with each water OES data set, so the storage volume of 5551 x4 for the Loadings (Q^ X4 ) does not have to be taken into account in determining an effective compression ratio for the OES wafer data The approximate Scores (T| 0 o x ) only require a storage volume of 100x4 Therefore, the effective compression ratio for the OES wafer data in this illustrative embodiment is about (5551 l 00)/(5551 χ l ) or about 100 to 1 ( 100 1 ) More precisely, the compression ratio in this illustrative embodiment is about (5551 χ l 00)/(5551 χ l + 100 χ 4) or about 93 3 to 1 (93 3 1 ) In another illustrative embodiment, n = 555 1 , m = 100, and p = 4, so that the raw OES data 120 (X 555 iχ ioo) requires a storage volume of 5551 x 100. and generates a means matrix (M 555 i X | 00 ) that only requires a storage volume of 793x 1 , since all the 100 columns of the means matrix (M 5 , 5 i X | 0 o) are identical and the rows are arranged into 793 sets of 7 identical rows (each of the 5551 rows of each column being the mean value tor that wavelength or channel over the 100 scans, averaged over the respective 7 wavelengths) The Loadings (Q, 35 ,χ ) are determined off-line from archived data sets (Ysssi x ioo) of OES wavelengths (or frequencies), tor example, by any of the techniques discussed above, and need not be separately stored with each water OES data set, so the storage volume of 5551 x4 for the Loadings (Q 555 . 1 ) does not have to be taken into account in determining an effective compression ratio for the OES water data The approximate Scores (T ]00x4 ) onlv require a storage volume of 100x4 Therefore, the effective compression ratio tor the OES water data in this illustrative embodiment is about (555 l χ l 00)/(793 χ l ) or about 700 to 1 ( 100 1 ) More precisely, the compression ratio in this illustrative embodiment is about (5551 χ l 00)/(793 χ l + 100 χ 4) or about 465 to 1 (465 1 ) In yet another illustrative embodiment n ^ 555 1 , m = 100, and p = 4. so that the raw OES data 120 (X 555 100 ) requires a storage volume of 5551 x 100. and generates a means matrix (M 555 | X | 0U ) that only requires a storage volume ot 5551 χ l . since all the 100 columns ot the means matrix (M 555 i x ιoo) are identical (each of the 5551 rows ot each column being the mean value tor that wavelength or channel over the 100 scans) The Loadings (Q 5551 X ) are determined off-line trom archived data sets ( 555 100 ) of OES wavelengths (or frequencies), for example, by anv of the techniques discussed above and require a storage volume ot 555 1 x4, in this alternative embodiment The approximate Scores (T| ()0 ) onlv require a storage volume of 100x4 Therefore, the effective compression ratio for the OES wafer data in this illustrative embodiment is about (5551 χ i00)/(5551 χ5 ) or about 20 to 1 (20 1 ) More precisely, the compression ratio in this illustrative embodiment is about (5551 χ l 00)/(555 1 x5+ 100x4) or about 19 7 to I ( 19 7 1 ) In still yet another illustrative embodiment, n - 555 1. m = 100, and p = 4. so that the raw OES data 120 (X 5551 X 100 ) requires a storage volume of 5551 x 100. and generates a means matrix (M^ ioo) that only requires a storage volume of less than or equal to about 555 1 \ 1 since all the 100 columns of the means matrix (M 555 | X |oo) are identical and means thresholding is used to decimate the rows (each of the 5551 rows of each column being the mean value for that wavelength or channel over the 100 scans, if that mean value is greater than or equal to a specified threshold value, such as a specified threshold value in a range of about 30-50, or zero, otherwise) The Loadings (Q 555 i x ) are determined off-line from archived data sets of OES wavelengths (or frequencies), for example, by any of the techniques discussed above and need not be separately stored with each wafer OES data set, so the storage volume of 5551 x4 for the Loadings (Q„ 5 i x4 ) does not have to be taken into account in determining an effective compression ratio for the OES wafer data The approximate Scores (T, 0 oχ 4 ) only require a storage volume of 100x4 Therefore, the effective compression ratio tor the OES wafer data in this illustrative embodiment is less than or equal to about (5551 χ l 00)/(5551 χ l ) or about 100 to 1 (100 1 ) As shown in Figure 30. a representative OES trace 3000 of a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against spectrometer counts plotted along the vertical axis As shown in Figure 30, by about 40 seconds into the etching process, as indicated by dashed line 3010, the OES trace 3000 of spectrometer counts "settles down to a range of values less than or about 300. tor example A representative reconstructed OES trace 3020 (corresponding to X„ m ), for times to the right of the dashed line 3010 (greater than or equal to about 40 seconds, for example), is schematically illustrated and compared with the corresponding noisv raw OES trace 3030 (corresponding to X n χ m ), also for times to the right of the dashed line 3010 The reconstructed OES trace 3020 (corresponding to X n%m ) is much smoother and less noisy than the raw OES trace 3030 (corresponding to X nxrπ ) As shown in Figure 7, in addition to. and or instead of, the feedback control signal 140. a feedback control signal 155 may be sent trom the save compressed PCA data step 150 to the etching step j + 1 1 10 to adjust the processing performed in the etching step j^ l 1 10 For example based on the determination ot the approximate Scores ( T„,. p ) calculated using the Loadings (Q„> p ) derived trom the model built from the archived mean-scaled data sets (Y„„„-N„ m ) of OES wavelengths (or frequencies), trom waters that had previously been plasma etched, the feedback control signal 155 may be used to signal the etch endpoint In another illustrative embodiment ot a method according to the present invention, as shown in Figures 8- 14 archived data sets (Y nxm ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, mav be processed and Loadings (Q n 4 ) for the first through fourth Principal Components determined from the archived OES data sets (Y nxm ) may be used as model Loadings (Q„χ 4 ) to calculate approximate Scores (T mx4 ) corresponding to newly acquired OES data (X„ <m ) These approximate Scores (T„,, ) may be used as an etch endpoint indicator to determine an endpoint tor an etch process As shown in Figure 8 a workpiece 800 such as a semiconducting substrate or water, having one or more process layers and/or semiconductor devices such as an MOS transistor disposed thereon, tor example is delivered to an etching preprocessing step j 805 w here j may have any value trom j - 1 to j = N- l T he total number N of processing steps, such as masking, etching, depositing material and the like, used to form the finished workpiece 800, mav range from N = 1 to about anv finite value As shown in Figure 9. the workpiece 800 is sent from the etching preprocessing step j 805 to an etching step j+ 1 810 In the etching step j+ 1 810. the workpiece 800 is etched to remove selected portions from one or more process layers tormed in any of the previous processing steps (such as etching preprocessing step j 805, where j may have anv value from j = 1 to ι = N- l ) As shown in Figure 9, if there is further processing to do on the workpiece 800 (if j < N- l ), then the workpiece 800 may be sent from the etching step j+ 1 810 and delivered to a postetching processing step j+2 815 for further postetch processing, and then sent on from the postetching processing step j+2 815 Alternatively, the etching step j+ 1 810 may be the final step in the processing of the workpiece 800 In the etching step j+ 1 810. OES spectra are measured m situ by an OES spectrometer (not shown), producing raw OES data 820 (X nxm ) indicative of the state of the workpiece 800 during the etching In one illustrative embodiment, about 5500 samples ot each wafer may be taken on wavelengths between about 240- 1 100 nm at a high sample rate of about one per second For example 5551 sampling points/spectrum/second (corresponding to 1 scan per wafer per second taken at 5551 wavelengths) may be collected in real time, during etching of a contact hole using an Applied Materials AMAT 5300 Centura etching chamber, to produce high resolution and broad band OES spectra As shown in Figure 10, the raw OES data 820 (X πxm ) is sent from the etching step j-H 810 and delivered to a mean-scaling step 825, producing a means matrix (M nxm ), whose m columns are each the column mean vector (ϋ nx i) of the raw OES data 820 (X mm ), and mean-scaled OES data (X n χ m -M nxrπ ) In the mean-scaling step 825. in various illustrative embodiments, the mean values are treated as part ot a model built from the archived data sets (Y nxm ) of OES wavelengths (or frequencies), from waters that had previously been plasma etched In other words, a means matrix (N„ xm ) previously determined from the archived data sets (Y nxm ) of OES wavelengths (or Irequencies), trom waters that had previously been plasma etched, is used to generate alternative mean-scaled OES data (X„ xm -N n m ) In various alternative illustrative embodiments, the mean values for each wafer and/or mean value for each wavelength for example, are determined as discussed above, and are used to generate the mean-scaled OES data (\„ m -M„ m ) As shown in Figure 1 1 the means matrix (M nxm ) and the mean-scaled OES data (X„ m -M„ ,„) 830 are sent trom the mean scaling step 825 to a Scores calculating step 835. producing approximate Scores ( T rτl p ) In the Scores calculating step 835 in various illustrative embodiments, the mean-scaled OES data ( X , m -M - πl ) are multiplied on the left bv the transpose of the Principal Component (Loadings) matrix Q„x P , with columns U , rjj, ( jp , that are the first p orthonormalized eigenvectors of the matrix product ( Y-N)(Y-N) T (T ) /) H = (Q ) n - n ^ ~~ l ) ικm ' producing the transpose ot the Scores matrix T τ of the approximate Scores ( T xp ) The columns ot the transpose of the Scores matrix T , or, equivalently, the rows of the approximate Scores matrix (T mxp ), are the coordinates of the OES data points referred to new approximate Principal Component axes In the Scores calculating step 835, in various illustrative embodiments, the approximate Scores (T mxp ) are calculated using the Loadings (Q nxp ) derived from the model built trom the archived mean-scaled data sets (Y n χ m -N n ,„) of OES wavelengths (or frequencies), from waters that had previously been plasma etched In other words the Loadings (Q n p ) previously determined from the archived mean-scaled data sets (Y„ m -N m ) of OES wavelengths (or frequencies) trom wafers that had previously been plasma etched, are used to generate the approximate Scores (T mxp ) corresponding to the mean-scaled OES data (X„ xm -M nxm ) derived from the raw OES data 820 (X nxm ) The Loadings (O n P ) are defined by the first through pth Principal Components Here p^r in various illustrative embodiments p is in a range of 1 -4 in various alternative illustrative embodiments p 2 The first through pth Principal Components may be determined off-line from the archived data sets (Y „ m ) of OES wavelengths (or frequencies) for example, by any of the techniques discussed above The values ot the rectangular nxm matrix Y (Y„ xm ) tor the archived data sets mav be counts representing the intensity of the archived OES spectrum, or ratios ot spectral intensities (normalized to a reference intensity), or logarithms ot such ratios, for example The rectangular nxm matrix Y (Y nx ) for the archived data sets may have rank r. where r mιn{m,n} is the maximum number ot independent variables in the matrix Y The use of PCA, for example, generates a set of Principal Component Loadings (Q„ xp ), representing contributing spectral components, an eigenmatrix (a matrix whose columns are eigenvectors) of the equation ((Y-N)(Y-N) T )Q=Λ"P, where N is a rectangular nxm matrix of the mean values of the columns of Y (the m columns of N are each the column mean vector u^ i ot Y n m ). A 2 is an nxn diagonal matrix of the squares of the eigenvalues λ„ ι= l 2. ,r ot the mean-scaled matrix Y-N and a Scores matrix. U, with Y-N=QU ' and )(UQ T ))Q an d ((QU T )(UQ 1 ))0=(Q(U r U)O r )Q=0(U T U)=A 2 Q The rectangular nxm matrix Y, also denoted Y-, m , may have elements y ψ where ι= l ,2, ,n, and |= 1 ,2, ,m, and the rectangular mxn matrix Y τ . the transpose of the rectangular nxm matrix Y, also denoted (Y ) mxn , may have elements y jn where ι= l ,2, ,n, and j= l ,2, ,m The nxn matrix (Y-N)(Y-N) T is (m- 1 ) times the covaπance matrix S πxn , having elements s,,, where ι= I ,2, ,n, and j= l ,2, .,n, 1∑ >',* , A - ∑ ', ∑ V A -l A=l A I defined so that A = . corresponding to the rectangular nxm matrix Y nx m(m - X) As shown in Figure 12. a feedback control signal 840 may be sent from the Scores calculating step 835 to the etching step j+ 1 810 to adjust the processing performed in the etching step j+ 1 810 For example, based on the determination of the approximate Scores (T mxp ) calculated using the Loadings (Q nxP ) derived trom the model built from the archived mean-scaled data sets (Y nxm -N n χ m ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, the feedback control signal 840 mav be used to signal the etch endpoint As shown in Figure 13 the approximate Scores (T mxp ) 845 are sent trom the Scores calculating step 835 and delivered to a use Scores as etch indicator step 850 In the use Scores as etch indicator step 850, the approximate Scores (T mxp ) 845 are used as an etch indicator For example, as shown in Figure 27, a representative Scores time trace 2700 corresponding to the second Principal Component during a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against Scores (in arbitrary units) plotted along the vertical axis As shown in Figure 27, the Scores time trace 2700 corresponding to the second Principal Component during a contact hole etch may start at a relatively high value initially, decrease with time, pass through a minimum value, and then begin increasing before leveling off We have found that the inflection point (indicated by dashed line 2710, and approximately where the second derivative of the Scores time trace 2700 with respect to time vanishes) is a robust indicator for the etch endpoint As shown in Figure 14. in addition to. and/or instead of, the feedback control signal 840 a teedback control signal 855 mav be sent trom the use Scores as etch indicator step 850 to the etching step j + 1 810 to adjust the processing performed in the etching step j+ 1 810 For example, based on the determination ot the approximate Scores (T mxp ) calculated using the Loadings (Q n x P ) derived from the model built from the archived mean-scaled data sets (Y n χ m -N n x m ) of OES wavelengths (or frequencies), trom wafers that had previously been plasma etched, the feedback control signal 855 may be used to signal the etch endpoint In yet another illustrative embodiment of a method according to the present invention, as shown in Figures 15-21. archived data sets (Y nxm ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched, mav be processed and Loadings (Q nx ) for the first through fourth Principal Components determined from the archived OES data sets (Y nxm ) may be used as model Loadings (Q nx4 ) to calculate approximate Scores (T mx4 ) corresponding to newly acquired OES data (X nxm ) These approximate Scores (T mx ), with or without the mean values for each wavelength (N nxm ), effectively the column mean vector (ϋ nX ι) of the archived OES data (Y nxm ), may then be stored as compressed OES data These approximate Scores (T mx4 ) may also be used as an etch endpoint indicator to determine an endpoint for an etch process As shown in Figure 15. a workpiece 1500, such as a semiconducting substrate or wafer, having one or more process layers and/or semiconductor devices such as an MOS transistor disposed thereon, tor example, is delivered to an etching preprocessing step j 1505. where j mav have anv value from j = 1 to j = N- l The total number N of processing steps, such as masking, etching, depositing material and the like, used to form the finished workpiece 1500, may range from N = 1 to about any finite value As shown in Figure 16, the workpiece 1500 is sent from the etching preprocessing step j 1505 to an etching step j+ 1 15 10 In the etching step j+ 1 1510, the workpiece 1500 is etched to remove selected portions from one or more process layers formed in any of the previous processing steps (such as etching preprocessing step j 1505, where j may have anv value from j = 1 toj = N- l ) As shown in Figure 16, if there is further processing to do on the workpiece 1500 (if j < N- l ), then the workpiece 1500 may be sent trom the etching step j+ 1 1510 and delivered to a postetching processing step j+2 1515 for further postetch processing, and then sent on from the postetching processing step j+2 15 15 Alternatively, the etching step j+ 1 15 10 may be the final step in the processing of the workpiece 1500 In the etching step j+ 1 1510 OES spectra are measured in situ by an OES spectrometer (not shown) producing raw OES data 1520 (X nxm ) indicative of the state ot the workpiece 1500 during the etching In one illustrative embodiment, about 5500 samples ot each water mav be taken on wavelengths between about 240- 1 100 nm at a high sample rate ot about one per second For example 555 1 sampling pomts/spectrum/second (corresponding to 1 scan per wafer per second taken at 555 1 wavelengths) mav be collected in real time, during etching ot a contact hole using an Applied Materials AMAT 5300 Centura etching chamber to produce high resolution and broad band OES spectra As shown in Figure 17 the raw OES data 1520 (X„ x , n ) is sent trom the etching step j+ 1 15 10 and delivered to a Scores calculating step 1525, where a means matrix (N„ „,) whose m columns are each the column mean vector (u_„ x |) of the archived OES data (Y nxm )' is used to produce alternative mean-scaled OES data (X n<n ,-N nxm ) In the Scores calculating step 1525, in various illustrative embodiments, the mean values are treated as part ot a model built from the archived data sets (Y n χ ) of OES wavelengths (or frequencies) from wafers that had previously been plasma etched In other words a means matrix (N„ m ) previously determined from the archived data sets (Y nxm ) of OES w avelengths (or frequencies) trom waters that had previously been plasma tched is used to generate the alternative mean-scaled OES data (X„ m N„ „,) As shown in Figure 17 the alternative mean-scaled OES data ( \ n „,-N 1Mm ) are used to produce alternative approximate Scores (T m x P ) In the Scores calculating step 1525 in various illustrative embodiments, the mean-scaled OES data (X nxm -N nxm ) are multiplied on the left by the transpose of the Principal Component (Loadings) matrix Q nxp , w ith columns g , 2 , , cj p , that are the first p orthonormalized eigenvectors of the matrix product (Y-N)(Y-N) T (7 " ) = (O ) „ \X — N) nxm , producing the transpose of the Scores matrix T τ of the alternative approximate Scores (T mxp ) The columns of the transpose of the Scores matrix T τ , or, equivalently the rows ot the alternative approximate Scores matrix (T mxp ), are the alternative coordinates of the OES data points referred to new approximate Principal Component axes In the Scores calculating step 1525, in various illustrative embodiments the alternative approximate Scores (T mxp ) are calculated using the Loadings (Q πxp ) derived from the model built from the archived mean-scaled data sets (Y π χ -N n χ m ) of OES wavelengths (or frequencies), from wafers that had previously been plasma etched In other words, the Loadings (Q nxp ) previously determined from the archived mean-scaled data sets (Y nxm -N nxm ) of OES wavelengths (or frequencies) trom wafers that had previously been plasma etched, are used to generate the alternative approximate Scores (T m p ) corresponding to the mean-scaled OES data (X nxm - nxm ) derived from the raw OES data 1520 (X nxrn ) The Loadings (Q„ xp ) are defined by the first through pth Principal Components Here p≤r, m various illustrative embodiments, p is in a range of 1-4, in various alternative illustrative embodiments, p = 2 The first through pth Principal Components may be determined off-line from the archived data sets (Y nxm ) of OES wavelengths (or frequencies), for example, by any of the techniques discussed above The values of the rectangular nxm matrix Y (Y nxm ) for the archived data sets may be counts representing the intensity of the archived OES spectrum, or ratios of spectral intensities (normalized to a reference intensity), or logarithms of such ratios, for example The rectangular nxm matrix Y (Y nxm ) for the archived data sets may have rank r, where r<mιn{m,n} is the maximum number of independent variables in the matrix Y The use of PCA, for example, generates a set of Principal Component Loadings (Q nxp ), representing contributing spectral components, an eigenmatrix (a matrix whose columns are eigenvectors) of the equation ((Y-N)(Y-N) T )Q=Λ 2 P, where N is a rectangular nxm matrix of the mean values of the columns of Y (the m columns of N are each the column mean vector u_„ x l of Y nxrn ), A 2 is an nxn diagonal matrix of the squares of the eigenvalues λ„ ι=l ,2, ,r, of the mean-scaled matrix Y-N, and a Scores matrix U, with Y-N=QU T and (Y-N) T =(QU T ) T =(U T ) T Q T =UQ T , so that ((Y-N)(Y-N) T )Q=((QU T )(UQ T ))Q and ((QU r )(UQ T ))Q=(Q(U T U)Q T )Q=Q(U r U)=A 2 Q The rectangular nxm matrix Y, also denoted Y nxm , may have elements y y , where ι= l ,2, ,n, and j= 1 ,2, ,m, and the rectangular mxn matrix Y r the transpose of the rectangular nxm matrix Y, also denoted (Y ) mxn , mav have elements y }l , where ι=l ,2, ,n, and j= l ,2, ,m The nxn matrix (Y-N)(Y-N) T is (m- l ) times the covaπance matrix S nxn , having elements s„, where ι= l ,2, ,n, and j= l ,2, ,n. m∑y lk y lk - ∑ .VΛ ∑J',* defined so that S = , corresponding to the rectangular nxm matrix Y nxm m(m - \) As shown in Figure 18, a feedback control signal 1530 may be sent from the Scores calculating step 1535 to the etching step j+1 1510 to adjust the processing performed in the etching step j+ 1 1510 For example, based on the determination of the alternative approximate Scores (T mxp ) calculated using the Loadings (Q„ xp ) derived from the model built from the archived mean-scaled data sets (Y nxm -N nxm ) of OES wavelengths (or frequencies), from waters that had previously been plasma etched, the feedback control signal 1530 may be used to signal the etch endpoint As shown in Figure 19, the alternative approximate Scores (T mxp ) 1535 are sent from the Scores calculating step 1525 and delivered to a save compressed PCA data step 1540 In the save compressed PCA data step 1540, the alternative approximate Scores (T mxp ) 1535 are saved and/or stored to be used in reconstructing A xm , the decompressed alternative approximation to the raw OES data 1520 (X„ xm ) The decompressed alternative approximation X„ xm to the raw OES data 1520 (X„ xm ) mav be reconstructed from the means matrix (N n , m ) and the alternative approximate Scores (T mxp ) 1545 as follows X nxm = O nxp [T 1 ) pxm + N llxm In one illustrative embodiment, n = 5551, m = 100, and p = 4, so that the raw OES data 1520 (Xsssi x ioo) requires a storage volume of 5551 x 100 The means matrix (N 555 i x l oo) is determined off-line from archived data sets of OES wavelengths (or frequencies), and need not be separately stored with each wafer OES data set Thus, the storage volume of 5551 x 1 for the means matrix (N 5551X 100 ), where all the 100 columns of the means matrix (N 555 i x ιoo) are identical (each of the 5551 rows of each column being the mean value for that wavelength or channel over the 100 scans), does not have to be taken into account in determining an effective compression ratio for the OES water data The Loadings (Q-, 5 ,| x4 ) are also determined off-line from archived data sets of OES wavelengths (or frequencies), tor example, by any of the techniques discussed above, and also need not be separatelv stored with each wafer OES data set so the storage volume of 5551 x4 for the Loadings (Q 555 4 also does not have to be taken into account in determining an effective compression ratio for the OES wafer data The approximate Scores (T| 0 oχ 4 ) only require a storage volume ot 100x4 Therefore, the effective compression ratio for the OES wafer data in this illustrative embodiment is about (5551 x 100)/( 100x4) or about 1387 75 to 1 (1387 75 1 ) As shown in Figure 30, a representative OES trace 3000 of a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against spectrometer counts plotted along the vertical axis As shown in Figure 30, by about 40 seconds into the etching process, as indicated by dashed line 3010, the OES trace 3000 of spectrometer counts ' settles down to a range of values less than or about 300, for example A representative reconstructed OES trace 3020 (corresponding to X nxm ), for times to the right of the dashed line 3010 (greater than or equal to about 40 seconds for example), is schematically illustrated and compared with the corresponding noisv raw OES trace 3030 (corresponding to X„ <m ), also for times to the right ot the dashed line 3010 The reconstructed OES trace 3020 ( corresponding to X„ ^ „, ) is much smoother and less noisv than the raw OES trace 3030 (corresponding to X„ ,„ ) As shown in Figure 20, the alternative approximate Scores (T mxp ) 1545 are sent trom the save compressed PCA data step 1540 and delivered to a use Scores as etch indicator step 1550 In the use Scores as etch indicator step 1550. the alternative approximate Scores ( r m p ) 1545 are used as an etch indicator For example, as shown in Figure 27, a representative Scores time trace 2700 corresponding to the second Principal Component during a contact hole etch is illustrated Time, measured in seconds (sec) is plotted along the horizontal axis against Scores (in arbitrary units) plotted along the vertical axis As shown in Figure 27. the Scores time trace 2700 corresponding to the second Principal Component during a contact hole etch mav start at a relatively high value initially, decrease with time, pass through a minimum value and then begin increasing before leveling otf We have found that the inflection point ( indicated bv dashed line 2710 and approximately where the second derivative ot the Scores time trace 2700 with respect to time vanishes) is a robust indicator for the etch endpoint As shown in Figure 21 in addition to and or instead of the feedback control signal 1530, a feedback control signal 1555 mav be sent trom the use Scores as etch indicator step 1550 to the etching step j+ 1 1510 to adjust the processing performed in the etching step j+ 1 1510 For example, based on the determination of the alternative approximate Scores (T m v P ) calculated using the Loadings (Q nxp ) derived from the model built from the archived mean-scaled data sets (Y n v m - „ m ) ot OES wavelengths (or frequencies), from wafers that had previouslv been plasma etched, the feedback control signal 1 55 may be used to signal the etch endpoint Figure 3 1 illustrates one particular embodiment of a method 3100 practiced in accordance with the present invention Figure 32 illustrates one particular apparatus 3200 with which the method 3100 may be practiced For the sake of clarity, and to further an understanding of the invention, the method 3100 shall be disclosed in the context of the apparatus 3200 However the invention is not so limited and admits wide variation, as is discussed further below Referring now to both Figures 3 1 and 32 a batch or lot of workpieces or wafers 3205 is being processed through an etch processing tool 3210 The etch processing tool 3210 may be any etch processing tool known to the art, such as Applied Materials AMAT 5300 Centura etching chamber, provided it includes the requisite control capabilities The etch processing tool 3210 includes an etch processing tool controller 3215 for this puφose The nature and function of the etch processing tool controller 3215 will be implementation specific For instance, the etch processing tool controller 3215 may control etch control input parameters such as etch recipe control input parameters and etch endpoint control parameters, and the like Four workpieces 3205 are shown in Figure 32, but the lot ot workpieces or wafers, / , the ' wafer lot,' may be any practicable number ot waters from one to any finite number The method 3 100 begins, as set forth in box 3 120, by measuring parameters such as OES spectral data characteristic of the etch processing performed on the workpiece 3205 in the etch processing tool 3210 The nature, identity, and measurement of characteristic parameters will be largely implementation specific and even tool specific For instance, capabilities tor monitoring process parameters vary, to some degree, trom tool to tool Greater sensing capabilities may permit wider latitude in the characteristic parameters that are identified and measured and the manner in which this is done Conversely, lesser sensing capabilities may restrict this latitude Turning to Figure 32, in this particular embodiment, the etch process characteristic parameters are measured and/or monitored by tool sensors (not shown) The outputs of these tool sensors are transmitted to a computer svstem 3230 over a line 3220 The computer system 3230 analyzes these sensor outputs to ldentifv the characteristic parameters Returning, to Figure 3 1. once the characteristic parameter is identified and measured, the method 3 100 proceeds by modeling the measured and identified characteristic parameter using PCA as set forth in box 3 130 The computer system 3230 in Figure 32 is, in this particular embodiment, programmed to model the characteristic parameter using PCA The manner in which this PCA modeling occurs will be implementation specific In the embodiment of Figure 32, a database 3235 stores a plurality of PCA models and/or archived PCA data sets that might potentially be applied, depending upon which characteristic parameter is identified This particular embodiment, therefore, requires some a priori knowledge of the characteristic parameters that might be measured The computer system 3230 then extracts an appropriate model from the database 3235 ot potential models to apply to the identified characteristic parameters If the database 3235 does not include an appropriate model, then the characteristic parameter may be ignored, or the computer system 3230 may attempt to develop one, if so programmed The database 3235 mav be stored on anv kind ot computer-readable, program storage medium, such as an optical disk 3240. a floppy disk 3245 or a hard disk drive (not shown ) ot the computer system 3230 The database 3235 mav also be stored on a separate computer system (not shown) that interfaces with the computer system 3230 Modeling of the identified characteristic parameter may be implemented differently in alternative embodiments For instance, the computer svstem 3230 may be programmed using some form of artificial intelligence to analvze the sensor outputs and controller inputs to develop a PCA model on-the-fly in a real-time PCA implementation This approach might be a useful adjunct to the embodiment illustrated in Figure 32, and discussed above where characteristic parameters are measured and identified for which the database 3235 has no appropriate model The method 3 100 of Figure 31 then proceeds by applying the PCA model to compress the OES data and/or determine an etch endpoint, as set forth in box 3 140 The OES data compressed using PCA according to any of the various illustrative embodiments of the present invention may be stored on any kind of computer-readable, program storage medium, such as an optical disk 3240. a floppy disk 3245, or a hard disk drive (not shown) of the computer svstem 3230. and/or together with the database 3235 The OES data compressed using PCA according to any ot the various illustrative embodiments of the present invention may also be stored on a separate computer system (not shown) that interfaces with the computer system 3230 Depending on the implementation, applying the PCA model may yield either a new value for the etch endpoint control parameter or a correction and/or update to the existing etch endpoint control parameter The new etch endpoint control parameter is then formulated from the value yielded by the PCA model and is transmitted to the etch processing tool controller 3215 over the line 3220 The etch processing tool controller 3215 then controls subsequent etch process operations in accordance with the new etch control inputs Some alternative embodiments may employ a form ot feedback to improve the PCA modeling of characteristic parameters The implementation ot this feedback is dependent on several disparate facts, including the tool s sensing capabilities and economics One technique for doing this would be to monitor at least one effect of the PCA model s implementation and update the PCA model based on the effect(s) monitored The update may also depend on the PCA model For instance, a linear model mav require a different update than would a non-linear model, all other factors being the same As is evident trom the discussion above, some features ot the present invention are implemented in software for instance the acts set forth in the boxes 3120-3 140 in Fιgure 31 are in the illustrated embodiment, sottware-implemented in whole or in part Thus some features ot the present invention are implemented as instructions encoded on a computer-readable, program storage medium The program storage medium may be ot any type suitable to the particular implementation However, the program storage medium w ill typically be magnetic, such as the floppy disk 3245 or the computer 3230 hard disk drive (not shown ) or optical, such as the optical disk 3240 When these instructions are executed by a computer, thev perform the disclosed functions The computer may be a desktop computer, such as the computer 3230 However, the computer might alternatively be a processor embedded in the etch processing tool 3210 The computer might also be a laptop, a workstation, or a mainframe in various other embodiments The scope of the invention is not limited bv the tvpe or nature of the program storage medium or computer with which embodiments ot the invention might be implemented Thus, some portions ot the detailed descriptions herein are, or mav be. presented in terms of algorithms lunctions, techniques and or processes These terms enable those skilled in the art most ettectivelv to convey the substance of their work to others skilled in the art These terms are here, and are general Iv conceived to be a self-consistent sequence ot steps leading to a desired result The steps are those requiring phv sical manipulations of physical quantities Usually, though not necessarily, these quantities take the form ot electromagnetic signals capable of being stored transferred, combined, compared and otherwise manipulated It has proven convenient at times, principally for reasons ot common usage, to reter to these signals as bits, values, elements sv mbols. characters, terms, numbers, and the like All of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities and actions Unless specifically stated otherwise, or as may be apparent from the discussion terms such as ' processing," ' computing "calculating," "determining," "displaying," and the like used herein refer to the actιon(s) and processes ot a computer system, or similar electronic and/or mechanical computing device, that manipulates and transforms data, represented as physical (electromagnetic) quantities withm the computer system's registers and/or memories, into other data similarly represented as physical quantities within the computer system s memories and/or registers and or other such information storage, transmission and/or displav devices Any of these illustrative embodiments may be applied in real-time etch processing Alternatively, either of the illustrative embodiments shown in Figures 8- 14 and 15-21 may be used as an identification technique when using batch etch processing, with archived data being applied statistically, to determine an etch endpoint for the batch In various illustrative embodiments, a process engineer may be provided with advanced process data monitoring capabilities, such as the ability to provide historical parametric data in a user-friendly format, as well as event logging, real-time graphical display of both current processing parameters and the processing parameters of the entire run, and remote, / e , local site and worldwide, monitoring These capabilities may engender more optimal control of critical processing parameters, such as throughput accuracy, stability and repeatability, processing temperatures, mechanical tool parameters, and the like This more optimal control of critical processing parameters reduces this variability This reduction in variability manifests itself as fewer within-run disparities, fewer run-to-run disparities and fewer tool-to- tool disparities This reduction in the number of these disparities that can propagate means fewer deviations in product quality and performance In such an illustrative embodiment of a method of manufacturing according to the present invention, a monitoring and diagnostics system may be provided that monitors this variability and optimizes control of critical parameters An etch endpoint determination signal as in any of the embodiments disclosed above may have a high signal-to-noise ratio and may be reproducible over the variations of the incoming waters and the state of the processing chamber, for example, whether or not the internal hardware in the processing chamber is worn or new, or whether or not the processing chamber is in a 'clean ' or a dirty condition Further, in particular applications, for example, the etching ot contact and/or via holes, an etch endpoint determination signal as in any of the embodiments disclosed above mav have a high enough signal-to-noise ratio to overcome the inherently very low signal-to-noise ratio that may arise simply by virtue of the small percentage ( 1 % or so) of surface area being etched In various illustrative embodiments, the etch endpoint signal becomes very stable, and may throughput may be improved by reducing the ma etch time from approximately 145 seconds, for example to approximately 90-100 seconds, depending on the thickness of the oxide In the absence of an etch endpoint determination signal as in any of the embodiments disclosed above, a longer etch time is conventionally needed to insure that all the material to be etched awav has been adequately etched away, even in vias and contact holes with high aspect ratios The presence of a robust etch endpoint determination signal as in anv of the embodiments disclosed above thus allows for a shorter etch time, and. hence, increased throughput, compared to conventional etching processes Thus, embodiments of the present invention fill a need in present day and future technology for optimizing selection ot wavelengths to monitor for endpoint determination or detection during etching Similarly, embodiments of the present invention fill a need in present day and future technology for being able to determine an etch endpoint expeditiously, robustly, rehablv and reproduciblv under a variety of different conditions, even in real-time processing Further, it should be understood that the present invention is applicable to any plasma etching system, including reactive ion etching (RIE), high-density, inductively coupled plasma (ICP) systems, electron cyclotron resonance (ECR) systems, radio frequency induction systems, and the like Data compression of OES spectra as in any of the embodiments disclosed above may solve the set of problems is posed by the sheer number of OES frequencies or wavelengths available to monitor The monitoring typically generates a large amount of data For example, a data file for each wafer monitored may be as large as 2-3 megabytes (MB), and each etcher can typically process about 500-700 wafers per day Conventional storage methods would require over a gigabytes (GB) of storage space per etcher per day and over 365 GB per etcher per year Further, the raw OES data generated in such monitoring is typically "noisy" and unenhanced Compression ratios of 100 1, as in various of the illustrative embodiments disclosed above, would only require tens of MB of storage per etcher per day and only about 4 GB per etcher per year In addition, data compression and reconstruction of OES spectra as in any of the embodiments disclosed above may smooth and enhance the otherwise noisy and unenhanced raw OES data generated in etch monitoring Data compression of OES spectra as in any of the embodiments disclosed above may feature high compression ratios for the raw OES data ranging from about 20 1 to about 400 1, provide for efficient noise removal from the raw OES data and preserve relevant features of raw OES data Data compression of OES spectra as in any of the embodiments disclosed above may also have fast computation characteristics, such as fast compression and fast reconstruction, so as to be suitable for robust on-line, real-time implementation Data compression of OES spectra as in any of the embodiments disclosed above may further be compatible with real-time, PCA-based fault detection and classification (FDC), improve the accuracy and efficiency of etch processing, simplify manufacturing, lower overall costs and increase throughput The particular embodiments disclosed above are illustrative only, as the invention may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein Furthermore, no limitations are intended to the details ot construction or design herein shown, other than as described in the claims below It is therefore evident that the particular embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the invention Accordingly, the protection sought herein is as set forth in the claims below

Claims: CLAIMS 1 A method for determining an etch endpoint (855), the method comprising collecting intensity data (820) representative of optical emission spectral wavelengths duπng a plasma etch process (810), calculating Scores (845) from at least a portion of the collected intensity data (820) using at most first, second, third and fourth Principal Components deπved from a model (835), and determining the etch endpoint (855) using Scores (845) corresponding to at least one of the first, second, third and fourth Principal Components as an indicator (835) for the etch endpoint (855) 2 The method of claim 1 , wherein determining the etch endpoint (855) using the Scores (845) corresponding to the at least one of the first, second, third and fourth Principal Components as an indicator (835) includes using a trace (2700) of the Scores (845) corresponding to the at least one of the first, second, third and fourth Principal Components along time to signal (2710) the etch endpoint (855) 3 The method of claim 2, wherein using the trace (2700) of the Scores (845) corresponding to the at least one of the first, second, third and fourth Principal Components along time to signal (2710) the etch endpoint (855) includes using the trace (2700) of the Scores (845) of the second Principal Component along time to signal (2710) the etch endpoint (855) 4 The method of claim 1 wherein determining the etch endpoint (855) using the Scores (845) corresponding to the at least one ot the first, second, third and fourth Principal Components as an indicator (835) further comprises selecting multiple wavelengths (2610, 2620. 2630) as indicator variables based on Loadings for at least one of the first, second, third and fourth Principal Components, the multiple wavelengths (2610, 2620, 2630) varying during the plasma process so that the etch endpoint (855) can be determined by monitoring the multiple wavelengths (2610 2620 2630) 5 The method of claim 2 wherein determining the etch endpoint (855) using the Scores (845) corresponding to the at least one of the first, second, third and fourth Principal Components as an indicator (835) further comprises selecting multiple wavelengths (2610, 2620, 2630) as indicator variables based on Loadings for the at least one of the first, second, third and fourth Principal Components, the multiple wavelengths (2610, 2620, 2630) varying during the plasma process so that the etch endpoint (855) can be determined by monitoring the multiple wavelengths (2610, 2620 2630) 6 The method of claim 3 wherein determining the etch endpoint (855) using the Scores (845) corresponding to the at least one of the first, second, third and fourth Principal Components as the indicator (835) further comprises selecting multiple wavelengths (2610. 2620, 2630) as indicator variables based on Loadings for the second Principal Component, the multiple wavelengths (2610. 2620, 2630) varying during the plasma process so that the etch endpoint (855) can be determined by monitoring the multiple wavelengths (2610, 2620, 2630) 7 The method of claim 1 , further comprising scaling the at least the portion of the collected intensity data (820) to generate mean values and mean-scaled values for the collected intensity data (820) 8 The method of claim 1 , further comprising building the model (835) by saving Loadings corresponding to at most first, second, third and fourth Principal Components derived from analyzing a plurality of intensity data representative of optical emission spectral wavelengths collected duπng a plurality of previous plasma etch processes (805), using at least one of an eigenanalysis method, a singular value decomposition method, and a power method 9 A computer-readable, program storage device encoded with instructions that, when executed by a computer, perform a method, the method comprising collecting intensity data (820) representative of optical emission spectral wavelengths during a plasma etch process (810), calculating Scores (845) from at least a portion of the collected intensity data (820) using at most first, second, third and fourth Principal Components derived from a model (835), and determining the etch endpoint (855) using Scores (845) corresponding to at least one of the first, second third and fourth Principal Components as an indicator (835) for the etch endpoint (855) 10 A computer programmed to perform a method, the method comprising collecting intensity data (820) representative of optical emission spectral wavelengths during a plasma etch process (810), calculating Scores (845) trom at least a portion of the collected intensity data (820) using at most first, second, third and fourth Principal Components deπved from a model (835), and determining the etch endpoint (855) using Scores (845) corresponding to at least one of the first, second, third and fourth Principal Components as an indicator (835) for the etch endpoint (855)

Inventors: TOPRAC ANTHONY JOHN, YUE HONGYU, TOPRAC, ANTHONY, JOHN, YUE, HONGYU
Applicants: ADVANCED MICRO DEVICES INC, ADVANCED MICRO DEVICES, INC.
Application year: 2000
Main classification: H01J37
IPC codes: A61K39, C07K14, G01N21, H01J37, H01L21, H05H1