Patent Publication Number: US-2012034845-A1

Title: Techniques for matching measured spectra to reference spectra for in-situ optical monitoring

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority from Provisional Application Ser. No. 61/371,405, filed Aug. 6, 2010, which is incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to optical monitoring, e.g., during chemical mechanical polishing of substrates. 
     BACKGROUND 
     An integrated circuit is typically formed on a substrate by the sequential deposition of conductive, semiconductive, or insulative layers on a silicon wafer. One fabrication step involves depositing a filler layer over a non-planar surface and planarizing the filler layer. For certain applications, the filler layer is planarized until the top surface of a patterned layer is exposed. A conductive filler layer, for example, can be deposited on a patterned insulative layer to fill the trenches or holes in the insulative layer. After planarization, the portions of the conductive layer remaining between the raised pattern of the insulative layer form vias, plugs, and lines that provide conductive paths between thin film circuits on the substrate. For other applications, such as oxide polishing, the filler layer is planarized until a predetermined thickness is left over the non planar surface. In addition, planarization of the substrate surface is usually required for photolithography. 
     Chemical mechanical polishing (CMP) is one accepted method of planarization. This planarization method typically requires that the substrate be mounted on a carrier head. The exposed surface of the substrate is typically placed against a rotating polishing pad. The carrier head provides a controllable load on the substrate to push it against the polishing pad. A polishing liquid, such as a slurry with abrasive particles, is typically supplied to the surface of the polishing pad. 
     One problem in CMP is determining whether the polishing process is complete, i.e., whether a substrate layer has been planarized to a desired flatness or thickness, or when a desired amount of material has been removed. Variations in the initial thickness of the substrate layer, the slurry composition, the polishing pad condition, the relative speed between the polishing pad and the substrate, and the load on the substrate can cause variations in the material removal rate. These variations cause variations in the time needed to reach the polishing endpoint. Therefore, it may not be possible to determine the polishing endpoint merely as a function of polishing time. 
     In some systems, a substrate is optically monitored in-situ during polishing, e.g., through a window in the polishing pad. However, existing optical monitoring techniques may not satisfy increasing demands of semiconductor device manufacturers. 
     SUMMARY 
     In some optical monitoring processes, a spectrum measured in-situ, e.g., during the polishing processes, is compared to a library of reference spectra to find the best matching reference spectrum. One technique to find the best match is to calculate the sum of squares difference between the measured spectrum and each reference spectrum in the library; the reference spectrum with the lowest sum of squared differences is the best match. However, for polishing of some substrates, e.g., where multiple dielectric layers are to be removed at the same platen, the matching algorithm may be unreliable. Without being limited to any particular theory, the sum of squared differences may be highly sensitive to the position of peaks in the spectra, and variations in the thickness of an underlying layer may cause shifts in the positions of the peaks. However, it may be possible to reduce or avoid these problems if another technique, such as cross-correlation, is used to find the best matching reference spectrum. 
     In one aspect, a method of controlling polishing includes storing a library having a plurality of reference spectra, polishing a substrate, measuring a sequence of spectra of light from the substrate during polishing, for each measured spectrum of the sequence of spectra, finding a best matching reference spectrum using a matching technique other than sum of squared differences to generate a sequence of best matching reference spectra, and determining at least one of a polishing endpoint or an adjustment for a polishing rate based on the sequence of best matching reference spectra. 
     Implementations can include one ore more of the following features. Finding a best matching reference spectrum may include performing a cross-correlation of the measured spectrum with each of two or more of the plurality of reference spectra from the library and selecting a reference spectrum with the greatest correlation to the measured spectrum as a best matching reference spectrum. Each reference spectrum of the plurality of reference spectra may have a stored associated index value, and the associated index value for each best matching spectrum from the sequence of best matching reference spectra may be determined to generate a sequence of index values, and a function may be fit to the sequence of index values. Polishing may be halted when the linear function matches or exceeds a target index. The substrate may include a second layer overlying a first layer, the first layer having a different composition than the second layer. The second layer may be a barrier layer and the first layer may be a dielectric layer. The barrier layer may be tantalum nitride or titanium nitride and the dielectric layer may be carbon doped silicon dioxide or may be formed from tetraethyl orthosilicate. The function may be fit to a portion of the sequence of index values corresponding to spectra measured after detection of exposure of the first layer. Finding a best matching reference spectrum may include performing a sum of enclidean vector distances between the measured spectrum and each of two or more of the plurality of reference spectra from the library and selecting a reference spectrum with the lowest sum as a best matching reference spectrum. Finding a best matching reference spectrum may include performing a sum of derivative differences between the measured spectrum and each of two or more of the plurality of reference spectra from the library and selecting a reference spectrum with the lowest sum as a best matching reference spectrum. Measuring the sequence of spectra of light from the substrate may include making a plurality of sweeps of a sensor across the substrate. Each spectrum from the sequence of spectra may correspond to a single sweep of the sensor from the plurality of sweeps. The substrate may include a plurality of zones, and a polishing rate of each zone may be independently controllable by an independently variable polishing parameter. A sequence of spectra may be measured from each zone during polishing. For each measured spectrum in the sequence of spectra for each zone, a cross-correlation of the measured spectrum with each of two or more of the plurality of reference spectra from the library may be performed, and a reference spectrum with the greatest correlation to the measured spectrum may be selected as a best matching reference spectrum from the library to generate a sequence of best matching reference spectra. The polishing parameter for at least one zone may be adjusted to adjust the polishing rate of the at least one zone such that the plurality of zones have a smaller difference in thickness at the polishing endpoint than without such adjustment. 
     In another aspect, a computer program product, tangibly embodied in a machine readable storage device, includes instructions to carry out the method. 
     In another aspect, a polishing apparatus includes a support to hold a polishing pad, a carrier head to hold a substrate against the polishing pad, a motor to generate relative motion between the carrier head and the support to polish the substrate, an optical monitoring system to measure a sequence of spectra of light from the substrate while the substrate is being polished, and a controller. The controller is configured to store a library having a plurality of reference spectra, each reference spectrum of the plurality of reference spectra having a stored associated index value, for each measured spectrum of the sequence of spectra, find a best matching reference spectrum using a matching technique other than sum of squared differences to generate a sequence of best matching reference spectra, and determine at least one of a polishing endpoint or an adjustment for a polishing rate based on the sequence of best matching reference spectra. 
     Implementations may optionally include one or more of the following advantages. The matching technique can be less sensitive to positions of peaks in the measured spectra, thereby reducing sensitivity to underlying thickness variation. Reliability of the endpoint system to detect a desired polishing endpoint can be improved, and within-wafer and wafer-to-wafer thickness non-uniformity (WTWNU and WTWNU) can be reduced. 
     The details of one or more embodiments are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages will become apparent from the description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A and 1B  are schematic cross-sectional views of a substrate before and after polishing. 
         FIG. 2  illustrates a schematic cross-sectional view of an example of a polishing apparatus. 
         FIG. 3  illustrates a schematic top view of a substrate having multiple zones. 
         FIG. 4  illustrates a top view of a polishing pad and shows locations where in-situ measurements are taken on a substrate. 
         FIG. 5  illustrates a measured spectrum from the in-situ optical monitoring system. 
         FIG. 6  illustrates a library of reference spectra. 
         FIG. 7  illustrates an index trace. 
         FIG. 8  illustrates an index trace having a linear function fit to index values collected after clearance of an overlying layer is detected. 
         FIG. 9  is a flow diagram of an example process for fabricating a substrate and detecting a polishing endpoint. 
         FIG. 10  illustrates a plurality of index traces. 
         FIG. 11  illustrates a calculation of a plurality of desired slopes for a plurality of adjustable zones based on a time that an index trace of a reference zone reaches a target index. 
         FIG. 12  illustrates a calculation of an endpoint for based on a time that an index trace of a reference zone reaches a target index. 
         FIG. 13  is a flow diagram of an example process for adjusting the polishing rate of a plurality of zones in a plurality of substrates such that the plurality of zones have approximately the same thickness at the target time. 
         FIG. 14  shows a flow chart for detecting clearance of an overlying layer. 
         FIG. 15A  shows a graph of spectra collected during a single sweep at the beginning of polishing. 
         FIG. 15B  shows a graph of spectra collected during a single sweep near barrier clearing. 
         FIG. 16  shows a graph of standard deviation of spectra as a function of polishing time. 
         FIG. 17  is a graph showing a comparison of different techniques for determining a best matching reference spectrum. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     One optical monitoring technique is to measure spectra of light reflected from a substrate during polishing, and identify a matching reference spectra from a library. The matching reference spectra provide a series of index values, and a function, e.g., a line, is fit to the series of index values. The projection of the function to a target value can be used to determine endpoint or to change a polishing rate. 
     One potential problem, for polishing of some types of substrates, e.g., substrates in which multiple layers of material are to be polished at the same platen, is that the technique used to find the best matching spectrum from the library, e.g., selecting the reference spectrum with the lowest sum of squared differences, can be unreliable. Without being limited to any particular theory, a sum of squares difference may be highly sensitive to the position of peaks in the spectra, and variations in the thickness of an underlying layer may cause shifts in the positions of the peaks. However, it may be possible to reduce or avoid these problems if another technique, e.g., cross-correlation, is used to find the best matching reference spectrum. 
     As an example, referring to  FIG. 1A , a substrate  10  can include a patterned first layer  12  (this layer can also be referred to as an underlying layer) of a first dielectric material, e.g., a low-k material, e.g., carbon doped silicon dioxide, e.g., Black Diamond™ (from Applied Materials, Inc.) or Coral™ (from Novellus Systems, Inc.). Disposed over the first layer  12  is a second layer  16  (this layer can also be referred to as an overlying layer) of a different second dielectric material, e.g., a barrier layer, e.g., a nitride, e.g., tantalum nitride or titanium nitride. Optionally disposed between the first layer and the second layer are one or more additional layers  14  of another dielectric material, different from both the first and second dielectric materials, e.g., a low-k capping material, e.g., a material formed from tetraethyl orthosilicate (TEOS). Together, the first layer  12  and the one or more additional layers  14  provide a layer stack below the second layer. Disposed over the second layer (and in trenches provided by the pattern of the first layer) is a conductive material  18 , e.g., a metal, e.g., copper. 
     Chemical mechanical polishing can be used to planarize the substrate until the first layer of the first dielectric material is exposed. For example, referring to  FIG. 1B , after planarization, the portions of the conductive material  18  remaining between the raised pattern of the first layer  12  form vias and the like. In addition, it is sometimes desired to remove the first dielectric material until a target thickness remains or a target amount of material has been removed. 
     One method of polishing is to polish the conductive material on a first polishing pad at least until the second layer, e.g., the barrier layer, is exposed. In addition, a portion of the thickness of the second layer can be removed, e.g., during an overpolishing step at the first polishing pad. The substrate is then transferred to a second polishing pad, where the second layer, e.g., the barrier layer is completely removed, and a portion of the thickness of the underlying first layer, e.g., the low-k dielectric, is also removed. In addition, if present, the additional layer or layers, e.g., the capping layer, between the first and second layer can be removed in the same polishing operation at the second polishing pad. 
       FIG. 2  illustrates an example of a polishing apparatus  100 . The polishing apparatus  100  includes a rotatable disk-shaped platen  120  on which a polishing pad  110  is situated. 
     The platen is operable to rotate about an axis  125 . For example, a motor  121  can turn a drive shaft  124  to rotate the platen  120 . The polishing pad  110  can be a two-layer polishing pad with an outer polishing layer  112  and a softer backing layer  114 . 
     The polishing apparatus  100  can include a port  130  to dispense polishing liquid  132 , such as a slurry, onto the polishing pad  110  to the pad. The polishing apparatus can also include a polishing pad conditioner to abrade the polishing pad  110  to maintain the polishing pad  110  in a consistent abrasive state. 
     The polishing apparatus  100  includes one or more carrier heads  140 . Each carrier head  140  is operable to hold a substrate  10  against the polishing pad  110 . Each carrier head  140  can have independent control of the polishing parameters, for example pressure, associated with each respective substrate. 
     In particular, each carrier head  140  can include a retaining ring  142  to retain the substrate  10  below a flexible membrane  144 . Each carrier head  140  also includes a plurality of independently controllable pressurizable chambers defined by the membrane, e.g., 3 chambers  146   a - 146   c,  which can apply independently controllable pressurizes to associated zones  148   a - 148   c  on the flexible membrane  144  and thus on the substrate  10  (see  FIG. 3 ). Referring to  FIG. 2 , the center zone  148   a  can be substantially circular, and the remaining zones  148   b - 148   e  can be concentric annular zones around the center zone  148   a . Although only three chambers are illustrated in  FIGS. 2 and 3  for ease of illustration, there could be one or two chambers, or four or more chambers, e.g., five chambers. 
     Returning to  FIG. 2 , each carrier head  140  is suspended from a support structure  150 , e.g., a carousel, and is connected by a drive shaft  152  to a carrier head rotation motor  154  so that the carrier head can rotate about an axis  155 . Optionally each carrier head  140  can oscillate laterally, e.g., on sliders on the carousel  150 ; or by rotational oscillation of the carousel itself. In operation, the platen is rotated about its central axis  125 , and each carrier head is rotated about its central axis  155  and translated laterally across the top surface of the polishing pad. 
     While only one carrier head  140  is shown, more carrier heads can be provided to hold additional substrates so that the surface area of polishing pad  110  may be used efficiently. Thus, the number of carrier head assemblies adapted to hold substrates for a simultaneous polishing process can be based, at least in part, on the surface area of the polishing pad  110 . 
     The polishing apparatus also includes an in-situ optical monitoring system  160 , e.g., a spectrographic monitoring system, which can be used to determine whether to adjust a polishing rate or an adjustment for the polishing rate as discussed below. An optical access through the polishing pad is provided by including an aperture (i.e., a hole that runs through the pad) or a solid window  118 . The solid window  118  can be secured to the polishing pad  110 , e.g., as a plug that fills an aperture in the polishing pad, e.g., is molded to or adhesively secured to the polishing pad, although in some implementations the solid window can be supported on the platen  120  and project into an aperture in the polishing pad. 
     The optical monitoring system  160  can include a light source  162 , a light detector  164 , and circuitry  166  for sending and receiving signals between a remote controller  190 , e.g., a computer, and the light source  162  and light detector  164 . One or more optical fibers can be used to transmit the light from the light source  162  to the optical access in the polishing pad, and to transmit light reflected from the substrate  10  to the detector  164 . For example, a bifurcated optical fiber  170  can be used to transmit the light from the light source  162  to the substrate  10  and back to the detector  164 . The bifurcated optical fiber an include a trunk  172  positioned in proximity to the optical access, and two branches  174  and  176  connected to the light source  162  and detector  164 , respectively. 
     In some implementations, the top surface of the platen can include a recess  128  into which is fit an optical head  168  that holds one end of the trunk  172  of the bifurcated fiber. The optical head  168  can include a mechanism to adjust the vertical distance between the top of the trunk  172  and the solid window  118 . 
     The output of the circuitry  166  can be a digital electronic signal that passes through a rotary coupler  129 , e.g., a slip ring, in the drive shaft  124  to the controller  190  for the optical monitoring system. Similarly, the light source can be turned on or off in response to control commands in digital electronic signals that pass from the controller  190  through the rotary coupler  129  to the optical monitoring system  160 . Alternatively, the circuitry  166  could communicate with the controller  190  by a wireless signal. 
     The light source  162  can be operable to emit white light. In one implementation, the white light emitted includes light having wavelengths of 200-800 nanometers. A suitable light source is a xenon lamp or a xenon mercury lamp. 
     The light detector  164  can be a spectrometer. A spectrometer is an optical instrument for measuring intensity of light over a portion of the electromagnetic spectrum. A suitable spectrometer is a grating spectrometer. Typical output for a spectrometer is the intensity of the light as a function of wavelength (or frequency). 
     As noted above, the light source  162  and light detector  164  can be connected to a computing device, e.g., the controller  190 , operable to control their operation and receive their signals. The computing device can include a microprocessor situated near the polishing apparatus, e.g., a programmable computer. With respect to control, the computing device can, for example, synchronize activation of the light source with the rotation of the platen  120 . 
     In some implementations, the light source  162  and detector  164  of the in-situ monitoring system  160  are installed in and rotate with the platen  120 . In this case, the motion of the platen will cause the sensor to scan across each substrate. In particular, as the platen  120  rotates, the controller  190  can cause the light source  162  to emit a series of flashes starting just before and ending just after the optical access passes below the substrate  10 . Alternatively, the computing device can cause the light source  162  to emit light continuously starting just before and ending just after each substrate  10  passes over the optical access. In either case, the signal from the detector can be integrated over a sampling period to generate spectra measurements at a sampling frequency. 
     In operation, the controller  190  can receive, for example, a signal that carries information describing a spectrum of the light received by the light detector for a particular flash of the light source or time frame of the detector. Thus, this spectrum is a spectrum measured in-situ during polishing. 
     As shown by in  FIG. 4 , if the detector is installed in the platen, due to the rotation of the platen (shown by arrow  204 ), as the window  108  travels below a carrier head, the optical monitoring system making spectra measurements at a sampling frequency will cause the spectra measurements to be taken at locations  201  in an arc that traverses the substrate  10 . For example, each of points  201   a - 201   k  represents a location of a spectrum measurement by the monitoring system (the number of points is illustrative; more or fewer measurements can be taken than illustrated, depending on the sampling frequency). The sampling frequency can be selected so that between five and twenty spectra are collected per sweep of the window  108 . For example, the sampling period can be between 3 and 100 milliseconds. 
     As shown, over one rotation of the platen, spectra are obtained from different radii on the substrate  10 . That is, some spectra are obtained from locations closer to the center of the substrate  10  and some are closer to the edge. Thus, for any given scan of the optical monitoring system across a substrate, based on timing, motor encoder information, and optical detection of the edge of the substrate and/or retaining ring, the controller  190  can calculate the radial position (relative to the center of the substrate being scanned) for each measured spectrum from the scan. The polishing system can also include a rotary position sensor, e.g., a flange attached to an edge of the platen that will pass through a stationary optical interrupter, to provide additional data for determination of which substrate and the position on the substrate of the measured spectrum. The controller can thus associate the various measured spectra with the controllable zones  148   b - 148   e  (see  FIG. 2 ) on the substrates  10   a  and  10   b . In some implementations, the time of measurement of the spectrum can be used as a substitute for the exact calculation of the radial position. 
     Over multiple rotations of the platen, for each zone, a sequence of spectra can be obtained over time. Without being limited to any particular theory, the spectrum of light reflected from the substrate  10  evolves as polishing progresses (e.g., over multiple rotations of the platen, not during a single sweep across the substrate) due to changes in the thickness of the outermost layer, thus yielding a sequence of time-varying spectra. Moreover, particular spectra are exhibited by particular thicknesses of the layer stack. 
     In some implementations, the controller, e.g., the computing device, can be programmed to compare a measured spectrum to multiple reference spectra and to determine which reference spectrum provides the best match. In particular, the controller can be programmed to compare each spectrum from a sequence of measured spectra from each zone to multiple reference spectra to generate a sequence of best matching reference spectra for each zone. 
     As used herein, a reference spectrum is a predefined spectrum generated prior to polishing of the substrate. A reference spectrum can have a pre-defined association, i.e., defined prior to the polishing operation, with a value representing a time in the polishing process at which the spectrum is expected to appear, assuming that the actual polishing rate follows an expected polishing rate. Alternatively or in addition, the reference spectrum can have a pre-defined association with a value of a substrate property, such as a thickness of the outermost layer. 
     A reference spectrum can be generated empirically, e.g., by measuring the spectra from a test substrate, e.g., a test substrate having a known initial layer thicknesses. For example, to generate a plurality of reference spectra, a set-up substrate is polished using the same polishing parameters that would be used during polishing of device wafers while a sequence of spectra are collected. For each spectrum, a value is recorded representing the time in the polishing process at which the spectrum was collected. For example, the value can be an elapsed time, or a number of platen rotations. The substrate can be overpolished, i.e., polished past a desired thickness, so that the spectrum of the light that reflected from the substrate when the target thickness is achieved can be obtained. 
     In order to associate each spectrum with a value of a substrate property, e.g., a thickness of the outermost layer, the initial spectra and property of a “set-up” substrate with the same pattern as the product substrate can be measured pre-polish at a metrology station. The final spectrum and property can also be measured post-polish with the same metrology station or a different metrology station. The properties for spectra between the initial spectra and final spectra can be determined by interpolation, e.g., linear interpolation based on elapsed time at which the spectra of the test substrate was measured. 
     In addition to being determined empirically, some or all of the reference spectra can be calculated from theory, e.g., using an optical model of the substrate layers. For example, and optical model can be used to calculate a reference spectrum for a given outer layer thickness D. A value representing the time in the polishing process at which the reference spectrum would be collected can be calculated, e.g., by assuming that the outer layer is removed at a uniform polishing rate. For example, the time Ts for a particular reference spectrum can be calculated simply by assuming a starting thickness D 0  and uniform polishing rate R (Ts=(D 0 −D)/R). As another example, linear interpolation between measurement times T 1 , T 2  for the pre-polish and post-polish thicknesses D 1 , D 2  (or other thicknesses measured at the metrology station) based on the thickness D used for the optical model can be performed (Ts=T 2 −T 1 *(D 1 −D)/(D 1 −D 2 )). 
     Referring to  FIGS. 5 and 6 , a measured spectrum  300  (see  FIG. 5 ) can be compared to reference spectra  320  from one or more libraries  310  (see  FIG. 6 ). As used herein, a library of reference spectra is a collection of reference spectra which represent substrates that share a property in common. However, the property shared in common in a single library may vary across multiple libraries of reference spectra. For example, two different libraries can include reference spectra that represent substrates with two different underlying thicknesses. For a given library of reference spectra, variations in the upper layer thickness, rather than other factors (such as differences in wafer pattern, underlying layer thickness, or layer composition), can be primarily responsible for the differences in the spectral intensities. 
     Reference spectra  320  for different libraries  310  can be generated by polishing multiple “set-up” substrates with different substrate properties (e.g., underlying layer thicknesses, or layer composition) and collecting spectra as discussed above; the spectra from one set-up substrate can provide a first library and the spectra from another substrate with a different underlying layer thickness can provide a second library. Alternatively or in addition, reference spectra for different libraries can be calculated from theory, e.g., spectra for a first library can be calculated using the optical model with the underlying layer having a first thickness, and spectra for a second library can be calculated using the optical model with the underlying layer having a different one thickness. 
     In some implementations, each reference spectrum  320  is assigned an index value  330 . In general, each library  310  can include many reference spectra  320 , e.g., one or more, e.g., exactly one, reference spectra for each platen rotation over the expected polishing time of the substrate. This index  330  can be the value, e.g., a number, representing the time in the polishing process at which the reference spectrum  320  is expected to be observed. The spectra can be indexed so that each spectrum in a particular library has a unique index value. The indexing can be implemented so that the index values are sequenced in an order in which the spectra of a test substrate were measured. An index value can be selected to change monotonically, e.g., increase or decrease, as polishing progresses. In particular, the index values of the reference spectra can be selected so that they form a linear function of time or number of platen rotations (assuming that the polishing rate follows that of the model or test substrate used to generate the reference spectra in the library). For example, the index value can be proportional, e.g., equal, to a number of platen rotations at which the reference spectra was measured for the test substrate or would appear in the optical model. Thus, each index value can be a whole number. The index number can represent the expected platen rotation at which the associated spectrum would appear. 
     The reference spectra and their associated index values can be stored in a reference library. For example, each reference spectrum  320  and its associated index value  330  can be stored in a record  340  of database  350 . The database  350  of reference libraries of reference spectra can be implemented in memory of the computing device of the polishing apparatus. 
     As noted above, for each zone of each substrate, based on the sequence of measured spectra or that zone and substrate, the controller  190  can be programmed to generate a sequence of best matching spectra. A best matching reference spectrum can be determined by comparing a measured spectrum to the reference spectra from a particular library. 
     In some implementations, the best matching reference spectrum can be determined by calculating, for each reference spectrum, a sum of squared differences between the measured spectrum and the reference spectrum. The reference spectrum with the lowest sum of squared differences has the best fit. Other techniques for finding a best matching reference spectrum are possible, e.g., lowest sum of absolute differences. 
     In some implementations, the best matching reference spectrum can be determined by using a matching technique other than sum of squared differences. In one implementation, for each reference spectrum, a cross-correlation between the measured spectrum and the reference spectrum is calculated, and the reference spectrum with the greatest correlation is selected as the matching reference spectrum. A potential advantage of cross-correlation is that it is less sensitive to lateral shift of a spectrum, and thus can be less sensitive to underlying thickness variation. In order to perform the cross-correlation, the leading and trailing ends of the measured spectrum can be padded with “zeros” to provide data to compare against the reference spectrum as the reference spectrum is shifted relative to the measured spectrum. Alternatively, the leading end of the measured spectrum can be padded with values equal to the value at the leading edge of the measured spectrum, and he trailing end of the measured spectrum can be padded with values equal to the value at the trailing edge of the measured spectrum. Fast Fourier transforms can be used to increase the speed of calculation of the cross-correlation for real-time application of the matching technique. 
     In another implementation, a sum of enclidean vector distances, e.g., D=1/(λa−λb)·[Σ λ=λa to λb ⊕I M (λ) 2 −I R (λ) 2 |], where λa to λb is wavelength summed over, calculated, I M (λ) is the measured spectrum, and I R (λ) is the reference spectrum. In another implementation, for each reference spectrum, a sum of derivative differences, e.g., D=1/(λa−λb)·[Σ λ=λa to λb |dI M (λ)/dλ−dI R (λ)/dλ|], and the reference spectrum with the lowest sum is selected as the matching reference spectrum. 
       FIG. 17  illustrates a comparison of index traces (indexes of best matching reference spectra as a function number of platen rotations) for spectra matching using cross-correlation and sum of squared differences methods for substrates with different thicknesses of the TEOS layer. The data was generated for product substrates having a stack of 1500 {acute over (Å)} thick layer of Black Diamond, a 130 {acute over (Å)} thick layer of Blok, and a TEOS layer that is 5200 {acute over (Å)}, 5100 {acute over (Å)} or 5000 {acute over (Å)} thick. A reference library was generated for a reference substrate having a TEOS layer that is 5200 {acute over (Å)} thick. As shown by trace  1702 , where the product substrate and the reference substrate have a TEOS layer of the same thickness, i.e., 5200 {acute over (Å)}, the two index traces overlap with no appreciable difference. However, where product substrate has a TEOS layer that is 5100 {acute over (Å)} thick and the reference substrate has a TEOS layer 5200 {acute over (Å)} thick, the index trace  1704  generated using sum of squared differences has some departure from linear behavior. In contrast, the index trace generated using cross-correlation overlaps the index trace  1702  (and is thus not visible in the graph). Finally, where product substrate has a TEOS layer that is 5000 {acute over (Å)} thick and the reference substrate has a TEOS layer 5200 {acute over (Å)} thick, the index trace  1706  generated using sum of squared differences has a significant departure from linear behavior and the trace  1702 , whereas the index trace  1708  generated using cross-correlation remains generally linear and much closer to the trace  1702 . In sum, this shows that using cross-correlation to determine the best matching spectrum results in a trace that better matches the ideal when there are variations in the thickness of the underlying layer. 
     A method that can be applied to decrease computer processing is to limit the portion of the library that is searched for matching spectra. The library typically includes a wider range of spectra than will be obtained while polishing a substrate. During substrate polishing, the library searching is limited to a predetermined range of library spectra. In some embodiments, the current rotational index N of a substrate being polished is determined. For example, in an initial platen rotation, N can be determined by searching all of the reference spectra of the library. For the spectra obtained during a subsequent rotation, the library is searched within a range of freedom of N. That is, if during one rotation the index number is found to be N, during a subsequent rotation which is X rotations later, where the freedom is Y, the range that will be searched from (N+X)−Y to (N+X)+Y. 
     Referring to  FIG. 7 , which illustrates the results for only a single zone of a single substrate, the index value of each of the best matching spectra in the sequence can be determined to generate a time-varying sequence of index values  212 . This sequence of index values can be termed an index trace  210 . In some implementations, an index trace is generated by comparing each measured spectrum to the reference spectra from exactly one library. In general, the index trace  210  can include one, e.g., exactly one, index value per sweep of the optical monitoring system below the substrate. 
     For a given index trace  210 , where there are multiple spectra measured for a particular zone in a single sweep of the optical monitoring system (termed “current spectra”), a best match can be determined between each of the current spectra and the reference spectra of one or more, e.g., exactly one, library. In some implementations, each selected current spectra is compared against each reference spectra of the selected library or libraries. Given current spectra e, f, and g, and reference spectra E, F, and G, for example, a matching coefficient could be calculated for each of the following combinations of current and reference spectra: e and E, e and F, e and G, f and E, f and F, f and G, g and E, g and F, and g and G. Whichever matching coefficient indicates the best match, e.g., is the smallest, determines the best-matching reference spectrum, and thus the index value. Alternatively, in some implementations, the current spectra can be combined, e.g., averaged, and the resulting combined spectrum is compared against the reference spectra to determine the best match, and thus the index value. 
     In some implementations, for at least some zones of some substrates, a plurality of index traces can be generated. For a given zone of a given substrate, an index trace can be generated for each reference library of interest. That is, for each reference library of interest to the given zone of the given substrate, each measured spectrum in a sequence of measured spectra is compared to reference spectra from a given library, a sequence of the best matching reference spectra is determined, and the index values of the sequence of best matching reference spectra provide the index trace for the given library. 
     In summary, each index trace includes a sequence  210  of index values  212 , with each particular index value  212  of the sequence being generated by selecting the index of the reference spectrum from a given library that is the closest fit to the measured spectrum. The time value for each index of the index trace  210  can be the same as the time at which the measured spectrum was measured. 
     An in-situ monitoring technique is used to detect clearing of the second layer and exposure of the underlying layer or layer structure. For example, exposure of the first layer at a time TC can be detected by a sudden change in the motor torque or total intensity of light reflected from the substrate, or from dispersion of the collected spectra as discussed in greater detail below. 
     As shown in  FIG. 8 , a function, e.g., a polynomial function of known order, e.g., a first-order function (e.g., a line  214 ) is fit to the sequence of index values of spectra collected after time TC, e.g., using robust line fitting. Index values for spectra collected before the time TC are ignored when fitting the function to the sequence of index values. Other functions can be used, e.g., polynomial functions of second-order, but a line provides ease of computation. Polishing can be halted at an endpoint time TE that the line  214  crosses a target index IT. 
       FIG. 9  shows a flow chart of a method of fabricating and polishing a product substrate. The product substrate can have at least the same layer structure and the same pattern, as the test substrates used to generate the reference spectra of the library. 
     Initially, the first layer is deposited on the substrate and patterned (step  902 ). As noted above, the first layer can be a dielectric, e.g., a low-k material, e.g., carbon doped silicon dioxide, e.g., Black Diamond™ (from Applied Materials, Inc.) or Coral™ (from Novellus Systems, Inc.). 
     Optionally, depending on the composition of the first material, one or more additional layers of another dielectric material, different from both the first and second dielectric materials, e.g., a low-k capping material, e.g., tetraethyl orthosilicate (TEOS), is deposited over the first layer on the product substrate (step  903 ). Together, the first layer and the one or more additional layers provide a layer stack. Optionally, patterning can occur after depositing of the one or more additional layers (so that the one or more additional layers do not extend into the trench in the first layer, as shown in  FIG. 1A ). 
     Next, the second layer of a different second dielectric material, e.g., a barrier layer, e.g., a nitride, e.g., tantalum nitride or titanium nitride, is deposited over the first layer or layer stack of the product substrate (step  904 ). In addition, a conductive layer, e.g., a metal layer, e.g., copper, can be deposited over the second layer of the product substrate (and in trenches provided by the pattern of the first layer) (step  906 ). Optionally, patterning of the first layer can occur after depositing of the second layer (in which case the second layer would not extend into the trench in the first layer). 
     The product substrate is polished (step  908 ). For example, the conductive layer and a portion of the second layer can be polished and removed at a first polishing station using a first polishing pad (step  908   a ). Then the second layer and a portion of the first layer can be polished and removed at a second polishing station using a second polishing pad (step  908   b ). However, it should be noted that for some implementations, there is no conductive layer, e.g., the second layer is the outermost layer when polishing begins. Of course, steps  902 - 906  can be performed elsewhere, so that the process for a particular operator of the polishing apparatus begins with step  908 . 
     An in-situ monitoring technique is used to detect clearing of the second layer and exposure of the first layer (step  910 ). For example, exposure of the first layer at a time TC (see  FIG. 8 ) can be detected by a sudden change in the motor torque or total intensity of light reflected from the substrate, or from dispersion of the collected spectra as discussed in greater detail below. 
     Beginning at least with detection of the clearance of second layer (and potentially earlier, e.g., from the beginning of polishing of the product substrate with the second polishing pad), a sequence of measured spectra are obtained during polishing (step  912 ), e.g., using the in-situ monitoring system described above. 
     The measured spectra are analyzed to generate a sequence of index values, and a function is fit to the sequence of index values. In particular, for each measured spectrum in the sequence of measured spectra, the index value for the reference spectrum that is the best fit is determined to generate the sequence of index values (step  914 ). A function, e.g., a linear function, is fit to the sequence of index values for the spectra collected after the time TC at which clearance of the second layer is detected (step  916 ). In other words, index values for spectra collected before the time TC at which clearance of the second layer is detected are not used in the calculation of the function. 
     Polishing can be halted once the index value (e.g., a calculated index value generated from the linear function fit to the new sequence of index values) reaches target index (step  918 ). The target thickness IT can be set by the user prior to the polishing operation and stored. Alternatively, a target amount to remove can be set by the user, and a target index IT can be calculated from the target amount to remove. For example, an index difference ID can be calculated from the target amount to remove, e.g., from an empirically determined ratio of amount removed to the index (e.g., the polishing rate), and adding the index difference ID to the index value IC at the time TC that clearance of the overlying layer is detected (see  FIG. 8 ). 
     It is also possible to use the function fit to the index values from spectra collected after clearance of the second layer is detected to adjust the polishing parameters, e.g., to adjust the polishing rate of one or more zones on a substrate to improve polishing uniformity. 
     Referring to  FIG. 10 , a plurality of index traces is illustrated. As discussed above, an index trace can be generated for each zone. For example, a first sequence  210  of index values  212  (shown by hollow circles) can be generated for a first zone, a second sequence  220  of index values  222  (shown by hollow squares) can be generated for a second zone, and a third sequence  230  of index values  232  (shown by hollow triangles) can be generated for a third zone. Although three zones are shown, there could be two zones or four or more zones. All of the zones can be on the same substrate, or some of the zones can be from different substrates being polished simultaneously on the same platen. 
     As discussed above, an in-situ monitoring technique is used to detect clearing of the second layer and exposure of the underlying layer or layer structure. For example, exposure of the first layer at a time TC can be detected by a sudden change in the motor torque or total intensity of light reflected from the substrate, or from dispersion of the collected spectra as discussed in greater detail below. 
     For each substrate index trace, a polynomial function of known order, e.g., a first-order function (e.g., a line) is fit to the sequence of index values of spectra collected after time TC for the associated zone, e.g., using robust line fitting. For example, a first line  214  can be fit to index values  212  for the first zone, a second line  224  can be fit to the index values  222  of the second zone, and a third line  234  can be fit to the index values  232  of the third zone. Fitting of a line to the index values can include calculation of the slope S of the line and an x-axis intersection time T at which the line crosses a starting index value, e.g., 0. The function can be expressed in the form I(t)=S·(t−T), where t is time. The x-axis intersection time T can have a negative value, indicating that the starting thickness of the substrate layer is less than expected. Thus, the first line  214  can have a first slope S 1  and a first x-axis intersection time T 1 , the second line  224  can have a second slope S 2  and a second x-axis intersection time T 2 , and the third line  234  can have a third slope S 3  and a third x-axis intersection time T 3 . 
     At some during the polishing process, e.g., at a time T 0 , a polishing parameter for at least one zone is adjusted to adjust the polishing rate of the zone of the substrate such that at a polishing endpoint time, the plurality of zones are closer to their target thickness than without such adjustment. In some embodiments, each zone can have approximately the same thickness at the endpoint time. 
     Referring to  FIG. 11 , in some implementations, one zone is selected as a reference zone, and a projected endpoint time TE at which the reference zone will reach a target index IT is determined. For example, as shown in  FIG. 11 , the first zone is selected as the reference zone, although a different zone and/or a different substrate could be selected. The target thickness IT is set by the user prior to the polishing operation and stored. Alternatively, a target amount to remove TR can be set by the user, and a target index IT can be calculated from the target amount to remove TR. For example, an index difference ID can be calculated from the target amount to remove, e.g., from an empirically determined ratio of amount removed to the index (e.g., the polishing rate), and adding the index difference ID to the index value IC at the time TC that clearance of the overlying layer is detected. 
     In order to determine the projected time at which the reference zone will reach the target index, the intersection of the line of the reference zone, e.g., line  214 , with the target index, IT, can be calculated. Assuming that the polishing rate does not deviate from the expected polishing rate through the remainder polishing process, then the sequence of index values should retain a substantially linear progression. Thus, the expected endpoint time TE can be calculated as a simple linear interpolation of the line to the target index IT, e.g., IT=S·(TE−T). Thus, in the example of  FIG. 11  in which the first zone is selected as the reference zone, with associated first line  214 , IT=S 1 ·(TE−T 1 ), i.e., TE=IT/S 1 −T 1 . 
     One or more zones, e.g., all zones, other than the reference zone (including zones on other substrates) can be defined as adjustable zones. Where the lines for the adjustable zones meet the expected endpoint time TE define projected endpoint for the adjustable zones. The linear function of each adjustable zone, e.g., lines  224  and  234  in  FIG. 11 , can thus be used to extrapolate the index, e.g., EI 2  and EI 3 , that will be achieved at the expected endpoint time ET for the associated zone. For example, the second line  224  can be used to extrapolate the expected index, EI 2 , at the expected endpoint time ET for the second zone, and the third line  234  can be used to extrapolate the expected index, EI 3 , at the expected endpoint time ET for the third zone. 
     As shown in  FIG. 11 , if no adjustments are made to the polishing rate of any of the zones after time T 0 , then if endpoint is forced at the same time for all zones, then each zone can have a different thickness (which is not desirable because it can lead to defects and loss of throughput). 
     If the target index will be reached at different times for different zones (or equivalently, the adjustable zones will have different expected indexes at the projected endpoint time of the reference zone), the polishing rate can be adjusted upwardly or downwardly, such that the zones would reach the target index (and thus target thickness) closer to the same time than without such adjustment, e.g., at approximately the same time, or would have closer to the same index value (and thus same thickness), at the target time than without such adjustment, e.g., approximately the same index value (and thus approximately the same thickness). 
     Thus, in the example of  FIG. 11 , commencing at a time T 0 , at least one polishing parameter for the second zone is modified so that the polishing rate of the zone is increased (and as a result the slope of the index trace  220  is increased). Also, in this example, at least one polishing parameter for the third zone is modified so that the polishing rate of the third zone is decreased (and as a result the slope of the index trace  230  is decreased). As a result the zones would reach the target index (and thus the target thickness) at approximately the same time (or if pressure to the zones halts at the same time, the zones will end with approximately the same thickness). 
     In some implementations, if the projected index at the expected endpoint time ET indicate that a zone of the substrate is within a predefined range of the target thickness, then no adjustment may be required for that zone. The range may be 2%, e.g., within 1%, of the target index. 
     The polishing rates for the adjustable zones can be adjusted so that all of the zones are closer to the target index at the expected endpoint time than without such adjustment. For example, a reference zone of the reference substrate might be chosen and the processing parameters for all of the other zone adjusted such that all of the zones will endpoint at approximately the projected time of the reference substrate. The reference zone can be, for example, a predetermined zone, e.g., the center zone  148   a  or the zone  148   b  immediately surrounding the center zone, the zone having the earliest or latest projected endpoint time of any of the zones of any of the substrates, or the zone of a substrate having the desired projected endpoint. The earliest time is equivalent to the thinnest substrate if polishing is halted at the same time. Likewise, the latest time is equivalent to the thickest substrate if polishing is halted at the same time. The reference substrate can be, for example, a predetermined substrate, a substrate having the zone with the earliest or latest projected endpoint time of the substrates. The earliest time is equivalent to the thinnest zone if polishing is halted at the same time. Likewise, the latest time is equivalent to the thickest zone if polishing is halted at the same time. 
     For each of the adjustable zones, a desired slope for the index trace can be calculated such that the adjustable zone reaches the target index at the same time as the reference zone. For example, the desired slope SD can be calculated from (IT−I)=SD*(TE−T0), where I is the index value (calculated from the linear function fit to the sequence of index values) at time T 0  polishing parameter is to be changed, IT is the target index, and TE is the calculated expected endpoint time. In the example of  FIG. 11 , for the second zone the desired slope SD 2  can be calculated from (IT−I 2 )=SD 2 *(TE−T 0 ), and for the third zone the desired slope SD 3  can be calculated from (IT−I 3 )=SD 3 *(TE−T 0 ). 
     Alternatively, in some implementations, there is no reference zone, and the expected endpoint time can be a predetermined time, e.g., set by the user prior to the polishing process, or can be calculated from an average or other combination of the expected endpoint times of two or more zones (as calculated by projecting the lines for various zones to the target index) from one or more substrates. In this implementation, the desired slopes are calculated substantially as discussed above, although the desired slope for the first zone of the first substrate must also be calculated, e.g., the desired slope SD 1  can be calculated from (IT−I 1 )=SD 1 *(TE′−T 0 ). 
     Alternatively, in some implementations, there are different target indexes for different zones. This permits the creation of a deliberate but controllable non-uniform thickness profile on the substrate. The target indexes can be entered by user, e.g., using an input device on the controller. For example, the first zone of the first substrate can have a first target index, the second zone of the first substrate can have a second target index, the first zone of the second substrate can have a third target index, and the second zone of the second substrate can have a fourth target index. 
     For any of the above methods described above, the polishing rate is adjusted to bring the slope of index trace closer to the desired slope. The polishing rates can be adjusted by, for example, increasing or decreasing the pressure in a corresponding chamber of a carrier head. The change in polishing rate can be assumed to be directly proportional to the change in pressure, e.g., a simple Prestonian model. For example, for each zone of each substrate, where zone was polished with a pressure Pold prior to the time T 0 , a new pressure Pnew to apply after time T 0  can be calculated as Pnew=Pold*(SD/S), where S is the slope of the line prior to time T 0  and SD is the desired slope. 
     For example, assuming that pressure Pold 1  was applied to the first zone of the first substrate, pressure Pold 2  was applied to the second zone of the first substrate, pressure Pold 3  was applied to the first zone of the second substrate, and pressure Pold 4  was applied to the second zone of the second substrate, then new pressure Pnew 1  for the first zone of the first substrate can be calculated as Pnew 1 =Pold 1 *(SD 1 /S 1 ), the new pressure Pnew 2  for the second zone of the first substrate clan be calculated as Pnew 2 =Pold 2 *(SD 2 /S 2 ), the new pressure Pnew 3  for the first zone of the second substrate clan be calculated as Pnew 3 =Pold 3 *(SD 3 /S 3 ), and the new pressure Pnew 4  for the second zone of the second substrate clan be calculated as Pnew 4 =Pold 4 *(SD 4 /S 4 ). 
     The process of determining projected times that the substrates will reach the target thickness, and adjusting the polishing rates, can be performed just once during the polishing process, e.g., at a specified time, e.g., 40 to 60% through the expected polishing time, or performed multiple times during the polishing process, e.g., every thirty to sixty seconds. At a subsequent time during the polishing process, the rates can again be adjusted, if appropriate. During the polishing process, changes in the polishing rates can be made only a few times, such as four, three, two or only one time. The adjustment can be made near the beginning, at the middle or toward the end of the polishing process. 
     Polishing continues after the polishing rates have been adjusted, e.g., after time T 0 , the optical monitoring system continues to collect spectra for at least the reference zone and determine index values for the reference zone. In some implementations, the optical monitoring system continues to collect spectra and determine index values for each zone. Once the index trace of a reference zone reaches the target index, endpoint is called and the polishing operation stops. 
     For example, as shown in  FIG. 12 , after time T 0 , the optical monitoring system continues to collect spectra for the reference zone and determine index values  312  for the reference zone. If the pressure on the reference zone did not change (e.g., as in the implementation of  FIG. 11 ), then the linear function can be calculated using data points from both before T 0  (but not before TC) and after T 0  to provide an updated linear function  314 , and the time at which the linear function  314  reaches the target index IT indicates the polishing endpoint time. On the other hand, if the pressure on the reference zone changed at time T 0 , then a new linear function  314  with a slope S′ can be calculated from the sequence of index values  312  after time T 0 , and the time at which the new linear function  314  reaches the target index IT indicates the polishing endpoint time. The reference zone used for determining endpoint can be the same reference zone used as described above to calculate the expected endpoint time, or a different zone (or if all of the zones were adjusted as described with reference to  FIG. 11 , then a reference zone can be selected for the purpose of endpoint determination). If the new linear function  314  reaches the target index IT slightly later (as shown in  FIG. 12 ) or earlier than the projected time calculated from the original linear function  214 , then one or more of the zones may be slightly overpolished or underpolished, respectively. However, since the difference between the expected endpoint time and the actual polishing time should be less than a couple seconds, this need not severely impact the polishing uniformity. 
     In some implementations, e.g., for copper polishing, after detection of the endpoint for a substrate, the substrate is immediately subjected to an overpolishing process, e.g., to remove copper residue. The overpolishing process can be at a uniform pressure for all zones of the substrate, e.g., 1 to 1.5 psi. The overpolishing process can have a preset duration, e.g., 10 to 15 seconds. 
     Where multiple index traces are generated for a particular zone, e.g., one index trace for each library of interest to the particular zone, then one of the index traces can be selected for use in the endpoint or pressure control algorithm for the particular zone. For example, the each index trace generated for the same zone, the controller  190  can fit a linear function to the index values of that index trace, and determine a goodness of fit of that linear function to the sequence of index values. The index trace generated having the line with the best goodness of fit its own index values can be selected as the index trace for the particular zone and substrate. For example, when determining how to adjust the polishing rates of the adjustable zones, e.g., at time T 0 , the linear function with the best goodness of fit can be used in the calculation. As another example, endpoint can be called when the calculated index (as calculated from the linear function fit to the sequence of index values) for the line with the best goodness of fit matches or exceeds the target index. Also, rather than calculating an index value from the linear function, the index values themselves could be compared to the target index to determine the endpoint. 
     Determining whether an index trace associated with a spectra library has the best goodness of fit to the linear function associated with the library can include determining whether the index trace of the associated spectra library has the least amount of difference from the associated robust line, relatively, as compared to the differences from the associated robust line and index trace associated with another library, e.g., the lowest standard deviation, the greatest correlation, or other measure of variance. In one implementation, the goodness of fit is determined by calculating a sum of squared differences between the index data points and the linear function; the library with the lowest sum of squared differences has the best fit. 
     Referring to  FIG. 13 , a summary flow chart  1300  is illustrated. A plurality of zones of a substrate are polished in a polishing apparatus simultaneously with the same polishing pad (step  1302 ) as described above. During this polishing operation, each zone has its polishing rate controllable independently of the other substrates by an independently variable polishing parameter, e.g., the pressure applied by the chamber in carrier head above the particular zone. During the polishing operation, the substrate is monitored (step  1304 ) as described above, e.g., with a sequence of measure spectra obtained from each zone. For each measured spectrum in the sequence, the reference spectrum that is the best match is determined (step  1306 ). The index value for each reference spectrum that is the best fit is determined to generate sequence of index values (step  1308 ). 
     Clearance of the second layer is detected (step  1310 ). For each zone, a linear function is fit to the sequence of index values for spectra collected after clearance of the second layer is detected (step  1302 ). In one implementation, an expected endpoint time that the linear function for a reference zone will reach a target index value is determined, e.g., by linear interpolation of the linear function (step  1314 ). In other implementations, the expected endpoint time is predetermined or calculated as a combination of expected endpoint times of multiple zones. If needed, the polishing parameters for the other zones are adjusted to adjust the polishing rate of that substrate such that the plurality of zones reach the target thickness at approximately the same time or such that the plurality of zones have approximately the same thickness (or a target thickness) at the target time (step  1316 ). Polishing continues after the parameters are adjusted, and for each zone, measuring a spectrum, determining the best matching reference spectrum from a library, determining the index value for the best matching spectrum to generate a new sequence of index values for the time period after the polishing parameter has been adjusted, and fitting a linear function to index values (step  1318 ). Polishing can be halted once the index value for a reference zone (e.g., a calculated index value generated from the linear function fit to the new sequence of index values) reaches target index (step  1330 ). 
     In some implementations, the sequence of index values is used to adjust the polishing rate of one or more zones of a substrate, but another in-situ monitoring system or technique is used to detect the polishing endpoint. 
     As discussed above, for some techniques and some layer stacks, detection of clearance of the overlying layer and exposure of the underlying layer can be difficult. In some implementations, a sequence of groups of spectra are collected, and a value of a dispersion parameter is calculated for a each group of spectra to generate sequence of dispersion values. The clearance of the overlying layer can be detected from the sequence of dispersion values. This technique can be used to detect clearing of the second layer and exposure of the first layer, e.g., in steps  910  or  1310  of the polishing operations described above. 
       FIG. 14  shows a method  1400  for detecting clearance of the second layer and exposure of the first layer. As the substrate is being polished (step  1402 ), a sequence of groups of spectra are collected (step  1404 ). As shown in  FIG. 4 , if the optical monitoring system is secured to a rotating platen, then in a single sweep of the optical monitoring system across the substrate, spectra can be collected from multiple different locations  201   b - 201   j  on the substrate. The spectra collected from a single sweep provides a group of spectra. As polishing progresses, multiple sweeps of the optical monitoring system provide a sequence of groups of spectra. One group of spectra can be collected for each platen rotation, e.g., the groups can be collected at frequency equal to the platen rotation rate. Typically, each group will include five to twenty spectra. The spectra can be collected using the same optical monitoring system that is used to collect spectra for the peak tracking technique discussed above. 
       FIG. 15A  provides an example of a group of measured spectra  1500   a  of light reflected from the substrate  10  at the beginning of polishing, e.g., when a significant thickness of the overlying layer remains over the underlying layer. The group of spectra  1500   a  can include spectra  202   a - 204   a  collected at different locations on the substrate in a first sweep of the optical monitoring system across the substrate.  FIG. 15B  provides an example of a group of measured spectra  1500   b  of light reflected from the substrate  10  at or near clearance of the overlying layer. The group of spectra  1500   b  can include spectra  202   b - 204   b  collected at different locations on the substrate in a different second sweep of the optical monitoring system across the substrate (the spectra  1500   a  can be collected from different locations on the substrate than the spectra  1500   b ). 
     Initially, as shown in  FIG. 15A , the spectra  1500   a  are fairly similar. However, as shown in  FIG. 15B , as the overlying layer, e.g., a barrier layer, is cleared, and the underlying layer, e.g., a low-k or capping layer, is exposed, differences between the spectra  1500   b  from different locations on the substrate tend to become more pronounced. 
     For each group of spectra, a value of a dispersion parameter of the spectra in the group is calculated (step  1406 ). This generates a sequence of dispersion values. 
     In one implementation, to calculate a dispersion parameter for a group of spectra, the intensity values (as a function of wavelength) are averaged together to provide an average spectrum. That is I AVE (λ)=(1/N)·[Σ i=1 to N  I i (λ)], where N is the number of spectra in the group and I i (λ) are the spectra. For each spectrum in the group, a total difference between the spectrum and the average spectrum can then be calculated, e.g., using a sum of squares difference or sum of absolute values difference, e.g., D i =[1/(λa−λb)·[Σ λ=λa to λb  [I i  (λ)−I AVE (λ)] 2 ]] 1/2  or D i =[1/(λa−λb)·[Σ λ=λa to λb )·|I i (λ)−I AVE (λ)|]], where λa to λb is the wavelength range being summed over. 
     Once a difference value has been calculated for each spectrum in the group of spectra, the value of the dispersion parameter can be calculated for the group from the difference values. A variety of dispersion parameters are possible, such as standard deviation, interquartile range, range (maximum value minus minimum value), mean difference, median absolute deviation and average absolute deviation. 
     The sequence of dispersion values can be analyzed and used to detect clearance of the overlying layer (step  1408 ). 
       FIG. 16  shows a graph  1600  of the standard deviation of the spectra as a function of polishing time (with each standard deviation calculated from the difference values of a group of spectra). Thus, each plotted point  1602  in the graph is a standard deviation for the difference values of the group of spectra collected at a given sweep of the optical monitoring system. As illustrated, the standard deviation values remain fairly low during a first time period  1610 . However, after time period  1610 , the standard deviation values become larger and more disperse. Without being limited to any particular theory, a thick barrier layer may tend to dominate the reflected spectrum, masking differences in thickness of the barrier layer itself and any underlying layer. As polishing progresses, the barrier layer becomes thinner or is completely removed, and the reflected spectrum becomes more sensitive to variations in the underlying layer thickness. As a result, the dispersion of the spectra will tend to increase as the barrier layer is cleared. 
     A variety of algorithms can be used to detect the change in behavior of the dispersion values when the overlying layer is clearing. For example, the sequence of dispersion values can be compared to a threshold, and if a dispersion value exceeds the threshold, then a signal is generated indicating that the overlying layer has cleared. As another example, a slope of a portion of the sequence of dispersion values within a moving window can be calculated, and if the slope exceeds a threshold value then a signal is generated indicating that the overlying layer has cleared. 
     As part of the algorithm to detect the increase in dispersion, the sequence of dispersion values can be subject to a filter, e.g., a low-pass or band filter, in order to remove high frequency noise. Examples of low-pass filters include moving average and Butterworth filters. 
     Although the discussion above focuses on detection of clearance of a barrier layer, the technique can be used detection clearance of an overlying layer in other contexts, e.g., clearance of an overlying layer in another type semiconductor process that uses dielectric layer stacks, e.g., interlayer dielectric (ILD), or clearance of a thin metal layer over a dielectric layer. 
     In addition to use as trigger for initiating feature tracking as discussed above, this technique for detecting clearance of an overlying layer can be used for other purposes in a polishing operation, e.g., to be used as the endpoint signal itself, to trigger a timer so that the underlying layer is polished for a predetermined duration following exposure, or as a trigger to modify polishing parameter, e.g., to change carrier head pressure or slurry composition upon exposure of the underlying layer. 
     In addition, although the discussion above assumes a rotating platen with an optical endpoint monitor installed in the platen, system could be applicable to other types of relative motion between the monitoring system and the substrate. For example, in some implementations, e.g., orbital motion, the light source traverses different positions on the substrate, but does not cross the edge of the substrate. In such cases, the collected spectra can still be grouped, e.g., spectra can be collected at a certain frequency and spectra collected within a time period can be considered part of a group. The time period should be sufficiently long that five to twenty spectra are collected for each group. 
     As used in the instant specification, the term substrate can include, for example, a product substrate (e.g., which includes multiple memory or processor dies), a test substrate, a bare substrate, and a gating substrate. The substrate can be at various stages of integrated circuit fabrication, e.g., the substrate can be a bare wafer, or it can include one or more deposited and/or patterned layers. The term substrate can include circular disks and rectangular sheets. 
     Embodiments of the invention and all of the functional operations described in this specification can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structural means disclosed in this specification and structural equivalents thereof, or in combinations of them. Embodiments of the invention can be implemented as one or more computer program products, i.e., one or more computer programs tangibly embodied in a machine-readable storage media, for execution by, or to control the operation of, data processing apparatus, e.g., a programmable processor, a computer, or multiple processors or computers. A computer program (also known as a program, software, software application, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file. A program can be stored in a portion of a file that holds other programs or data, in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub-programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers at one site or distributed across multiple sites and interconnected by a communication network. 
     The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit). 
     The above described polishing apparatus and methods can be applied in a variety of polishing systems. Either the polishing pad, or the carrier heads, or both can move to provide relative motion between the polishing surface and the substrate. For example, the platen may orbit rather than rotate. The polishing pad can be a circular (or some other shape) pad secured to the platen. Some aspects of the endpoint detection system may be applicable to linear polishing systems, e.g., where the polishing pad is a continuous or a reel-to-reel belt that moves linearly. The polishing layer can be a standard (for example, polyurethane with or without fillers) polishing material, a soft material, or a fixed-abrasive material. Terms of relative positioning are used; it should be understood that the polishing surface and substrate can be held in a vertical orientation or some other orientation. 
     Particular embodiments of the invention have been described. Other embodiments are within the scope of the following claims.