Systems and methods for processing silicon wafers - Patents.com

Abstract

A method for processing silicon wafers, the method including: cutting an ingot to form a wafer; extracting a cross-sectional profile from measured shape data, the cross-sectional profile passing through a center of the wafer and aligned with a cutting direction of the ingot; interpolating the shape data with a fixed predetermined step size; fitting a first second order polynomial to the cross-sectional profile; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second second order polynomial to the residual profile using a sliding window of a predetermined width to determine a position, height, and curvature of each peak and valley in the residual profile; determining waviness parameters based on the position, height, and curvature of each peak and valley in the residual profile; and further processing the wafer based on the waviness parameters and a predetermined waviness threshold.

Description

[Technical field]

[0001] Field The present invention relates to a system and method for processing semiconductor wafers. [Background technology]

[0002] background Semiconductor wafers are commonly used in a variety of applications, including the manufacture of circuits and solar cells. With the ubiquitous use of integrated circuits in electrical and electromechanical devices, the production of high quality semiconductor wafers is increasingly important for optimal product quality and performance. This is especially true when constraints in the semiconductor industry, such as supply chain constraints, material shortages, high demand, and other local or global economic forces, require semiconductors to be manufactured efficiently with minimal errors.

[0003] The role of wafer geometry in wafer chucking was studied by K. Turner et al. [KT Turner, R. Ramkhalawon, JK Sinha, “Role of wafer geometry in wafer chucking”, Journal of Micro / Nanolithography, MEMS, and MOEMS, 12(2), 023007(2013)]. They simulated the wafer shape with sinusoids of various spatial frequencies and amplitudes and used finite element analysis to calculate the maximum amplitude of waves that can be completely flattened with a standard clamping pressure. They repeated the calculations for different spatial wavelengths of the waves. They found an approximately linear correlation between the threshold wavelength and the threshold amplitude on a log-log scale. Specifically, waves with longer wavelengths can be flattened on the chuck when their amplitudes are relatively high, while waves with shorter wavelengths can be flattened only when the amplitudes of the waves are very small. For example, according to their calculations, waves with a spatial wavelength of 100 mm could be flattened with an amplitude of up to about 2 mm. For a wave with a wavelength of 10 mm, the threshold amplitude at the wafer was calculated to be 0.2 microns, and for a wavelength of 1 mm, the threshold was found to be only about 0.5 nm.

[0004] In reality, features can have any shape, not just sinusoidal, so the sharpness of the peaks, or the curvature of their shapes near the apex of the peak or valley, is more appropriate to characterize their impact on device yield.

[0005] Wafers have shape irregularities due to inherent imperfections in the wafer manufacturing process, and wafer shape is one of the important parameters of silicon wafers. Wafer shape is defined as the geometry variation of the mid-surface of a wafer in the free state [“SEMI M49-0312 Guide for specifying geometry measurement systems for silicon wafers for the 130 nm to 22 nm technology generations,”(2012);Semiconductor Equipment and Materials International;http: / / www.semi.org]. Wafer shape can affect the yield of device manufacturing lines. One of the wafer irregularities called waviness is a measure of the wafer thickness variation or wafer surface height variation. Waviness describes wafer shape irregularities with spatial frequencies that fall between the local geometry and bow / warp. Waviness shape features are formed during sawing of silicon ingots and persist through grinding and / or polishing.

[0006] If a metric of waviness is not available, it may be difficult to resolve technical problems caused by wavy wafer shapes. Wafer shape may be a critical parameter for process steps that require the wafer to be planarized by vacuum, electrostatic, or semi-flexible chucks. Process steps in device manufacturing that use wafer chucking and therefore may be affected by waviness include photolithography and chemical-mechanical planarization / polishing (CMP). Wafer bonding may also see effects from wafer waviness.

[0007] Silicon wafers can, within a limited range, plastically deform and adhere to the shape of the chuck under clamping pressure. In the case of photolithography, if the shape deviation is not suitable for planarization under clamping pressure, parts of the exposure area may be out of focus, or the lithography scanner may have difficulty exposing the die accurately.

[0008] In addition to photolithography, a known potential area for waviness-related failures is CMP. CMP is a touch polishing process that can be utilized to remove a layer of oxide deposited on top of a nitride layer. A possible mechanism for failures in CMP caused by waviness was discussed by K. Turner [“Role of process-induced wafer geometry changes in advanced semiconductor manufacturing”, presentation on July 9, 2014 at Semicon West 2014 conference, San Francisco, CA, 2014]. If the wafer has a large waviness, the oxide is not completely removed in the wafer valleys, or the nitride is over-polished at the high points of the wafer. In addition, the depth of the shallow trench isolation wells can be variable. Wafer waviness can result in variable nitride thickness after CMP and variable shallow trench isolation (STI) depth. The failure risk is exacerbated by the fact that CMP typically uses a semi-flexible chuck that applies downward pressure on the wafer against the polishing pad. Such chucks have limited ability to planarize the wafer.

[0009] Traditional parameters used to characterize the attributes of waviness include wafer surface height, bow, and warp. Wafer surface height represents the thickness of the wafer at a specific location within the wafer. Bow is a parameter that represents the difference in mid-plane height at the center of the wafer compared to the height of a plane intersecting three mid-plane points around the edge of the wafer. Warp is similar to bow, but instead compares the mid-plane across the entire wafer instead of just the center of the wafer. The technique also utilizes thickness measurements to provide a curvature profile of the wafer by taking the second derivative of a data curve that represents thickness measurements along the radius of the wafer.

[0010] There have been previous attempts to detect and quantify waviness reported in the patent literature. For example, WO 2003 / 077309 describes using a bandpass Gaussian filter in the frequency domain to eliminate surface profile features with too short or too long spatial frequencies, and calculating the "warp" of the filtered surface (the distance between the highest peak and deepest valley, measured relative to a best-fit plane). This filtered warp metric is used as a characteristic of waviness.

[0011] WO 2021 / 084939 calculates the first or second derivative of the wafer shape and characterizes the waviness as the maximum difference value within predefined regions of the wafer (cut-in region, wafer center, cut-out region). Separate limits for the shape derivative can be set for these three regions in order to grade the wafer. Summary of the Invention [Problem to be solved by the invention]

[0012] The first of the two patents is sensitive to the height of the undulation features, but is not sensitive to the "sharpness" or "curvature" at the apexes of the peaks and valleys. The second patent is sensitive to the "sharpness" of the features, but does not consider their amplitude. Thus, there is a need in the art for improved systems and methods to address these shortcomings. [Means for solving the problem]

[0013] overview In one embodiment, the present disclosure provides a method for processing silicon wafers, the method including: cutting an ingot to form a wafer; extracting a cross-sectional profile from measured shape data, the cross-sectional profile passing through a center of the wafer and aligned with a cutting direction of the ingot; interpolating the shape data with a fixed predetermined step size; fitting a first second order polynomial to the cross-sectional profile; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second second order polynomial to the residual profile using a sliding window of a predetermined width to determine a position, height, and curvature of each peak and valley in the residual profile; determining a waviness parameter based on the position, height, and curvature of each peak and valley in the residual profile; and further processing the wafer based on a comparison of the waviness parameter to a predetermined waviness threshold.

[0014] The subject matter of the present disclosure will be described in more detail below based on the exemplary drawings. All features described and / or illustrated herein can be used alone or in different combinations. The features and advantages of various embodiments will become apparent by reading the following detailed description with reference to the accompanying drawings, which show: [Brief description of the drawings]

[0015] [Figure 1] FIG. 1 illustrates a graphical representation of a wafer profile and corresponding fit according to one embodiment of the present invention. [Diagram 2] FIG. 13 illustrates a graphical representation of a residual profile according to one embodiment of the present invention. [Diagram 3] FIG. 2 illustrates a two-dimensional map of a wafer according to one embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0016] Detailed Description Embodiments of the present invention provide a method for processing a wafer based on data processing different from that known in the prior art. In some embodiments, a parabolic fit to the linear cross-sectional profile of the wafer is first determined to eliminate bowl-like shapes caused by distortion from the backside layer of the wafer. The parabolic fit within a moving window is then used to identify and quantify peaks and valleys.

[0017] Embodiments of the present invention provide important advantages over prior art systems and methods. For example, the disclosed embodiments provide waviness measurements that are sensitive to both wafer height and sharpness. The disclosed embodiments combine contributions from all features on the wafer into a single dimensionless parameter.

[0018] Metrology tool manufacturers have recognized that there is a correlation between the typical amplitudes of topological features on a wafer and their spatial wavelengths. A whole range of wafer geometry metrics have been developed. For example, bow and warp metrics are sensitive to topological features with wavelengths similar to the radius of the wafer. Bow and warp metrics have relatively high amplitudes, from a few microns to tens of microns or even over hundreds of microns, and relatively large spatial wavelengths, ranging from about tens of centimeters to hundreds of centimeters. Local geometry metrics (e.g., site flatness quality requirements (SFQR)) characterize features with sizes on the scale of a few centimeters, but the amplitudes of those features are much smaller than the amplitudes of bow and warp features, which can be as small as a few microns. Nanotopography (NT) characterizes even shorter spatial wavelengths (approximately 1-2 cm or less in size) and even smaller amplitudes. Finally, even finer features are captured by roughness metrics, which start with millimeter-sized features with very small amplitudes and extend into the micron range. Due to how local geometry, NT, and roughness are calculated, they are strictly speaking flatness metrics rather than shape metrics, but in theory NT and local geometry metrics can be applied to wafer shape data.

[0019] The central region of the spatial wavelength range of approximately 3-10 cm (but can be extended to 2-20 cm) is characterized as "waviness" throughout this disclosure and is not covered by commercially available metrology tools. The warp metric is capable of detecting shape vibrations within the "waviness" spatial frequency range. However, because the amplitude of the waviness features is typically smaller than that of the warp features, the rejection limits that are appropriate for bow and warp are too high for waviness and cannot be used to reject non-compliant wafers. The bow and warp metric does not have the ability to distinguish shape features based on spatial wavelength.

[0020] An embodiment of the present invention develops metrology to characterize and quantify the waviness of wafer topography. Specifically, the metrology to characterize the waviness includes a dimensionless parameter that takes into account both the amplitude of the peaks and valleys and the sharpness of the valley shapes. The waviness, and the parameters used to quantify the waviness, can be used to screen wafers for defects and / or quality control.

[0021] As the sawing wire deviates laterally, it is expected that the waves will also traverse the entire wafer, as they do so across the ingot being sawed. Wafer waviness is often caused by multi-wire wafer sawing, a process in which ingot segments are sawed into individual wafers by a thin wire. Specifically, wafer height variations are typically introduced in a vibratory fashion perpendicular to the extent of the sawing wire. The lateral movement or deviation of the wire as it travels through the ingot causes these wavy variations in the sawing position. As the sawing wire tends to deviate laterally, the wire cutting the ingot tends to deviate along the entire ingot being sawed, thus creating wavy variations that extend across the wafer. The amplitude of the wavy features is measured in microns and is invisible to the human eye. Such small amplitude waves produce only very weak signals in bow and warp metrics. This signal is usually very small compared to typical bow and warp rejection limits and cannot be used to identify and screen such wafers.

[0022] Waviness, as defined in this disclosure, can be mathematically characterized as the sum of the absolute values ​​of the products of height and curvature of all peaks and valleys detected in a wafer. Specifically, waviness can be characterized by the following equation, where WavPar is the waviness parameter, height is the i-th height measurement at a location on the wafer, and curvature is the i-th curvature measurement at the same location on the wafer:

[0023]

number

[0024] The waviness parameter is dimensionless. The more waves in the wafer, the larger the amplitude of the waves, or the sharper and narrower the peaks and valleys, the higher the value of the waviness parameter. To bring the WavPar value into a readable range, typically above 1, a factor of 1000 is introduced.

[0025] Therefore, in some embodiments of the present invention, it is sufficient to analyze the wafer shape along a linear scan across the center of the wafer along the cut direction of the ingot.

[0026] The starting point for waviness evaluation is to extract a linear scan along the cut direction from the shape measurement. Some embodiments described in this disclosure use data measured on a 200 mm tool, but the same methods are equally applicable to measurements made on tools using 300 mm technology. Both size tools are designed to measure the geometry of the wafer using either capacitive or optical methods. Both tools are capable of calculating the dimensions of the wafer's surface profile. The cut direction during wire sawing of the wafer can be determined relative to the wafer's notch. The cut direction is determined for each production lot from data recorded in a production database when a particular ingot segment is mounted for slicing.

[0027] The spatial resolution of the wafer scan may be determined by the measurement mode of the scanner. To reduce the complexity of the data processing and make it independent of the measurement mode of the scanner, the data is linearly interpolated to a predefined step size, which can be any number. In one embodiment, this is a 1 mm step.

[0028] Silicon wafers, especially in 200 mm technologies, often have a bowl-like shape. This shape is primarily formed by stresses from the backside polysilicon and low temperature oxide (LTO) layers, but can also be influenced by stresses from epitaxial layers. In some embodiments of the invention, the bowl-like shape of the wafer is reduced or eliminated to improve detection of peaks and valleys associated with waviness. This is accomplished by fitting the shape profile of the wafer with a second order polynomial, such as a parabola. The parabolic fit is then subtracted from the raw linear profile (the "shape curve"), leaving what is referred to herein as the "residual."

[0029] FIG. 1 shows a wafer profile 10 and a corresponding parabolic fit 12. It is understood that profile 10 is only exemplary, since every wafer has an individually determinable profile that may vary and will have a different profile and therefore a different corresponding parabolic fit. Profile 10 represents the profile of the wafer surface, measured in the cutting direction in which the wafer was cut from the ingot. In the exemplary profile 10 shown in FIG. 1, profile 10 has a very pronounced wavy feature caused by multi-wire sawing, as indicated by the vibration profile 10 on the vertical axis. Parabolic fit 12 is a second order polynomial calculated based on the individual data points that make up profile 10.

[0030] Figure 2 shows residuals 20, which represent data points correlating to the difference between profile 10 and parabolic fit 12 for the example wafer of Figure 1. Specifically, residuals 20 are calculated by subtracting parabolic fit 12 from profile 10.

[0031] The subsequent step of assessing waviness requires identifying the peaks and valleys in the residuals 20. This can be accomplished by a variety of mathematical methods. In some embodiments, it is accomplished by running a sliding window 22 from one end of the residual to the other end of the residual and fitting the residual curve within the window with a second order polynomial according to the following equation:

[0032]

number

[0033] The coefficient a in this parabolic equation is a measure of the curvature of the crest. Both the amplitude of the crest and its location / height can also be easily determined from this equation.

[0034] Mountain location, P:

[0035]

number

[0036] and peak / valley height, H:

[0037]

number

[0038] 2 provides an illustration of a sliding window 22, which is merely illustrative and captures only the window that has already progressed from the left-most portion of the residual 20 to the center of the residual 20. A parabolic fit is performed using the data points within this sliding window 22.

[0039] The window must be narrow enough to allow the algorithm to separate the individual peaks and valleys and achieve a good quality fit. Various window widths may yield suitable results. In some embodiments, the width of the sliding window 22 is set to 20 mm. The criteria for selecting the width of the sliding window 22 is based on the ability of the algorithm to separate wafers with significant waviness from those with small waviness.

[0040] The sliding window 22 moves or "slides" one data point at a time along the residual 20. For example, if the interpolation is performed with a step size of 1 mm, the sliding window 22 moves 1 mm at a time. Peaks and valleys in the residual may be identified by a software algorithm based on detecting peaks and valleys at multiple adjacent positions of the sliding window as the sliding window 22 moves. In some embodiments, the software algorithm separates peaks and valleys with amplitudes above a predetermined threshold to help eliminate noise. In some embodiments, the curvature and amplitude of each peak are averaged by the software algorithm.

[0041] Once the peaks and valleys are identified by the software algorithm, the respective height 24 of each peak and valley relative to the horizontal axis is determined. Since the horizontal axis corresponds to a wafer having perfect flatness with no height variations, the respective height 24 of each peak and valley corresponds to a shape defect of the wafer that, when removed, will form a flat wafer. Using the peak height 24 and the curvature a at the apex, the waviness parameter equation is used to calculate the total contribution of all peaks and valleys to the waviness parameter. For example, the wafer represented by the residual 20 in FIG. 2 has approximately two peaks and two valleys, resulting in WavPar, which is the sum of four components. It will be readily appreciated that wafers of various shapes will have various numbers of peaks and valleys in various patterns. In each situation, the above equation and method will result in a WavPar value that is the sum of the identified peaks and valleys, which represents the height variations of the wafer, thus providing advantageous insight into the shape of the wafer over traditional wafer metrics.

[0042] In some embodiments, the waviness analysis calculates a linear profile from a two-dimensional shape map of the wafer. Such an embodiment allows for excluding data points within a certain radius from the center of the wafer (approximately 40-45 mm, but up to the radius of the wafer). This method can be particularly useful, for example, in 200 mm technology, where older metrology tools are equipped with small diameter chucks to hold the wafer during measurements. The wafer is first chucked at the center and scanned using a capacitance probe from the wafer edge to the edge of the chuck. Then, the wafer is chucked again near the edge and the center area is scanned. Wafer chucking, even with a small chuck, results in some deformation of the wafer shape, since the wafer is partially flattened in the chuck area. Older generation tool software attempts to stitch together wafer maps collected at two chucking locations. There is no perfect overlap between these two maps, since the wafer shape was distorted by chucking at two chucking locations. The process of stitching the maps can create artifacts in the wafer shape, for example because the circular regions of the chuck area do not smoothly follow the outer regions of the chuck, and in rare cases can even result in spurious peaks in the wafer waviness profile. Removing data points from the center of the wafer from a central circle with a radius equal to or greater than the chuck size allows for the determination of a cleaner linear "shape profile" free of such artifacts.

[0043] In some embodiments, the first step to obtain the cleaner linear profile described above is to fit the three-dimensional shape of the wafer with a paraboloid (a three-dimensional surface obtained by a rotating parabola). The equation for a symmetric paraboloid is:

[0044]

number

[0045] Or, in the case of the more general hyperbolic paraboloid (which can form a shape similar to that of a potato chip),

[0046]

number

[0047] This fit allows removal of wafer shape variations, e.g., the bowl or potato chip shape of the wafer. Subtracting the fitted paraboloid from the measured wafer shape gives a two-dimensional residual.

[0048] The second step is to exclude data points within a predefined radius of the center of the wafer. This allows excluding areas with potential artifacts in the measurement data. FIG. 3 shows an exemplary two-dimensional map 30 of a wafer profile 32 with a center region 34. A linear profile is calculated by averaging data points 38 along a vertical line 36 in the wafer profile 32 for each x-axis value rounded to a predefined step size. The vertical line 36 and the corresponding data points 38 provide a schematic visualization of the data point averaging to eliminate artifacts in or near the wafer center region 34. The x-axis indicates the cutting direction when cutting the wafer, and the y-axis indicates the direction of the range of the cutting wire.

[0049] Once the linear profile has been calculated, subsequent evaluation of the peaks and valleys can be performed using the processes and algorithms previously described.

[0050] Analysis of the waviness at the end of the wafering process can be used to reject wafers having waviness that exceeds a predefined rejection limit determined based on customer needs and sensitivity. Measurement of the waviness after multi-wire slicing allows wafer screening to be performed after multi-wire sawing to reject wafers that are unlikely to pass end of line inspection early in the process. Additionally, waviness data after multi-wire sawing can be utilized to monitor the performance of the multi-wire saw and adjust the sawing process as needed. Post-saw processes such as grinding, polishing, and / or epitaxy can also be performed based on the determination of the waviness parameter. For example, post-saw processes may be performed only on wafers that do not fit within a predetermined threshold of acceptable waviness. In other examples, the degree and / or extent of post-processing techniques may also be determined based on the determined waviness parameter. Thus, in each case, the determination of the waviness parameter may improve both the quality control process and the quality of the individual wafers themselves.

[0051] In some embodiments, the waviness parameters of one wafer can be used to predict the waviness of other wafers, thereby enabling efficient quality control decisions for production lots. For example, the determined waviness parameters can be used to predict the waviness of other wafers produced by the same ingot and / or set of multi-wires used in the sawing process. Depending on the predicted waviness parameters, the wafers may then need to be sorted and / or post-processed in different ways depending on the predicted waviness parameters. In this way, predicted waviness parameters that are within quality control limits can enable efficient manufacturing of wafers without the time-consuming and resource-consuming measurement process of independently determining the waviness of each wafer.

[0052] In some embodiments, the waviness parameters allow for feedback of the performance of the multi-wire saw and / or other equipment. For example, a baseline waviness parameter may be established by statistical means (e.g., average or median of past waviness parameters). The waviness parameters associated with a particular set of multi-wires may then be compared to the baseline waviness parameters. Such a comparison would allow for a quick determination of whether the multi-wire (or other equipment) is operating within acceptable tolerances. Equipment maintenance decisions can then be made based on the comparison of the waviness parameters to the baseline waviness parameters.

[0053] Although the subject matter of the present disclosure has been illustrated and described in detail in the drawings and the foregoing description, such illustration and description should be considered as explanatory or exemplary, and not restrictive. Since the present invention is defined by the claims, any statements made herein that characterize the present invention should also be considered as explanatory or exemplary, and not restrictive. Those skilled in the art will understand that changes and modifications may be made within the scope of the following claims, which may include any combination of features from the different embodiments described above.

[0054] The terms used in the claims should be construed to have the broadest reasonable interpretation consistent with the foregoing description. For example, the use of the article "a" or "the" in introducing an element should not be construed as excluding a plurality of elements. Similarly, "or" enumerations should be construed as inclusive, and the enumeration of "A or B" does not exclude "A and B" unless it is clear from the context or the foregoing description that only one of A and B is intended. Furthermore, a statement "at least one of A, B, and C" should be construed as one or more of the group of elements consisting of A, B, and C, regardless of whether A, B, and C are related as a category, and should not be construed as requiring at least one of each of the listed elements A, B, and C. Furthermore, a statement "A, B, and / or C" or "at least one of A, B, or C" should be construed as including any singular entity from the listed elements, e.g., A, any subset from the listed elements, e.g., A and B, or the entire list of elements A, B, and C.

Claims

1. 1. A method for processing a silicon wafer, the method comprising: cutting the ingot to form wafers; extracting a cross-sectional profile from the measured shape data, the cross-sectional profile passing through a center of the wafer and aligned with a cut direction of the ingot; Interpolating the shape data with a fixed, predetermined step size; fitting a first second order polynomial to the cross-sectional profile; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second second order polynomial to the residual profile using a sliding window of a predetermined width to determine the location, height, and curvature of each peak and valley of the residual profile; determining waviness parameters based on the position, height, and curvature of each peak and valley of the residual profile; further processing the wafer based on a comparison of the waviness parameter to a predetermined waviness threshold. A method comprising:

2. 2. The method of claim 1 , wherein the waviness parameter is determined based on the following formula, where WavPar is the waviness parameter, Height i is the height of the ith peak or valley, and Curvature i is the curvature of the ith peak or valley: [0010]

3. 2. The method of claim 1, wherein the waviness threshold is a critical limit, and further processing the wafer comprises rejecting the wafer if the waviness parameter exceeds the critical limit.

4. The method of claim 1 , wherein further processing the wafer comprises polishing or grinding the wafer.

5. The method of claim 1 further comprising determining an extent of further processing required to reduce wafer waviness to within the waviness threshold.

6. The method of claim 1 , further comprising determining predicted wafer waviness parameters of an as yet uncut wafer based on the determined waviness parameters.

7. 7. The method of claim 6, further comprising comparing the predicted wafer waviness parameters to the determined waviness parameters to determine whether a wafer sort should be performed.

8. Determining baseline waviness parameters; comparing the waviness parameters to the baseline waviness parameters to determine if maintenance of the multi-wire saw is required; performing maintenance on the multi-wire saw based on a comparison of the waviness parameters with the baseline waviness parameters; The method of claim 1 further comprising:

9. The method of claim 8 , wherein the baseline waviness parameters are determined statistically from previously determined waviness parameters.

10. 1. A method for processing a silicon wafer, the method comprising: cutting the ingot to form wafers; Extracting a two-dimensional shape profile from the measured wafer data; fitting a paraboloid to the two-dimensional shape to approximate the shape of the wafer; calculating a two-dimensional residual by subtracting the paraboloid from the two-dimensional shape profile; calculating a linear profile along a cut direction of the wafer by averaging data points corresponding to incremental abscissa values ​​rounded to predefined increments, the abscissa values ​​corresponding to the cut direction; fitting a second order polynomial to the residual profile using a sliding window of a predetermined width to determine the location, height, and curvature of each peak and valley in the residual profile; determining waviness parameters based on the position, height, and curvature of each peak and valley of the residual profile; further processing the wafer based on a comparison of the waviness parameter to a predetermined waviness threshold. A method comprising:

11. The method of claim 10 , further comprising excluding a center region of the wafer from the two-dimensional shape profile.

12. The method of claim 11 , wherein the central region corresponds to a wafer chuck region of a wafer metrology tool.

13. 11. The method of claim 10, wherein the waviness parameter is determined based on the following formula, where WavPar is the waviness parameter, Height i is the height of the i-th peak or valley, and Curvature i is the curvature of the i-th peak or valley: [0025]

14. 11. The method of claim 10, wherein the waviness threshold is a critical limit, and further processing the wafer comprises rejecting the wafer if the waviness parameter exceeds the critical limit.

15. The method of claim 10 , wherein further processing the wafer comprises grinding or polishing the wafer.

16. The method of claim 1 , further comprising determining predicted wafer waviness parameters of an as yet uncut wafer based on the determined waviness parameters.

17. 17. The method of claim 16, further comprising comparing the predicted wafer waviness parameters to the determined waviness parameters to determine whether a wafer sort should be performed.

18. Determining baseline waviness parameters; comparing the waviness parameters to the baseline waviness parameters to determine if maintenance of the multi-wire saw is required; performing maintenance on the multi-wire saw based on a comparison of the waviness parameters with the baseline waviness parameters; The method of claim 1 further comprising:

19. 20. The method of claim 18, wherein the baseline waviness parameters are determined statistically from previously determined waviness parameters.

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