System and method for processing silicon wafer

Abstract

A method for processing a silicon wafer, the method comprising dicing an ingot to form a wafer; extracting a cross-sectional profile from the measured shape data, the cross-sectional profile passing through the center of the wafer and aligned with the cutting direction of the ingot; interpolating the shape data with a fixed and predetermined step size; fitting a first quadratic polynomial for the cross section contour; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second quadratic polynomial to the residual profile using a sliding window having a predetermined width to determine the position, height, and curvature of each peak and valley of the residual profile; determining ripple parameters based on the position, height, and curvature of each peak and valley of the residual profile; and further processing the wafer based on a comparison of the corrugation parameter to a predetermined corrugation threshold.

Description

Technical Field The present invention relates to systems and methods for processing semiconductor wafers. Background Art

[0001] Semiconductor wafers are commonly used in a variety of applications, including in the manufacture of circuits and solar cells. Due to the ubiquitous use of integrated circuits in electrical and electromechanical devices, producing high-quality semiconductor wafers is increasingly important to optimize product quality and performance. This is especially true when constraints in the semiconductor industry such as supply chain limitations, material shortages, high demand, and other local or global economic forces require semiconductors to be produced efficiently with minimal error rates.

[0002] K. Turner et al. studied the role of wafer geometry in wafer chucking [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 using sinusoids with various spatial frequencies and amplitudes, and used finite element analysis to calculate the maximum amplitude of the wave that can be completely flattened using a standard clamping pressure. They repeated the calculation for different spatial wavelengths of the wave. They found that there is an almost linear correlation between the threshold wavelength and the threshold amplitude on a logarithmic scale. Specifically, when the amplitude of the wave is relatively high, waves with longer wavelengths can be flattened on the chuck, while waves with shorter wavelengths can only be flattened when their amplitude is very small. For example, according to their calculations, a wave with a spatial wavelength of 100 mm can be flattened with an amplitude of up to about 2 mm; for a wave with a wavelength of 10 mm, the threshold amplitude of the wafer was calculated to be 0.2 microns; and at a wavelength of 1 mm, the threshold was found to be only about 0.5 nm.

[0003] In reality, the shape feature is not a sinusoid and can have any shape. Therefore, the sharpness of the peak, or the curvature of the shape near the end of the peak or valley is more appropriate to describe its impact on device yield.

[0004] Due to inherent defects in the wafer manufacturing process, wafers have irregularities in shape, and wafer shape is one of the key parameters of silicon wafers. Wafer shape is defined as the geometric variation of the medial plane of the wafer in a free state [“SEMI M49-0312 Guide for specifying geometry measurement systems for silicon wafers for the 130nm to 22nm technology generations”(2012); Semiconductor Equipment and Materials International; http: / / www.semi.org]. Wafer shape may have an impact on the yield of the equipment manufacturing production line. One type of wafer irregularity known as waviness is a measure of the variation in the thickness of the wafer or the variation in the height of the wafer surface. Waviness represents irregularities in the shape of a wafer with a spatial frequency between local geometry and bowing / wrapping. The waviness shape feature is formed during sawing of the silicon ingot and is retained after grinding and / or polishing.

[0005] Without metrics for waviness, technical issues caused by wavy wafer shape can be difficult to troubleshoot. Wafer shape can be a critical parameter for process steps that require flattening the wafer via vacuum, electrostatic chucks, or semi-flexible chucks. In device manufacturing, process steps that use wafer chucks and therefore can be affected by waviness include photolithography and chemical mechanical planarization / polishing (CMP). Arguably, wafer bonding can also see the effects of waviness from the wafer.

[0006] Silicon wafers can deform plastically within a limited range and adhere to the shape of the chuck under clamping pressure. In the case of photolithography, if shape deviations do not make them suitable for being flattened under clamping pressure, some parts of the area being exposed may be out of focus or the photolithography scanner may have difficulty exposing the die correctly.

[0007] In addition to lithography, CMP is also a known potential failure area related to ripple. CMP is a touch polishing process that can be used to remove the oxide layer deposited on top of the nitride layer. K. Turner discussed the possible failure mechanisms of CMP caused by ripple in his speech "Role of Process-induced wafer geometry changes in advanced semiconductor manufacturing" on July 9, 2014 [Semicon West 2014 conference, San Francisco, CA, 2014]. If the wafer has significant ripples, either the oxide will not be completely removed in the wafer valley, or the nitride will be over-polished at the high points of the wafer. In addition, the depth of the shallow trench isolation wells may become variable. Wafer ripples may lead to varying nitride layer thickness and varying shallow trench isolation (STI) depth after CMP. The fact that CMP typically uses a semi-flexible chuck that applies downward pressure on the wafer against the polishing pad exacerbates the risk of failure. Such chucks have only limited ability to flatten the wafer.

[0008] Conventional parameters used to characterize the properties of wafer 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 height difference of the middle surface (or middle face) at the center of the wafer compared to the height of a plane that intersects three middle surface points around the edge of the wafer. Warp is similar to bow, but instead compares the middle surface of the entire wafer instead of just the middle surface at the center of the wafer. Current technology also utilizes thickness measurements to provide a curvature profile of the wafer by taking the second derivative of a data curve representing thickness measurements along the radius of the wafer.

[0009] Previous attempts to detect and quantify waviness have been reported in the patent literature. For example, International Patent Publication No. WO2003 / 077309 A2 describes the use of a bandpass Gaussian filter in the frequency domain to eliminate features of the surface profile that have spatial frequencies that are too short or too long, and to calculate the "warp" of the filtered surface (the distance between the highest peak and the deepest valley measured relative to the best fit plane). This filtered warp metric can be used as a characterization of waviness.

[0010] Patent WO 2021 / 084939 calculates the first or second derivative of the wafer shape and characterizes the ripple as the maximum difference within predefined areas of the wafer (cut-in area, center of the wafer, and cut-out area). Separate limit values ​​for the shape derivative can be set for these three areas to grade the wafer.

[0011] The first of the two patents mentioned above is sensitive to the height of the corrugation features, but is not sensitive to the "sharpness" or "curvature" of the peaks and valleys at the apex of the corrugation features. The second patent is sensitive to the "sharpness" of the features, but does not consider their amplitude. Therefore, there is a need in the art for improved systems and methods to address these deficiencies. Summary of the invention

[0012] In one embodiment, the present disclosure provides a method for processing a silicon wafer, the method comprising cutting an ingot to form wafers; extracting a cross-sectional profile from measured shape data, the cross-sectional profile passing through the center of the wafer and aligned with (or consistent with) the cutting direction of the ingot; interpolating the shape data with a fixed and predetermined step size; fitting a first quadratic polynomial to the cross-sectional profile; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second quadratic polynomial to the residual profile using a sliding window with a predetermined width to determine the position, height, and curvature of each peak and valley of the residual profile; determining a ripple parameter based on the position, height, and curvature of each peak and valley of the residual profile; and further processing the wafer based on a comparison of the ripple parameter with a predetermined ripple threshold. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The subject matter of the present disclosure will be described in more detail below based on exemplary drawings. All features described and / or shown herein may be used alone or in combination 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:

[0014] Figure 1 depicts a graphical representation of a wafer profile (graph) and a corresponding fit according to an embodiment of the present invention;

[0015] Figure 2 depicts a graphical representation of a residual profile according to an embodiment of the present invention; and

[0016] Figure 3 A two-dimensional image of a wafer according to an embodiment of the present invention is depicted. DETAILED DESCRIPTION

[0018] Embodiments of the present invention provide methods for processing wafers that are different from those based on data processing known in the prior art. In some embodiments, first, a parabolic fit of the linear cross-sectional profile of the wafer is determined to eliminate the bowl-like shape caused by the strain of the backside layer of the wafer. Then, the parabolic fit within a moving window is used to identify and quantify peaks and valleys.

[0019] Embodiments of the present invention provide important advantages over prior art systems and methods. For example, the disclosed embodiments provide a waviness measurement that is sensitive to both the height and sharpness of the wafer. The disclosed embodiments combine contributions from all features on the wafer into a single dimensionless parameter.

[0020] Metrology tool manufacturers have recognized that there is a correlation between the typical amplitude of shape features on a wafer and their spatial wavelength. A series (or set) of wafer geometry metrics have been developed. For example, bend and warp metrics are very sensitive to shape features with wavelengths similar to the wafer radius. Bend and warp metrics have relatively high amplitudes (from a few microns to tens of microns or even more than hundreds of microns) and have relatively large spatial wavelengths, which are approximately in the range of tens of centimeters to hundreds of centimeters. Local geometry metrics (e.g., point flatness quality requirements (SFQR)) characterize features with sizes on the scale of a few centimeters, but the amplitudes of these features are much smaller than the amplitudes of bend and warp features, and can be as low as a few microns. Nanotopography (NT) characterizes even shorter spatial wavelengths (about 1 cm to 2 cm or less in size) with even smaller amplitudes. Finally, roughness metrics capture even finer features, starting from millimeter-sized features with extremely small amplitudes, all the way to the micrometer range. Due to the way local geometry, NT, and roughness are calculated, strictly speaking, they are plane metrics rather than shape metrics. However, in theory, one can apply NT and local geometry metrics to wafer shape data.

[0021] In the middle of the spatial wavelength range (about 3 cm to 10 cm (but can extend to 2 c m to 20 c m)) is characterized as "waviness" in this disclosure and is not covered by commercially available metrology tools. Warp metrology can detect oscillations of shapes within the spatial frequency range of the "waviness". However, since the amplitude of the waviness feature is generally lower than the amplitude of the warp feature, the rejection limit suitable for bow and warp will be too high for waviness and cannot be used to reject non-compliant wafers. Bow metrology and warp metrology do not have the ability to distinguish shape features based on their spatial wavelength.

[0022] Embodiments of the present invention develop a metrology to characterize and quantify waviness in wafer shape. Specifically, the metrology used for characterization of waviness includes dimensionless parameters that take into account the amplitude of both peaks and valleys and the sharpness of the shape of the valleys. The waviness and the parameters used to quantify the waviness can be used to screen wafers for round defects and / or quality control.

[0023] Since the cutting line wanders across the entire ingot being cut as it wanders laterally, it is expected that the ripples will also appear across the wafer. Wafer ripples are often caused by multi-wire wafer sawing, a process in which ingot sections are cut into individual wafers by thin wires. Specifically, wafer height variations are usually introduced in an oscillatory manner orthogonal to the extent of the cutting line. As the wire passes through the ingot, the lateral movement or wandering of the wire causes wavy variations at the cutting location. Since the cutting line tends to wander laterally, the wire that is cutting the ingot tends to wander along the entire ingot being cut, resulting in wavy variations that extend across the entire wafer. The amplitude of the wavy features is measured in microns and is not visible to the human eye. Such small amplitude ripples will only produce very weak signals in the bow and warp metrics. This signal is often very small compared to typical bow and warp rejection limits and cannot be used to identify and screen out such wafers.

[0024] The waviness defined in the present disclosure can be mathematically characterized as the sum of the absolute values ​​of the product of the height and curvature of all peaks and valleys detected in the wafer. Specifically, the waviness can be characterized by the following equation, where WavPar is the waviness parameter, Height is the i-th height measurement at a position on the wafer, and Curvatur e is the i-th curvature measurement at the same position on the wafer:

[0025] WavPar=1000×∑i|Height i ×Curvature i |

[0026] The WavPar parameter is dimensionless. A greater number of waves in the wafer, a greater amplitude of the waves, or sharper and narrower peaks and valleys will result in a larger WavPar value. The factor of 1000 is introduced to keep the WavPar value in an easily readable range, usually above 1.

[0027] Therefore, in some embodiments of the present invention, it is sufficient to analyze the wafer shape along a linear scan through the center of the wafer along the cutting direction of the ingot.

[0028] The starting point for the evaluation of the waviness is to extract a linear scan along the cutting direction from the shape measurement. Some embodiments described in the present disclosure use data measured on a 200mm tool, although the same methods can also be applied to measurements completed on tools using 300mm technology. Both sizes of tools are designed to measure the geometry of the wafer using capacitive or optical methods. Both tools can calculate the dimensions of the surface shape of the wafer. The cutting direction during wire cutting of the wafer can be determined relative to the notch on the wafer. The cutting direction is determined for each production batch based on data recorded in the production database when a specific ingot section is mounted for slicing.

[0029] The spatial resolution of the wafer scan can be determined by the measurement mode of the scanner. In order to reduce the complexity of data processing and make it independent of the measurement mode of the scanner, the data is interpolated to a predefined step size (which can be any number or value) in the linear scan. In one embodiment, the predefined step size is a 1 mm step size.

[0030] Silicon wafers, especially in 200mm technology, typically have a bowl-like shape. This shape is primarily caused by stress from the backside polysilicon and low temperature oxide (LTO) layers, but may also be affected by stress from the epitaxial layers. In some embodiments of the present invention, the bowl-like shape of the wafer is reduced or eliminated in order to improve the detection of peaks and valleys associated with ripples. This is done by fitting the shape profile of the wafer using a second order polynomial (e.g., a parabola). The parabolic fit is then subtracted from the original linear shape profile (the "shape curve"), leaving what are referred to herein as "residuals."

[0031] Figure 1 A profile 10 and a corresponding parabola fit 12 of a wafer are depicted. Profile 10 is to be understood as exemplary only, as all wafers will have individually determinable profiles that may vary and have different profiles and thus different corresponding parabola fits. Profile 10 represents the profile of the surface of the wafer measured in the cutting direction along which the wafer is cut from the ingot. Figure 1 In the exemplary profile 10 depicted in FIG. 1 , the profile 10 has very obvious wavy features caused by multi-wire sawing, as shown by the profile 10 oscillating on the vertical axis. The parabola fit 12 is a quadratic polynomial calculated based on the various data points that make up the profile 10 .

[0032] Figure 2 The residual 20 is plotted, which represents the difference with Figure 1 The data points are related to the difference between the profile 10 and the parabolic fit 12 of an exemplary wafer. Specifically, the residual 20 is calculated by subtracting the parabolic fit 12 from the profile 10.

[0033] The subsequent step of evaluating the ripple requires identifying the peaks and valleys of the residual 20. This can be done by a variety of mathematical methods. In some embodiments, it can be done by running a sliding window 22 from one end of the residual to the other and fitting the residual curve within the window using a quadratic polynomial according to the following equation: y=ax 2 +bx+c

[0034] The coefficient a of the parabola equation is a measure of the curvature of the peak. The amplitude of the peak and its position / height can also be easily determined from this equation: Peak position, P: and the peak / valley height, H: H=aP 2 +bP+c

[0035] Figure 2 A schematic diagram of a sliding window 22 is provided which is exemplary only and captures only a window which has moved from the leftmost side of the residual 20 to the center of the residual 20. Using the data points within the sliding window 22, a parabola fit is completed.

[0036] The window should be narrow enough to enable the algorithm to separate the peaks and valleys and obtain a good fit quality. Various window widths may provide suitable results. In some embodiments, the width of the sliding window 22 is set to 20 mm. The criteria for the selection of the width of the sliding window 22 is based on the ability of the algorithm to separate wafers with significant ripples from wafers with (relatively) low ripples.

[0037] The sliding window 22 moves or "slides" one data point at a time along the residual 20. For example, if the interpolation is done using a step size of 1 mm, the sliding window 22 moves 1 mm at a time. Peaks and valleys in the residual can be identified by the software algorithm based on detecting the peaks and valleys at multiple adjacent positions of the sliding window 22 as the sliding window 22 moves. In some embodiments, the software algorithm separates peaks and valleys whose amplitude exceeds a predetermined threshold to help eliminate noise. In some embodiments, the software algorithm averages (means) the curvature and amplitude of each peak.

[0038] Once the software algorithm identifies the peaks and valleys, the corresponding height 24 of each peak and valley relative to the horizontal axis is determined. Since the horizontal axis corresponds to a wafer with perfect flatness and no height variation, the corresponding height 24 of each peak and valley corresponds to a wafer shape defect that, if removed, would result in a flat wafer. Using the height 24 of the peak at the top of the peak and the curvature a, the corrugation parameter equation can be used to calculate the total contribution of all peaks and valleys to the corrugation parameter. For example, Figure 2 The wafer represented by the residual 20 in will have approximately two peaks and two valleys, resulting in WavPar being the sum of four components. It will be readily appreciated that wafers of different shapes will have different numbers and shapes of peaks and valleys. In each case, the above equation and method will produce a WavPar value that is the sum of the identified peaks and valleys representing the height variations in the wafer, thus providing the advantage of deep insight into the shape of the wafer over traditional wafer metrology.

[0039] In some embodiments, the ripple analysis calculates a linear shape profile from a two-dimensional shape map of the wafer. This embodiment makes it possible to exclude data points within a certain radius (approximately 40mm to 45mm, but up to the radius of the wafer) starting from the center of the wafer. For example, in 200mm technology (where older generation metrology tools are equipped with small diameter chucks to hold the wafer during measurement), this method may be particularly useful. The wafer is initially clamped at the center and scanned from the edge of the wafer to the edge of the chuck using a capacitive probe. The wafer is then re-clamped closer to the edge and the center area is scanned. Wafer clamping (even on a small chuck) will cause a certain degree of deformation in the shape of the wafer because the wafer is partially flattened in the chuck area. The older generation tool software attempts to splice the wafer images collected at the two clamping positions. Because the wafer shape is deformed due to clamping at two clamping positions, there is no perfect overlap between the two images. The process of stitching the images may produce artifacts in the wafer shape, for example because a circular area in the chuck area does not smoothly follow the area outside the chuck and may, in rare cases, even lead to false peaks in the wafer waviness profile. Removing data points from the center of the wafer, from a circle in the middle with a radius equal to or greater than the chuck size, enables a clearer linear "shape profile" to be determined without such artifacts.

[0040] In some embodiments, the first step to obtain the above-mentioned sharper linear shape profile is to fit the three-dimensional shape of the wafer using a parabola (a three-dimensional surface obtained by rotating a parabola). The equation of the symmetric parabola is a(x 2 +y 2 )+bz+c=0 Or, in the more general case of a hyperbolic paraboloid (which can form a shape similar to a potato chip), ax 2 +by 2 +cz+d=0

[0041] This fitting can remove wafer shape variations, such as the bowl or potato chip shape of the wafer. The fitted parabola is subtracted from the measured wafer shape to obtain a two-dimensional residual.

[0042] The second step is to exclude data points within a predefined radius at the wafer center. This enables areas with potential artifacts to be excluded from the measured data. Figure 3 An exemplary two-dimensional map 30 of a wafer outline 32 having a central region 34 is shown. rounThe linear shape profile is calculated by averaging the data points 38 along the vertical line 36 within the wafer profile at each x-axis value with a predefined step size (de). The vertical line 36 and the corresponding data points 38 provide a visual representation of the averaging of the data points (to eliminate artifacts in or near the wafer center region 34). The x-axis represents the cutting direction during wafer dicing, and the y-axis represents the extension of the cutting line ( e xtent) direction.

[0043] Once the linear shape profile is calculated, subsequent evaluation of peaks and valleys can be performed using the previously described steps and algorithms.

[0044] Analysis of the ripple at the end of the wafer processing process can be used to reject wafers with ripples that exceed a predefined rejection limit determined based on customer needs and sensitivity. Measurement of ripples after multi-wire slicing enables wafer screening after multi-wire sawing to eliminate wafers that are unlikely to pass end-of-line inspection early in the process. In addition, ripple data after multi-wire sawing can be utilized to monitor the performance of the multi-wire saw and make adjustments to the sawing process as needed. Post-cutting processes (such as grinding, polishing and / or epitaxy) can also be performed based on ripple parameter determination. For example, post-cutting processes can be implemented only for wafers that are not within a predetermined (or predetermined) threshold of acceptable ripples. In other examples, the number of post-processing techniques and / or the extent of post-processing techniques can also be determined based on the determined ripple parameters. In each case, the ripple parameter determination can therefore enhance the quality control process and the quality of the individual wafers themselves.

[0045] In some embodiments, the corrugation parameters of one wafer can be used to predict the corrugation of other wafers, thereby enabling efficient quality control decisions about production batches. For example, the determined corrugation parameters can be used to predict the corrugation of other wafers that will be produced from the same ingot and / or multi-wire group used in the dicing process. The wafers may then need to be sorted and / or post-processed in different ways, depending on the predicted corrugation parameters. In this way, predicted corrugation parameters within quality control limits can allow efficient production of wafers without the need for a time- and resource-consuming measurement process to independently determine the corrugation of each wafer.

[0046] In some embodiments, the ripple parameters enable feedback to be provided for the performance of the multi-wire saw and / or other equipment. For example, a baseline ripple parameter may be established by statistical means (e.g., an average or median of historical ripple parameters). The ripple parameters associated with a particular set of multi-wire saws may then be compared to the baseline ripple parameters. Such a comparison will enable a quick determination of whether the multi-wire (or other equipment) is operating within acceptable tolerances. Equipment maintenance decisions may then be made based on the comparison of the ripple parameters to the baseline ripple parameters.

[0047] Although the subject matter of the present disclosure has been illustrated and described in detail in the drawings and the foregoing description, these illustrations and descriptions should be considered illustrative or exemplary, rather than restrictive. Any statements given herein to characterize the invention should also be considered illustrative or exemplary, rather than restrictive, because the invention is defined by the claims. It will be understood that changes and modifications may be made by those of ordinary skill in the art within the scope of the following claims, and these changes and modifications may include any combination of the features of the different embodiments described above.

[0048] The terms used in the claims should be interpreted as having the broadest reasonable interpretation consistent with the above description. For example, the use of the article "a" or "an" when introducing an element should not be interpreted as excluding multiple elements. Similarly, the expression "or" should also be interpreted as inclusive, for example, the expression "A or B" does not exclude "A and B" unless it is clear from the context or the preceding description that only one of A and B is intended to be used. In addition, the expression "at least one of A, B and C" should be interpreted as one or more of a group of elements consisting of A, B and C, and should not be interpreted as requiring at least one of each of the listed elements A, B and C, regardless of whether A, B and C are associated as a category or otherwise. In addition, the expression "A, B and / or C" or "at least one of A, B or C" should be interpreted as including any single entity (such as A) in the listed elements, any subset (such as A and B) in the listed elements, or the entirety of the elements A, B and C.

Claims

1. A method for processing a silicon wafer, the method comprising: Slicing 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 cutting direction of the ingot; interpolating the shape data with a fixed and predetermined step size; fitting a first quadratic polynomial to the cross-sectional profile; determining a residual profile by subtracting the polynomial from the cross-sectional profile; fitting a second quadratic polynomial to the residual profile using a sliding window having a predetermined width to determine the location, height, and curvature of each peak and valley of the residual profile; determining a corrugation parameter based on the position, the height, and the curvature of each peak and valley of the residual profile; as well as The wafer is further processed based on a comparison of the ripple parameter to a predetermined ripple threshold value.

2. The method according to claim 1, wherein: The corrugation parameters are determined based on the following equation: WavPar=1000×∑ i |Height i ×Curvature i | Among them, WavPar is the ripple parameter, Height i is the height of the ith peak or valley, Curvature i is the curvature of the ith peak or valley.

3. The method according to claim 1, wherein: The ripple threshold is a reject limit, and further processing the wafer includes rejecting the wafer if the ripple parameter exceeds the reject limit.

4. The method according to claim 1, wherein: Further processing the wafer may include polishing or grinding the wafer. 5 . The method of claim 1 , further comprising determining an extent of further processing required to reduce wafer ripple to within the ripple threshold. 6 . The method of claim 1 , further comprising determining predicted wafer waviness parameters of uncut wafers based on the determined waviness parameters.

7. The method of claim 6, further comprising comparing the predicted wafer corrugation parameter with the determined corrugation parameter to determine whether wafer sorting should be performed.

8. The method according to claim 1, further comprising: determining baseline ripple parameters; comparing the corrugation parameter to the baseline corrugation parameter to determine whether the multi-wire saw requires maintenance; The multi-wire saw is maintained based on a comparison of the corrugation parameter to the baseline corrugation parameter.

9. The method according to claim 8, wherein: The baseline ripple parameters are statistically determined from historically determined ripple parameters.

10. A method for processing a silicon wafer, the method comprising: Slicing the ingot to form wafers; extracting a 2D shape profile from measured wafer data; fitting a parabola to the two-dimensional shape to approximate the shape of the wafer; calculating a two-dimensional residual by subtracting the parabola from the two-dimensional shape profile; calculating a linear shape profile along a cutting direction of the wafer by averaging data points corresponding to incremental horizontal axis values ​​rounded to predefined increments, wherein the horizontal axis values ​​correspond to the cutting direction; fitting a quadratic polynomial to the residual profile using a sliding window having a predetermined width to determine the location, height, and curvature of each peak and valley of the residual profile; determining a corrugation parameter based on the position, the height, and the curvature of each peak and valley of the residual profile; as well as The wafer is further processed based on a comparison of the ripple parameter to a predetermined ripple threshold value. 11 . The method of claim 10 , further comprising excluding a center region of the wafer from the two-dimensional shape outline.

12. The method according to claim 11, wherein: The central area corresponds to a wafer chuck area of ​​a wafer measurement tool.

13. The method according to claim 10, wherein: The corrugation parameters are determined based on the following equation: WapPar=1000×Σ i |Height i ×Curvature i | Among them, WavPar is the ripple parameter, Height i is the height of the ith peak or valley, Curvature i is the curvature of the ith peak or valley.

14. The method according to claim 10, wherein: The ripple threshold is a reject limit, and further processing the wafer includes rejecting the wafer if the ripple parameter exceeds the reject limit.

15. The method according to claim 10, wherein: Further processing the wafer may include polishing or grinding the wafer. 16 . The method of claim 1 , further comprising determining predicted wafer waviness parameters for uncut wafers based on the determined waviness parameters.

17. The method of claim 16, further comprising comparing the predicted wafer corrugation parameter with the determined corrugation parameter to determine whether wafer sorting should be performed.

18. The method according to claim 1, further comprising: determining baseline ripple parameters; comparing the corrugation parameter to the baseline corrugation parameter to determine whether the multi-wire saw requires maintenance; as well as Based on the comparison of the corrugation parameter to the baseline corrugation parameter, maintenance is performed on the multi-wire saw.

19. The method according to claim 18, wherein: The baseline ripple parameters are statistically determined based on historically determined ripple parameters.

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