Method and system for on-line checking thickness distribution in wafer in-situ thinning process

Abstract

The application provides a kind of thickness distribution online verification method and system in wafer in-situ thinning process, the method includes based on process model and real-time thickness data to construct processing perception anisotropic tensor diagram, to measure point as node, and through second-order processing influence tensor representation to measure the material removal coupling relationship between points;Further establish the multi-objective optimization function of fusion path length, fidelity weighted pedigree deviation and information coverage gain, and combine probe motion dimensionless parameter to construct measurement error covariance tensor, to tabu constraint to screen out the path of larger motion error;Then, according to the geometric accuracy factor, select the optimal calibration path from the pareto optimal solution set, and based on dimensionless geodesic distance, the global thickness distribution is reconstructed by graph kernel regression, to realize online comparison and verification with target thickness distribution.

Description

Technical Field

[0001] This application belongs to the field of verification, and in particular relates to an online verification method and system for thickness distribution in wafer in-situ thinning process. Background Technology

[0002] In wafer thinning processes such as chemical mechanical polishing (CMP) and back-side grinding, the material removal process is affected by factors such as processing pressure, relative motion speed, contact state, and local process disturbances, which can easily lead to non-uniform removal phenomena, resulting in local thickness deviations or surface anomalies on the wafer surface. To monitor wafer thickness changes in real time during processing and provide a basis for adjusting process parameters, in-situ online thickness measurement technology is widely used in advanced semiconductor manufacturing equipment. Compared with traditional offline measurement, in-situ online measurement can obtain real-time thickness information without interrupting the processing flow, which is beneficial for improving process control capabilities and production efficiency. However, due to the limited space of the processing chamber, the complex measurement environment, and the tight process cycle, the measurement probe can usually only perform sparse sampling along a limited trajectory, making it difficult to directly obtain a complete, high-resolution global thickness distribution.

[0003] Existing wafer thickness distribution verification methods mostly rely on sampling along pre-set fixed geometric trajectories, failing to fully consider the spatial differences in material removal rates. This easily leads to the omission of critical areas with drastic thickness variations or strong processing coupling, resulting in insufficient global representativeness of the collected data. Furthermore, during in-situ measurement, the probe is in dynamic motion, and factors such as acceleration changes and normal displacement fluctuations may introduce additional measurement disturbances, affecting data reliability. In subsequent global thickness reconstruction, traditional methods typically employ ordinary spatial interpolation, which struggles to simultaneously consider information coverage of unknown areas, approximation of higher-order features of complex surfaces, and the anisotropic effects of the processing. Therefore, it is necessary to propose an online verification method that integrates processing characteristics, measurement motion constraints, and multi-objective path optimization to improve the reliability of wafer global thickness distribution reconstruction and comparison verification under limited measurement points. Summary of the Invention

[0004] To address the problems of insufficient representativeness of limited measurement points, the impact of probe motion disturbance on measurement reliability, and the difficulty of accurately reconstructing the global thickness distribution using ordinary spatial interpolation in existing technologies, this invention provides an online verification method and system for thickness distribution in wafer in-situ thinning processes.

[0005] In a first aspect, the present invention provides a method for online verification of thickness distribution in a wafer in-situ thinning process, comprising the following steps:

[0006] A processing-aware anisotropic tensor diagram representing the thickness state is constructed based on the wafer process model and real-time thickness data. The nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model. An optimization objective function is established, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain. The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients. The information coverage gain quantifies the degree to which the path reduces uncertainty in unknown regions. Using taboo constraints, the tangential acceleration and normal displacement rate of change generated by the measurement probe motion are predicted and mapped to a second-order measurement error covariance tensor after dimensionless processing. When the maximum eigenvalue exceeds a threshold, the optimization operation is defined as a taboo movement. The optimal verification path is selected from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor. The optimal verification path is then used to reconstruct the global thickness on the processing-sensing anisotropic tensor using the graph kernel regression method. The thickness of any point is expressed as a normalized value of the weighted sum of the path point thickness based on the kernel function of the dimensionless geodesic distance. The comparison and verification are then completed.

[0007] Optionally, the construction of a processing-aware anisotropy tensor representing the thickness state based on the wafer process model and real-time thickness data includes: Extract the set of three-dimensional absolute coordinate points of the wafer on the measuring device, and divide the set of points into a uniformly distributed set of initial grid nodes; Read the actual processing pressure parameters set in the current process control system and extract the mathematical equation for the local material removal rate corresponding to the specified relative motion linear velocity; The second-order partial derivative of the mathematical equation for the local material removal rate is performed with respect to the spatial coordinate system to obtain the second-order processing influence tensor matrix representing the mutual influence relationship of the thickness at geometrically adjacent positions; The eigenvalues ​​and corresponding eigenvectors of the tensor matrix are calculated. The thickness measurement points are used as graph nodes. The average absolute value of the largest eigenvalue of the second-order processing influence tensor corresponding to two adjacent nodes is used as the non-negative scalar edge weight between the nodes to establish a connected processing-aware anisotropic tensor graph.

[0008] Optionally, the fidelity-weighted spectral bias utilizes fidelity coefficients to perform a weighted summation of the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution, including: The measurement signal-to-noise ratio is calculated based on the feedback signal and noise floor amplitude at each measurement location, and the measurement signal-to-noise ratio is mapped to information fidelity coefficients, which are then converted into spectral fidelity coefficients corresponding to each order of Zernike polynomial. The two-dimensional thickness distribution surface generated from the actual verification path thickness data and the ideal target thickness distribution are respectively subjected to Zernike polynomial expansion to obtain the corresponding Zernike expansion coefficient vector; Calculate the difference of the same order between two Zernike expansion coefficient vectors, and then perform a weighted summation of the difference of the same order based on the system fidelity coefficient to obtain the fidelity-weighted system bias.

[0009] Optionally, the degree to which the information coverage gain quantization path reduces uncertainty in unknown regions includes: The wafer space region to be inspected is divided into multiple detection units, and the unknown mesh sets before and after the addition of the candidate verification path are determined respectively; Calculate the distance between each unknown grid and the nearest measured node, and determine the corresponding unknown degree weight based on the dimensionless distance; Substituting the unknown degree weight into a preset uncertainty function, the unknown region uncertainty before and after the addition of the candidate verification path is obtained, and the difference between the two is used as the information coverage gain of the candidate verification path.

[0010] Optionally, the step of using tabu constraints to predict the tangential acceleration and normal displacement rate of change generated by the motion of the measurement probe, and mapping them to a second-order measurement error covariance tensor after dimensionless processing, includes: Extract the single-step motion time array of each candidate verification path segment, calculate the second derivative of the single-step spatial displacement function with respect to the time variable, and obtain the tangential running acceleration of the measuring probe; Based on the deformation parameters of the servo sensor allocated by the probe's main control program, the normal micro-indentation value within the same millisecond time window is calculated and converted into the normal displacement rate of change. The tangential acceleration and the normal displacement rate of change are dimensionless by using preset reference acceleration and reference velocity to obtain kinematic parameters with uniform dimensions. A kinematic state space column vector is constructed, and the kinematic parameters with uniform dimensions are assigned values ​​according to vector. The second-order measurement error covariance tensor representing the transient measurement fluctuation uncertainty is constructed by mapping the outer product of the column vector and the transpose of the column vector, which serves as the constraint monitoring benchmark for determining whether taboo movement is triggered.

[0011] Optionally, the step of selecting the optimal verification path from the Pareto optimal solution set based on the criterion of minimizing the geometric precision factor includes: Extract the three-dimensional spatial coordinate parameters of the detection points in each group of candidate optimal verification paths, and perform centering and dimensionless processing with the centroid of the detection points as a reference. Construct the observation configuration matrix based on the processed relative coordinate parameters. The error covariance matrix is ​​calculated based on the observed configuration matrix, and the geometric precision factor is determined by the main diagonal elements of the error covariance matrix. The optimal verification path is selected based on the criterion of minimizing the geometric accuracy factor.

[0012] Optionally, the step of reconstructing the global thickness on the processing-aware anisotropic tensor using a graph kernel regression method based on the optimal verification path, expressing the thickness of any point as a normalized value of the path point thickness based on a kernel function weighted by a dimensionless geodesic distance, and completing the comparison verification, includes: Acquire and store the first thickness measurement feedback sequence obtained after the probe performs scanning detection according to the optimal verification path; Based on the connectivity structure of the processing-aware anisotropic tensor graph, the edge weights between nodes are normalized according to the preset benchmark edge weights to obtain the dimensionless processing influence weights, and the reciprocal of the dimensionless processing influence weights is taken to obtain the dimensionless propagation impedance between nodes. Based on the dimensionless propagation impedance, calculate the shortest dimensionless geodesic distance between each node to be predicted that is not within the range of the measurement feedback and the existing measurement nodes in the storage list. The Gaussian distance decay kernel function is used to transform the calculated shortest dimensionless geodesic distance into a distance weight parameter. The measured thickness values ​​of existing measurement nodes are multiplied by the distance weight parameters, and then globally weighted and summed. The sum is then divided by the global sum of the distance weight parameters for normalization. The predicted thickness value of the node to be predicted is calculated, and a global thickness distribution map of the wafer is generated and output to complete the comparison and verification.

[0013] On the other hand, the present invention also provides an online thickness distribution verification system for in-situ wafer thinning processes, comprising the following modules: The module is used to construct a processing-aware anisotropic tensor diagram representing the thickness state based on the wafer process model and real-time thickness data. The nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model. A module is established to define the optimization objective function, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain. The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients. The information coverage gain quantifies the degree to which the path reduces uncertainty in unknown regions. Using taboo constraints, the tangential acceleration and normal displacement rate of change generated by the measurement probe motion are predicted and mapped to a second-order measurement error covariance tensor after dimensionless processing. When the maximum eigenvalue exceeds a threshold, the optimization operation is defined as a taboo move. The verification module is used to select the optimal verification path from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor; and to reconstruct the global thickness on the processing-sensing anisotropic tensor map using the optimal verification path by using the graph kernel regression method. The thickness of any point is expressed as a normalized value of the thickness of the path point based on the kernel function weighted sum of the dimensionless geodesic distance. The comparison and verification are then completed.

[0014] Preferably, the step of constructing a processing-aware anisotropy tensor map representing the thickness state based on the wafer process model and real-time thickness data includes: Extract the set of three-dimensional absolute coordinate points of the wafer on the measuring device, and divide the set of points into a uniformly distributed set of initial grid nodes; Read the actual processing pressure parameters set in the current process control system and extract the mathematical equation for the local material removal rate corresponding to the specified relative motion linear velocity; The second-order partial derivative of the mathematical equation for the local material removal rate is performed with respect to the spatial coordinate system to obtain the second-order processing influence tensor matrix representing the mutual influence relationship of the thickness at geometrically adjacent positions; The eigenvalues ​​and corresponding eigenvectors of the tensor matrix are calculated. The thickness measurement points are used as graph nodes. The average absolute value of the largest eigenvalue of the second-order processing influence tensor corresponding to two adjacent nodes is used as the non-negative scalar edge weight between the nodes to establish a connected processing-aware anisotropic tensor graph.

[0015] Preferably, the fidelity-weighted spectral bias utilizes fidelity coefficients to perform a weighted summation of the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution, including: The measurement signal-to-noise ratio is calculated based on the feedback signal and noise floor amplitude at each measurement location, and the measurement signal-to-noise ratio is mapped to information fidelity coefficients, which are then converted into spectral fidelity coefficients corresponding to each order of Zernike polynomial. The two-dimensional thickness distribution surface generated from the actual verification path thickness data and the ideal target thickness distribution are respectively subjected to Zernike polynomial expansion to obtain the corresponding Zernike expansion coefficient vector; Calculate the difference of the same order between two Zernike expansion coefficient vectors, and then perform a weighted summation of the difference of the same order based on the system fidelity coefficient to obtain the fidelity-weighted system bias.

[0016] Preferably, the degree to which the information coverage gain quantization path reduces uncertainty in unknown regions includes: The wafer space region to be inspected is divided into multiple detection units, and the unknown mesh sets before and after the addition of the candidate verification path are determined respectively; Calculate the distance between each unknown grid and the nearest measured node, and determine the corresponding unknown degree weight based on the dimensionless distance; Substituting the unknown degree weight into a preset uncertainty function, the unknown region uncertainty before and after the addition of the candidate verification path is obtained, and the difference between the two is used as the information coverage gain of the candidate verification path.

[0017] Preferably, the step of using tabu constraints to predict the tangential acceleration and normal displacement rate of change generated by the motion of the measuring probe, and mapping them to a second-order measurement error covariance tensor after dimensionless processing, includes: Extract the single-step motion time array of each candidate verification path segment, calculate the second derivative of the single-step spatial displacement function with respect to the time variable, and obtain the tangential running acceleration of the measuring probe; Based on the deformation parameters of the servo sensor allocated by the probe's main control program, the normal micro-indentation value within the same millisecond time window is calculated and converted into the normal displacement rate of change. The tangential acceleration and the normal displacement rate of change are dimensionless by using preset reference acceleration and reference velocity to obtain kinematic parameters with uniform dimensions. A kinematic state space column vector is constructed, and the kinematic parameters with uniform dimensions are assigned values ​​according to vector. The second-order measurement error covariance tensor representing the transient measurement fluctuation uncertainty is constructed by mapping the outer product of the column vector and the transpose of the column vector, which serves as the constraint monitoring benchmark for determining whether taboo movement is triggered.

[0018] Preferably, the step of selecting the optimal verification path from the Pareto optimal solution set according to the geometric precision factor minimization criterion includes: Extract the three-dimensional spatial coordinate parameters of the detection points in each group of candidate optimal verification paths, and perform centering and dimensionless processing with the centroid of the detection points as a reference. Construct the observation configuration matrix based on the processed relative coordinate parameters. The error covariance matrix is ​​calculated based on the observed configuration matrix, and the geometric precision factor is determined by the main diagonal elements of the error covariance matrix. The optimal verification path is selected based on the criterion of minimizing the geometric accuracy factor.

[0019] Preferably, the step of reconstructing the global thickness on the processing-aware anisotropic tensor using a graph kernel regression method based on the optimal verification path, expressing the thickness of any point as a normalized value of the path point thickness based on a kernel function weighted sum of dimensionless geodesic distances, and completing the comparison verification includes: Acquire and store the first thickness measurement feedback sequence obtained after the probe performs scanning detection according to the optimal verification path; Based on the connectivity structure of the processing-aware anisotropic tensor graph, the edge weights between nodes are normalized according to the preset benchmark edge weights to obtain the dimensionless processing influence weights, and the reciprocal of the dimensionless processing influence weights is taken to obtain the dimensionless propagation impedance between nodes. Based on the dimensionless propagation impedance, calculate the shortest dimensionless geodesic distance between each node to be predicted that is not within the range of the measurement feedback and the existing measurement nodes in the storage list. The Gaussian distance decay kernel function is used to transform the calculated shortest dimensionless geodesic distance into a distance weight parameter. The measured thickness values ​​of existing measurement nodes are multiplied by the distance weight parameters, and then globally weighted and summed. The sum is then divided by the global sum of the distance weight parameters for normalization. The predicted thickness value of the node to be predicted is calculated, and a global thickness distribution map of the wafer is generated and output to complete the comparison and verification.

[0020] This invention constructs a processing-aware anisotropic tensor map and utilizes a second-order processing influence tensor to characterize the strength and direction of material decoupling between measurement points, thus improving the rationality of wafer thickness state modeling. In path planning, multi-objective optimization is performed by integrating total path length, fidelity-weighted spectral bias, and information coverage gain. Taboo constraints are also set using the probe motion error covariance tensor to reduce the impact of motion disturbances on measurement results. Furthermore, a verification path with a superior geometric accuracy factor is selected from the Pareto optimal solution set, and kernel regression reconstruction is performed based on dimensionless geodesic distance, thereby improving the reliability of global thickness distribution reconstruction and online comparison verification under sparse sampling conditions. Attached Figure Description

[0021] Figure 1 A flowchart of an online thickness distribution verification method in a wafer in-situ thinning process; Figure 2 This is a schematic diagram of a tensor mesh oriented towards the processing procedure. Detailed Implementation

[0022] In this application, the terms "exemplarily" or "for example" are used to indicate that something is being described as an example, illustration, or illustration. Any embodiment or design described as "exemplarily" or "for example" in this application should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of terms such as "exemplarily" or "for example" is intended to present the relevant concepts in a specific manner.

[0023] This specification provides one or more embodiments of a method for online verification of thickness distribution in an in-situ wafer thinning process, such as... Figure 1 As shown, it includes the following steps: S1, a processing-aware anisotropic tensor diagram representing the thickness state is constructed based on the wafer process model and real-time thickness data; the nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model.

[0024] Real-time thickness data of the wafer surface is acquired using an optical thickness sensor, and the corresponding spatial coordinates are recorded as node data. This data is then input into the NetworkX graph computation library to initialize tensor graph nodes, and the real-time thickness data is stored as the thickness attribute of the corresponding node. The optical thickness sensor is a spectral interferometric thickness sensor, a white light interferometric thickness sensor, or a spectral confocal thickness sensor. The output data of each sensor are uniformly calibrated in coordinates and thickness to form thickness data in the same coordinate system. A material removal rate model is established based on the Preston equation. A local material removal rate function is obtained according to the processing pressure and relative motion linear velocity. The second-order partial derivative of the local material removal rate function with respect to the wafer plane coordinates is performed to construct a Hessian matrix as a second-order processing influence tensor. The processing pressure is polishing pressure or grinding pressure, and the relative motion linear velocity includes the relative linear velocity between the processing tool and the wafer, hereinafter referred to as processing pressure and relative motion linear velocity. The second-order processing influence tensor is subjected to eigenvalue decomposition to calculate the eigenvalue and eigenvector at each measurement point, thereby characterizing the strength and direction of the material removal coupling effect in the neighborhood of the measurement point. Spatially adjacent measurement points are connected by edges in the NetworkX graph computation library, and non-negative scalar edge weights are calculated based on the second-order processing influence tensor corresponding to adjacent nodes to complete the construction of the processing-aware anisotropic tensor graph.

[0025] As an optional implementation, the construction of a processing-aware anisotropy tensor representing the thickness state based on the wafer process model and real-time thickness data includes: Extract the set of three-dimensional absolute coordinate points of the wafer on the measuring device, and divide the set of points into a uniformly distributed set of initial grid nodes; Read the actual processing pressure parameters set in the current process control system and extract the mathematical equation for the local material removal rate corresponding to the specified relative motion linear velocity; The second-order partial derivative of the mathematical equation for the local material removal rate is performed with respect to the spatial coordinate system to obtain the second-order processing influence tensor matrix representing the mutual influence relationship of the thickness at geometrically adjacent positions; The eigenvalues ​​and corresponding eigenvectors of the tensor matrix are calculated. The thickness measurement points are used as graph nodes. The average absolute value of the largest eigenvalue of the second-order processing influence tensor corresponding to two adjacent nodes is used as the non-negative scalar edge weight between the nodes to establish a connected processing-aware anisotropic tensor graph.

[0026] The absolute coordinates (x, y, z) of the 12-inch wafer surface are acquired using an optical thickness sensor and divided into a uniform initial grid of, for example, 120 × 120 pixels according to a Cartesian coordinate system. The spacing between adjacent nodes is preferably set to 2.5 mm. The real-time processing pressure distribution P(x, y) is read from the machine's main control PLC system at a millisecond sampling rate. Its local reference value is typically within the range of 450-550 hPa, preferably set to 500 hPa. Combined with the local relative motion linear velocity distribution V(x, y) calculated from the wafer's current rotation center, its local reference value is within the range of 15-25 m / s. For example, if the specified point velocity is 20 m / s, the local material removal rate is extracted using the Preston modified model equation MRR(x, y) = K·P(x, y)·V(x, y), where the Preston scaling factor K is set to approximately 2.5 × 10⁻⁶. -13 Pa -1 .

[0027] Based on the above equations, second-order partial derivatives are performed on the Hessian matrix of MRR(x,y) in the x and y directions of the two-dimensional plane. This 2×2 Hessian matrix represents the second-order processing influence tensor matrix of the material removal rate spatial curvature and the cross-coupling effect of adjacent regions. Eigenvalue decomposition is performed on this matrix to extract two sets of eigenvalues ​​and their corresponding eigenvectors. For adjacent nodes i and j, the absolute values ​​of the largest eigenvalues ​​of their corresponding second-order processing influence tensors are obtained, and the average of the two largest eigenvalues ​​is used as the non-negative scalar edge weight between nodes i and j. For example, the non-negative scalar edge weight of the connection between two adjacent meshes is 1.85 μm / s·mm. -2 By filtering out edge weights less than 0.05 μm / s·mm -2Weakly correlated edges of the threshold are identified, and the Delaunay triangulation algorithm is used to construct a connected processing-aware anisotropic tensor graph that integrates physical space and processing interference factors. Since P(x,y) and V(x,y) are spatial distribution functions, and MRR(x,y) has non-zero spatial curvature in local regions, a non-zero second-order processing influence tensor and its corresponding edge weights can be obtained. A schematic diagram of the tensor mesh oriented towards the processing process is shown below. Figure 2 As shown.

[0028] S2, establish the optimization objective function, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain; the fidelity-weighted spectral bias uses the absolute value of the difference between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients; the information coverage gain quantifies the degree to which the path reduces the uncertainty of the unknown region; using taboo constraints, predict the tangential acceleration and normal displacement rate of change generated by the movement of the measurement probe, and map them into a second-order measurement error covariance tensor after dimensionless processing; when the maximum eigenvalue exceeds the threshold, the optimization operation is defined as a taboo movement.

[0029] The NSGA-III non-dominated sorting genetic algorithm is used to iteratively optimize the candidate verification path. The cumulative Euclidean distance formed by connecting the nodes in the current path is calculated as the target for the total path length. Zernike polynomial basis functions are generated, and the Zernike expansion coefficient vectors of the interpolated thickness surface of the current path and the ideal target thickness distribution surface are obtained by least squares fitting. The absolute value of the difference between the coefficients of the same order is calculated, and the weighted sum is combined with the preset fidelity weight as the target for fidelity weighted spectral bias. The uncertainty of the unknown region before and after the candidate verification path is added is calculated, and the difference between the two is used as the information coverage gain. Maximizing the information coverage gain is used as one of the path optimization objectives, or its negative value is converted into a minimization objective to participate in multi-objective optimization.

[0030] When generating a new path node in each iteration, the tangential acceleration and normal displacement rate of change of the probe executing the path are calculated based on the cubic spline interpolation kinematic model. Using preset reference acceleration and velocity, the tangential acceleration and normal displacement rate of change are dimensionless and formed into a kinematic state space column vector. A second-order measurement error covariance tensor is constructed by the cross product of this kinematic state space column vector and its transpose. The maximum eigenvalue of this second-order measurement error covariance tensor is calculated. If the maximum eigenvalue is greater than a preset threshold constant, a taboo constraint is triggered; the path solution is marked as a constraint violation solution and discarded in subsequent selection processes.

[0031] As an optional implementation, the fidelity-weighted spectral bias utilizes fidelity coefficients to weight and sum the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution, including: The measurement signal-to-noise ratio is calculated based on the feedback signal and noise floor amplitude at each measurement location, and the measurement signal-to-noise ratio is mapped to information fidelity coefficients, which are then converted into spectral fidelity coefficients corresponding to each order of Zernike polynomial. The two-dimensional thickness distribution surface generated from the actual verification path thickness data and the ideal target thickness distribution are respectively subjected to Zernike polynomial expansion to obtain the corresponding Zernike expansion coefficient vector; Calculate the difference of the same order between two Zernike expansion coefficient vectors, and then perform a weighted summation of the difference of the same order based on the system fidelity coefficient to obtain the fidelity-weighted system bias.

[0032] The average amplitude returned from each spatial grid measurement point is acquired by an optical thickness sensor, for example, in the range of 1.2-1.8V. A pre-calibrated noise floor amplitude, such as a constant 0.15V, is subtracted from this average amplitude under static, uncalibrated conditions. The resulting signal amplitude is then used to calculate the local measurement signal-to-noise ratio (SNR), for example, an SNR of 4.0. A nonlinear mapping model using the Logistic function is then applied. The signal-to-noise ratio is mapped to a spatial information fidelity coefficient between 0 and 1, such as 0.82, and multiplied by a pre-calculated Zernike basis function inner product projection matrix. The coefficients are then transformed to the frequency domain spectral space, and the spectral fidelity coefficients corresponding to the order of each polynomial term are extracted. For example, the fidelity weight corresponding to the fourth defocus term is 0.85.

[0033] The discrete actual scan thickness data is reconstructed into a continuous surface using cubic B-Spline spatial spline interpolation. The Zernike orthogonal polynomial expansion truncation number N is preferably set to 36 terms. Least squares orthogonal basis fitting is used to extract the 36×1 dimensional first expansion coefficient vector of the actual surface and the second expansion coefficient vector of the ideal target distribution, such as a flat 775μm uniform thickness matrix. These two vectors with the same dimension and corresponding terms are subtracted row-by-row to generate a 36×1 dimensional deviation column vector containing morphological deviation information for each feature. The absolute value of each element in this column vector, for example, the absolute deviation of the 5th astigmatism term is extracted as 0.12μm, and then multiplied term-by-term with the corresponding spectral fidelity coefficient, resulting in 0.102μm. All weighted values ​​of the first 36 terms are summed, for example, the total sum is 2.35μm. This total sum is output as a fidelity-weighted spectral deviation that takes into account different spatial frequency error sensitivities and measurement confidence.

[0034] As an optional implementation, the degree to which the information coverage gain quantization path reduces uncertainty in unknown regions includes: The wafer space region to be inspected is divided into multiple detection units, and the unknown mesh sets before and after the addition of the candidate verification path are determined respectively; Calculate the distance between each unknown grid and the nearest measured node, and determine the corresponding unknown degree weight based on the dimensionless distance; Substituting the unknown degree weight into a preset uncertainty function, the unknown region uncertainty before and after the addition of the candidate verification path is obtained, and the difference between the two is used as the information coverage gain of the candidate verification path.

[0035] Using the geometric center of a 300mm standard wafer as the origin of the coordinate system, an equidistant square detector array with a grid step size of 1.5mm is constructed, forming a global spatial set of approximately 31,400 detector units. Before adding candidate verification paths, a first unknown grid set is determined based on existing measured nodes. After adding candidate verification paths, the scan spot coverage area corresponding to the candidate verification path is excluded from the first unknown grid set, resulting in a second unknown grid set. The scan spot coverage radius tolerance can be set to within 0.5mm, and detector units within this coverage area are considered valid sampling areas. Using the KD-Tree spatial index nearest neighbor search algorithm, the first and second unknown grid sets are traversed respectively, and the Euclidean distance d between the coordinates of any grid center and the nearest measured node is calculated. For example, the nearest distance is found to be 15mm.

[0036] The distance value is dimensionless by performing a division operation using the total wafer diameter D, and the dimensionless distance is converted into an unknown degree weight using a preset monotonic mapping function. In one example, the unknown degree weight is calculated using Q = 1 - exp(-5.0 × d / D), where the greater the distance, the higher the unknown degree weight; when d is 15 mm and D is 300 mm, the corresponding unknown degree weight is approximately 0.22. Based on the unknown degree weight, the independent uncertainty of each unknown grid is calculated, and the first unknown grid set and the second unknown grid set are accumulated respectively to obtain the uncertainty of the first unknown region and the uncertainty of the second unknown region; the difference between the uncertainty of the first unknown region and the uncertainty of the second unknown region is used as the information coverage gain of the candidate verification path. Therefore, the larger the information coverage gain, the greater the degree to which the candidate verification path reduces the uncertainty of the unknown region.

[0037] As an optional implementation, the step of using tabu constraints to predict the tangential acceleration and normal displacement rate of change generated by the movement of the measurement probe, and mapping them to a second-order measurement error covariance tensor after dimensionless processing, includes: Extract the single-step motion time array of each candidate verification path segment, calculate the second derivative of the single-step spatial displacement function with respect to the time variable, and obtain the tangential running acceleration of the measuring probe; Based on the deformation parameters of the servo sensor allocated by the probe's main control program, the normal micro-indentation value within the same millisecond time window is calculated and converted into the normal displacement rate of change. The tangential acceleration and the normal displacement rate of change are dimensionless by using preset reference acceleration and reference velocity to obtain kinematic parameters with uniform dimensions. A kinematic state space column vector is constructed, and the kinematic parameters with uniform dimensions are assigned values ​​according to vector. The second-order measurement error covariance tensor representing the transient measurement fluctuation uncertainty is constructed by mapping the outer product of the column vector and the transpose of the column vector, which serves as the constraint monitoring benchmark for determining whether taboo movement is triggered.

[0038] The trajectory planning instructions of the thickness gauge servo controller are analyzed via the EtherCAT high-speed bus with a sampling period of 1ms. The single-step motion time sequence and interpolated spatial displacement function of each trajectory segment are extracted. The second time derivative of this displacement function is calculated using the central difference algorithm to obtain the instantaneous value of the probe's tangential acceleration along the planned trajectory. The acceleration is approximately 1.2 m / s² during stable operation. 2 The piezoelectric ceramic displacement or cantilever elastic feedback data within the closed-loop normal force system of the thickness probe is read, and the corresponding normal micro-indentation displacement within the aforementioned 1ms time window is calculated, such as 0.3μm. The instantaneous rate of change of the normal displacement is then calculated by dividing the displacement by the time window length, for example, 0.0003m / s. The mechanical vibration safety threshold of the probe, calibrated at the machine tool's factory, is selected as the nominal reference, for example, setting the reference acceleration to 5.0m / s². 2 The base speed is 0.01 m / s.

[0039] Divide the calculated acceleration and normal rate of change by the corresponding reference values ​​to obtain dimensionless kinematic parameters, such as a tangential dimensionless value of 0.24 and a normal dimensionless value of 0.03. These are then assembled into a 2×1 dimension kinematic state space column vector. The column vector U is compared with its own transpose. Perform matrix outer product multiplication to construct a 2×2 second-order measurement error covariance tensor containing coupled mechanical jitter terms. Solve for the maximum eigenvalue of this covariance tensor and compare it with the limiting fluctuation threshold, preferably 0.08. When the eigenvalue is greater than the threshold, it is determined that the current transient kinematic conditions may lead to a decrease in measurement focusing stability. In the global optimization process, taboo movement blocking is activated and executed.

[0040] S3. Select the optimal verification path from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor; use the optimal verification path to reconstruct the global thickness on the processing-sensing anisotropic tensor map using the graph kernel regression method, and express the thickness of any point as a value of the path point thickness based on the kernel function weighted sum of the dimensionless geodesic distance and normalized, and complete the comparison and verification.

[0041] All non-dominated path solutions are extracted from the Pareto optimal solution set output by the NSGA-III algorithm. An observation configuration matrix is ​​constructed based on the centered and dimensionless relative coordinates of the probe points in each path solution. The error covariance matrix and geometric accuracy factor are then calculated from this matrix. The path solution with the smallest geometric accuracy factor is selected as the optimal verification path. The non-negative scalar edge weights between nodes are normalized according to a preset benchmark edge weight to obtain the dimensionless processing influence weight. The reciprocal of this weight is used to form the dimensionless propagation impedance. Using this dimensionless propagation impedance as the pathfinding weight, the shortest path length from the unmeasured node to all measured nodes in the optimal verification path is calculated and taken as the dimensionless geodesic distance.

[0042] The calculated dimensionless geodesic distance is substituted into the Gaussian radial basis function to calculate the kernel weight value corresponding to each path node. The actual measured thickness of each path node is multiplied by its respective kernel weight value, summed, and then divided by the sum of all kernel weight values ​​to achieve normalization. This allows the reconstruction of the thickness value for any unmeasured node, completing the reconstruction of the global thickness distribution surface. The reconstructed global thickness distribution surface is then compared point-by-point with the preset standard process thickness tolerance upper and lower limit matrix to output the set of coordinates of abnormal areas exceeding the tolerance range, completing the online verification process.

[0043] As an optional implementation, the step of selecting the optimal verification path from the Pareto optimal solution set according to the geometric precision factor minimization criterion includes: Extract the three-dimensional spatial coordinate parameters of the detection points in each group of candidate optimal verification paths, and perform centering and dimensionless processing with the centroid of the detection points as a reference. Construct the observation configuration matrix based on the processed relative coordinate parameters. The error covariance matrix is ​​calculated based on the observed configuration matrix, and the geometric precision factor is determined by the main diagonal elements of the error covariance matrix. The optimal verification path is selected based on the criterion of minimizing the geometric accuracy factor.

[0044] After completing the multi-objective heuristic optimization algorithm iteration, the obtained Pareto front non-dominated solution set is traversed, and the three-dimensional coordinate sequence of all M probe nodes corresponding to each candidate verification path is analyzed, such as M=150 points. First, the centroids of the coordinates of the M probe nodes are calculated, and the three-dimensional coordinates of each probe node are subtracted from the centroids to obtain the centered coordinates. Then, the centered planar coordinates and thickness direction coordinates are dimensionless based on the wafer radius and thickness reference, respectively, to obtain the relative coordinate sequence. An observation configuration matrix H is constructed based on the relative coordinate sequence. Each row of the observation configuration matrix H corresponds to a probe node, and each column corresponds to a relative coordinate combination feature. The relative coordinate combination feature includes at least one of the first-order term, second-order term, and cross term of the relative coordinates. The matrix dimension is M×K, where K represents the number of selected relative coordinate combination features.

[0045] Performing the matrix multiplication operation between the transpose of matrix H and itself constructs a K×K dimension positive definite normal equation reference matrix N that reflects the geometric characteristics of the intersection configuration of the current probe lattice topology across the entire wafer surface. To reduce the matrix singularity and multicollinearity risks that may be induced by dense sampling, a small penalty parameter γ is input to the main diagonal of the reference matrix, preferably set to 10. -6 After regularizing the identity matrix γI, an inversion operation is performed. The resulting inverse matrix is ​​multiplied by the scalar of the white noise variance of the probe hardware intrinsic measurement obtained through prior statistics, such as the noise variance of 0.04 μm² obtained from static calibration, to inversely derive a K×K inverse error covariance matrix C. K variance elements representing the uncertainty of the independent state estimation in each dimension are extracted one by one from the diagonal of this covariance matrix, and the absolute values ​​of these elements are linearly summed and accumulated. The square root of the sum is then taken to obtain the GDOP geometric precision factor, such as 1.85. Numerical comparisons are performed in the candidate library of all Pareto solutions, and the arrangement scheme with the smallest GDOP factor is output as the optimal verification path.

[0046] As an optional implementation, the step of reconstructing the global thickness on the processing-aware anisotropic tensor using a graph kernel regression method on the optimal verification path, expressing the thickness of any point as a normalized value of the path point thickness based on a kernel function weighted sum of dimensionless geodesic distances, and completing the comparison verification includes: Acquire and store the first thickness measurement feedback sequence obtained after the probe performs scanning detection according to the optimal verification path; Based on the connectivity structure of the processing-aware anisotropic tensor graph, the edge weights between nodes are normalized according to the preset benchmark edge weights to obtain the dimensionless processing influence weights, and the reciprocal of the dimensionless processing influence weights is taken to obtain the dimensionless propagation impedance between nodes. Based on the dimensionless propagation impedance, calculate the shortest dimensionless geodesic distance between each node to be predicted that is not within the range of the measurement feedback and the existing measurement nodes in the storage list. The Gaussian distance decay kernel function is used to transform the calculated shortest dimensionless geodesic distance into a distance weight parameter. The measured thickness values ​​of existing measurement nodes are multiplied by the distance weight parameters, and then globally weighted and summed. The sum is then divided by the global sum of the distance weight parameters for normalization. The predicted thickness value of the node to be predicted is calculated, and a global thickness distribution map of the wafer is generated and output to complete the comparison and verification.

[0047] The thickness measurement device drives the motor and servo platform according to the output optimal verification path, sequentially scanning and sampling on the actual physical wafer surface. Thickness data is extracted using an optical thickness sensor, and discrete point coordinates are paired with corresponding thickness response values ​​after low-pass filtering, storing and generating a first measurement thickness feedback sequence such as 150 sets (x, y, thickness 775.2 μm). Based on the previously constructed tensor graph network topology with physical anisotropy constraints, the non-negative scalar edge weights representing the interaction of processing influences are normalized according to preset benchmark edge weights to obtain dimensionless processing influence weights. The reciprocal of these dimensionless processing influence weights is then taken to obtain the dimensionless propagation impedance, which is set as the pathfinding weight for mesh topology propagation. For tens of thousands of unknown mesh points in the entire graph to be predicted, Dijkstra's algorithm is used to search and optimize in the weighted graph, calculating the shortest graph theory distance from each point to all existing thickness measurement nodes. This distance represents the dimensionless geodesic distance propagating along the processing influence direction.

[0048] All dimensionless geodesic distance parameters are uniformly substituted into the Gaussian distance decay function, where both distance and bandwidth parameters are dimensionless. A bandwidth smoothing coefficient σ is optimized through input cross-validation, typically set within the range of 1.0 to 2.5 (e.g., 1.5). The dimensionless geodesic distance is transformed into a distance weight parameter W representing the radiative support intensity of surrounding known sampling points; the weight increases exponentially with shorter distances. Using the Nadaraya-Watson kernel regression operator, the actual thickness data of measured points within the search radius of the point to be predicted is multiplied item by item with their corresponding weight parameters and then globally accumulated. This is then divided uniformly by the sum of these associated weights to achieve probability normalization, calculating the predicted thickness value at the current grid, such as 775.05 μm. After mapping and covering all blind zone grids one by one, smooth rendering is performed to generate a 12-inch full-size 3D thickness topography distribution matrix. This matrix is ​​then compared with the process tolerance settings for residual analysis and verification.

[0049] This specification also provides, through one or more embodiments, an online thickness distribution verification system for in-situ wafer thinning processes, comprising the following modules: The module is used to construct a processing-aware anisotropic tensor diagram representing the thickness state based on the wafer process model and real-time thickness data. The nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model. A module is established to define the optimization objective function, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain. The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients. The information coverage gain quantifies the degree to which the path reduces uncertainty in unknown regions. Using taboo constraints, the tangential acceleration and normal displacement rate of change generated by the measurement probe motion are predicted and mapped to a second-order measurement error covariance tensor after dimensionless processing. When the maximum eigenvalue exceeds a threshold, the optimization operation is defined as a taboo move. The verification module is used to select the optimal verification path from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor; and to reconstruct the global thickness on the processing-sensing anisotropic tensor map using the optimal verification path by using the graph kernel regression method. The thickness of any point is expressed as a normalized value of the thickness of the path point based on the kernel function weighted sum of the dimensionless geodesic distance. The comparison and verification are then completed.

[0050] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for online verification of thickness distribution in an in-situ wafer thinning process, characterized in that, Includes the following steps: A processing-aware anisotropic tensor diagram representing the thickness state is constructed based on the wafer process model and real-time thickness data. The nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model. An optimization objective function is established, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain. The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients. The information coverage gain quantifies the degree to which the path reduces uncertainty in unknown regions. Using taboo constraints, the tangential acceleration and normal displacement rate of change generated by the measurement probe motion are predicted and mapped to a second-order measurement error covariance tensor after dimensionless processing. When the maximum eigenvalue exceeds the threshold, the optimization operation is defined as a taboo move. The optimal verification path is selected from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor. The optimal verification path is then used to reconstruct the global thickness on the processing-sensing anisotropic tensor using the graph kernel regression method. The thickness of any point is expressed as a normalized value of the weighted sum of the path point thickness based on the kernel function of the dimensionless geodesic distance. The comparison and verification are then completed.

2. The method according to claim 1, characterized in that, The construction of a processing-aware anisotropy tensor representing the thickness state based on the wafer process model and real-time thickness data includes: Extract the set of three-dimensional absolute coordinate points of the wafer on the measuring device, and divide the set of points into a uniformly distributed set of initial grid nodes; Read the actual processing pressure parameters set in the current process control system and extract the mathematical equation for the local material removal rate corresponding to the specified relative motion linear velocity; The second-order partial derivative of the mathematical equation for the local material removal rate is performed with respect to the spatial coordinate system to obtain the second-order processing influence tensor matrix representing the mutual influence relationship of the thickness at geometrically adjacent positions; The eigenvalues ​​and corresponding eigenvectors of the tensor matrix are calculated. The thickness measurement points are used as graph nodes. The average absolute value of the largest eigenvalue of the second-order processing influence tensor corresponding to two adjacent nodes is used as the non-negative scalar edge weight between the nodes to establish a connected processing-aware anisotropic tensor graph.

3. The method according to claim 2, characterized in that, The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients, including: The measurement signal-to-noise ratio is calculated based on the feedback signal and noise floor amplitude at each measurement location, and the measurement signal-to-noise ratio is mapped to information fidelity coefficients, which are then converted into spectral fidelity coefficients corresponding to each order of Zernike polynomial. The two-dimensional thickness distribution surface generated from the actual verification path thickness data and the ideal target thickness distribution are respectively subjected to Zernike polynomial expansion to obtain the corresponding Zernike expansion coefficient vector; Calculate the difference of the same order between two Zernike expansion coefficient vectors, and then perform a weighted summation of the difference of the same order based on the system fidelity coefficient to obtain the fidelity-weighted system bias.

4. The method according to claim 1, characterized in that, The degree to which the information coverage gain quantization path reduces uncertainty in unknown regions includes: The wafer space region to be inspected is divided into multiple detection units, and the unknown mesh sets before and after the addition of the candidate verification path are determined respectively; Calculate the distance between each unknown grid and the nearest measured node, and determine the corresponding unknown degree weight based on the dimensionless distance; Substituting the unknown degree weight into a preset uncertainty function, the unknown region uncertainty before and after the addition of the candidate verification path is obtained, and the difference between the two is used as the information coverage gain of the candidate verification path.

5. The method according to claim 1, characterized in that, The method of using tabu constraints to predict the tangential acceleration and normal displacement rate of change generated by the motion of the measurement probe, and mapping them to a second-order measurement error covariance tensor after dimensionless processing, includes: Extract the single-step motion time array of each candidate verification path segment, calculate the second derivative of the single-step spatial displacement function with respect to the time variable, and obtain the tangential running acceleration of the measuring probe; Based on the deformation parameters of the servo sensor allocated by the probe's main control program, the normal micro-indentation value within the same millisecond time window is calculated and converted into the normal displacement rate of change. The tangential acceleration and the normal displacement rate of change are dimensionless by using preset reference acceleration and reference velocity to obtain kinematic parameters with uniform dimensions. A kinematic state space column vector is constructed, and the kinematic parameters with uniform dimensions are assigned values ​​according to vector. The second-order measurement error covariance tensor representing the transient measurement fluctuation uncertainty is constructed by mapping the outer product of the column vector and the transpose of the column vector, which serves as the constraint monitoring benchmark for determining whether taboo movement is triggered.

6. The method according to claim 1, characterized in that, The selection of the optimal verification path from the Pareto optimal solution set based on the criterion of minimizing the geometric precision factor includes: Extract the three-dimensional spatial coordinate parameters of the detection points in each group of candidate optimal verification paths, and perform centering and dimensionless processing with the centroid of the detection points as a reference. Construct the observation configuration matrix based on the processed relative coordinate parameters. The error covariance matrix is ​​calculated based on the observed configuration matrix, and the geometric precision factor is determined by the main diagonal elements of the error covariance matrix. The optimal verification path is selected based on the criterion of minimizing the geometric accuracy factor.

7. The method according to claim 5, characterized in that, The process involves reconstructing the global thickness on the processing-aware anisotropic tensor using a graph kernel regression method based on the optimal verification path. The thickness at any point is expressed as a normalized value representing the weighted sum of path point thicknesses based on a dimensionless geodesic distance kernel function. This process is then used to complete the comparison and verification, including: Acquire and store the first thickness measurement feedback sequence obtained after the probe performs scanning detection according to the optimal verification path; Based on the connectivity structure of the processing-aware anisotropic tensor graph, the edge weights between nodes are normalized according to the preset benchmark edge weights to obtain the dimensionless processing influence weights, and the reciprocal of the dimensionless processing influence weights is taken to obtain the dimensionless propagation impedance between nodes. Based on the dimensionless propagation impedance, calculate the shortest dimensionless geodesic distance between each node to be predicted that is not within the range of the measurement feedback and the existing measurement nodes in the storage list. The Gaussian distance decay kernel function is used to transform the calculated shortest dimensionless geodesic distance into a distance weight parameter. The measured thickness values ​​of existing measurement nodes are multiplied by the distance weight parameters, and then globally weighted and summed. The sum is then divided by the global sum of the distance weight parameters for normalization. The predicted thickness value of the node to be predicted is calculated, and a global thickness distribution map of the wafer is generated and output to complete the comparison and verification.

8. An online thickness distribution verification system for in-situ wafer thinning processes, characterized in that, Includes the following modules: The module is used to construct a processing-aware anisotropic tensor diagram representing the thickness state based on the wafer process model and real-time thickness data. The nodes of the processing-aware anisotropic tensor diagram are measurement points, and the connecting edges are defined by a second-order processing influence tensor. The eigenvalues ​​and eigenvectors of the tensor represent the strength and direction of the mutual influence between nodes derived from the material removal rate model. A module is established to define the optimization objective function, including minimizing the total path length, minimizing the fidelity-weighted spectral bias, and maximizing the information coverage gain. The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients. The information coverage gain quantifies the degree to which the path reduces uncertainty in unknown regions. Using taboo constraints, the tangential acceleration and normal displacement rate of change generated by the measurement probe motion are predicted and mapped to a second-order measurement error covariance tensor after dimensionless processing. When the maximum eigenvalue exceeds the threshold, the optimization operation is defined as a taboo move. The verification module is used to select the optimal verification path from the Pareto optimal solution set according to the criterion of minimizing the geometric accuracy factor; and to reconstruct the global thickness on the processing-sensing anisotropic tensor map using the optimal verification path by using the graph kernel regression method. The thickness of any point is expressed as a normalized value of the thickness of the path point based on the kernel function weighted sum of the dimensionless geodesic distance. The comparison and verification are then completed.

9. The system according to claim 8, characterized in that, The construction of a processing-aware anisotropy tensor representing the thickness state based on the wafer process model and real-time thickness data includes: Extract the set of three-dimensional absolute coordinate points of the wafer on the measuring device, and divide the set of points into a uniformly distributed set of initial grid nodes; Read the actual processing pressure parameters set in the current process control system and extract the mathematical equation for the local material removal rate corresponding to the specified relative motion linear velocity; The second-order partial derivative of the mathematical equation for the local material removal rate is performed with respect to the spatial coordinate system to obtain the second-order processing influence tensor matrix representing the mutual influence relationship of the thickness at geometrically adjacent positions; The eigenvalues ​​and corresponding eigenvectors of the tensor matrix are calculated. The thickness measurement points are used as graph nodes. The average absolute value of the largest eigenvalue of the second-order processing influence tensor corresponding to two adjacent nodes is used as the non-negative scalar edge weight between the nodes to establish a connected processing-aware anisotropic tensor graph.

10. The system according to claim 8, characterized in that, The fidelity-weighted spectral bias is calculated by weighting the absolute values ​​of the differences between the Zernike polynomial coefficient vectors of the interpolation surface and the target distribution using fidelity coefficients, including: The measurement signal-to-noise ratio is calculated based on the feedback signal and noise floor amplitude at each measurement location, and the measurement signal-to-noise ratio is mapped to information fidelity coefficients, which are then converted into spectral fidelity coefficients corresponding to each order of Zernike polynomial. The two-dimensional thickness distribution surface generated from the actual verification path thickness data and the ideal target thickness distribution are respectively subjected to Zernike polynomial expansion to obtain the corresponding Zernike expansion coefficient vector; Calculate the difference of the same order between two Zernike expansion coefficient vectors, and then perform a weighted summation of the difference of the same order based on the system fidelity coefficient to obtain the fidelity-weighted system bias.

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