A wafer thinning process full-field stress online sensing method and system based on modal sparse reconstruction

Abstract

The application discloses a wafer thinning process full-field stress online sensing method and system based on modal sparse reconstruction, and the method comprises the following steps: extracting a spatial base function from historical data and constructing a base function matrix by using intrinsic orthogonal decomposition; collecting sparse stress observation values in real time through a small number of sensors during the thinning process; solving a sparse reconstruction optimization problem to obtain dominant modal coefficients by taking the minimum linear combination error of the observation values and the base function submatrix and the sparsity of the coefficients as constraints; linearly combining the coefficients and the complete base function matrix to instantaneously reconstruct a full-field stress distribution; and performing abnormality determination and early warning based on the reconstructed stress and uncertainty quantification information. The application can realize real-time sensing of the full-field stress under a small number of sensors, simultaneously output confidence evaluation, and effectively support online monitoring and closed-loop control of the thinning process.

Description

Technical Field

[0001] This invention belongs to the field of intelligent monitoring technology in semiconductor manufacturing, specifically relating to a method and system for online sensing of full-field stress in the wafer thinning process based on modal sparse reconstruction. Background Technology

[0002] As semiconductor devices continue to evolve towards higher integration and higher-density interconnects, wafer thinning has become a key process step in 3D integration and advanced packaging. To meet structural requirements such as exposed through-silicon vias (TSVs), multi-chip stacking, and compact packaging, wafers often need to be thinned to ultra-thin thicknesses of tens or even several micrometers. Under this extreme thinning condition, the wafer's bending stiffness decreases sharply. Mechanical grinding forces, process thermal effects, and mismatches in the coefficients of thermal expansion between multilayer materials can all lead to significant warping deformation and residual stress concentration. This can affect subsequent bonding alignment accuracy or even cause wafer cracking or fragmentation, directly restricting product yield and manufacturing costs.

[0003] In response to the aforementioned issues, the industry has placed increasingly higher demands on stress state sensing during wafer thinning processes. However, existing technologies still have significant limitations in terms of real-time performance, spatial coverage, and the reliability of sensing results.

[0004] In numerical simulations, the finite element method (FEM) can provide reference information on the stress distribution across the entire wafer, but its accuracy depends on a series of prior assumptions, such as material constitutive parameters, boundary conditions, and contact models. During actual thinning processes, process parameters fluctuate dynamically, and material properties vary between batches, resulting in persistent, difficult-to-quantify deviations between simulation predictions and the actual physical state. More critically, the FEM struggles to provide confidence assessments of its predictions, thus failing to offer a quantitative basis for subsequent process decisions.

[0005] In experimental measurements, methods such as Raman spectroscopy, laser interferometry, and X-ray diffraction can acquire stress or warpage data at specific locations on the wafer surface, offering high single-point measurement accuracy. However, these methods are essentially point-by-point or small-range scanning measurements with limited spatial sampling density. Obtaining the full-field stress distribution requires a significant amount of scanning time, making it difficult to match the online cycle time of thinning processes. If only a very small number of measurement points are retained in pursuit of real-time performance, directly inferring the full-field stress distribution from sparse measurement points faces the problem of insufficient information, compromising the accuracy of the reconstruction results.

[0006] More importantly, regardless of whether the approach is based on simulation or actual measurement, current methods can only output point estimates of stress distribution, without simultaneously providing quantitative information on the reliability of these estimates, such as the degree of uncertainty, confidence interval, or credibility index of the reconstructed values. This leaves field operators without objective evidence to determine whether the reconstructed stress results are sufficiently reliable, whether alarms should be triggered, or whether process parameters should be adjusted. Without quantitative information on uncertainty, anomaly thresholds are often conservatively set based on experience, easily leading to false alarms or missed alarms, thus limiting the in-depth application of online stress sensing technology in actual production lines.

[0007] In the current technology, there is a lack of a method and system that can both use sparse sensor data to realize real-time reconstruction of full-field stress and simultaneously output information on the uncertainty quantification of reconstruction results, so as to fill the shortcomings of the existing technology in both full-field perception and credibility assessment, and support the actual needs of adaptive control and yield improvement in the thinning process. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method and system for online sensing of full-field stress in the wafer thinning process based on modal sparse reconstruction.

[0009] This invention is implemented as follows:

[0010] In a first aspect, the present invention provides a method for online sensing of full-field stress in a wafer thinning process based on modal sparse reconstruction, comprising the following steps:

[0011] S1. Extract spatial basis function modes from historical data of wafer stress field using intrinsic orthogonal decomposition, and construct basis function matrix;

[0012] S2. During the wafer thinning process, a small number of sensors arranged on the wafer surface or stage are used to collect sparse stress observation values ​​in real time, forming an observation vector.

[0013] S3. Based on the basis function matrix and the observation vector, construct and solve the sparse reconstruction optimization problem. The sparse reconstruction optimization problem takes minimizing the observation error as the data fidelity objective and the sparseness of the coefficient vector as the constraint objective, and obtains the optimal sparse coefficient vector.

[0014] S4. The optimal sparse coefficient vector is linearly combined with the complete basis function matrix to reconstruct the full-field stress distribution at the current moment;

[0015] S5. Based on the reconstructed full-field stress distribution and the quantitative information of prediction uncertainty generated during the reconstruction process, anomaly judgment and early warning are performed on the wafer thinning process.

[0016] Preferably, the construction of the basis function matrix in step S1 specifically includes:

[0017] Obtain the historical dataset of the wafer mechanical field obtained through finite element simulation or historical measurement. The mechanical field is a full-field distribution corresponding to the type of physical quantity collected by the sensor in step S2, covering various physical quantities that can directly or indirectly reflect the wafer stress state, including but not limited to warping field, displacement field or stress field.

[0018] The historical dataset is subjected to intrinsic orthogonal decomposition to obtain a set of spatial basis function modes and the energy contribution of each mode;

[0019] Sort the modes by energy contribution from largest to smallest, retain the dominant modes whose cumulative energy contribution meets a preset threshold, and form the basis function matrix from all retained dominant modes.

[0020] Preferably, the number of sensors deployed is less than the total number of grid points and less than the number of spatial basis function modes extracted in step S1.

[0021] The sensor can be a displacement sensor, an optical profile sensor, a piezoelectric sensor, or a strain gauge; the collected observations include wafer warpage, out-of-plane displacement, or stress components.

[0022] As a preferred option, step S3 specifically involves:

[0023] Based on the spatial basis function matrix constructed in step S1 and the observation vectors collected in step S2, the following sparse reconstruction optimization problem is constructed and solved:

[0024]

[0025] Obtain the optimal sparse coefficient vector .in, The observation vectors collected in step S2, Let Ψ be the submatrix of the basis function matrix Ψ at the sensor location, and A be the vector of basis function coefficients to be solved. The adjustment parameter for sparse constraints, This represents the square of the L2 norm (data fidelity term). This represents the L1 norm (sparse constraint term).

[0026] The above optimization problem takes minimizing the linear combination error of the observation and basis functions at the sensor position submatrix as its primary objective, while using the sparsity of the coefficient vector as a constraint, so that even under the underdetermined condition that the number of sensors is significantly less than the number of basis function modes, the few dominant modes corresponding to the current stress state and their coefficients can still be uniquely determined.

[0027] The sparse reconstruction optimization problem can be solved using the LASSO algorithm, the orthogonal matching pursuit algorithm, or the Bayesian sparse learning algorithm. The adjustment parameters... It can be automatically determined through cross-validation or L-curve method to balance reconstruction error and coefficient sparsity.

[0028] Preferably, step S4 is expressed as follows:

[0029]

[0030] in, Let Ψ be the optimal sparse coefficient vector, and Ψ be the basis function matrix. Since the basis function matrix Ψ contains modal information for all spatial locations, while the coefficient vector... If the number of non-zero elements is less than the total number of spatial basis function modes, and the sum of the energy contributions of the dominant modes corresponding to the non-zero elements to the full-field stress reconstruction exceeds a preset ratio threshold, the reconstruction operation can cover all grid points in the entire field with minimal computational cost and meets real-time requirements.

[0031] Preferably, the uncertainty quantification information of the prediction in step S5 is obtained by at least one of the following methods:

[0032] Pointwise uncertainty estimation based on Bayesian posterior distribution: When solving the problem using the Bayesian sparse learning algorithm, the posterior distribution of the basis function coefficient vector A is obtained simultaneously, and the variance or confidence interval of the reconstructed field at any spatial location is calculated accordingly.

[0033] Global uncertainty index based on observation residuals: Calculate the observation vector in step S3. With reconstructed observations residual modulus between Or relative residuals, serving as a quantitative indicator of the global uncertainty of the entire reconstructed field;

[0034] Statistical uncertainty estimation based on multiple sampling: Multiple observation vectors are obtained by performing multiple time-domain samplings on the sensor, and each vector is solved and reconstructed independently. The statistical discrete measure of the reconstructed value at each spatial location is calculated as a point-by-point uncertainty estimate.

[0035] Preferably, the process anomaly determination and early warning in step S5 includes:

[0036] When the warping, displacement, or stress value in the reconstructed full-field stress distribution exceeds the preset safety threshold, it is determined to be a stress anomaly.

[0037] When the quantified uncertainty information of the prediction exceeds a preset uncertainty threshold, it is determined that the reconstruction confidence is insufficient.

[0038] When an abnormality is detected, an alarm is triggered or a command to adjust the thinning process parameters is output.

[0039] Secondly, the present invention provides an online full-field stress sensing system for wafer thinning process based on modal sparse reconstruction for implementing the above method, comprising:

[0040] The basis function construction and storage module is used to extract spatial basis function modes from historical data of wafer stress field using intrinsic orthogonal decomposition, and to construct and store the basis function matrix.

[0041] A sparse sensing module is used to collect sparse stress observation values ​​in real time through a small number of deployed sensors during the wafer thinning process, and to construct an observation vector from the sparse stress observation values.

[0042] The sparse reconstruction calculation module, connected to the basis function construction and storage module and the sparse sensing module, is used to construct and solve the sparse reconstruction optimization problem with the constraint of minimizing the error between the linear combination of the observation vector and the submatrices of the basis function matrix at the sensor position and the sparsity of the coefficient vector to be solved, so as to obtain the optimal dominant mode coefficient vector.

[0043] The full-field stress reconstruction module, connected to the basis function construction and storage module and the sparse reconstruction calculation module, is used to linearly combine the optimal dominant mode coefficient vector with the complete basis function matrix to reconstruct the full-field stress distribution at the current moment.

[0044] The anomaly detection and early warning module is connected to the full-field stress reconstruction module and the sparse reconstruction calculation module. It is used to detect and warn of anomalies in the wafer thinning process based on the reconstructed full-field stress distribution and the quantitative information of prediction uncertainty generated during the reconstruction process.

[0045] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, characterized in that the program, when executed by a processor, implements the above-described method.

[0046] The beneficial effects of this invention are at least as follows:

[0047] First, it significantly reduces the number of sensors and system complexity. By combining intrinsic orthogonal decomposition dimensionality reduction with sparse reconstruction theory, the number of sensors can be less than both the number of grid points and the number of basis function modes. Full-field stress sensing can be achieved with a very small number of sensors, greatly reducing hardware costs and process space requirements, which is especially suitable for confined space scenarios such as wafer thinning.

[0048] Second, it enables real-time online reconstruction of the stress distribution across the entire field. Traditional offline finite element simulations cannot track fluctuations in process parameters in real time. However, this invention employs a two-stage strategy of "offline library construction + online sparse reconstruction," shifting the main computational burden to the offline stage. Online reconstruction requires only a small number of matrix operations, thus meeting real-time requirements.

[0049] Third, it provides the ability to assess the confidence level of the reconstruction results. This invention not only outputs the stress distribution across the entire field, but also quantifies the uncertainty information of the reconstruction results through methods such as Bayesian posterior variance, observation residuals, or multiple sampling statistics. This provides a basis for the reliability judgment of subsequent anomaly detection and effectively avoids false alarms or missed alarms caused by insufficient reconstruction accuracy.

[0050] Fourth, a complete online closed-loop sensing and early warning scheme is formed. This invention covers the entire closed loop process from offline basis function library construction, sparse signal acquisition, sparse reconstruction solution, full-field stress recovery to uncertainty quantification and anomaly early warning, providing complete online sensing technology support for adaptive control and yield improvement of wafer thinning process. Attached Figure Description

[0051] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0052] Figure 1 This is an overall flowchart of the online sensing method and system for full-field stress in wafer thinning process based on modal sparse reconstruction provided in the embodiments of the present invention.

[0053] Figure 2 This is provided by the embodiments of the present invention. Figure 1 A detailed flowchart of the offline construction of the spatial basis function matrix in step S1.

[0054] Figure 3 This is a schematic diagram comparing the placement of the sparse sensor on the wafer surface with the grid points across the entire field, as provided in an embodiment of the present invention.

[0055] Figure 4 This is provided by the embodiments of the present invention. Figure 1 The logical flowchart of step S5, which is based on the full-field stress distribution and uncertainty quantification information, for anomaly judgment and early warning.

[0056] Figure 5 This is a block diagram of the module structure of the online sensing system for full-field stress in the wafer thinning process based on modal sparse reconstruction provided in an embodiment of the present invention. Detailed Implementation

[0057] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0058] like Figure 1As shown, this embodiment provides a method for online sensing of full-field stress in the wafer thinning process based on modal sparse reconstruction, which includes the following five steps:

[0059] Step S1: Extract spatial basis function modes from historical data of wafer stress field using intrinsic orthogonal decomposition and construct basis function matrix;

[0060] Specifically, firstly, it is necessary to understand the physical application scenario of this invention. Wafer thinning equipment mainly includes a grinding head and a wafer stage, with the wafer stage used to support and fix the wafer to be thinned. Sensors are arranged inside the wafer stage. It should be noted that arranging sensors inside the wafer stage to monitor the wafer status online is a mature technology with existing practical applications in this field. For example, published technical documents have reported that embedding micro-force measuring units or displacement sensor arrays in the wafer stage of thinning equipment can acquire real-time information on the force or deformation of the wafer at different discrete positions, for process monitoring and feedback control. Therefore, the "arrangement of sensors inside the wafer stage" described in this invention has clear engineering feasibility and is not a conjecture divorced from the reality of the production line.

[0061] In this embodiment, the sensors are multiple displacement sensors or optical profile sensors, which measure the out-of-plane displacement or warpage of the wafer surface in real time in a non-contact manner. The number of sensors, M, is typically 5 to 20, far fewer than the total number of spatial discrete points N (typically thousands to tens of thousands) and the number of spatial basis function modes K required to describe the stress distribution across the entire wafer. The spatial positions of the sensors on the wafer surface remain fixed or known, and their layout preferably covers the central and edge regions of the wafer to fully capture various typical deformation modes that may occur during wafer thinning, such as central bulges, edge warpage, and saddle-shaped deformation.

[0062] The signals collected by the sensors are converted from analog to digital and synchronously acquired by the data acquisition unit before being transmitted to the data processing and computing unit. The data processing and computing unit can be a device with computing capabilities, such as an industrial computer, embedded processor, or cloud server. This unit houses the functional modules of the full-field stress online sensing system provided by this invention, used to execute the online calculation tasks in steps S1 to S5.

[0063] The output of the data processing and calculation unit is connected to the display and alarm unit to display the reconstructed full-field stress distribution cloud map and uncertainty quantification index in real time, and to issue an audible and visual alarm when an anomaly is detected.

[0064] Based on the physical scenario described above, this step constructs the spatial basis function matrix. Figure 2The detailed process of offline construction of the spatial basis function matrix in step S1 is shown. Step S1 is the premise and foundation of the method of the present invention. Its purpose is to pre-construct a low-dimensional basis function space that can compactly represent the stress distribution of the entire wafer using historical data before the wafer thinning process officially begins.

[0065] S1.1 Obtain historical datasets of wafer stress fields, specifically:

[0066] First, it is necessary to acquire a batch of historical data that can fully cover the typical stress states during the wafer thinning process. This data can come from finite element simulation results, measured data from historical batch production processes, or a combination of both.

[0067] Preferably, an experimentally validated finite element simulation model is used to simulate the stress field distribution of the wafer under different thinning stages (such as initial thickness, intermediate thickness, and target thickness), different process parameters (such as pressure, rotation speed, and cooling conditions), and different material properties (such as batch fluctuation range of thermal expansion coefficient and elastic modulus), generating a snapshot dataset containing S samples.

[0068] In another embodiment, historical data is acquired by deploying a one-time dense-grid sensor array on the thinning device. This dense-grid data is used only in the offline database construction phase, while only a small number of sensors are needed in the online monitoring phase.

[0069] S1.2, Intrinsic orthogonal decomposition and mode extraction, specifically:

[0070] S historical stress field sample data are organized into a snapshot matrix, with each sample being an N-dimensional column vector representing the mechanical quantity value at N grid points on the wafer surface. This mechanical quantity is of the same type as the physical quantity acquired by the sensor in step S2. The snapshot matrix is ​​then subjected to eigenorthogonal decomposition to obtain a set of spatial basis function modes (i.e., POD modes) and the singular values ​​(or eigenvalues) corresponding to each mode. The square of the singular value reflects the contribution of each mode to the total energy (variance) of the dataset.

[0071] S1.3, Dominant mode selection and basis function matrix construction, specifically:

[0072] The obtained modes are sorted from largest to smallest according to their energy contribution. Preferably, a cumulative energy contribution threshold T is set, for example, retaining the top K modes that achieve a cumulative energy contribution rate in the range of 90% to 99%, especially above 95%. Taking a total of 30 modes as an example, the cumulative energy contribution rate of the first 6 to 12 modes is usually sufficient to exceed 95%, so the value of K is significantly smaller than the total number of modes and the total number of grid points N.

[0073] In another embodiment, the value of K can also be determined by combining the cross-validation results of the reconstruction error. The retained K dominant modes constitute a basis function matrix Ψ (with dimensions N×K), and each column of it is a normalized spatial basis function mode. Therefore, the overall stress distribution can be approximately expressed as a linear combination of these K modes:

[0074]

[0075] in, The coefficient vector of the basis functions. When the modes are extracted from displacement field data, the above linear combination gives the full-field displacement distribution. Since there is a definite elastic mechanical relationship between the displacement field and the stress field, those skilled in the art can further convert it to obtain the full-field stress distribution expressed in stress units as needed.

[0076] S1.4 Storage of the basis function matrix, specifically:

[0077] The constructed basis function matrix Ψ is stored in the storage module of the data processing and computation unit for use in the online monitoring phase. Since K is much smaller than N, the storage space occupied by the basis function matrix is ​​much smaller than the original full-field grid data volume.

[0078] Step S2: During the thinning process, sparse stress observation values ​​are collected in real time using only a small number of sensors;

[0079] Specifically, in the online monitoring stage of the wafer thinning process, step S2 uses sensors to collect stress observation values ​​on the wafer surface in real time at a preset sampling frequency.

[0080] Figure 3 This diagram illustrates a comparison between the sparse sensor placement on the wafer surface and the total number of grid points. The diagram uses a fine grid to represent all grid points (total N), and star-shaped markers to highlight the sensor placement locations (number M). It is clearly visible from the diagram that the sensor placement is extremely sparse, with M being much smaller than N.

[0081] The number of sensors M satisfies the following dual constraints: First, M is less than the total number of grid points N; second, M is less than the number of spatial basis function modes K extracted in step S1. Since M < K, the online monitoring stage faces an "underdetermined" problem, that is, the number of unknowns K is greater than the number of equations M. The traditional least squares method cannot obtain a definite solution under this condition, which is precisely the necessity for introducing sparse constraints in this invention.

[0082] In one specific embodiment, eight laser displacement sensors (M=8) are used to measure out-of-plane displacement at specific locations on the wafer surface in a non-contact manner. The total number of grid points N is approximately 5000 (derived from a finite element mesh), and the number of dominant modes K retained by the POD is 20. At each sampling time, the eight displacement values ​​acquired constitute the observation vector. (8×1).

[0083] In another embodiment, the sensor may also be a piezoelectric sensor or a strain gauge to directly measure the stress components in a localized area of ​​the wafer. The choice of sensor type depends on the process conditions and measurement accuracy requirements, and this invention does not impose specific limitations on this.

[0084] It should be noted that regardless of the type of physical quantity collected by the sensor in step S2, it should be ensured that it is the same type of physical quantity as the historical data used to construct the basis function matrix in step S1, so as to ensure the observation vector in the optimization problem in step S3. with basis function submatrix They have consistent physical dimensions. For example, when the sensor collects out-of-plane displacement, the historical dataset in step S1 should be displacement field data, and the extracted basis function modes should be displacement modes. and Both are in the dimension of length; when the sensor collects stress components, the historical dataset should be stress field data, and the extracted basis function modes are stress modes, both of which are in the dimension of stress.

[0085] Step S3: With the constraint of minimizing the linear combination error of the observation and basis functions in the sensor position submatrix and the sparse coefficient vector, construct and solve the sparse reconstruction optimization problem to obtain the dominant mode coefficients;

[0086] Specifically, step S3 is the core technical step of the present invention, the purpose of which is to accurately recover the dominant mode coefficients corresponding to the stress field at the current moment from sensor observations that are far fewer than the number of modes.

[0087] S3.1 The construction of the optimization problem is as follows:

[0088] Based on the basis function matrix Ψ constructed in step S1 and the observation vectors collected in step S2 Construct the following sparse reconstruction optimization problem:

[0089] (1)

[0090] In equation (1), Let Ψ be a submatrix formed by the corresponding rows of the basis function matrix Ψ at the M sensor installation locations, with dimensions M×K. Each column corresponds to the value of the spatial basis function mode of the same physical quantity at the sensor location, and is related to the observation vector. They have the same physical dimensions; A is the K×1 dimensional coefficient vector to be solved; This is a regularization parameter used to balance the weight relationship between data fidelity terms and sparse constraint terms; It represents the square of the L2 norm, used to measure the residual modulus between the reconstructed observations and the actual observations; L1 norm is represented by the sum of the absolute values ​​of all elements in the coefficient vector A. Its function is to guide the optimization process to tend to produce sparse coefficient solutions, that is, to push most of the elements in the coefficient vector A to approach zero or be exactly zero.

[0091] S3.2, The selection of the solution algorithm, specifically:

[0092] In a preferred embodiment, the LASSO algorithm (Least Absolute Shrinkage and Selection Operator) is used to solve the above optimization problem. The LASSO algorithm can efficiently search for the optimal sparse solution using coordinate descent or least angle regression (LARS) algorithms.

[0093] In another embodiment, the Orthogonal Matching Pursuit (OMP) algorithm is employed. OMP is a greedy algorithm that selects the basis function mode with the highest correlation to the current residual in each iteration and includes it in the support set. Then, the coefficient estimates are updated on the support set using least squares until the residual is reduced to below a preset error threshold or the size of the support set reaches a preset maximum sparsity.

[0094] In another embodiment, a Bayesian sparse learning algorithm is employed. Specifically, each element in the coefficient vector A is assigned an independent Gaussian prior distribution, and the variance of each prior distribution is determined by a hyperparameter. Independent control is employed. Bayesian inference of hyperparameters and coefficient vectors is performed using either maximum likelihood estimation (evidence approximation) of the second type or Markov chain Monte Carlo sampling (MCMC). In this method, the hyperparameters controlling sparsity are automatically determined during inference, eliminating the need for additional pre-set regularization parameters. A unique advantage of the Bayesian method is that it not only outputs point estimates of the coefficient vectors but also simultaneously outputs the posterior variance of the coefficients, which can be directly used for subsequent uncertainty quantification.

[0095] S3.3 Automatic determination of regularization parameters, specifically:

[0096] For using explicit regularization parameters The solution algorithm (such as LASSO) requires automatic determination of λ. In one embodiment, K-fold cross-validation is used to automatically determine λ. Divide the data from M sensors into a training set and a validation set, and use different alternatives... Values ​​(such as taking equally spaced sequences on a logarithmic scale) Under the condition that the coefficient vector is solved using the training set, and the reconstruction error is calculated on the validation set; the set with the smallest validation error is selected. The value is used as the final parameter.

[0097] In another embodiment, the L-curve method is used to determine... : Draw different The log-log curve of the data fidelity term and the sparse term corresponding to the value is selected, and the point of maximum curvature on the curve is chosen. The L-curve method does not require a validation set and can make full use of all sensor data.

[0098] For algorithms that employ Bayesian sparse learning, the hyperparameters controlling sparsity are automatically optimized and determined based on the data during the Bayesian inference process, without the need for additional independent parameter selection steps.

[0099] S3.4 Analysis of the sparsity characteristics of the solution results, specifically:

[0100] After the solution is completed, the optimal coefficient vector is obtained. Due to the effect of L1 regularization, The vast majority of elements have values ​​of zero (or absolute values ​​less than a preset threshold, which can be considered zero), with only a few elements having significantly non-zero absolute values. The basis function modes corresponding to these non-zero elements are the "dominant modes" of the stress state at the current moment. The number of dominant modes is usually 3 to 8, much smaller than the total number of modes K (e.g., K = 20), demonstrating the effectiveness of sparse constraints.

[0101] Step S4: The full-field stress distribution at the current moment is instantly reconstructed by the linear combination of the coefficient vector and the complete basis function matrix;

[0102] Specifically, this involves obtaining the dominant mode coefficient vector that characterizes the stress state at the current moment. Then, step S4 instantly reconstructs the stress distribution across the entire field through a simple matrix-vector multiplication operation, i.e., a linear combination:

[0103] (2)

[0104] in, The reconstructed N×1 dimensional full-field stress distribution vector contains the reconstructed stress value at each grid point on the wafer surface; Ψ is the N×K dimensional complete basis function matrix constructed and stored in step S1. The K×1 dimensional sparse coefficient vector obtained in step S3.

[0105] It should be noted that the full-field mechanical field distribution obtained from the reconstruction in step S4... The types of physical quantities are consistent with the historical data types used in extracting the basis function modes in step S1 and the types of observations collected by the sensor in step S2.

[0106] In a preferred embodiment, when the sensor collects out-of-plane displacement (warpage) of the wafer surface and the historical data in step S1 corresponds to displacement field data, the extracted basis function modes are displacement modes. In this case, the linear combination in step S4 directly outputs the full-field displacement distribution. Those skilled in the art know that a definite correspondence exists between the displacement field and the stress field through the elastic constitutive relationship of the wafer material (such as thin-plate bending theory). The reconstructed full-field displacement distribution can be further converted to a full-field stress distribution expressed in stress units (such as megapascals) according to actual engineering needs. This invention does not specifically limit this aspect.

[0107] In another embodiment, when the sensor directly or indirectly measures the stress component and the historical data in step S1 corresponds to the stress field data, the extracted basis function modes are stress modes, and at this time, the linear combination in step S4 directly outputs the full-field stress distribution.

[0108] Since the basis function matrix Ψ contains modal information for all spatial locations, while the coefficient vector... The number of non-zero elements is much smaller than the total number of modes K, with only a few coefficients corresponding to the dominant mode of the current stress state being significantly non-zero. This reconstruction operation can cover all grid points in the entire field, with minimal computational cost, meeting real-time requirements. In implementations using ordinary industrial computers, the time taken for a single reconstruction is typically on the order of milliseconds or less, perfectly matching the online monitoring cycle of the wafer thinning process.

[0109] Step S5: Based on the reconstructed full-field stress distribution and the quantitative information on the predicted uncertainty, perform process anomaly judgment and early warning;

[0110] Specifically, Figure 4 The logical flow of anomaly detection and early warning based on full-field stress distribution and uncertainty quantification information in step S5 is shown.

[0111] S5.1 Obtaining quantitative information on prediction uncertainty, specifically:

[0112] The uncertainty quantification information of the prediction can be obtained through at least one of the following methods:

[0113] Method 1, uncertainty estimation based on Bayesian posterior distribution: When step S3 is solved using the Bayesian sparse learning algorithm, the coefficient vector can be obtained simultaneously. The posterior probability distribution of the reconstructed field. At any spatial location i, its variance can be derived from... Calculation, where For vectors The i-th element, Let be the i-th row of the basis function matrix Ψ. Let be the posterior covariance matrix of the coefficient vector. The index of the coefficient vector A indicates that the covariance matrix belongs to the posterior covariance of the coefficient vector A. This yields a point-by-point uncertainty estimate (variance or confidence interval) of the reconstructed field at each grid point.

[0114] Method 2, based on the global uncertainty index of observation residuals: Under any sparse solution method, the residual modulus between the reconstructed observations and the actual observations can be calculated. Alternatively, the relative residual can be used as a global indicator reflecting the overall reliability of the current reconstruction. The smaller the residual modulus, the better the reconstruction result fits the observed data, and the higher the overall reliability of the reconstruction.

[0115] Method 3, based on statistical uncertainty estimation using multiple sampling: In step S2, the sensor undergoes multiple consecutive time-domain samplings (e.g., collecting observation vectors at T consecutive times within a thinning cycle). Steps S3 and S4 are executed independently for each step, yielding T sets of full-field reconstruction results. At each spatial location, the variance or range of the T reconstructed values ​​is calculated as a point-by-point uncertainty estimate for that location. This method is particularly effective in applications where the sensor itself exhibits significant measurement noise.

[0116] S5.2, Anomaly detection rules, specifically:

[0117] Based on the reconstructed full-field stress distribution Based on the aforementioned uncertainty quantification information, the following parallel decision logic is executed:

[0118] Stress anomaly detection: Traversing the stress distribution across the entire field For each grid point, it is determined whether there is a location where the absolute value of the stress exceeds a preset safety threshold. The specific value of the preset safety threshold depends on the mechanical properties of the wafer material (such as the fracture strength of silicon) and the requirements of the thinning process window. In one embodiment, the out-of-plane warpage safety threshold is set to ±50 μm, and the stress threshold is set to 70% of the material's yield strength. If an over-limit point exists, it is determined to be a stress anomaly.

[0119] Reconstruction confidence assessment: The acquired uncertainty quantification information is compared with a preset uncertainty threshold. In one embodiment, when using the Bayesian method, if the posterior standard deviation of any grid point exceeds a preset value (e.g., 15% of the maximum reconstructed value across the entire field), or if the confidence interval width is too large, the reconstruction confidence is deemed insufficient. When using the observation residual method, if the relative residual exceeds a preset proportion (e.g., 10%), the reconstruction confidence is deemed insufficient. When using the multiple sampling statistical method, if the pointwise variance at a certain location exceeds a preset threshold, the reconstruction confidence is also deemed insufficient.

[0120] It should be noted that insufficient reconstruction confidence does not necessarily mean that the stress state of the wafer is abnormal. Rather, it indicates that the current sensor observation data is insufficient to support a high-confidence full-field reconstruction judgment. This is an important online early warning signal, suggesting that the sensor status may need to be checked, the sampling frequency increased, or the process parameters adjusted.

[0121] S5.3, Early Warning, specifically:

[0122] When any of the above determination results is abnormal, an alarm action is triggered. Alarm actions include, but are not limited to: audible and visual alarm signals, displaying alarm information on the human-machine interface and highlighting the spatial location of the stress exceeding the limit or the area with excessive uncertainty.

[0123] Figure 5 The block diagram of the module structure of the online sensing system for full-field stress in wafer thinning process based on modal sparse reconstruction provided in an embodiment of the present invention is shown.

[0124] The system includes a basis function construction and storage module, a sparse sensing module, a sparse reconstruction calculation module, a full-field stress reconstruction module, and an anomaly detection and early warning module. Each module can be deployed in the data processing and computing unit.

[0125] The basis function construction and storage module is used to perform the operations in step S1 above, and completes the extraction of spatial basis function modes, the selection of dominant modes, and the construction and persistent storage of the basis function matrix Ψ in the offline stage.

[0126] The sparse sensing module, used to perform the operation in step S2 above, is connected to the sensor and data acquisition unit and is responsible for real-time acquisition of sparse stress observations during the online phase. .

[0127] The sparse reconstruction computation module, connected to the basis function construction and storage module and the sparse sensing module, is used to perform the operation in step S3 above, construct and solve the sparse reconstruction optimization problem equation (1), and obtain the optimal coefficient vector at the current time. It also outputs quantitative information on uncertainties during the reconstruction process.

[0128] The full-field stress reconstruction module, connected to the basis function construction and storage module and the sparse reconstruction calculation module, is used to perform the operation of step S4 above and calculate the complete full-field stress distribution at the current moment according to equation (2). .

[0129] The anomaly detection and early warning module is connected to the full-field stress reconstruction module and the sparse reconstruction calculation module. It is used to perform the operation of step S5 above, to detect anomalies based on the full-field stress distribution and uncertainty quantification information, and to issue alarms or process adjustment instructions according to the detection results.

[0130] The specific data flow between the above modules has been described. Figure 5 The functions of each module are indicated by arrows. The functions of each module can be implemented by computer program instructions to related hardware, and the computer program can be stored in a computer-readable storage medium.

[0131] As can be seen from the above embodiments, this invention, through its overall architecture combining offline library construction and online sensing, and the technical solution comprised of steps S1 to S5, achieves a closed-loop process from dimensionality reduction characterization of historical wafer stress field data, online sampling by sparse sensors, sparse reconstruction solution, real-time recovery of full-field stress, to uncertainty quantification and process risk early warning. This invention effectively overcomes key technical problems in traditional solutions, such as insufficient data collaborative utilization, lack of quantitative evaluation methods for reconstruction confidence, and difficulty in meeting process cycle time requirements for real-time online computation. It provides a technical solution with high precision, high real-time performance, and high reliability for online monitoring of stress state and adaptive process control during wafer thinning.

[0132] It should be understood that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any equivalent substitutions, modifications, and improvements made to some of the technical features within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction, characterized in that, Includes the following steps: S1. Extract spatial basis function modes from historical data of wafer stress field using intrinsic orthogonal decomposition, and construct basis function matrix; S2. During the wafer thinning process, a small number of sensors arranged on the wafer surface or stage are used to collect sparse stress observation values ​​in real time, forming an observation vector. S3. Based on the basis function matrix and the observation vector, construct and solve the sparse reconstruction optimization problem. The sparse reconstruction optimization problem takes minimizing the observation error as the data fidelity objective and the sparseness of the coefficient vector as the constraint objective, and obtains the optimal sparse coefficient vector. S4. The optimal sparse coefficient vector is linearly combined with the complete basis function matrix to reconstruct the full-field stress distribution at the current moment; S5. Based on the reconstructed full-field stress distribution and the quantitative information of prediction uncertainty generated during the reconstruction process, anomaly judgment and early warning are performed on the wafer thinning process.

2. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 1, characterized in that, The construction of the basis function matrix in step S1 specifically includes: Obtain the historical dataset of the wafer mechanical field obtained through finite element simulation or historical measurement. The mechanical field is a full-field distribution corresponding to the type of physical quantity collected by the sensor in step S2, covering various physical quantities that can directly or indirectly reflect the wafer stress state, including warp field, displacement field or stress field. The historical dataset is subjected to intrinsic orthogonal decomposition to obtain a set of spatial basis function modes and the energy contribution of each mode; Sort the modes by energy contribution from largest to smallest, retain the dominant modes whose cumulative energy contribution meets a preset threshold, and form the basis function matrix from all retained dominant modes.

3. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 1, characterized in that, The number of sensors deployed is less than the total number of grid points and less than the number of spatial basis function modes extracted in step S1.

4. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 3, characterized in that, The sensors include displacement sensors, optical profile sensors, piezoelectric sensors, or strain gauges; the collected observations include wafer warpage, out-of-plane displacement, or stress components.

5. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 1, characterized in that, Step S3 is as follows: Based on the spatial basis function matrix and the observation vector, construct and solve the following sparse reconstruction optimization problem: Obtain the optimal sparse coefficient vector ;in, For the observation vector, Let Ψ be the submatrix of the basis function matrix Ψ at the sensor location, and A be the vector of basis function coefficients to be solved. The adjustment parameter for sparse constraints, Denotes the square of the L2 norm. This represents the L1 norm.

6. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 5, characterized in that, The sparse reconstruction optimization problem can be solved using any one of the LASSO algorithm, orthogonal matching pursuit algorithm, or Bayesian sparse learning algorithm.

7. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 1, characterized in that, The uncertainty quantification information of the prediction mentioned in step S5 is obtained through at least one of the following methods: Pointwise uncertainty estimation based on Bayesian posterior distribution: When solving the problem using the Bayesian sparse learning algorithm, the posterior distribution of the basis function coefficient vector A is obtained simultaneously, and the variance or confidence interval of the reconstructed field at any spatial location is calculated accordingly. Global uncertainty index based on observation residuals: Calculate the observation vector With reconstructed sensor values residual modulus between Or relative residuals, serving as a quantitative indicator of the global uncertainty of the entire reconstructed field; Statistical uncertainty estimation based on multiple sampling: Multiple observation vectors are obtained by performing multiple time-domain samplings on the sensor, and each vector is solved and reconstructed independently. The statistical discrete measure of the reconstructed value at each spatial location is calculated as a point-by-point uncertainty estimate.

8. The method for online sensing of full-field stress in wafer thinning process based on modal sparse reconstruction according to claim 7, characterized in that, The process anomaly determination and early warning in step S5 includes: When the warping, displacement, or stress value in the reconstructed full-field stress distribution exceeds the preset safety threshold, it is determined to be a stress anomaly. When the quantified uncertainty information of the prediction exceeds a preset uncertainty threshold, it is determined that the reconstruction confidence is insufficient. When an abnormality is detected, an alarm is triggered or a command to adjust the thinning process parameters is output.

9. A full-field online stress sensing system for wafer thinning process based on modal sparse reconstruction, implementing the method as described in any one of claims 1-8, characterized in that, include: The basis function construction and storage module is used to extract spatial basis function modes from historical data of wafer stress field using intrinsic orthogonal decomposition, and to construct and store the basis function matrix. A sparse sensing module is used to collect sparse stress observation values ​​in real time through a small number of deployed sensors during the wafer thinning process, and to construct an observation vector from the sparse stress observation values. The sparse reconstruction calculation module, connected to the basis function construction and storage module and the sparse sensing module, is used to construct and solve the sparse reconstruction optimization problem with the constraint of minimizing the error between the linear combination of the observation vector and the submatrices of the basis function matrix at the sensor position and the sparsity of the coefficient vector to be solved, so as to obtain the optimal dominant mode coefficient vector. The full-field stress reconstruction module, connected to the basis function construction and storage module and the sparse reconstruction calculation module, is used to linearly combine the optimal dominant mode coefficient vector with the complete basis function matrix to reconstruct the full-field stress distribution at the current moment. The anomaly detection and early warning module is connected to the full-field stress reconstruction module and the sparse reconstruction calculation module. It is used to detect and warn of anomalies in the wafer thinning process based on the reconstructed full-field stress distribution and the quantitative information of prediction uncertainty generated during the reconstruction process.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the method as described in any one of claims 1-8.

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