Semiconductor manufacturing process parameter optimization method based on bayesian average kriging and evidence theory

Abstract

A method of semiconductor manufacturing process parameter optimization based on Bayesian average Kriging and evidence theory is proposed. By introducing Bayesian model averaging into the basic Kriging model, different variance functions are adaptively weighted, which effectively improves the robust estimation of spatial correlation structure and alleviates the bias caused by model mis-specification under small sample conditions. At the same time, the introduction of evidence theory not only realizes the fusion of multi-source model prediction results, but also explicitly describes the cognitive uncertainty at the model level, enhancing the interpretability of the prediction results and the value of process diagnosis. The method shows good applicability in high-cost and data-scarce manufacturing systems, and can provide reliable proxy modeling support for complex processes such as semiconductor etching.

Description

Technical Field

[0001] This invention belongs to the field of semiconductor manufacturing and process parameter optimization, specifically relating to a method for optimizing semiconductor manufacturing process parameters based on Bayesian average kriging and evidence theory. Background Technology

[0002] Semiconductor etching, a crucial step in micro / nano manufacturing, directly impacts device linewidth control, sidewall morphology, and structural consistency, making it a core step determining integrated circuit performance and yield. As device feature sizes continue to shrink to deep submicron and even nanometer scales, the number of parameters in advanced etching processes has increased significantly, including RF power, gas flow rate, chamber pressure, and pulse timing. Simultaneously, the etching process is often accompanied by complex interactions of physical and chemical mechanisms, exhibiting strong nonlinear behavior and multi-parameter coupling characteristics. Furthermore, the preparation of test samples for process development in semiconductor manufacturing is time-consuming and costly, resulting in typically limited available experimental data. Therefore, effectively modeling and optimizing etching process parameters under small sample conditions has become a key challenge for improving process control accuracy and R&D efficiency.

[0003] To address this challenge efficiently and effectively, researchers have gradually shifted their focus from physical experiments and computer simulations to data-driven modeling strategies. Among these, surrogate models, as an efficient mathematical approximation method, construct a mapping relationship between input parameters and output responses based on limited simulation or experimental data. This significantly reduces computational overhead while supporting process sensitivity analysis, parameter optimization, and uncertainty assessment. In recent years, surrogate model technology has made significant progress, giving rise to various classic surrogate model methods, including Response Surface Methodology (RSM), Kriging, Radial Basis Function (RBF), Support Vector Regression (SVR), and Inverse Distance Weighting (IDW).

[0004] Among numerous surrogate models, the Kriging model has received widespread attention in recent years due to its combination of accurate interpolation capabilities and strong fitting characteristics to local nonlinear relationships. As a classic geostatistical method, its various implementations, such as simple Kriging, ordinary Kriging, generalized Kriging, and co-Kriging, are widely used in engineering optimization and reliability analysis. To further improve the robustness and predictive reliability of the model in handling complex engineering problems, researchers have proposed many improvement and optimization methods to address challenges such as model uncertainty quantification, small sample learning efficiency, and high-dimensional nonlinear fitting. Cai et al. proposed in "Cai X, Qiu H, Gao L, et al. Metamodeling for high dimensional design problems by multi-fidelity simulations[J]. Structural and multidisciplinary optimization, 2017, 56(1): 151-166." that by integrating the Cut-HDMR decomposition strategy and co-Kriging, a multi-fidelity surrogate model was constructed, and combined with an improved sampling method, the modeling efficiency of multi-source data in high-dimensional complex problems was effectively improved. In their paper "Wu P, Li Y. Adaptive kriging model-based structural reliability analysis under interval uncertainty with incomplete data[J]. Structural and Multidisciplinary Optimization, 2023, 66(1): 22.), Wu et al. proposed a dual-model method based on adaptive kriging (AK). By constructing two cascaded AK models and introducing an improved IH learning function, the time-consuming two-layer Monte Carlo decomposition is decomposed into efficient surrogate model iterations, which significantly improves the efficiency and confidence of structural reliability analysis under incomplete input information.In their paper "Bai J, Hu B, Xue Z. EMC uncertainty simulation method based on improved Kriging model[J]. IEEE Letters on Electromagnetic Compatibility Practice and Applications, 2023, 5(4): 127-130.", Bai et al. proposed introducing an active sampling strategy based on a stochastic reduced-order model in Kriging modeling, which overcomes the computational redundancy and accuracy fluctuation problems caused by traditional passive Latin hypercube sampling in electromagnetic compatibility analysis. In their paper "Nan H, Liang H, Di H, et al. A gradient-assisted learning strategy of Kriging model for robust design optimization[J]. Reliability Engineering & System Safety, 2024, 244: 109944.", Nan et al. designed a novel learning function that integrates gradient information, prediction uncertainty, and spatial distance, and combined it with a gradient-guided stopping criterion, effectively improving the search efficiency for robust optimal solutions in highly nonlinear problems. Furthermore, in his paper "Zhang Sen. Research on Temperature Compensation Method for Conversion Force Based on Kriging Interpolation [D]. Dalian University of Technology, 2023," Zhang Sen employed an adaptive mutated chaotic particle swarm optimization algorithm to optimize the range and smoothing parameters of the Kriging model, constructing a high-precision temperature compensation model that successfully suppressed the nonlinear output deviation of the switch machine sensor caused by temperature drift. However, current research on the Kriging model is mostly based on a single model framework or a deterministic weighted fusion strategy, focusing on sampling optimization and local accuracy improvement, while paying insufficient attention to the explicit modeling and dynamic fusion mechanism of cognitive uncertainty between multi-source Kriging models. This makes it difficult to apply to modeling and learning under small sample conditions, such as the optimization of semiconductor manufacturing process parameters. Summary of the Invention

[0005] To address the problems or shortcomings of existing technologies, this invention aims to provide a semiconductor manufacturing process parameter optimization method based on Bayesian average kriging and evidence theory. This method introduces Bayesian model averaging into the basic kriging model, adaptively weighting different variogram functions to effectively improve robust estimation of spatially correlated structures and mitigate biases caused by model misspecification under small sample conditions. Simultaneously, the introduction of evidence theory not only achieves the fusion of prediction results from multiple models but also explicitly characterizes cognitive uncertainty at the model level, enhancing the interpretability and process diagnostic value of the prediction results. This method demonstrates good applicability in high-cost, data-scarce manufacturing systems, providing reliable surrogate modeling support for complex processes such as semiconductor etching, and achieving rapid and accurate parameter optimization under small sample conditions.

[0006] The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory includes the following steps: Step 1: Based on the semiconductor manufacturing process, determine the key input parameters and output response indicators that affect the process; Step 2: Using a full factorial experimental design method, sample within the space of key process input parameters and obtain the corresponding output response data through simulation software to construct an input-output dataset; Step 3: Based on the input and output datasets from Step 2, construct a basic Kriging proxy model based on Bayesian model averaging and evidence theory. After optimization, perform cross-model fusion to output the final predicted value and the expected upper and lower bounds of the predicted value; that is, realize the quality prediction part of semiconductor manufacturing process parameter optimization.

[0007] Step 3 is implemented as follows: Step 3.1, BMA optimization of the variation function of the basic kriging model: The basic kriging includes three types: simple kriging, ordinary kriging, and generalized kriging. Bayesian model averaging is performed independently within each basic kriging model, that is, three variation functions are fitted to each type of kriging: Gaussian function, exponential function, and Matérn function. The dataset consisting of the process parameters and response data in Step 2 is used as the input to the three basic kriging models. Finally, each of the three basic kriging models outputs a BMA-optimized predicted value and a calculated residual, which serves as the input source for DS evidence fusion in Step 3.2. Step 3.2, DS Evidence Theory Fusion Across Kriging Types: Based on evidence theory, the three optimized Kriging models obtained in Step 3.1 are fused across models. Specifically, first, basic probability assignments are constructed based on model residuals; then, the Yager combination rule is used to sequentially fuse the probability assignments of simple Kriging, ordinary Kriging, and generalized Kriging to obtain a comprehensive probability assignment; based on the comprehensive probability assignment, the belief degree and likelihood are calculated, and dynamic weights of each model are constructed to generate the final predicted value and the upper and lower bounds of the expected predicted value. The final predicted value represents the nonlinear relationship between the input parameters and the output response, and the upper and lower bounds of the expected predicted value are used to quantify the uncertainty information of the prediction results. The smaller the error between the predicted and actual values, the better the constructed Kriging proxy model based on Bayesian model averaging and evidence theory performs. The difference between the upper bound and the lower bound of the expected value represents the degree of uncertainty of the sample point at that point.

[0008] The process involves fitting three variogram functions to each type of Kriging: Gaussian function fitting... , Exponential function fitting Matérn function fitting The fitting results for the three functions are as follows: (twenty two) (twenty three) (twenty four) in, For the nugget effect, For the partial sill value, For variable range, parameters Fitting the empirical variation function using the nonlinear least squares method To better reflect the characteristics of real-world etching physical systems, the nu value of the Matérn function is set to 1.5. h It is the independent variable, representing the distance between two points.

[0009] The specific implementation method for constructing the basic probability assignment based on model residuals is as follows: Set up a recognition framework Let Kriging represent the mutually exclusive and exhaustive set of propositions consisting of three candidate Krigings, each model... Each stage of processing yields a predicted value. and a residual index reflecting its internal goodness of fit. These two parameters will be used as input for step 3.2; We employ a basic probability assignment construction method based on the inverse of the error, and let the set of effective models be... First, calculate the inverse residual weight: (1) Normalization yields: (2) m' represents the temporary variable that is taken sequentially through all models in set M during the summation of the denominator; Assign trust to a subset of single elements, while introducing unknowns to characterize uncertainties that are not fully represented: (3) (4) in, express The basic probability distribution, Preset unknown weights are used to allocate a portion of the confidence score to the entire set. This indicates a conservative judgment that there is no complete trust in any Kriging model.

[0010] The comprehensive probability allocation is specifically as follows: Treating the three types of Kriging models as three independent sources of evidence, their corresponding basic probability assignments are as follows: The overall probability allocation is synthesized stepwise using a fixed fusion order of simple kriging, ordinary kriging, and pan-kriging, with the initial fusion result as follows: (5) The second step is to integrate ordinary Kriging evidence: (6) The third step involves fusing pan-Kriging evidence to ultimately obtain the comprehensive probability allocation after fusion of the three sources: (7) in, Let Yager be the combinatorial operator, defined as follows: for any proposition... and : (8) For the complete series : (9).

[0011] The calculation of belief degree and likelihood degree based on comprehensive probability allocation is specifically as follows: According to DS theory, for any proposition Its belief function With likelihood function Defined as: (10) (11).

[0012] The specific implementation method for constructing dynamic weights for each model to generate the final predicted value and the expected upper and lower bounds of the predicted value is as follows: Using a dynamically weighted average based on local consistency as the final output, for the effective model set M, the deviation between its predicted values ​​and the mean of the training set is calculated: (12) And construct the weights: (13) The final predicted value is: (14) Likelihood With belief These represent the upper and lower bounds of the expected value, respectively.

[0013] Calculate the root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) of the basic Kriging surrogate model constructed in step 3. 2 This is used to evaluate the model's prediction accuracy and generalization ability on simulation data. The specific implementation method is as follows: Root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) 2 The calculation formulas are as follows: (15) (16) (17) in, Let i be the true value of the i-th sample. These are the model's predicted values. The average of the true values; RMSE reflects the overall degree of deviation of the predicted values; MAE measures the average error level; R 2 RSE indicates the model's ability to explain the variability of the target variable. A smaller RMSE indicates less bias in the model's predictions, while a smaller MAE indicates better generalization ability. 2 The closer the value is to 1, the better the fit.

[0014] The uncertainty quantification information of the prediction results of the optimized base Kriging surrogate model in step 3 is calculated by quantifying the difference between the likelihood and belief for each sample point. The model uncertainty interval is obtained, which reflects the degree of conflict between pieces of evidence and the level of uncertainty in the model when fusing multi-source information; the specific implementation method is as follows: Build a recognition framework , recorded as In the recognition framework Above, define the basic probability assignment function: (18) function value This is known as the basic reliability of set A. It represents the overall uncertainty or unknown information regarding all fundamental propositions; This represents three different propositions.

[0015] Based on the basic probability assignment, the trust function is derived. With likelihood function : (19) (20) Indicates the proposition A Overall level of trust Indicates the proposition A The maximum possible support; and Together they form a confidence interval Its length This reflects the degree of uncertainty regarding proposition A. The larger the value, the greater the evidence conflict and the higher the uncertainty of the model at that point.

[0016] The present invention also includes: A system including a processor capable of running the semiconductor manufacturing process parameter optimization method.

[0017] An apparatus comprising: a memory for storing a computer program for the semiconductor manufacturing process parameter optimization method; Processor: Used to implement the semiconductor manufacturing process parameter optimization method when executing the computer program.

[0018] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the semiconductor manufacturing process parameter optimization method.

[0019] In summary, the present invention has at least the following beneficial technical effects: 1. In step 3.1, this invention introduces Bayesian model averaging into the three basic Kriging models to achieve robustness of variation function selection and adaptive modeling of spatially related structures, effectively improving robust estimation of spatially related structures and mitigating bias caused by model misspecification under small sample conditions.

[0020] 2. In step 3.2, this invention performs cross-model fusion of three types of optimized Kriging models based on evidence theory. Through basic probability allocation and evidence synthesis mechanisms, it outputs the expected upper and lower bounds of the model prediction values ​​and the final prediction values. This not only realizes the fusion of prediction results from multiple sources, but also explicitly characterizes the cognitive uncertainty at the model level, enhancing the interpretability of the prediction results and the value of process diagnosis.

[0021] In summary, this invention is applicable to both simulation data and actual experimental data, and can be extended to the optimization of other semiconductor manufacturing processes. It demonstrates good applicability in high-cost, data-scarce manufacturing systems and can provide reliable proxy modeling support for complex processes such as semiconductor etching. Attached Figure Description

[0022] Figure 1 is a flowchart of the BMA optimization process for the basic Kriging model.

[0023] Figure 2 is a flowchart of the DS evidence theory fusion process across Kriging types.

[0024] Figure 3. Structure diagram of the BMA-DSK model.

[0025] Figure 4(a) is a scatter plot of the BMA-DSK model's predictions for the simulation data; Figure 4(b) is a scatter plot of the SK model's predictions for the simulation data; Figure 4(c) is a scatter plot of the OK model's predictions for the simulation data; Figure 4(d) is a scatter plot of the UK model's predictions for the simulation data; Figure 4(e) is a scatter plot of the GPR model's predictions for the simulation data. Figure 4(f) is a scatter plot of the SVR model's predictions for the simulation data.

[0026] Figure 5 shows the prediction results of each model for the simulation data.

[0027] Figure 6 shows the uncertainty interval distribution of BMA-DSK for simulation data. Detailed Implementation

[0028] The specific embodiments of the present invention will be fully described below with reference to the accompanying drawings and examples. The embodiments described below are only individual applications of the present invention, and not all applications, and should not be used to limit or define the scope of protection of the present invention.

[0029] This invention first optimizes semiconductor manufacturing process parameters through a systematic process: Based on clearly defined key input parameters and output response indicators, a full factorial experimental design method is used to sample within the parameter space, and corresponding response data is obtained using simulation tools to construct a complete input-output dataset. Based on this dataset, Bayesian model averaging (BMA) weighted fusion is independently applied to three basic models—simple kriging, ordinary kriging, and generalized kriging—to generate their respective optimal predicted values ​​and residuals, which serve as multi-source evidence input. Subsequently, Dempster-Shafer (DS) evidence theory is introduced. Cross-model fusion is performed on the prediction results of three types of Kriging models optimized by BMA. Through basic probability allocation and evidence synthesis mechanisms, the expected value of the prediction with upper and lower bounds and the final integrated prediction result are output. On this basis, the root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination R² of the surrogate model are calculated to evaluate its fitting accuracy and generalization ability. Furthermore, uncertainty quantification analysis is carried out to characterize the degree of conflict and ambiguity of the model in the multi-source information fusion process by the difference between likelihood and belief, so as to provide prediction support with both high accuracy and reliability for subsequent robust process decisions.

[0030] The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory includes the following steps: Step 1: Based on the semiconductor manufacturing process, determine the key input parameters and output response indicators that affect the process; Definition of process parameters and response indicators: In this embodiment, plasma etching process is used to process the semiconductor wafer surface. The key input parameters affecting the process are two-dimensional variables, including chamber gas pressure and radio frequency power; the output response indicator is the etching rate.

[0031] Step 2: Using a full factorial experimental design method, sample within the space of key process input parameters and obtain the corresponding output response data through simulation software to construct an input-output dataset; This invention uses CFD-based plasma etching simulation data for model verification. The plasma chamber discharge process is numerically simulated using the software CFD-ACE+. This simulation yields key parameters such as ion flux distribution, energy distribution, incident angle distribution, and neutral particle flux distribution at several specified locations on the semiconductor wafer surface. In the micro-etching process simulation, the CFD-TOPO module receives and utilizes the calculation results from CFD-ACE+, employing the level set method to track the evolution of the etched surface, thereby obtaining the dynamically changing microstructure over time.

[0032] The chamber pressure and radio frequency power were set to seven levels, namely (10, 15, 20, 25, 30, 35, 40) mTorr and (1500, 2000, 2500, 3000, 3500, 4000, 4500) W. A total of 49 sample points were selected using a full factorial experimental design. The specific values ​​are shown in Table 1.

[0033] Table 1. Plasma Etching Simulation Dataset Serial Number Chamber pressure (mT) Radio frequency power (W) Simulated etching rate (nm / min) 1 10 1500 73.03 2 10 2000 81.23 3 10 2500 91.75 4 10 3000 98.6 5 10 3500 100.89 6 10 4000 92.06 7 10 4500 87.74 8 15 1500 82.45 9 15 2000 94.12 10 15 2500 106.06 11 15 3000 120.51 12 15 3500 99.31 13 15 4000 93.31 14 15 4500 93.75 15 20 1500 83.57 16 20 2000 95.24 17 20 2500 107.78 18 20 3000 110.52 19 20 3500 110.4 20 20 4000 100.35 21 20 4500 84.86 22 25 1500 76.46 23 25 2000 99.31 24 25 2500 109.06 25 25 3000 116.6 26 25 3500 114.4 27 25 4000 103.93 28 25 4500 91.28 29 30 1500 74.19 30 30 2000 98.8 31 30 2500 102.51 32 30 3000 116.36 33 30 3500 112.77 34 30 4000 102.68 35 30 4500 83.72 36 35 1500 74.13 37 35 2000 99.15 38 35 2500 117.21 39 35 3000 114.85 40 35 3500 108.61 41 35 4000 92.01 42 35 4500 72.86 43 40 1500 71.66 44 40 2000 96.53 45 40 2500 115.2 46 40 3000 111.03 47 40 3500 105.51 48 40 4000 87.88 49 40 4500 68.09 Step 3: Based on the input and output datasets constructed in Step 2, construct a basic Kriging agent model based on Bayesian model averaging and evidence theory. After optimization, perform cross-model fusion to output the final predicted value of the model and the expected upper and lower bounds of the predicted value.

[0034] This method employs a two-stage fusion strategy: In the first stage, Bayesian model averaging is introduced into the three basic kriging models to achieve robustness in variogram selection and adaptive modeling of spatial correlation structure. In the second stage, cross-model fusion is performed on the three optimized kriging models based on evidence theory. Through basic probability assignment and evidence synthesis mechanisms, the expected upper and lower bounds of the model predictions and the final predictions are output. Through this two-stage fusion mechanism, a comprehensive surrogate model with both high-precision prediction capabilities and interpretable uncertainty is ultimately constructed. Steps 3.1 and 3.2 will be discussed in detail from the perspectives of BMA optimization and the DS fusion framework, respectively.

[0035] Step 3.1: BMA optimization of the variogram function in the basic kriging model. In spatial modeling, the choice of variogram function has a decisive impact on the accuracy of kriging predictions. However, under small sample conditions, empirical variogram function estimation has significant uncertainty, and choosing a single variogram function model can easily lead to overfitting or prediction bias. To address this, this invention introduces Bayesian model averaging within the basic kriging model. The basic kriging model includes three types: simple kriging, ordinary kriging, and generalized kriging. Bayesian model averaging (BMA) is performed independently within each basic kriging model, that is, three variogram functions are fitted to each type of kriging: Gaussian function, exponential function, and Matérn function. Finally, each of the three basic kriging models outputs a BMA-optimized prediction value and a calculated residual, which serves as the input source for the DS evidence fusion in the subsequent step 3.2. Figure 1 The flowchart of the BMA optimization process for the basic Kriging model is shown, which can be summarized in four steps: (1) Data preparation. Load and standardize spatial data.

[0036] (2) Empirical variation function estimation. The distance between paired points and the corresponding empirical variation value are calculated based on the observation data.

[0037] (3) Candidate model fitting. Select a candidate variogram model, fit the model parameters and output the prediction results.

[0038] (4) Weighting and weighted prediction. Calculate the weight coefficient of each model based on the sum of squared residuals of each candidate model, and then average the prediction results of each model according to their weights to obtain the final BMA prediction value and equivalent residual, and output them.

[0039] The following are the detailed mathematical implementation steps.

[0040] First, the empirical variation function is calculated using the Cressie weight estimator. : (twenty one) Where h represents the spatial distance interval. For spatial sample pairs within the distance interval h Quantity, and and represent the target variable values ​​of two observation points located within a distance h. This estimator suppresses the influence of outliers through fourth-order moment normalization, making it more robust than the classic Matheron estimation.

[0041] This invention selects three variogram models—Gaussian model, exponential model, and Matérn model—as candidate models. The three models are chosen for their favorable theoretical properties and practical adaptability in spatial modeling: the Gaussian model is suitable for highly smooth spatial processes, the exponential model characterizes moderate spatial dependence, and the Matérn model provides differentiability between the two, more realistically reflecting the continuous but not infinitely smooth characteristics of physical fields. Together, they cover the spectrum of common spatially correlated structures, and all three have positive definite covariance matrices in any dimension, exhibiting numerical stability. The fitting results of the three candidate models are as follows: (twenty two) (twenty three) (twenty four) in, For the nugget effect, For the partial sill value, For variable range, parameters Fitting the empirical variation function using the nonlinear least squares method To better reflect the characteristics of real etching physical systems, the nu value of the Matérn model in this invention is set to 1.5.

[0042] For each candidate model Construct the corresponding Kriging model and predict test points. The predicted mean is Prediction variance An uncertainty-guided mean contraction mechanism is introduced to suppress overfitting in high-variance regions, resulting in adjusted predicted values. for: (25) (26) in, This is the contraction coefficient, which decreases as prediction uncertainty increases. The smaller the parameter is, the better. The stronger the contraction.

[0043] Define Model The sum of squared residuals from the fit is: (27) Where H is the total number of distance intervals. Let be the center value of the i-th distance interval. In distance The empirical variation function value at that location, For the model In distance The theoretical value of the variation function at that point. Therefore, the model... The weighting coefficients are: (28) Where K is the number of candidate models. To prevent small constants from being divided by zero, this weight reflects the model. The smaller the residual, the higher the weight given to the ability to characterize spatial structure. The final BMA optimization prediction value is calculated as follows: (29) To support the reliability assessment in subsequent evidence theory, the equivalent residuals of the Kriging global best-fit variation function are recorded: (30) The smaller the value, the better the fit of the Kriging internal variation function, and the more reliable the model. After introducing BMA into the three types of basic Kriging through the above process, three optimized prediction values ​​are finally output. and three model residual indices These outputs will serve as the basic input source for the evidence theory fusion stage, enabling uncertainty integration across models.

[0044] Step 3.2, Dempster-Shafer evidence theory fusion across kriging types. After completing the average optimization of Bayesian models based on the set of variation functions within the three basic kriging types, this paper adopts Dempster-Shafer evidence theory to achieve cross-model fusion in order to further integrate information from heterogeneous spatial modeling strategies and improve prediction robustness and uncertainty quantification capabilities. This method does not rely on the mutually exclusive completeness assumption in traditional probability theory, allows for the assignment of quality to "unknown" or "uncertain" elements, and is particularly suitable for multi-source heterogeneous model integration scenarios. The flowchart of the cross-kriging type DS evidence theory fusion proposed in this section is as follows: Figure 2 As shown, the process can be summarized in four steps: 1) Obtain input data. The predicted values ​​and fitting residuals of the three types of Kriging models after BMA optimization are used as the input evidence sources for DS.

[0045] 2) Constructing basic probability assignments. Basic probability assignments are constructed based on the model residuals.

[0046] 3) Evidence fusion. The Yager combination rule is used to sequentially fuse the three types of Kriging probability assignments to obtain a comprehensive probability assignment.

[0047] 4) Dynamic weighted average. Belief degree and likelihood are calculated based on comprehensive probability allocation, and dynamic weights for each model are constructed to generate the final predicted value.

[0048] The following are the detailed mathematical implementation steps.

[0049] Set up a recognition framework Let represent the mutually exclusive and exhaustive set of propositions formed by the three candidate Krigings. Each model Each stage of processing yields a predicted value. and a residual index reflecting its internal goodness of fit. These two parameters will serve as inputs for the second stage.

[0050] Based on this, the present invention employs a basic probability allocation construction method based on the inverse of error, allowing the effective model set to be... First, calculate the inverse residual weight: (31) Normalization yields: (32) The confidence level is then assigned to a subset of single elements, while an unknown is introduced to characterize the uncertainty that is not fully represented: (33) (34) in, Preset unknown weights are used to allocate a portion of the confidence score to the entire set. This indicates a conservative judgment that there is no complete trust in any Kriging model.

[0051] Traditional Dempster's combinatorial rule can lead to counterintuitive results when there is a high degree of conflict in the evidence. This invention uses an improved combinatorial rule proposed by Yager for evidence fusion. Its core idea is to directly assign all conflict quality to the entire set. Instead of renormalizing, this enhances the system's robustness to model inconsistencies. The three types of Kriging models are treated as three independent sources of evidence, with their corresponding basic probability assignments being... The overall probability allocation is synthesized stepwise using a fixed fusion order of simple kriging, ordinary kriging, and generalized kriging. Let the initial fusion result be: (35) The second step is to integrate ordinary Kriging evidence: (36) The third step involves fusing pan-Kriging evidence to ultimately obtain the comprehensive probability allocation after fusion of the three sources: (37) in, Let Yager be the combinatorial operator, defined as follows: for any proposition... and : (38) For the complete series : (39) According to DS theory, for any proposition Its belief function and likelihood function are defined as follows: (40) (41) This paper uses a dynamic weighted average based on local consistency as the final output. For the effective model set M, the deviation between its predicted values ​​and the mean of the training set is calculated: (42) And construct the weights: (43) The final predicted value is: (44) Through the above two-stage integration strategy, a Kriging model integrating Bayesian model averaging and evidence theory is finally formed. The specific structure and engineering application scheme of the proposed BMA-DSK model are as follows: Figure 3As shown, this model collects process parameters and electron microscopy observation data based on experimental design. It improves the performance of individual models through BMA optimization and then achieves cross-model fusion using DS evidence theory, outputting prediction results with high accuracy, strong robustness, and clear uncertainty characterization. This model can effectively address the nonlinearity, strong coupling, and multi-source uncertainty problems in semiconductor etching processes, providing reliable data support and intelligent decision-making tools for subsequent process parameter optimization design.

[0052] The smaller the error between the predicted and actual values, the better the constructed Kriging proxy model based on Bayesian model averaging and evidence theory performs. The difference between the upper bound and the lower bound of the expected predicted value represents the degree of uncertainty of the sample point at that point.

[0053] To better illustrate the superiority of the proposed method, three basic kriging models and two other mainstream surrogate models were tested, and the predictive performance of each model was compared and analyzed. The five comparison models are as follows: 1) Simple Kriging (SK). Simple Kriging is a Kriging interpolation method that assumes a known and constant global mean, and is suitable for modeling stationary random fields. It obtains the optimal linear unbiased prediction by minimizing the estimated variance, and has good interpolation performance when spatial correlation is strong and the prior mean is reliable.

[0054] 2) Ordinary Kriging (OK). Ordinary Kriging relaxes the assumption of known global mean in simple Kriging, requiring only that the mean is constant within a local region. By estimating the mean as an unknown constant during the prediction process, it is more suitable for situations in practical engineering where the mean is unknown but spatially stationary. It is one of the most commonly used Kriging variants in geostatistics.

[0055] 3) Universal Kriging (UK). Universal Kriging further generalizes from ordinary Kriging, allowing the trend term to be explicitly modeled for spatial nonstationarity, meaning the mean of the response variable can systematically drift with changes in input parameters. This method is more flexible and adaptable in modeling process data with obvious trends or drifts.

[0056] 4) Gaussian Process Regression (GPR). GPR is a nonparametric regression method based on a Bayesian framework, suitable for modeling uncertainty and learning nonlinear relationships. This method can naturally provide prediction intervals and give an estimate of the uncertainty of the predicted values.

[0057] 5) Support Vector Regression (SVR). Support Vector Regression is a supervised learning algorithm that evolved from Support Vector Machines. SVR attempts to find an optimal hyperplane that minimizes the distance of all training samples to this hyperplane while controlling the error term within a set threshold.

[0058] To accommodate small sample training and ensure fairness in experimental comparisons, all methods employ a leave-one-out cross-validation (LOOCV) training strategy. Specifically, for a dataset containing n samples, each sample is used as a test sample, and the remaining n-1 samples constitute the training set. This process is repeated n times to ensure that each sample is predicted once and a predicted value is output.

[0059] Figures 4(a)-4(f) are scatter plots of sample predictions for the proposed model and five comparative surrogate models, respectively. As can be seen from the figures, the proposed BMA-CDS model has the smallest residual.

[0060] Step 4: Calculate the root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) of the basic Kriging surrogate model constructed in Step 3. 2 This is used to evaluate the model's prediction accuracy and generalization ability on simulation data.

[0061] To comprehensively evaluate the prediction accuracy and generalization ability of each model on the simulation data, three evaluation metrics were set: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination R. 2 .

[0062] Root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) 2 The calculation formulas are as follows: (45) (46) (47) in, Let i be the true value of the i-th sample. These are the model's predicted values. The mean of the true values; RMSE is sensitive to large errors and reflects the overall degree of deviation of the predicted values; MAE measures the average error level and has good robustness; R 2RSE indicates the model's ability to explain the variability of the target variable. A smaller RMSE indicates less bias in the model's predictions, while a smaller MAE indicates better generalization ability. 2 The closer the value is to 1, the better the fit.

[0063] Table 2 and Figure 5 The paper details the evaluation metrics of each model on the simulation data across three metrics. The comparison shows that the proposed BMA-DSK model achieves the best results in all three metrics. Specifically, its RMSE is reduced to 4.269, approximately 15.0% lower than the second-best model UK, indicating stronger suppression of extreme errors; its MAE is only 3.131, further demonstrating its advantage in global prediction consistency; and R... 2 A value of 0.907 indicates that the model effectively captures the complex nonlinear relationship between input and output, demonstrating good explanatory and predictive capabilities. In contrast, while the GP and SVR models possess strong nonlinear fitting capabilities, their generalization performance deteriorates in this simulation scenario due to a higher risk of overfitting. Therefore, the BMA-DSK model exhibits significant advantages in accuracy, stability, and fitting ability, providing strong support for high-precision modeling.

[0064] Table 2. Prediction and evaluation metrics of each model for simulation data Model RMSE MAE R2 BMA-DSK 4.269 3.131 0.907 UK 5.019 3.913 0.871 SK 4.892 3.811 0.877 OK 4.977 3.873 0.873 GPR 6.800 5.398 0.763 SVR 6.046 4.511 0.813 Step 5: Calculate the uncertainty quantification information of the prediction results of the basic Kriging surrogate model constructed in Step 3, that is, by calculating the difference between the likelihood and the belief of each sample point. The uncertainty interval of the model is obtained. The value of the uncertainty interval reflects the degree of conflict between the evidence and the level of uncertainty of the model when integrating multi-source information.

[0065] Based on the DS evidence theory, the proposed model BMA-DSK can directly output quantitative information on the uncertainty of the prediction results. The uncertainty interval of the model is obtained by calculating the difference between the likelihood and the belief of each sample point. This value reflects the degree of conflict between the evidence and the level of ambiguity of the model when fusing multi-source information.

[0066] The foundation of evidence theory is the construction of an identification framework, denoted as... This is a finite set of all mutually exclusive and complete basic propositions, representing an exhaustive description of all possible answers to a given problem, where only one proposition is true.

[0067] In the identification framework Above, define the basic probability assignment function: (48) function value The fundamental reliability of set A is specifically referred to as the reliability of set A. It represents the overall uncertainty or unknown information regarding all fundamental propositions.

[0068] Based on the basic probability assignment, the trust function is derived. With likelihood function : (49) (50) Indicates the proposition A Overall level of trust Indicates the proposition A The maximum possible support. and Together they form a confidence interval Its length This value reflects the degree of uncertainty regarding proposition A, indicating the level of conflict between pieces of evidence and the ambiguity of the model when fusing multi-source information. The larger the value, the greater the evidence conflict and the higher the uncertainty of the model at that point.

[0069] like Figure 6 As shown, the uncertainty range for most sample points is small, indicating that the model has high confidence in these areas. However, at sample points #7, #43, and #49, the uncertainty increases significantly, indicating that there is a large discrepancy in the model outputs from different evidence sources, leading to a decrease in the credibility of the overall prediction result. This mechanism enables the model to have self-diagnostic capabilities, effectively identifying high-risk prediction scenarios and providing reliable support for subsequent decision-making.

[0070] Through steps 4 and 5, this invention verifies that the proposed BMA-DSK surrogate model has higher prediction accuracy and stronger generalization ability, and can effectively quantify cognitive uncertainty, providing a new modeling paradigm with high accuracy, strong robustness and interpretability for high-cost, small-sample semiconductor manufacturing processes.

[0071] The present invention also includes: A system including a processor capable of running the semiconductor manufacturing process parameter optimization method.

[0072] An apparatus comprising: a memory for storing a computer program for the semiconductor manufacturing process parameter optimization method; Processor: Used to implement the semiconductor manufacturing process parameter optimization method when executing the computer program.

[0073] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the semiconductor manufacturing process parameter optimization method.

[0074] In summary, this invention addresses the insufficient attention paid to explicit modeling and dynamic fusion mechanisms of cognitive uncertainty among multi-source kriging models. It proposes a systematic solution integrating Bayesian Model Averaging (BMA) and Dempster-Shafer Theory (DS Theory), specifically a kriging method integrating Bayesian Model Averaging (BMA) and Dempster-Shafer Theory (DS Theory) (BMA-DSK). The proposed method first introduces Bayesian model averaging into three basic kriging models: simple kriging, ordinary kriging, and generalized kriging. Robust estimation of spatial autocorrelation structure is achieved through weighted fusion of multiple candidate variograms. Secondly, a DS fusion framework is constructed to generate basic probability assignments for the basic kriging models and perform multi-source evidence fusion, achieving an effective expression of cross-model cognitive uncertainty. Finally, a dynamic weighting strategy is used to generate a comprehensive predicted value. The proposed method was evaluated using leave-one-out cross-validation on an etching simulation dataset. The results show that the proposed method not only improves prediction accuracy but also effectively quantifies cognitive uncertainty, providing a novel surrogate model that combines accuracy and interpretability for small-sample modeling of high-value semiconductor etching processes.

Claims

1. A method for optimizing semiconductor manufacturing process parameters based on Bayesian average kriging and evidence theory, characterized in that, Includes the following steps: Step 1: Based on the semiconductor manufacturing process, determine the key input parameters and output response indicators that affect the process; Step 2: Using a full factorial experimental design method, sample within the space of key process input parameters and obtain the corresponding output response data through simulation software to construct an input-output dataset; Step 3: Based on the input and output datasets from Step 2, construct a basic Kriging proxy model based on Bayesian model averaging and evidence theory. After optimization, perform cross-model fusion to output the final predicted value and the expected upper and lower bounds of the predicted value; that is, realize the quality prediction part of semiconductor manufacturing process parameter optimization.

2. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 1, characterized in that, Step 3 is implemented as follows: Step 3.1, BMA optimization of the variation function of the basic kriging model: The basic kriging includes three types: simple kriging, ordinary kriging, and generalized kriging. Bayesian model averaging is performed independently within each basic kriging model, that is, three variation functions are fitted to each type of kriging: Gaussian function, exponential function, and Matérn function. The dataset consisting of the process parameters and response data in Step 2 is used as the input to the three basic kriging models. Finally, each of the three basic kriging models outputs a BMA-optimized predicted value and a calculated residual, which serves as the input source for DS evidence fusion in Step 3.

2. Step 3.2, DS Evidence Theory Fusion Across Kriging Types: Based on evidence theory, the three optimized Kriging models obtained in Step 3.1 are fused across models. Specifically, first, basic probability assignments are constructed based on model residuals; then, Yager's combination rule is used to sequentially fuse the probability assignments of simple Kriging, ordinary Kriging, and generalized Kriging to obtain a comprehensive probability assignment; based on the comprehensive probability assignment, belief degree and likelihood are calculated, and dynamic weights of each model are constructed to generate the final predicted value and the upper and lower bounds of the expected predicted value. The final predicted value represents the nonlinear relationship between the input parameters and the output response, and the upper and lower bounds of the expected predicted value are used to quantify the uncertainty information of the prediction results. The smaller the error between the predicted and actual values, the better the constructed Kriging proxy model based on Bayesian model averaging and evidence theory performs. The difference between the upper bound and the lower bound of the expected value represents the degree of uncertainty of the sample point at that point.

3. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 2, characterized in that, The method involves fitting three variogram functions to each type of Kriging: Gaussian function, etc. Exponential function Matérn function The fitting results for the three functions are as follows: (22) (23) (24) in, For the nugget effect, For the partial sill value, For variable range, parameters Fitting the empirical variation function using the nonlinear least squares method To better reflect the characteristics of real-world etching physical systems, the nu value of the Matérn function is set to 1.

5. h It is the independent variable, representing the distance between two points.

4. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 2, characterized in that, The specific implementation method for constructing the basic probability assignment based on model residuals is as follows: Set up a recognition framework Let Kriging represent the mutually exclusive and exhaustive set of propositions consisting of three candidate Krigings, each model... Each stage of processing yields a predicted value. and a residual index reflecting its internal goodness of fit. These two parameters will be used as input for step 3.2; We employ a basic probability assignment construction method based on the inverse of the error, and let the set of effective models be... First, calculate the inverse residual weight: (1) Normalization yields: (2) m' represents the temporary variable that is taken sequentially through all models in set M during the summation of the denominator; Assign trust to a subset of single elements, while introducing unknowns to characterize uncertainties that are not fully represented: (3) (4) in, express The basic probability distribution, Preset unknown weights are used to allocate a portion of the confidence score to the entire set. This indicates a conservative judgment that there is no complete trust in any Kriging model.

5. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 2, characterized in that, The comprehensive probability allocation is specifically as follows: Treating the three types of Kriging models as three independent sources of evidence, their corresponding basic probability assignments are as follows: The overall probability allocation is synthesized stepwise using a fixed fusion order of simple kriging, ordinary kriging, and pan-kriging, with the initial fusion result as follows: (5) The second step is to integrate ordinary Kriging evidence: (6) The third step involves fusing pan-Kriging evidence to ultimately obtain the comprehensive probability allocation after fusion of the three sources: (7) in, Let Yager be the combinatorial operator, defined as follows: for any proposition... and : (8) For the complete series : (9)； The calculation of belief degree and likelihood degree based on comprehensive probability allocation is specifically as follows: According to DS theory, for any proposition Its belief function With likelihood function Defined as: (10) (11)； The specific implementation method for constructing dynamic weights for each model to generate the final predicted value and the expected upper and lower bounds of the predicted value is as follows: Using a dynamically weighted average based on local consistency as the final output, for the effective model set M, the deviation between its predicted values ​​and the mean of the training set is calculated: (12) And construct the weights: (13) The final predicted value is: (14) Likelihood With belief These represent the upper and lower bounds of the expected value, respectively.

6. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 1, characterized in that, Calculate the root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) of the basic Kriging surrogate model constructed in step 3. 2 This is used to evaluate the model's prediction accuracy and generalization ability on simulation data. The specific implementation method is as follows: Root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) 2 The calculation formulas are as follows: (51) (52) (53) in, Let i be the true value of the i-th sample. These are the model's predicted values. The average of the true values; RMSE reflects the overall degree of deviation of the predicted values; MAE measures the average error level; R 2 RSE indicates the model's ability to explain the variability of the target variable. A smaller RMSE indicates less bias in the model's predictions, while a smaller MAE indicates better generalization ability. 2 The closer the value is to 1, the better the fit.

7. The semiconductor manufacturing process parameter optimization method based on Bayesian average Kriging and evidence theory according to claim 1, characterized in that, The uncertainty quantification information of the prediction results of the optimized base Kriging surrogate model in step 3 is calculated by quantifying the difference between the likelihood and belief for each sample point. The model uncertainty interval is obtained, which reflects the degree of conflict between pieces of evidence and the level of uncertainty in the model when fusing multi-source information; the specific implementation method is as follows: Build a recognition framework , recorded as In the recognition framework Above, define the basic probability assignment function: (54) function value This is known as the basic reliability of set A. It represents the overall uncertainty or unknown information regarding all fundamental propositions; This represents three different propositions. Based on the basic probability assignment, the trust function is derived. With likelihood function : (55) (56) Indicates the proposition A Overall level of trust Indicates the proposition A The maximum possible support; and Together they form a confidence interval Its length This reflects the degree of uncertainty regarding proposition A. The larger the value, the greater the evidence conflict and the higher the uncertainty of the model at that point.

8. A system comprising a processor, characterized in that, It is capable of running the semiconductor manufacturing process parameter optimization method according to any one of claims 1-4.

9. A device, characterized in that, include: Memory: for storing the computer program of the semiconductor manufacturing process parameter optimization method according to any one of claims 1-4; Processor: Used to implement the semiconductor manufacturing process parameter optimization method according to any one of claims 1-4 when executing the computer program.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the semiconductor manufacturing process parameter optimization method according to any one of claims 1-4.

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