Magnetron sputtering film process parameter optimization method and system based on ARD-GPR model
By optimizing magnetron sputtering process parameters using Latin hypercube sampling and the ARD-GPR model, the problems of empirical dependence and poor reproducibility in traditional methods are solved, achieving efficient optimization of film uniformity and deposition rate, and reducing R&D costs and time.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-22
- Publication Date
- 2026-04-03
AI Technical Summary
Traditional methods for optimizing magnetron sputtering process parameters rely on experience, making it difficult to cope with new material systems or equipment state drift. They also have poor reproducibility and struggle to achieve a balance between film quality and optimization under multiple objective constraints, resulting in long R&D cycles and high costs.
The experimental parameter combinations were designed using the Latin hypercube sampling method, and the data were modeled and preprocessed using the ARD-GPR model. The process parameters were optimized using the Gaussian process regression model, and multi-objective optimization was achieved through the non-dominated sorting genetic algorithm. An uncertainty quantification mechanism was introduced to identify process schemes that need to be experimentally verified first.
It improves experimental reproducibility, shortens the R&D cycle, achieves synergistic optimization of film uniformity and deposition rate, reduces the number of experiments and costs, and provides high-precision process parameter recommendations.
Smart Images
Figure CN121787250A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of thin film preparation technology, specifically to a method and system for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model. Background Technology
[0002] Magnetron sputtering, a key technology in physical vapor deposition, is widely used in semiconductor chip manufacturing, optical coatings, flat panel displays, and functional materials. Thin film uniformity and deposition rate are core indicators for evaluating sputtering process quality, directly impacting device performance and production efficiency. However, the magnetron sputtering process involves multi-physics coupling, including plasma physics, gas dynamics, and particle transport. Thin film quality is strongly influenced by the nonlinear coupling of multiple process parameters such as sputtering power, gas pressure, gas flow rate, target-substrate distance, and substrate temperature.
[0003] Traditional process parameter optimization methods rely heavily on operator experience and numerous orthogonal experiments. This "experience-based" method is highly dependent on engineer experience, making it difficult to address challenges posed by new material systems or equipment state drift, and it also suffers from poor reproducibility. While the orthogonal experimental method is systematic, the number of experiments increases exponentially with the increase in parameter dimensions; for example, for four process parameters with five levels of conditions each, the number of experiments can reach as high as 500. 4 =625 times, resulting in long R&D cycles and high material and time costs.
[0004] Furthermore, traditional process parameter optimization methods typically only address a single objective, such as increasing the deposition rate or improving film thickness uniformity, making it difficult to find a balanced solution under multi-objective constraints. Summary of the Invention
[0005] Based on this, the purpose of this invention is to provide a method and system for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model, which can improve experimental reproducibility, shorten the R&D cycle, and achieve adaptive identification of operating conditions and multi-objective collaborative optimization.
[0006] In a first aspect, the present invention provides a method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model, comprising the following steps: Step S11: The Latin hypercube sampling method is used to design the combination of experimental parameters in the multidimensional process parameter space, and the original data is preprocessed to obtain the dataset. The dataset is divided into training set and test set in a 7:3 ratio. The input features in the dataset include power supply power, power supply frequency, power supply duty cycle, process gas flow rate, radial distance and angular position. The output variable in the dataset is the single-point deposition rate. Step S12: Fit and model the training set using different regression models to obtain multiple target prediction models. Then, evaluate the multiple target prediction models using a test set to select the Gaussian process regression model with the best performance as the final target prediction model. Step S13: Construct the final Gaussian process regression model based on the Matérn5 / 2 kernel which includes an automatic correlation determination mechanism. At the same time, optimize the hyperparameters and noise terms of the kernel function by maximizing the log marginal likelihood so that the final Gaussian process regression model can predict the deposition rate of a single point and the corresponding spatial distribution, and output the arithmetic mean and standard deviation of the deposition rate of all predicted points. Step S14: Using the final Gaussian process regression model as the fitness function, and taking maximizing the average deposition rate and maximizing film uniformity as the dual objective functions, a non-dominated sorting genetic algorithm with an elitist strategy is employed to optimize the dual objective functions to generate a Pareto optimal solution set. The maximum film uniformity U is: , In the formula, CV is the coefficient of variation of the deposition rate at all measurement points predicted in the final Gaussian process regression model, which is equal to the standard deviation of the deposition rate at all predicted points divided by the arithmetic mean of the deposition rates at all measurement points.
[0007] Preferably, the optimization method of the present invention further includes quantifying the uncertainty of the process scheme to identify process schemes that need to be experimentally verified first, and selecting at least one representative combination of process parameters from them for output, specifically including: Using the probability output characteristics of the final Gaussian process regression model, the predicted mean, predicted standard deviation and 95% confidence interval of the deposition rate at each prediction point in the spatial sampling grid are calculated to draw a spatial distribution map of the uncertainty in the deposition rate prediction. Extrapolate the input features of the process parameter combination to be evaluated to the distribution of training data. Calculate the extrapolation factor of the prediction points that exceed the range of the training data distribution based on the degree of deviation, and amplify the prediction standard deviation of the prediction points that exceed the range of the training data distribution using a multiplicative gain method to mark the reliability of the extrapolation area. In the multi-objective optimization process, the process schemes are screened based on reliability labels to identify the process schemes that need to be experimentally verified first, and at least one representative combination of process parameters is selected and output.
[0008] Preferably, the method of preprocessing the raw data in step S11 specifically includes: The 3σ criterion is used to detect outliers for each feature variable in order to remove outliers that are outside the statistical range; Missing values were imputed using median imputation to ensure sample integrity. Z-score standardization was performed on all feature variables and the target variable to unify the units of measurement.
[0009] Preferably, in step S11, the angular position is represented using Cartesian coordinates, as follows: , In the formula, r Radial distance, θ It is the azimuth angle.
[0010] Preferably, in step S12, the different regression models include artificial neural network models, random forest models, gradient boosting tree models, and Gaussian process regression models.
[0011] Preferably, in step S12, the evaluation method is as follows: using the coefficient of determination, root mean square error, mean absolute error, and mean absolute percentage error as evaluation indicators, the five-fold stratified cross-validation method is used to evaluate the prediction accuracy, stability, and generalization ability of different regression models.
[0012] Preferably, in step S13, when performing kernel function optimization, a Bayesian optimization strategy is used to perform a global search on the initial values to obtain a highly stable combination of kernel parameters.
[0013] Secondly, the present invention provides a magnetron sputtering thin film process parameter optimization system based on the ARD-GPR model, used to implement the above-mentioned magnetron sputtering thin film process parameter optimization method based on the ARD-GPR model. The optimization system includes: The data acquisition module is used to design experimental parameter combinations in a multidimensional process parameter space using the Latin hypercube sampling method, and to preprocess the raw data to obtain a dataset. The dataset is divided into a training set and a test set in a 7:3 ratio. The input features in the dataset include power supply power, power supply frequency, power supply duty cycle, process gas flow rate, radial distance and angular position. The output variable in the dataset is the single-point deposition rate. The model creation module is used to fit and model the training set using different regression models. Multiple target prediction models are obtained after training, and the test set is used to evaluate the multiple target prediction models respectively, so as to select the Gaussian process regression model with the best performance as the final target prediction model. The model optimization module is used to construct the final Gaussian process regression model based on the Matérn5 / 2 kernel, which includes an automatic correlation determination mechanism. At the same time, it optimizes the hyperparameters and noise terms of the kernel function by maximizing the log marginal likelihood, so that the final Gaussian process regression model can predict the deposition rate of a single point and the corresponding spatial distribution, and output the arithmetic mean and standard deviation of the deposition rate of all predicted points. The objective optimization module uses the final Gaussian process regression model as the fitness function, with the dual objective functions of maximizing the average deposition rate and maximizing film uniformity. A non-dominated sorting genetic algorithm with an elitist strategy is employed to optimize these dual objective functions to generate a Pareto optimal solution set. The goal of maximizing film uniformity U is: , In the formula, CV is the coefficient of variation of the deposition rate at all measurement points predicted in the final Gaussian process regression model, which is equal to the standard deviation of the deposition rate at all predicted points divided by the arithmetic mean of the deposition rates at all measurement points.
[0014] Preferably, the optimization system further includes a predictive analysis module for quantifying the uncertainty of the process scheme, identifying process schemes that need to be experimentally verified first, and selecting at least one representative combination of process parameters for output.
[0015] Preferably, the optimization system also includes a visualization analysis module for generating Pareto optimal solution front curves, deposition rate spatial contour plots, three-dimensional surface plots, and uncertainty spatial distribution maps, and providing recommended process parameters for uniformity-first, efficiency-first, and balance schemes.
[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: During the data acquisition phase, a Latin hypercube sampling strategy was adopted to ensure uniform coverage of the parameter space. A quartz crystal balance was used to collect the deposition rate at multiple spatial locations on the substrate. The collected data was preprocessed by standardization, anomaly detection, missing value repair, and redundant feature removal to form a seven-dimensional feature space for model training. In the modeling phase, a Matérn5 / 2 kernel Gaussian process regression (GPR) model with an automatic correlation determination (ARD) mechanism was adopted. Key hyperparameters such as signal variance, length scale, and noise variance were optimized using maximum likelihood estimation and Bayesian optimization algorithms. The model's generalization performance was verified through five-fold cross-validation. The ARD mechanism enables the model to automatically learn the length scale of different input features, thereby identifying the most critical process parameters and spatial variables. The importance of each feature can be quantified by analyzing the normalized vector of the inverse of the length scale. Combined with Pearson correlation analysis, the linear coupling structure between process parameters and deposition rate is revealed, thus significantly outperforming traditional black-box models in terms of model interpretability. In terms of spatial prediction, this invention employs a 48-point verification scheme, which involves constructing a spatial sampling grid at four fixed radial layers and 12 equally spaced angles to verify the model's predictive capability. This invention further utilizes the probability output capability of Gaussian process regression to quantify uncertainty, obtaining the mean, variance, and 95% confidence interval for each prediction point. Furthermore, it uses extrapolation detection and uncertainty enhancement mechanisms to perform reliability correction on prediction points that exceed the training range. In the optimization stage, this invention constructs two conflicting key indicators, thin film uniformity and deposition rate, into a multi-objective function, where uniformity is characterized by the coefficient of variation (CV%) and deposition efficiency is characterized by the average deposition rate. Based on a surrogate model constructed from a Gaussian process regression model, a non-dominated sorting genetic algorithm (NSGA-II) is used to search within the feasible region of process parameters to generate a Pareto optimal solution set between process conditions and performance indicators. Through the Pareto front curve, this invention can automatically identify the optimal uniformity solution, the optimal efficiency solution, and a balanced solution that takes both into account, thereby forming a decision support system for practical process optimization. Attached Figure Description
[0017] Figure 1 This is a schematic flowchart of the magnetron sputtering thin film process parameter optimization method based on the ARD-GPR model of the present invention.
[0018] Figure 2 This is a schematic diagram of the off-target magnetron sputtering device and substrate coordinate system of the present invention; wherein the geometric center of the circular substrate stage is the origin of the coordinate system, the X-axis is along the sputtering target offset direction, and the Y-axis is orthogonal to the X-axis; the Al target is arranged above the substrate stage at a position offset towards the positive X direction to form unilateral off-target sputtering; the surface of the substrate stage is schematically arranged with quartz crystal film thickness gauge measuring points for collecting deposition rate data at different spatial positions.
[0019] Figure 3 The parallel coordinate plot shows the distribution of experimental samples in the multidimensional process parameter space. The parallel coordinate plot displays the range and combination of input features such as power, duty cycle, frequency, gas flow rate and spatial position, demonstrating that the training data of this invention has good coverage of the process space.
[0020] Figure 4 The graph shows a comparison of the prediction performance of the Gaussian Process Regression (GPR) model, Artificial Neural Network (ANN) model, Random Forest (RF) model, and Gradient Boosting Tree (GB) model on the same training dataset. Through the scatter plot of predicted and measured values, residuals and root mean square error (RMSE), symmetric mean absolute percentage error (SMAPE), mean absolute error (MAE), and accuracy (R²) radar statistics, it is shown that the Gaussian Process Regression model selected in this invention is superior to other models in terms of prediction accuracy and stability.
[0021] Figure 5A schematic diagram of feature importance analysis obtained by using a Gaussian process regression (GPR) model combined with an automatic correlation determination (ARD) kernel function is presented. The normalized results of the inverse of each feature length scale show the relative contribution of process parameters and spatial features to the prediction of deposition rate.
[0022] Figure 6 The Pearson correlation matrix heatmap between process parameters and deposition rate reflects the degree of linear correlation between input features such as power, duty cycle, frequency, gas flow rate, and X, Y, and radial positions and deposition rate, providing statistical basis for feature selection and physical mechanism analysis.
[0023] Figure 7 This is a schematic diagram of the Pareto optimal solution set curve between the film thickness uniformity index (CV%) and the average deposition rate obtained based on the NSGA-II multi-objective optimization algorithm. Representative process schemes such as the optimal uniform solution, the optimal deposition rate solution, and the equilibrium solution are distinguished by different colors and markers.
[0024] Figure 8 This is a schematic diagram of the spatial distribution of deposition rates on a substrate for three representative process schemes predicted based on the GPR model. It includes two-dimensional contour plots and three-dimensional surface plots to show the spatial variation and differences of deposition rates under different process conditions.
[0025] Figure 9 The diagram shows the uncertainty quantification results of three representative process schemes. The upper part shows the predicted mean and 95% confidence interval of the sampling points along the radial direction, and the lower part shows the corresponding spatial prediction standard deviation distribution map, which is used to illustrate the prediction reliability of the model of this invention in different spatial regions.
[0026] Figure 10 The spatial relative error distribution verification diagrams for three representative optimization schemes are shown. By comparing the relative errors of the model prediction values and the measured deposition rates on the substrate, it is verified that the Gaussian process regression model and its multi-objective optimization results of this invention have high spatial prediction accuracy under different process conditions.
[0027] The following detailed description, in conjunction with the accompanying drawings, will further illustrate the present invention. Detailed Implementation
[0028] The following description is intended to disclose the invention and enable those skilled in the art to implement it. The preferred embodiments described below are merely examples, and other obvious variations will occur to those skilled in the art.
[0029] The first aspect of this invention is the optimization method for magnetron sputtering thin film process parameters based on the ARD-GPR model. This method integrates data acquisition, predictive modeling, mechanism analysis, multi-objective optimization, and uncertainty quantification into an intelligent prediction and process optimization method for magnetron sputtering thin film thickness uniformity. In a specific embodiment, it is applied to optimize the thickness uniformity of aluminum magnetron sputtered thin films on a circular substrate. The embodiments of this invention provide a complete implementation process for experimental parameter setting, data acquisition and feature construction, Gaussian process regression model training, multi-objective optimization, uncertainty analysis, and experimental verification.
[0030] It should be noted that within a constant process time window, the film thickness and deposition rate exhibit an approximately linear proportional relationship at the same spatial sampling point. This invention uses the deposition rate measured by a quartz crystal balance (QCM) as a proxy variable for film thickness. By predicting the spatial distribution and characterizing the uniformity of the deposition rate, it effectively models and optimizes the uniformity of film thickness.
[0031] Step 1: Experimental Parameter Range and Data Generation Please see Figure 2 This embodiment focuses on the off-target magnetron sputtering deposition process of aluminum target material on a circular substrate with a diameter of 18 cm. To cover the main process variables within a typical process window, and to eliminate meaningless or unstable parameters based on experimental constraints and physical mechanism analysis, this invention selects power supply power, power supply frequency, power supply duty cycle, and argon flow rate as process parameter dimensions, as shown in Table 1.
[0032] In terms of spatial positioning, this embodiment arranges a quartz crystal thickness gauge (QCM) near the substrate, establishing a polar coordinate system with the substrate center as the origin. Each QCM position is calibrated by radial distance and azimuth angle. During the experiment, the substrate stage rotates according to preset rules, and instantaneous deposition rate data can be obtained at multiple spatial locations after each set of parameter conditions is completed, forming an original sample set containing process parameters and spatial position information.
[0033] Approximately 3,000 experimental cases were generated using the Latin hypercube sampling method for subsequent feature construction and screening.
[0034] Table 1 Process Parameter Table Step 2: Data Preprocessing and Spatial Feature Construction To improve data quality and ensure the numerical and physical validity of the model input, this embodiment performs the following preprocessing steps on the collected samples: (1) Outlier removal: The 3σ criterion is used to detect outliers for each feature variable and output variable, and outliers that deviate significantly from the overall distribution are removed to avoid the adverse effects of extreme noise on model training.
[0035] (2) Missing value handling: For a small number of missing records, the median imputation strategy is used to ensure the integrity of the sample sequence and avoid the bias introduced by the mean.
[0036] (3) Feature standardization: Perform Z-score standardization on all input features and deposition rate output to bring features of different dimensions to the same numerical scale and avoid unreasonable amplification or reduction during model training. Standardization is only used for model training and numerical optimization; before calculating the uniformity index (CV% / U) and average deposition rate, the model output should be inversely standardized back to physical dimensions.
[0037] (4) Redundant feature deletion: Preliminary modeling and feature correlation analysis results show that the power supply voltage is highly linearly correlated with the power supply power, and its contribution to the deposition rate prediction is less than 1%. Therefore, the voltage is regarded as a redundant feature and deleted.
[0038] (5) Invalid measurement removal: For data points with extremely low deposition rates and film thickness increments calculated from the crystal frequency difference that are lower than the preset resolution threshold, they are determined to be invalid measurements mainly affected by the instrument's background noise and removed, so as to avoid the error amplification caused by the measurement resolution limitation from having an adverse effect on the subsequent model training and spatial verification results.
[0039] (6) Spatial Feature System Construction: In terms of spatial representation, this embodiment adopts a combined feature system of "radial coordinates + Cartesian coordinates". Using radial distance... r and azimuth θ Given the original data, X and Y are constructed using the following relationship: , θ is input in radians. Generally, angular variables are discontinuous at 0° / 360°. To avoid this problem, this embodiment uses X and Y coordinates to encode azimuth information.
[0040] Please see Figure 3 As shown in Table 2, the input features constructed in this embodiment consist of four process parameters: power, frequency, duty cycle, and gas flow rate, and three spatial features: r, X, and Y. The corresponding output target is the single-point deposition rate. This feature system balances physical interpretability and numerical stability, capable of characterizing both radial decay trends and reflecting asymmetric distribution characteristics caused by device assembly errors, magnetic field asymmetry, and cavity geometry.
[0041] Table 2. Examples of partial samples from the dataset Step 3: Building a Machine Learning Prediction Model Please see Figure 4After completing data preprocessing and feature construction, this embodiment compares the performance of various regression models, including Artificial Neural Network (ANN), Random Forest (RF), Gradient Boosting Tree (GB), and Gaussian Process Regression (GPR). The evaluation is conducted using indicators such as coefficient of determination R², root mean square error RMSE, and mean absolute error MAE, combined with a five-fold hierarchical cross-validation method, in terms of prediction accuracy, stability, and generalization ability.
[0042] The comparison results show that, under the conditions of limited samples and high-dimensional features, the GPR model can achieve higher R² and lower RMSE on the test set, and the prediction variance is relatively small, making it more suitable as a surrogate model for spatial prediction of magnetron sputtering deposition rates.
[0043] Please see Figure 5 This embodiment uses the Matérn5 / 2 kernel Gaussian process regression model with an automatic correlation determination (ARD) mechanism. Let the training dataset be... ,in, Represents the 7-dimensional input features of N samples. This represents the corresponding deposition rate. Model assumptions: , in, For Matérn5 / 2 kernel functions with independent length scale parameters, To observe the noise variance, for function, when hour, ,otherwise This is used to write independent and identically distributed Gaussian observation noise into the diagonal terms of the covariance matrix.
[0044] By maximizing the logarithmic marginal likelihood function and combining it with a Bayesian optimization search strategy, hyperparameters such as signal variance, length scales of each dimension, and noise variance are jointly optimized.
[0045] During training, this embodiment employs a hierarchical five-fold cross-validation method based on deposition rate quantiles to ensure consistency in data distribution between the training and validation sets at each fold. Validation results show that the GPR model maintains stable high prediction accuracy and small error fluctuations across different folds, demonstrating its robustness and reliability under the conditions of this embodiment.
[0046] Step 4: Interpretability Analysis To reveal the relationship between process parameters and deposition rate, this embodiment introduces a dual interpretability analysis mechanism at the model level: Please see Figure 2 and Figure 5In this embodiment, the magnetron sputtering target is not positioned directly below the center of the substrate, but rather offset approximately 20 mm relative to the substrate edge along the positive X-axis. A planar coordinate system is established with the geometric center of the substrate as the origin, the X-axis along the target offset direction, the Y-axis orthogonal to it, and the radial distance is defined as... In this arrangement, the angular distribution of sputtered atoms emitted from the target surface and the finite target-substrate distance together determine the spatial morphology of the deposition flux field. The flux peak shifts in the positive X direction along with the target center, thus forming an asymmetric distribution with "unilateral enhancement" on the substrate plane. Statistical results show that the deposition rate has the most significant positive correlation with the X coordinate, a weak negative correlation with the Y coordinate, and a linear correlation coefficient with the radial distance r close to zero. This indicates that the deposition rate field in this embodiment is mainly dominated by the offset and gradient along a specific direction in the X-Y plane, rather than an ideal axisymmetric pure radial monotonic decay. This phenomenon is consistent with the eccentric arrangement of the target relative to the substrate and the imperfect symmetry of the cavity and magnet structures. Meanwhile, the nonlinear feature importance analysis based on ARD shows that the radial distance r and spatial features such as X and Y still have high importance in the Gaussian process regression model, indicating that r still plays a key role in nonlinear coupling terms and spatial scale control, and its influence cannot be fully characterized by linear Pearson correlation alone.
[0047] 1. Importance analysis of ARD length scale In the GPR model, each input dimension corresponds to an independent length scale parameter. Normalizing the reciprocals of each length scale allows for a quantitative ranking of the contributions of each input feature to the deposition rate prediction. Analysis results show that the Y-coordinate has the greatest impact on the deposition rate, followed by radial position and the X-coordinate, reflecting the radial attenuation law during deposition and the spatial offset effect caused by device structural asymmetry. Among the process parameters, power, frequency, duty cycle, and gas flow rate, although their individual contributions are relatively small, together determine the overall morphology of the deposition rate field through coupling with spatial features.
[0048] 2. Pearson correlation analysis Please see Figure 6To further verify the rationality of the ARD feature importance ranking, this embodiment calculated the Pearson correlation coefficient between each input feature and the deposition rate. The results show that the deposition rate is generally positively correlated with power, duty cycle, and frequency, with relatively high correlations between duty cycle and power. The most significant positive correlation exists with the X-coordinate, while a weak negative correlation exists with the Y-coordinate. The linear correlation with gas flow rate and radial distance is not significant, with correlation coefficients close to zero. These results indicate that, under the equipment structure and process window used in this embodiment, the thin film deposition rate field exhibits an asymmetric shift and gradient along a specific direction on the substrate plane, mainly dominated by positional changes. This phenomenon is consistent with the plasma cloud shift caused by factors such as actual magnet assembly eccentricity and incomplete cavity structure asymmetry.
[0049] Although the linear correlation coefficient between radial distance and deposition rate is not significant, the nonlinear feature importance analysis based on ARD shows that radial distance and spatial features such as X and Y are still key input variables affecting the spatial distribution of deposition rate in Gaussian process regression models. This indicates that Pearson correlation can only characterize linear monotonic relationships, while GPR can further capture nonlinear and coupling effects.
[0050] The above analysis shows that the GPR model constructed in this invention not only has high prediction accuracy, but also good interpretability, enabling engineers to identify key process factors and carry out targeted process optimization based on feature importance and correlation results.
[0051] Step 5: Multi-objective optimization and process scheme selection After obtaining the GPR model with high-precision prediction capability and uncertainty output, this embodiment further optimizes the two performance indicators "film thickness uniformity" and "average deposition rate" within the feasible domain of process parameters to obtain a combination of process parameters that is engineering-significant between film quality and process efficiency.
[0052] 1. Definition of film thickness uniformity and deposition efficiency indicators Given process parameters, assuming there are M sampling points in a pre-defined spatial sampling grid on the substrate, the deposition rate prediction values of the GPR model at these locations are as follows: R 1, ..., R M Its arithmetic mean with standard deviation They are respectively: , The coefficient of variation is defined as: , CV%, a dimensionless quantity, is used to measure the consistency of the spatial distribution of deposition rates. A smaller CV% indicates that the deposition rate at each sampling point is closer to the overall average, corresponding to better film thickness uniformity. For ease of engineering description, a film thickness uniformity index can be defined: , A larger U indicates better film thickness uniformity. In the optimization algorithm implementation, this example directly minimizes CV%, which is equivalent to maximizing U. The average deposition rate index S is defined as the arithmetic mean of the deposition rates at all spatial sampling points: , Used to characterize the deposition efficiency under these process conditions.
[0053] 2. Problem Formulation and NSGA-II Solution for Multi-Objective Optimization Based on the above definitions, this embodiment integrates "film thickness uniformity priority" and "deposition efficiency priority" into a multi-objective optimization framework. Objective 1: Minimize CV% (i.e., maximize film thickness uniformity); Objective 2: Maximize S (i.e., maximize deposition efficiency).
[0054] To facilitate solving within a unified minimization framework, this embodiment equivalently transforms objective 2 into minimizing -S. A 4-dimensional decision vector u is constructed using power, duty ratio, frequency, and gas flow as decision variables. The upper and lower bounds of each variable are determined by the actual value range of the training data.
[0055] Please see Figure 7 In this embodiment, the NSGA-II algorithm is implemented using the built-in gamultiobj function in MATLAB. The specific configuration is as follows: the population size is set to 200, the maximum number of iterations is set to 150, and the maximum number of stalled generations is set to 30.
[0056] The optimization solution process includes: (1) Population initialization: Randomly generate an initial population within the upper and lower limits of each decision variable; (2) Fitness evaluation: For each candidate solution u in the population, combine the spatial sampling grid (4 layers of radial positions R=[2,4,6,8]cm × 12 angles = 48 sampling points) and call the trained GPR model to predict the deposition rate of each grid point, and calculate the coefficient of variation CV% and the average deposition rate S according to the prediction results, so as to obtain the target vector F(u)=[CV%(u), -S(u)] of the candidate solution; (3) Fast non-dominated sorting: Sort the population in layers according to the Pareto dominance relationship; (4) Crowding distance calculation: Calculate the crowding distance in the same non-dominated layer to maintain the diversity of the solution set; (5) Selection, crossover and mutation: Use binary tournament selection, simulated binary crossover (SBX) and polynomial mutation operator to generate offspring; (6) Elite retention: Merge the parent generation and offspring generation and re-sort them, and prioritize the retention of individuals with low rank and large crowding distance into the next generation.
[0057] After iterative optimization, the Pareto optimal solution set is output. To support engineering decisions, this embodiment extracts three representative solutions from the Pareto front: (a) the optimal homogeneity solution—the solution with the minimum CV%; (b) the optimal deposition efficiency solution—the solution with the maximum average deposition rate S; and (c) the compromise solution—using the normalized Euclidean distance method, the solution closest to the ideal point (CV_min, S_max) is selected as the recommended solution that takes into account both objectives.
[0058] 3. Selection of representative process schemes To facilitate practical engineering applications, this embodiment selects three representative combinations of process parameters from the Pareto optimal solution set: (1) Optimal uniform solution: The solution with the smallest CV% among all Pareto solutions corresponds to the best film thickness uniformity; (2) Optimal sedimentation rate solution: The solution with the largest S among all Pareto solutions corresponds to the highest sedimentation efficiency; (3) Equilibrium solution: The objective vector of each Pareto solution F ( x ) is denoted as: ， In the Pareto solution set, CV% and - S Find the minimum value to reach the ideal point F min : ， By calculating the objective vector for each Pareto solution F ( x (and ideal point) Fmin The Euclidean distance between the two is used to select the solution with the smallest distance as a balance solution that takes into account both film thickness uniformity and deposition efficiency.
[0059] Please see Figure 8 When the above representative schemes are input into the GPR model, a spatial distribution prediction map of deposition rate is generated on the preset verification grid. It can be observed that the distribution is relatively flat and the radial gradient is weakened under the optimal uniform solution, and the central region shows obvious peak concentration under the optimal deposition rate solution, while the equilibrium solution maintains a good balance between the two.
[0060] Step Six: Uncertainty Quantification, Extrapolation Detection, and Experimental Verification To improve the reliability of process optimization results and identify high-risk areas with a limited number of experiments, this embodiment constructs an uncertainty quantification and extrapolation detection mechanism based on the probability output characteristics of the GPR model, and forms a closed loop by combining it with space experimental verification.
[0061] 1. Uncertainty Quantification Based on GPR The GPR model, given any input x The posterior distribution of the predicted deposition rate can be given at each location: , in, For input x The predicted posterior mean of the deposition rate at that location. For input x The predicted posterior standard deviation at that location, To predict the posterior variance, this embodiment calculates the variance for each candidate combination of process parameters and its corresponding spatial sampling grid points during the multi-objective optimization process. and And based on this, a 95% confidence interval is obtained: , By plotting the distribution of prediction standard deviation and confidence intervals across the entire spatial domain, the spatial distribution of the uncertainty in sedimentation rate prediction can be obtained.
[0062] 2. Extrapolation detection and uncertainty enhancement mechanism Please see Figure 9 To identify extrapolated regions that exceed the distribution range of the training data, this embodiment measures the distance between the point to be evaluated and the training samples in the input feature space and constructs an extrapolation factor. E ( x When a prediction point has a relatively small normalized distance across power, frequency, duty cycle, gas flow rate, radial position, and X and Y dimensions, and falls within a dense region of training data, it is identified as an "interpolation region".E ( x When the predicted point is far from the training sample cluster, or when some or all features exceed the existing data distribution boundary, it is determined to be an "extrapolation region". E ( x The value is greater than 1 and increases monotonically with increasing deviation. Based on this, this embodiment uses a multiplicative gain method to amplify the predicted standard deviation of the extrapolated region: , Furthermore, the extrapolated regions are prominently marked on the spatial uncertainty distribution map to indicate a higher level of uncertainty. Through this mechanism, the system can automatically identify high-risk areas that deviate from the coverage of the training data in both process parameter space and spatial location.
[0063] 3. Application of uncertainty in multi-objective optimization and decision-making In the NSGA-II optimization process, this embodiment introduces prediction uncertainty as a penalty term into the objective function calculation, that is, in the CV% and S The evaluation is superimposed with The relevant weights or penalty factors automatically reduce the weight or eliminate process options with high overall prediction uncertainty during the selection process. This avoids unreliable predictions in the extrapolation region from adversely shifting the Pareto front, ensuring that the optimal solution set is concentrated in the region with "sufficient data support and high prediction reliability".
[0064] From an engineering application perspective, this uncertainty enhancement mechanism enables the present invention to identify data sparse regions that require focused attention and supplementary experiments in advance without having to repeatedly conduct a large number of iterative experiments to correct errors, thus significantly reducing the number of experiments and time costs required for process development.
[0065] 4. Spatial verification grid design and experimental verification Please see Figure 10 To quantitatively evaluate the effectiveness of the model in spatial prediction and multi-objective optimization, this embodiment designed a 48-point spatial verification grid comprising four radial layers, each with twelve equiangular positions. This grid covers the areas with the most significant changes in deposition rate while ensuring the representativeness of the overall spatial structure. Deposition rates were measured at the verification locations using QCM and compared with the prediction results of representative process schemes.
[0066] Experimental results show that, under this verification scheme, the average relative errors of the optimal uniform solution, the optimal deposition rate solution, and the equilibrium solution of this embodiment are approximately 4.32%, 0.34%, and 0.37%, respectively. The optimal uniform solution corresponds to a lower power setting, with the overall deposition rate at the lower limit of the process window in this embodiment. The increase in film thickness at a single point is close to the resolution limit of the quartz crystal balance, resulting in some sampling points having frequency differences close to zero or equivalent to background noise before and after data acquisition. These were deemed invalid measurements and removed during data preprocessing, leading to a significantly fewer effective verification points for statistical analysis compared to the other two conditions. This condition is more sensitive to local measurement noise, resulting in a slightly larger average relative error. Nevertheless, the error level is still controlled within approximately 5%, while the prediction errors of the optimal deposition rate solution and the equilibrium solution are both less than 1%. Overall, this verifies that the GPR-based spatial prediction model for deposition rate and its multi-objective optimization results have high accuracy and stability under different process conditions.
[0067] Through the above-mentioned uncertainty quantification, extrapolation detection and experimental verification closed loop, this invention achieves reliable prediction of film thickness uniformity and deposition efficiency of magnetron sputtering thin films and optimization of multi-objective process parameters under limited data conditions, providing a general method and system implementation path for intelligent process design in semiconductor manufacturing, optical coating and functional material preparation scenarios.
[0068] Secondly, the magnetron sputtering thin film process parameter optimization system based on the ARD-GPR model provided by this invention is used to implement the above-mentioned magnetron sputtering thin film process parameter optimization method based on the ARD-GPR model. The optimization system includes a data acquisition module, a model creation module, a model optimization module, and a target optimization module. The modules are interconnected through a standard communication protocol to form a complete solution that can be deployed in both laboratory and industrial production environments. The data acquisition module is used to design experimental parameter combinations in a multidimensional process parameter space using the Latin hypercube sampling method, and to preprocess the raw data to obtain a dataset. The dataset is divided into a training set and a test set in a 7:3 ratio. The input features in the dataset include power supply power, power supply frequency, power supply duty cycle, process gas flow rate, radial distance and angular position. The output variable in the dataset is the single-point deposition rate. The model creation module is used to fit and model the training set using different regression models. Multiple target prediction models are obtained after training, and the test set is used to evaluate the multiple target prediction models respectively, so as to select the Gaussian process regression model with the best performance as the final target prediction model. The model optimization module is used to construct the final Gaussian process regression model based on the Matérn5 / 2 kernel, which includes an automatic correlation determination mechanism. At the same time, it optimizes the hyperparameters and noise terms of the kernel function by maximizing the log marginal likelihood, so that the final Gaussian process regression model can predict the deposition rate of a single point and the corresponding spatial distribution, and output the arithmetic mean and standard deviation of the deposition rate of all predicted points. The objective optimization module uses the final Gaussian process regression model as the fitness function, with the dual objective functions of maximizing the average deposition rate and maximizing film uniformity. A non-dominated sorting genetic algorithm with an elitist strategy is employed to optimize these dual objective functions to generate a Pareto optimal solution set. The goal of maximizing film uniformity U is: , In the formula, CV is the coefficient of variation of the deposition rate at all measurement points predicted in the final Gaussian process regression model, which is equal to the standard deviation of the deposition rate at all predicted points divided by the arithmetic mean of the deposition rates at all measurement points.
[0069] In a preferred embodiment of the present invention, the optimization system further includes a predictive analysis module for quantifying the uncertainty of the process scheme, identifying process schemes that need to be experimentally verified first, and selecting at least one representative combination of process parameters and outputting it.
[0070] In a preferred embodiment of the present invention, the optimization system further includes a visualization analysis module for generating Pareto optimal solution front curves, deposition rate spatial contour plots, three-dimensional surface maps, and uncertainty spatial distribution maps, and providing recommended process parameter schemes for uniformity-first, efficiency-first, and balanced schemes. Users can select uniformity-first, efficiency-first, or balanced schemes in the interface and export the corresponding sputtering parameters for direct use in subsequent process verification or production line operation.
[0071] In terms of data structure, this invention supports storing model parameters, feature importance results, uncertainty distribution data, and multi-objective optimization solutions as reproducible engineering files for subsequent analysis and version management.
[0072] This invention can be widely applied in semiconductor manufacturing, optoelectronic device fabrication, ITO and AZO transparent conductive film deposition, metal tool coating, and optical interference film preparation, and is particularly suitable for production environments with high requirements for multi-batch process development and film uniformity. In practical deployment, this invention can run as a standalone software module or be embedded in existing MES systems or control software provided by equipment manufacturers to achieve functions such as process parameter recommendation, uniformity analysis, and deposition efficiency evaluation.
[0073] Traditional thin film thickness uniformity optimization typically requires extensive experimental measurements to obtain representative process parameters. However, this invention, through a GPR model and multi-objective optimization algorithm, achieves high-precision spatial prediction results and a Pareto-optimal process set with only a limited number of experiments, significantly reducing the data volume and time cost required for magnetron sputtering process development. Furthermore, by introducing an uncertainty quantification mechanism, this invention automatically enhances risk warnings in high-risk areas of the process space, thereby avoiding wasted experimental resources and incorrect process selection, and improving the reliability of process development. The feature importance analysis mechanism constructed in this invention provides engineers with intuitive judgments of process influencing factors, supporting the diagnosis of process deviations and equipment failures. In summary, this invention significantly improves prediction accuracy, interpretability, optimization efficiency, and reliability, effectively promoting the transformation of magnetron sputtering thin film manufacturing processes from an experience-driven to a data-driven model.
[0074] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.
Claims
1. A method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model, characterized in that, Includes the following steps: Step S11: The Latin hypercube sampling method is used to design the combination of experimental parameters in the multidimensional process parameter space, and the original data is preprocessed to obtain the dataset. The dataset is divided into training set and test set in a 7:3 ratio. The input features in the dataset include power supply power, power supply frequency, power supply duty cycle, process gas flow rate, radial distance and angular position. The output variable in the dataset is the single-point deposition rate. Step S12: Fit and model the training set using different regression models to obtain multiple target prediction models. Then, evaluate the multiple target prediction models using a test set to select the Gaussian process regression model with the best performance as the final target prediction model. Step S13: Construct the final Gaussian process regression model based on the Matérn5 / 2 kernel which includes an automatic correlation determination mechanism. At the same time, optimize the hyperparameters and noise terms of the kernel function by maximizing the log marginal likelihood so that the final Gaussian process regression model can predict the deposition rate of a single point and the corresponding spatial distribution, and output the arithmetic mean and standard deviation of the deposition rate of all predicted points. Step S14: Using the final Gaussian process regression model as the fitness function, and taking maximizing the average deposition rate and maximizing film uniformity as the dual objective functions, a non-dominated sorting genetic algorithm with an elitist strategy is employed to optimize the dual objective functions to generate a Pareto optimal solution set. The maximum film uniformity U is: ， In the formula, CV is the coefficient of variation of the deposition rate at all predicted points in the final Gaussian process regression model, which is equal to the standard deviation of the deposition rate at all predicted points divided by the arithmetic mean of the deposition rates at all predicted points.
2. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, It also includes quantifying the uncertainty of the process scheme to identify process schemes that need to be experimentally verified first, and selecting at least one representative combination of process parameters for output, specifically including: Using the probability output characteristics of the final Gaussian process regression model, the predicted mean, predicted standard deviation and 95% confidence interval of the deposition rate at each prediction point in the spatial sampling grid are calculated to draw a spatial distribution map of the uncertainty in the deposition rate prediction. Extrapolate the input features of the process parameter combination to be evaluated to the distribution of training data. Calculate the extrapolation factor of the prediction points that exceed the range of the training data distribution based on the degree of deviation, and amplify the prediction standard deviation of the prediction points that exceed the range of the training data distribution using a multiplicative gain method to mark the reliability of the extrapolation area. In the multi-objective optimization process, the process schemes are screened based on reliability labels to identify the process schemes that need to be experimentally verified first, and at least one representative combination of process parameters is selected and output.
3. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, In step S11, the preprocessing of the raw data specifically includes: The 3σ criterion is used to detect outliers for each feature variable in order to remove outliers that are outside the statistical range; Missing values were imputed using median imputation to ensure sample integrity. Z-score standardization was performed on all feature variables and the target variable to unify the units of measurement.
4. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, In step S11, the angular position is represented using Cartesian coordinates, as follows: ， In the formula, r Radial distance, θ This is the azimuth angle.
5. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, In step S12, the different regression models include artificial neural network models, random forest models, gradient boosting tree models, and Gaussian process regression models.
6. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, In step S12, the evaluation method is as follows: using the coefficient of determination, root mean square error, mean absolute error, and mean absolute percentage error as evaluation indicators, the five-fold stratified cross-validation method is used to evaluate the prediction accuracy, stability, and generalization ability of different regression models.
7. The method for optimizing magnetron sputtering thin film process parameters based on the ARD-GPR model according to claim 1, characterized in that, In step S13, when performing kernel function optimization, a Bayesian optimization strategy is used to perform a global search on the initial values to obtain a highly stable combination of kernel parameters.
8. A magnetron sputtering thin film process parameter optimization system based on the ARD-GPR model, characterized in that, For implementing the magnetron sputtering thin film process parameter optimization method based on the ARD-GPR model as described in any one of claims 1 to 7, the optimization system comprises: The data acquisition module is used to design experimental parameter combinations in a multidimensional process parameter space using the Latin hypercube sampling method, and to preprocess the raw data to obtain a dataset. The dataset is then divided into a training set and a test set in a 7:3 ratio. The input features in the dataset include power supply power, power supply frequency, power supply duty cycle, process gas flow rate, radial distance, and angular position. The output variable in the dataset is the single-point deposition rate. The model creation module is used to fit and model the training set using different regression models. Multiple target prediction models are obtained after training, and the test set is used to evaluate the multiple target prediction models respectively, so as to select the Gaussian process regression model with the best performance as the final target prediction model. The model optimization module is used to construct the final Gaussian process regression model based on the Matérn5 / 2 kernel, which includes an automatic correlation determination mechanism. At the same time, it optimizes the hyperparameters and noise terms of the kernel function by maximizing the log marginal likelihood, so that the final Gaussian process regression model can predict the deposition rate of a single point and the corresponding spatial distribution, and output the arithmetic mean and standard deviation of the deposition rate of all predicted points. The objective optimization module uses the final Gaussian process regression model as the fitness function, with the dual objective functions of maximizing the average deposition rate and maximizing film uniformity. A non-dominated sorting genetic algorithm with an elitist strategy is employed to optimize these dual objective functions to generate a Pareto optimal solution set. The goal of maximizing film uniformity U is: ， In the formula, CV is the coefficient of variation of the deposition rate at all measurement points predicted in the final Gaussian process regression model, which is equal to the standard deviation of the deposition rate at all predicted points divided by the arithmetic mean of the deposition rates at all measurement points.
9. The magnetron sputtering thin film process parameter optimization system based on the ARD-GPR model according to claim 8, the optimization system further includes a predictive analysis module for quantifying the uncertainty of the process scheme, identifying the process schemes that need to be experimentally verified first, and selecting at least one representative combination of process parameters for output.
10. The magnetron sputtering thin film process parameter optimization system based on the ARD-GPR model according to claim 8, the optimization system further includes a visualization analysis module for generating Pareto optimal solution set front curves, deposition rate spatial contour plots, three-dimensional surface plots, and uncertainty spatial distribution plots, and providing process parameter recommendations based on uniformity priority, efficiency priority, and balance schemes.