Neural network model for spectral prediction and applications thereof
Patent Information
- Application Number
- CN202610734164.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-05-26
- Publication Date
- 2026-08-21
AI Technical Summary
[0009]针对现有技术的以上缺陷或改进需求，本发明提供了一种用于光谱预测的神经网络模型及其应用，解决现有技术中存在的物理一致性差、反演精度低，数据量大的问题
1. 本发明用于光谱预测的神经网络模型，该模型的采用复合损失函数对模型进行优化，该函数包含数据误差、物理梯度一致性误差
、光谱形态的高阶微分误差
以及物理谱值一致性误差
，物理梯度一致性误差给神经网络引入了物理一致性约束，使得神经网络具有一定的物理一致性，从而减少一定数据量，反演时依赖梯度优化，而本发明训练时引入了梯度监督，使得训练后的模型天然对梯度更敏感，提高反演精度，解决现有技术中存在的物理一致性差、反演精度低，数据量大的问题。
Smart Images
Figure CN122616632A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of semiconductor integrated circuit manufacturing and testing, and more specifically, relates to a neural network model for spectral prediction and its application. Background Technology
[0002] As semiconductor manufacturing processes continue to evolve towards 7nm, 5nm, and even smaller dimensions, the precision requirements for controlling the critical dimension (CD) in photolithography are becoming increasingly stringent. To monitor the quality of photolithography and etching processes, precise measurement of the geometry of nanostructures on the wafer (such as gratings and FinFETs) is essential. Traditional measurement techniques, such as scanning electron microscopy (SEM), while offering high resolution, are destructive, have low throughput, and can only acquire surface information. Transmission electron microscopy (TEM), while providing cross-sectional information, involves complex sample preparation and cannot be used for online full inspection.
[0003] Optical scattering (OCD) measurement, also known as optical critical dimension measurement, has become a mainstream metrology technology on semiconductor production lines as a non-contact, non-destructive, and high-throughput measurement method. OCD technology measures the reflectivity or polarization state changes (such as ellipsometric parameters) produced when a broadband light beam illuminates a periodic nanostructure. and The geometric parameters of the structure (such as top width, bottom width, side wall angle, height, etc.) are calculated using an inversion algorithm.
[0004] The core of OCD measurement lies in solving the "inverse problem," that is, inferring geometric parameters from the measured spectral data. Since the interaction between light and nanostructures follows Maxwell's equations, numerical algorithms such as Rigorous Coupled-Wave Analysis (RCWA) are typically used as forward solvers. However, RCWA has high computational complexity and long simulation times, making it difficult to meet real-time measurement requirements. Existing solutions mainly include library matching and machine learning regression methods.
[0005] The library matching method pre-builds a large "parameter-spectrum" database, and during measurement, it searches for the best-matching spectrum by looking up the table. However, with increasing structural complexity, the parameter space explodes exponentially, making database construction and search extremely inefficient. In recent years, deep learning-based methods have been introduced into OCD measurement, using trained neural networks to fit the nonlinear mapping relationship between parameters and spectra, thus replacing RCWA as a positive surrogate model. However, existing purely data-driven neural network methods have significant drawbacks: 1. Poor physical consistency: The traditional mean squared error (MSE) loss function only focuses on numerical approximation and ignores the inherent laws of physical processes, resulting in weak generalization ability of the model outside the training data and easy to produce predictions that violate physical common sense.
[0006] 2. Gradient information distortion: OCD inverse solving usually uses gradient-based optimization algorithms (such as Levenberg-Marquardt). If the neural network only fits the numerical values and does not accurately fit the derivatives of the physical quantities with respect to the geometric parameters (i.e., the Jacobian matrix), the inverse optimization process may get stuck in local minima or fail to converge.
[0007] 3. Reliance on large amounts of data: Traditional machine learning methods rely on extremely large training sets, which often leads to excessive time and computational costs in generating data, and in some scenarios it is difficult to obtain a large amount of data.
[0008] Therefore, there is an urgent need for a physical neural network measurement method that can integrate physical mechanisms, ensure gradient accuracy, and solve problems such as large data volume. Summary of the Invention
[0009] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a neural network model for spectral prediction and its application, which solves the problems of poor physical consistency, low inversion accuracy and large data volume in the existing technology.
[0010] To achieve the above objectives, according to one aspect of the present invention, the neural network model is provided to take the geometric parameters of the periodic grating under test as input and to output the theoretical spectrum of the periodic grating under test as output. The loss function of this neural network model is:
[0011] in, It is the loss between the spectrum predicted by the neural network and the spectrum calculated by the physical model. It is the loss between the predicted gradient and the theoretical gradient of the neural network. It is a higher-order differential form loss. The loss between the predicted spectrum and the theoretical spectrum of the neural network, , , , These are the weights corresponding to the loss function.
[0012] More preferably, the loss between the neural network predicting the spectrum and the physical model calculating the spectrum... The formula is as follows:
[0013] in, The total number of training samples, The spectral values predicted by the neural network. Spectral values calculated for the physical model.
[0014] More preferably, the higher-order differential shape loss The formula is as follows:
[0015] in, , For the first and second order values of the spectrum predicted by the neural network, , These are the first and second order values of the theoretical spectrum. , These are the corresponding weighting coefficients.
[0016] More preferably, the loss between the neural network's predicted gradient and the theoretical gradient... The formula is as follows:
[0017]
[0018]
[0019] in, The gradient value predicted by the neural network. -grad represents the theoretical gradient value. To predict the spectral values of the corresponding parameters for the neural network, The components of the theoretical gradient value are obtained using the RCWA method with finite differences. These are theoretical parameters. For small perturbations, It is a unit vector.
[0020] More preferably, the neural network model is a multilayer perceptron or a one-dimensional residual convolutional network.
[0021] According to another aspect of the present invention, a training method using the neural network model described above is provided, the training method comprising the following steps: The theoretical spectrum corresponding to the geometric parameters is calculated using the geometric parameters of the periodic grating under test; The geometric parameters and theoretical parameters of the periodic grating under test are used as input and output, respectively, to train the neural network model, thereby obtaining the trained neural network model.
[0022] According to another aspect of the present invention, a method for predicting the actual geometric parameters of a periodic grating is provided, the steps of which are as follows: The theoretical geometric parameters of the periodic grating to be tested are input into the trained neural network model to obtain the spectral prediction value; Obtain the actual spectrum of the periodic grating under test, and calculate the geometric parameters corresponding to the minimum distance between the actual spectrum and the predicted spectrum. These geometric parameters are the actual geometric parameters of the periodic grating under test.
[0023] More preferably, the formula for minimizing the distance between the actual spectrum and the predicted spectrum is as follows:
[0024] in, These are spectral prediction values. is the actual spectrum, and x is the actual geometric parameter.
[0025] More preferably, the LM algorithm is used to minimize the distance between the actual spectrum and the predicted spectrum. The formula for solving the actual geometric parameters by this algorithm is as follows:
[0026] in, The parameter values are the initial values for the iteration. For the next parameter value in the iteration, It is the identity matrix. The weights of the identity matrix, To predict spectra for neural networks, Its transpose, These are spectral prediction values. That is the actual spectrum.
[0027] More preferably, the actual spectrum is obtained by adding noise to the theoretical spectrum.
[0028] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art: 1. This invention provides a neural network model for spectral prediction, which employs a composite loss function to optimize the model, including data error. Physical gradient consistency error Higher-order differential errors in spectral morphology and the consistency error of physical spectrum values Physical gradient consistency error introduces physical consistency constraints to neural networks, enabling them to have a certain degree of physical consistency and thus reducing the amount of data. Inversion relies on gradient optimization. However, this invention introduces gradient supervision during training, making the trained model naturally more sensitive to gradients, improving inversion accuracy, and solving the problems of poor physical consistency, low inversion accuracy, and large data volume in existing technologies.
[0029] 2. The physical gradient consistency error in the loss function of this invention differs from that of traditional neural networks, where training only minimizes the output value. The error is due to neglecting the derivative. In terms of accuracy, sensitivity (i.e., the rate of change of the spectrum with respect to CD value) is the key factor determining measurement accuracy in spectral measurements. The physical gradient consistency error of the point in this invention utilizes a physical model to calculate the true physical Jacobian matrix and uses it as a supervisory signal to constrain the gradient field of the neural network. This ensures that the neural network not only predicts accurate values but also makes the changing trend conform to physical laws, thereby significantly improving the mathematical fidelity and convergence stability of the model in subsequent inverse optimization tasks.
[0030] 3. This invention explicitly incorporates first-order difference (slope) and second-order difference (curvature) penalty terms into the loss function of the spectral curve, i.e., higher-order differential shape constraints. Spectral data is essentially a continuous function of wavelength or energy, possessing physical smoothness. Traditional MSE training is prone to overfitting, generating high-frequency oscillations. This application, by penalizing large fluctuations in the second-order difference, forces the network to generate smooth curves that conform to the principle of physical continuity, eliminating "high-frequency oscillation artifacts."
[0031] 4. This invention incorporates the error between the spectrum predicted by the neural network and the physical spectrum calculated by the physical model into the loss function, namely, the physical spectrum value consistency loss. The physical spectrum serves as the "teacher signal" for the neural network, adding physical constraints to the neural network so that it can find a correct target when making predictions at the beginning, thereby reducing the dependence on training data and improving the physical consistency of the network. Attached Figure Description
[0032] Figure 1 This is a schematic diagram of the overall process of an optical critical dimension measurement method based on a physical neural network provided in an embodiment of this application.
[0033] Figure 2 This is a block diagram illustrating the principle of the hybrid training framework and physical gradient alignment mechanism in the embodiments of this application.
[0034] Figure 3 This is a structural block diagram of an optical critical dimension measurement system based on a physical neural network provided in an embodiment of this application.
[0035] Figure 4 This is a comparison chart of the errors of the physical neural network and the traditional machine learning model provided in the embodiments of this application under different training set sizes.
[0036] Figure 5 This is a distribution diagram of the predicted spectral error of the physical neural network provided in the embodiments of this application on a simple grating test set.
[0037] Figure 6 This is a distribution diagram of the forward calculation spectral accuracy results (mean absolute error of the predicted spectrum) of the multilayer film structure grating provided in the embodiments of this application. Detailed Implementation
[0038] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0039] In semiconductor manufacturing, optical critical dimension (OCD) measurement aims to infer the geometry of a structure by analyzing the interaction between broadband polarized light and nanostructures on a wafer surface. The structure under test is typically a periodic grating, whose geometric parameters (denoted as vectors) are... This includes, but is not limited to: grating top CD, bottom CD, grating height / thickness, sidewall angle (SWA), and underlying film thickness. The spectral response (denoted as a vector) acquired by the measurement system (such as an ellipsometer) is also considered. This typically includes ellipticity parameters that vary with wavelength. ) or reflectivity ( ).
[0040] The forward problem can be described as a physical mapping. This mapping is governed by Maxwell's equations. The inverse problem is to solve... .because Given its highly nonlinear and complex nature, and the absence of an explicit analytic inverse function, it is typically transformed into an optimization problem:
[0041] Build a physical neural network To replace computationally expensive And through special physical constraint training strategies, ensure Not only numerically close Moreover, in terms of derivative (gradient) properties and physical form, it is similar to... High consistency, thus ensuring subsequent utilization Accuracy and convergence when performing optimization solutions.
[0042] Please see Figure 1 The following is a flowchart of the overall process for measuring optical critical dimensions based on physical neural networks, as provided in the embodiments of this application. The main steps are detailed below: Construction of the S1 Neural Network Model (1) Construction of physical model and data preparation First, simulation software is used to model the selected grating structure, with the parameter space varying as follows: top width [10nm, 20nm], bottom width [20nm, 30nm], and thickness [70nm, 90nm]. A training sample set is then generated through random sampling. Theoretical spectrum Then, a spectral calculator based on rigorous coupled-wave analysis was written in PyTorch as the physical model.
[0043] For some or all samples, the physical gradient (Jacobi matrix) is calculated using the Finite Difference Method (FDM). For the input vector... The Each component Its corresponding spectral output The physical partial derivatives are approximately:
[0044] in For small perturbations, It is a unit vector. Input vector The set of physical gradients generated by all components in the equation is denoted as . .
[0045] (2) Constructing a physical neural network model Construct a multilayer perceptron (MLP) or a one-dimensional residual convolutional network as the forward proxy model. Introduce a physical model module. The input layer dimension corresponds to geometric parameters. The number of sampling points corresponds to the number of spectral data points in the output layer dimension (e.g., 69 sampling points for a wavelength range of 250nm-800nm). The intermediate layers of the network use the LeakyReLU activation function to maintain good gradient propagation properties. The final layer uses the Sigmoid function.
[0046] (3) Hybrid training framework aligned with physical gradients Please see Figure 2 This is the core innovation of this invention. Traditional training only compares... and This application introduces physical gradient alignment, the specific logic of which is as follows: During the forward propagation phase of training, the input... Predicted spectra obtained via network Using the automatic differentiation feature of deep learning frameworks, the gradient of the network output with respect to the input is calculated:
[0047] A matrix describes the network's perceived "sensitivity." Simultaneously, the actual physical gradients are obtained from the pre-computed dataset. Calculate the gradient consistency loss:
[0048] By minimizing This forces the neural network to learn not only function values but also the local linearization behavior of the physical system. This is crucial for inverse optimization because the commonly used Levenberg-Marquardt algorithm heavily relies on accurate Jacobian matrices to update the search step size. Error, reverse reasoning will lead to divergence.
[0049] (4) Introduce higher-order differential form constraints Spectral data It's about wavelength. The function has physical continuity and smoothness. Purely data-driven training can easily lead to independent noise predictions between wavelengths, forming "glitch" in the network. This application introduces morphological constraint loss. Let the spectral vector be... First-order difference (slope): Second-order difference (curvature): The morphological loss function is defined as:
[0050] in The weighting coefficients, especially the second-order difference constraint, can effectively suppress high-frequency oscillation artifacts and ensure that the generated spectral curves conform to the physical continuity principle of light-matter interaction.
[0051] (5) Using physical spectral value loss To ensure that neural network predictions conform to physical laws, the input will be... The physical spectrum is obtained by substituting into the physical model. Calculate the consistency loss of physical spectrum values:
[0052] Using physical spectra as "teacher signals" to guide a neural network to a correct answer, by minimizing... This forces the neural network to learn towards the correct results.
[0053] (6) Optimization of composite loss function Based on the above modules, the overall training objective function is:
[0054] in The loss between the predicted spectrum and the theoretical spectrum for the neural network. Weights to It can be dynamically adjusted according to the training phase (e.g., initial focus). and Later focus ).
[0055] Step S2: Reverse measurement execution Once the physical neural network is trained, it is stored and deployed to the measurement system. The measurement process is as follows: 1. Add a certain amount of noise to the theoretical spectrum to obtain the actual spectrum. .
[0056] 2. Initialize the guessed values of geometric parameters .
[0057] 3. Define the optimization goal: .
[0058] 4. Use the LM algorithm for iterative updates:
[0059] In this step, because After physical gradient alignment training, its calculated It can accurately reflect the sensitivity of the physical system, thereby guiding the algorithm to converge to the true geometric parameters quickly and accurately.
[0060] Figure 3 The complete measurement process of the present invention is shown in the structural block diagram of the critical dimension measurement system.
[0061] The present invention will be further described below with reference to specific embodiments.
[0062] To verify the effectiveness of the method proposed in this application, the following experiments were conducted: 1. Robustness verification with small datasets Figure 4The diagram shows a comparison of the mean absolute error (spec_mae) of the predicted spectra of the proposed Physical Neural Network (PINN) and traditional machine learning (ML) models under different training set sizes. As can be seen from the diagram, in scenarios with very limited data (e.g., training set samples of 500 or 1000), the error of the traditional ML model increases sharply, exhibiting severe overfitting and a decline in generalization ability. In contrast, the PINN model of this application, benefiting from the physical gradient alignment mechanism and physical spectrum value consistency constraints, maintains an extremely low prediction error (stable below 0.001). This fully demonstrates the high robustness and practical application value of the proposed method under conditions of limited datasets.
[0063] 2. High-precision model verification Figure 5 The Spec_MAE distribution of the proposed model on the test set samples is shown. As shown in the figure, for 127 random test samples, the overall distribution of the prediction error of the proposed model is as follows: to Within an extremely low order of magnitude range. This extremely high fitting accuracy effectively avoids the high-frequency oscillation artifacts of conventional deep learning models in complex nonlinear mappings, further verifying that the model in this application possesses extremely high measurement accuracy in OCD forward proxy and reverse inference tasks.
[0064] 3. Applicability and high-precision verification of multilayer film grating structures The method proposed in this application is applicable not only to simple gratings without thin-film structures, but also highly suitable for gratings containing complex multilayer film structures. For multilayer film gratings, we verified the spectral accuracy of the model's forward calculations: Spectral accuracy results from forward calculation: Figure 6 The figure shows the distribution of the mean absolute error (Spec_MAE) of the forward predicted spectrum of the proposed model for multilayer film gratings on the test set samples. As shown in the figure, for 127 random test samples, the error between the forward calculated predicted spectrum and the true spectrum of the proposed model is generally distributed in an extremely low order of magnitude range of 5.00×10⁻⁶ to 4.5×10⁻⁵. This fitting accuracy effectively avoids the high-frequency oscillation artifacts of conventional deep learning models in complex nonlinear mappings, further verifying that the proposed model, as a forward surrogate model, has extremely excellent fitting ability when dealing with complex multilayer film structures.
[0065] In summary, this application successfully solves the problems of poor physical interpretability and insufficient accuracy of deep learning models in optical critical dimension measurement by constructing a physical neural network and employing innovative physical gradient alignment, physical spectral value consistency constraints, and high-order morphological constraints. This method not only retains the high-speed inference advantage of neural networks but also endows them with mathematical fidelity at the level of physical simulation, possessing significant industrial application value.
[0066] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A neural network model for spectral prediction, characterized in that, The input to this neural network model is the geometric parameters of the periodic grating under test, and the output is the theoretical spectrum of the periodic grating under test. The loss function of this neural network model is: in, It is the loss between the spectrum predicted by the neural network and the spectrum calculated by the physical model. It is the loss between the predicted gradient and the theoretical gradient of the neural network. It is a higher-order differential form loss. The loss between the predicted spectrum and the theoretical spectrum of the neural network, , , , These are the weights corresponding to the loss function.
2. The neural network model for spectral prediction as described in claim 1, characterized in that, The loss between the neural network predicting the spectrum and the physical model calculating the spectrum. The formula is as follows: in, The total number of training samples, The spectral values predicted by the neural network. Spectral values calculated for the physical model.
3. A neural network model for spectral prediction as described in claim 1 or 2, characterized in that, The higher-order differential form loss The formula is as follows: in, , For the first and second order values of the spectrum predicted by the neural network, , These are the first and second order values of the theoretical spectrum. , These are the corresponding weighting coefficients.
4. A neural network model for spectral prediction as described in claim 3, characterized in that, The loss between the predicted gradient and the theoretical gradient of the neural network The formula is as follows: in, The gradient value predicted by the neural network. -grad represents the theoretical gradient value. To predict the spectral values of the corresponding parameters for the neural network, The components of the theoretical gradient value are obtained using the RCWA method with finite differences. These are theoretical parameters. For small perturbations, It is a unit vector.
5. A neural network model for spectral prediction as described in claim 1, characterized in that, The neural network model is a multilayer perceptron or a one-dimensional residual convolutional network.
6. A training method using the neural network model according to any one of claims 1-5, characterized in that, The training method includes the following steps: The theoretical spectrum corresponding to the geometric parameters is calculated using the geometric parameters of the periodic grating under test; The geometric parameters and theoretical parameters of the periodic grating under test are used as input and output, respectively, to train the neural network model, thereby obtaining the trained neural network model.
7. A method for predicting the actual geometric parameters of a periodic grating, characterized in that, The steps of this method are as follows: The theoretical geometric parameters of the periodic grating to be tested are input into the neural network model trained according to claim 6 to obtain the spectral prediction value; Obtain the actual spectrum of the periodic grating under test, and calculate the geometric parameters corresponding to the minimum distance between the actual spectrum and the predicted spectrum. These geometric parameters are the actual geometric parameters of the periodic grating under test.
8. The prediction method as described in claim 7, characterized in that, The formula that minimizes the distance between the actual spectrum and the predicted spectrum is as follows: in, These are spectral prediction values. is the actual spectrum, and x is the actual geometric parameter.
9. The prediction method as described in claim 7 or 8, characterized in that, The LM algorithm is used to minimize the distance between the actual spectrum and the predicted spectrum. The formula for solving the actual geometric parameters by this algorithm is as follows: in, The parameter values are the initial values for the iteration. For the next parameter value in the iteration, It is the identity matrix. The weights of the identity matrix, To predict spectra for neural networks, Its transpose, These are spectral prediction values. That is the actual spectrum.
10. The prediction method as described in claim 1, characterized in that, The actual spectrum is obtained by adding noise to the theoretical spectrum.