Mask optimization method, electronic device, and computer-readable storage medium

Abstract

The application discloses a mask optimization method, an electronic device and a computer readable storage medium, and belongs to the field of semiconductors. In order to solve the nonlinear and trade-off problem between local and global performance, the technical key points are as follows: a rule set is determined; a sampled metal layer segment is processed by an ILT engine to generate a quasi-optimized continuous transmission mask (CTM) which is used for rule distribution value initialization; an exponential natural evolution strategy with lithography perception is used to optimize the rule distribution value; and the optimized rule distribution value is used for generating sub-resolution assist features (SRAF) and model-based optical proximity correction (OPC) of a test pattern segment. As a result, the lithography performance and the calculation efficiency can be improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of semiconductors, and relates to a mask optimization method, an electronic device and a computer readable storage medium, in particular to an OPC rule extraction method based on an inverse lithography technology engine. BACKGROUND

[0002] Model-based optical proximity correction (OPC) and its sub-resolution assist feature (SRAF) generation are key standard methods to compensate for lithographic distortion in integrated circuit manufacturing at advanced technology nodes. Typical model-based OPC and SRAF algorithms involve user-controlled rule parameter selection. Traditionally, these rules are usually determined by heuristic methods and applied globally within the entire correction area, which is both time-consuming and requires expert knowledge of the tool. In addition, the relationship between rule parameters and optimization objectives is highly nonlinear. All these factors make it a very complex task to develop a high-performance OPC engine for complex metal designs.

[0003] Over the past few decades, the progress of semiconductor manufacturing technology has continuously pushed the boundaries of chip design, leading to an increasing demand for more complex computational lithography techniques. Therefore, applying resolution enhancement techniques (RET), such as sub-resolution assist feature (SRAF) insertion and optical proximity correction (OPC), to adjust layout patterns has become a key to achieving high pattern fidelity and mask manufacturability.

[0004] As an important resolution enhancement technique, optical proximity correction (OPC) addresses the problem of unnecessary wafer image distortion by pre-correcting pattern distortion on the lithography mask. The current rapidly developing OPC methods mainly include model-based OPC, inverse lithography technology (ILT), and deep learning-based OPC. Model-based OPC methods rely on complex simulations to precisely adjust mask layouts, mainly through edge splitting and moving, combined with strategically placing sub-resolution assist features (SRAFs) to optimize the lithography process. ILT methods treat mask optimization as a pixel-based inverse imaging problem and continuously optimize mask layouts through iterations until the simulated wafer image is highly consistent with the target pattern. It is particularly good at handling complex design patterns and generating masks that can consider optical and process effects. Finally, deep learning-based OPC predicts an initial mask solution through a deep learning model, and then uses an ILT engine for subsequent optimization, accelerating the mask optimization process.

[0005] However, these techniques face challenges when dealing with complex large-scale industrial patterns. The complexity of ILT methods requires a large amount of computational resources, making their runtime in large-scale designs too long. Moreover, the masks generated by ILT are difficult to manufacture due to their pixel-based nature. For deep learning-based OPC, the accuracy of the generated models highly depends on the quantity and quality of the training data, which can not generalize well across different pattern regions, and their black-box nature makes it difficult to understand and diagnose OPC results.

[0006] Currently, model-based OPC methods are still the mainstream in industry, mainly because they can handle complex pattern geometries and adapt to advanced lithography techniques. The key lies in the optimization of mask edge segmentation and fragment movement rules, which control how to segment edges into different fragments and the displacement of fragments in each OPC iteration. In the SRAF insertion process, the rules are mainly divided into two categories: the distance between SRAF and main features, and the size of SRAF. Optimizing these rule parameters is crucial for achieving an excellent OPC solution that should have outstanding lithography performance and high efficiency. In industrial applications, the current common method mainly relies on OPC engineers to manually configure rule values, which requires rich domain knowledge and consumes a lot of manpower. However, fine-tuning these parameters in high-dimensional space is complex and challenging, mainly due to their indirect and non-analytical impact on lithography performance, and the subtle trade-off between local and global performance. To address these challenges, genetic algorithms have been used to adjust SRAF generation rules and OPC parameters, which model the rule parameter adjustment as a complete black-box optimization problem and do not consider the information in the lithography process. This optimization process can cause dramatic changes in gene combinations, making the search algorithm prone to local optimal solutions. SUMMARY

[0007] To solve the above problems, in a first aspect, according to the mask optimization method in some embodiments of the present application, comprising

[0008] determining a rule set;

[0009] The sampled metal layer fragments are processed by the ILT engine to generate a quasi-optimized continuous transmission mask (CTM) for rule distribution value initialization;

[0010] Optimizing the rule distribution values using lithography-aware exponential natural evolution strategy;

[0011] The optimized rule distribution values are used to generate sub-resolution assist features (SRAF) and model-based optical proximity correction (OPC) for test pattern fragments.

[0012] According to the mask optimization method in some embodiments of this application, the optimization target of the ILT engine includes the wafer image under nominal conditions and the target image. The deviation between them, and the area of ​​the process variation zone; a given regular distribution vector. loss function It is a mask The function is represented by the following formula:

[0013]

[0014] In the formula, Indicates a printed image. Represents the target image. This indicates the mask used to generate the printed image. These are weighting coefficients. Indicates the outermost contour. Indicates the innermost contour;

[0015] Iterative updates of unconstrained intermediate variables To minimize the loss function The quasi-optimized continuous transmission mask (CTM) is generated, as expressed by the following formula:

[0016]

[0017] In the formula, Represents unconstrained intermediate variables. This indicates that the step size parameter is updated iteratively; where, , Indicates from arrive The gradient parameters of the transformation.

[0018] According to the mask optimization method in some embodiments of this application, the method for iteratively updating the continuous transmission mask (CTM) includes...

[0019] After average pooling downsampling, fast optimization is performed at low resolution r_L;

[0020] Perform double upsampling for detailed optimization at a higher resolution r_H;

[0021] The optimization is represented by the following formula:

[0022]

[0023] The constraints are: ;

[0024] In the formula, This represents the final optimized continuous projection mask (CTM). denotes the initial high-resolution continuous transmission mask obtained by upsampling, denotes the initial high-resolution continuous transmission mask obtained by upsampling, denotes the continuous transmission mask (CTM) obtained by iterative update at low resolution, denotes the continuous transmission mask (CTM) obtained by iterative update at high resolution; denotes the mask scaling factor upsampling.

[0025] According to the mask optimization method in some embodiments of the present application, the rule distribution is optimized using a lithography-aware exponential natural evolution strategy, including

[0026] optimizing the rule distribution parameters by a lithography-aware natural gradient method; wherein the natural gradient update is performed by natural exponential expansion;

[0027] shaping an adaptive function based on relative ranking; wherein adaptive sampling is performed on test graphic segments.

[0028] According to the mask optimization method in some embodiments of the present application, wherein in the optimization of the rule distribution parameters by the lithography-aware natural gradient method, the update of the rule distribution parameters is performed in the direction of minimizing the expected lithography cost, while imposing a constraint on the information gain at each step;

[0029] The natural gradient is formalized as a solution to a constrained lithography rule value optimization problem:

[0030]

[0031] wherein, denotes the expected lithography cost after changing the parameters denotes the expected lithography cost under the rule distribution parameters, denotes the expected lithography cost under the rule distribution parameters, denotes the parameter change amount, denotes the gradient of the loss function with respect to the parameters;

[0032] The constraint condition is:

[0033]

[0034] wherein, denotes the distance between the distribution with as the parameter and the distribution with as the parameter, denotes an arbitrarily small amount;

[0035] In In the case of natural distance measure,

[0036]

[0037] where, denotes the parameter distribution and the KL divergence, is the Fisher information matrix for a given family of regularizing parameters, which is expressed as:

[0038]

[0039] where, denotes the probability distribution, E[·] denotes the average over the regularizing vector m

[0040] The update step is obtained by substituting the direction of the natural gradient into the Lagrange multiplier method of the constrained optimization problem, and is expressed as:

[0041] .

[0042] where, denotes the scaling factor;

[0043] The natural gradient is determined as the same direction as , and is calculated as:

[0044]

[0045] where, denotes the natural gradient of the loss function with respect to the regularizing distribution parameter , denotes the inverse of the Fisher information matrix; The updated parameters are:

[0046]

[0047]

[0048] where, denotes the update step of the regularizing distribution parameter.

[0049] According to the mask optimization method in some embodiments of the application, the natural gradient update is performed through natural exponential expansion, and the update formula is as follows:

[0050]

[0051] where, and denote the direction of the natural gradient update, denotes the updated distribution represented by the mean,​ denotes the updated distribution denotes the standard deviation of the representative;

[0052] In the natural gradient coordinate system, the Fisher matrix is the identity matrix, the current search distribution is encoded as The logarithmic expression of the probability density is:

[0053]

[0054] where denotes the trace operation on the matrix Q, denotes the regular parameters to be optimized, denotes the mean of the distribution of m;

[0055] For the sample , where denotes the loss function, denotes the identity matrix;

[0056] The sample generation follows the following formula:

[0057] The calculation of the natural gradient of

[0058]

[0059] where denotes the gradient of the loss function of the current distribution state with respect to the distribution parameters , (0,0) represents the current distribution state, n denotes the nth sampling sample, and N denotes the total number of samples for each cycle to update the distribution parameters, denotes the gradient of the orthogonal basis ;

[0060] The calculation of the natural gradient of

[0061]

[0062] where denotes the natural gradient of , and denotes the gradient of the loss function with respect to the orthogonal basis W;

[0063] The transformation matrix is decomposed into a step size and a normalized matrix that satisfies ; ​​​

[0064] gradient Represented as:

[0065]

[0066] gradient Represented as:

[0067]

[0068] In the formula, Indicates the total dimension of the parameters;

[0069] Natural gradient The complexity is calculated.

[0070] According to the mask optimization method in some embodiments of this application, a relative ranking-based fitness function is shaped to correct the gradient estimation, so as to maintain scale invariance and ranking invariance among different graphic fragments, and to convert lithography cost into utility value. Individuals are ranked according to their cost, where the i-th highest-ranking individual is the one whose cost is in ascending order. ,and It is the rule parameter with the lowest cost. It is the parameter with the highest cost, expressed as:

[0071]

[0072] In the formula, This represents the number of population samples taken per cycle to update the distribution parameters. Indicates the distribution parameter Next, the nth parameter takes The posterior distribution of;

[0073] Select utility value The utility function is derived from the following formula. express:

[0074]

[0075] According to the mask optimization method in some embodiments of this application, in the adaptive sampling on the test graphic fragment, the layout is divided into overlapping tiles, and the width and height of each tile are calculated:

[0076]

[0077] In the formula, and These represent the width and height of the complete layout, respectively. and denote the width and height of the cropped tile, and denote the step size in the width and height directions, denote the number of tiles, respectively.

[0078] In a second aspect, the embodiments of the present application also provide an electronic device, comprising: one or more processors, a memory, and one or more programs; wherein the one or more programs are stored in the memory, and the one or more programs include instructions, when the instructions are executed by the electronic device, causing the electronic device to perform the first aspect and any possible technical solutions of the first aspect.

[0079] In a third aspect, the embodiments of the present application also provide a computer-readable storage medium, comprising a computer program, when the computer program is run on an electronic device, causing the electronic device to perform the first aspect and any possible technical solutions of the first aspect.

[0080] Beneficial effects: the present application proposes a comprehensive mask optimization system and method named RuleLearner, which is specially used for SRAF generation and model-based OPC application in real industrial scenarios. The framework learns through an information-enhanced inverse lithography technology (ILT) engine, although this engine can handle complex designs, it is costly to generate optimized masks for the entire set of design fragments. Considering the non-linear and trade-off between local and global performance, the extracted rule value distribution is further optimized through a customized natural gradient. Complex SRAF generation, edge segmentation, and shifting are all guided by rule parameters. Experimental results show that RuleLearner can be applied in different complex design patterns and achieve the best lithography performance and computational efficiency. Additional aspects and advantages of the present application will be partially given in the following description, partially will become obvious from the following description, or will be understood by practicing the present application. BRIEF DESCRIPTION OF DRAWINGS

[0081] Figure 1 is a continuous transmittance mask generation flow for efficient update of multi-resolution hierarchy.

[0082] Figure 2 is a flowchart of parameter distribution update.

[0083] Figure 3 is a mask optimization result visualization diagram of four samples, each row corresponds to a sample, wherein (a) is the target pattern, (b) is the optimized mask, (c) is the pattern printed on the wafer, and (d) is the process variation bandwidth. DETAILED DESCRIPTION

[0084] Embodiments of the present application are described in detail below with reference to the attached drawing figures, wherein the implementations of the application are shown by way of examples. The application provides a method and an electronic device. The method and the device are based on the same technical concept, and since the principles by which the method and the device solve problems are similar, the implementation of the device can be mutually referred to the implementation of the method, and repeated parts will not be described in detail.

[0085] I. SUMMARY

[0086] The present application optimizes the rule parameters of complex 2D metal designs in conjunction with the information acquired from the ILT engine. To this end, the present application proposes a comprehensive mask optimization system named RuleLearner, which aims to improve lithography performance by extracting and optimizing rule parameters and can be applied to various mask fragments. The present application develops a customized ILT engine that utilizes hierarchical augmented information to quickly generate quasi-optimized continuous transmission masks (CTM) for small-scale design fragments in fewer iterations, from which the derived expression intensity map can provide guidance for the initial rule value distribution. Unlike binary masks that use binary values to represent each grid location as either completely transparent or completely blocked, CTM assigns a floating intensity value to each location, providing a spectrum from complete blocking (value of 0) to complete transparency (value of 1). The intensity value gradient in CTM provides a more refined representation than the traditional binary encoding. To address the lack of a direct analytical relationship between rule values and lithography performance results and to achieve a better balance between local and global performance, the present application introduces an exponentially expanding index of lithography-aware natural gradients. The present application uses this parameterized search distribution to generate a batch of rule parameters for the graphic fragments, and then creates SRAF based on the sampled rule parameters and applies model-based OPC to these fragments. The lithography performance of each graphic fragment is evaluated as fitness, and the natural gradient is estimated to update the rule parameter distribution accordingly. By identifying reusable lithography-aware mask updates and SRAF generation criteria, the present application can simplify the complexity of mask generation to the application of these rules, thereby achieving efficient, scalable, and manufacturable OPC correction while maintaining accuracy.

[0087] II. PRELIMINARY KNOWLEDGE

[0088] A. Compact Rule Set

[0089] In this study, the present application focuses on a compact rule set applicable to industrial-level SRAF generation and model-based OPC. It is worth noting that this framework can be easily extended to consider other rule parameters.

[0090] SRAF generation rule. SRAFs are strategically placed near isolated target patterns to increase spatial frequency without being actually printed on the wafer. This technique significantly improves the imaging fidelity of target patterns. This study follows an industry-level SRAF generation procedure, whose rule is as follows when SRAFs are deployed at the edge of LLL:

[0091] 1. If the main pattern is within the vertical distance range, no SRAF is inserted, which is called SRAF insertion forbidden zone.

[0092] 2. Otherwise, if the main pattern is within the distance range, a rectangular SRAF with width and length is inserted at the center between the nearest opposite edges.

[0093] 3. If there is no other main pattern within the distance range, two SRAFs with width and and length and are inserted at the distance and respectively.

[0094] Therefore, the rule parameters of SRAF generation are defined as a vector containing the above parameters:

[0095] .

[0096] Model-based OPC rule. The model-based OPC considered contains two steps: edge segmentation and segment movement.

[0097] 4. In the segmentation stage, a projection segment is generated when there is another main pattern within the projection area and its vertical distance is within the projection threshold distance range. Subsequently, if the edge length L exceeds the threshold , a corner segment with length is generated at the corner to precisely control at the corner, and the rest of the edge is uniformly segmented into segments with length . Conversely, if L is less than or equal to , the edge is uniformly segmented without considering the corner case. After segmentation, the segment length is fixed.

[0098] 5. In the subsequent movement phase, each iteration adjusts the segments inward or outward to correct the pattern distortion printed on the wafer. If the printed contour matches the target shape well, the corresponding segment remains unchanged. The movement step of the projection segment, corner segment, and uniform segment is , and , respectively.

[0099] Therefore, the rule set that defines the OPC process is:

[0100]

[0101] The complete rule set is the combination of SRAF and OPC rules, whose rule vector is represented as the concatenation of these two parts:

[0102]

[0103] B. Search Gradient Method

[0104] The basic idea of the search gradient method is to use the sampled gradient of the expected loss as the search gradient to update the search distribution parameters. Without loss of generality, the present invention assumes that the rule parameter search process follows a multivariate normal distribution. Let denote the parameters of the probability density function , and let denote the lithography cost of the sampled rule vector . The specific details will be described in Section III-A. The expected cost under the search distribution is:

[0105]

[0106] In order to update the distribution parameters to reduce the expected cost, the direct derivative of the expected loss with respect to the distribution parameters can be derived by the log-likelihood trick:

[0107]

[0108]

[0109] In the last step, the search gradient is estimated by the Monte Carlo sampling strategy using a sample size of NNN. The gradient of the expected cost makes the direct gradient descent method available for the iterative search distribution update:

[0110]

[0111] where is the learning rate.

[0112] C. Formalization of the Problem

[0113] In this research, a comprehensive mask optimization system is constructed to focus on extracting and optimizing the rule values to improve the lithography performance of SRAF and model-based OPC. To evaluate the quality of the final lithography simulation results, the edge placement error (EPE) and the process variation band (PVBand) are adopted as the criteria:

[0114] • Definition 1 (Edge Placement Error EPE): The edge placement error refers to the vertical or horizontal misalignment between the lithography profile and the target pattern profile under the metrology condition, which is usually measured by the Manhattan distance between the two. The measurement points are evenly distributed on the profile of the target pattern. Two metrics are adopted by the present invention to comprehensively evaluate the fidelity of the pattern:

[0115] o Total EPE length (TEPE): The sum of the EPE distances of all the measurement points, which provides a comprehensive quantification of the overall mask fidelity.

[0116] o Maximum EPE (MEPE): The maximum EPE value among all the measurement points, which is used to indicate the most severe pattern distortion.

[0117] • Definition 2 (Process Variation Band PVBand): In the actual lithography process, process variations can cause deviations in the printed image, which can lead to printing failure. The process variation band (PVBand) is composed of the exclusive-or region between multiple profiles under different process conditions.

[0118] Based on the above lithography performance evaluation metrics, the present invention can formally propose the SRAF and model-based OPC rule optimization problem.

[0119] Problem 1 (Mask optimization problem based on rule learning): Given a small portion of metal layer segments from a large-scale design and a specific rule pattern, the goal is to design a mask optimization system that can extract and optimize the rule values , and apply them to other unseen metal segments to minimize the total EPE length (TEPE), the maximum EPE (MEPE), and the area of the process variation band (PVBand), while shortening the OPC running time as much as possible.

[0120] III. RuleLearner framework

[0121] ​​​​Workflow of RuleLearner framework. After determining the considered rule set, a sampled portion of metal layer segments are processed by a customized ILT engine and a quasi-optimized continuous transmission mask (CTM) is generated for rule value initialization. Then, rules are evaluated according to the corrected mask quality and optimized using lithography-aware exponential natural evolution strategy. Finally, the optimized rules can be used to guide SRAF and model-based OPC for other test pattern segments.

[0122] A. Potential information extraction

[0123] Compared with binary mask that usually represents each grid point as either fully transparent or fully blocked with binary values, information-enhanced CTM provides more complex spatial frequency indication, allowing higher frequency diffraction components to participate in the imaging process. In addition, it preserves the potential information of SRAF that is usually unstructured. In the framework of the present invention, the generation of CTM relies on a customized ILT engine.

[0124] Forward lithography simulation converts mask into wafer image where light passing through the mask produces an aerial image intensity map on the wafer. The aerial intensity is converted into wafer image by comparing the aerial intensity with the intensity threshold of photoresist:

[0125]

[0126] where is the intensity threshold that controls the binary image on the wafer plane. This forward lithography process can be generally expressed as . Where, is the lithography engine. ILT then considers the pattern on the target wafer and the parameters of the lithography system to calculate the optimal transmission value for each mask grid point. The optimization objective considers the deviation between the wafer image and the target image under nominal conditions, as well as the area of the process variation band. For a given rule vector , its loss function is also a function of binary mask :

[0127]

[0128] In the formula, represents the printed image, represents the target image, represents the mask that produces the printed image, is the weight coefficient, denotes the outermost contour, denotes the innermost contour.

[0129] To update the binary mask using gradient descent, an unconstrained intermediate variable :

[0130]

[0131] where, denotes the gradient parameter of the conversion from to .

[0132] Due to the foundry manufacturing limitation, only binary masks can be produced. Previous work uses a large steepness to achieve a sharp binary conversion. By using a smaller to smooth the conversion, it helps to better express the light transmission distribution and enrich the gradient details. The information-enhanced CTM is optimized by iteratively updating to minimize the loss function:

[0133]

[0134] where, denotes the unconstrained intermediate variable, denotes the iterative update step parameter.

[0135] To better balance the accuracy and efficiency of the spatial transmission calculation, the information-enhanced CTM adopts a hierarchical way to update. The target design is first down-sampled by an 8x8 average pooling to a low resolution r_L for fast optimization, and then up-sampled by a factor of two for detailed optimization at a higher resolution r_H. The optimization process can be represented as:

[0136]

[0137] The constraint condition is:

[0138]

[0139] where, denotes the final optimized continuous transmission mask (CTM), denotes the initial high-resolution continuous transmission mask obtained by up-sampling , denotes the continuous transmission mask (CTM) obtained by iterative update at low resolution, denotes the continuous transmission mask (CTM) obtained by iterative update at high resolution; the function denotes using nearest neighbor interpolation to convert the mask Scaling factor Up-sampling is performed.

[0140] The generated CTM significantly enhances the potential information. The transmission distribution pattern near the main pattern edge shows dynamic shrinkage and expansion, closely matching the segment-level variation. For the i-th segment, the initialization of the OPC parameters will be close to the generated CTM shape. Accordingly, the SRAF parameters are initially adjusted to match the corresponding high-transmission region in the CTM. The transmission distribution guide rule value initialization greatly improves the efficiency of the optimization process.

[0141] B. Rule parameter optimization based on litho-aware natural evolution strategy

[0142] The generated CTM is a quasi-optimized mask, but the potential information it provides can be implicit and may not be generalized across different pattern segments, which requires further optimization. In addition, the lack of a direct analytical relationship between the rule parameters and the final lithography performance is one of the main challenges. The impact of the rule parameter values is non-differentiable, discrete, and highly random, so it is not possible to directly update each rule vector using a gradient descent-based algorithm.

[0143] However, directly implementing a conventional search gradient, as discussed in Section II-B, even in a simple quadratic case, leads to unstable and undesirable performance. It seeks the steepest descent direction in the rule parameter space, treating the parameter space as a Euclidean space and using the Euclidean metric to measure the distance between parameter vectors. This metric depends on the parameterization, which means that in complex rule optimization problems, different reparameterizations will lead to different gradients and update directions.

[0144] To address this issue, the natural gradient method provides a more powerful alternative that effectively addresses these challenges by optimizing the distribution parameters of the rule values. In the following section, we will introduce how to incorporate the guidance of lithography performance into the update process of rule parameters to further improve the efficiency of the mask optimization task.

[0145] B1 Litho-aware natural gradient: The key idea is to eliminate the dependence on the parameterization by relying on a more "natural" distance between and . The update of the parameters should be in the direction that maximally reduces the expected lithography cost while imposing a constraint on the information gain at each step. The natural gradient can be formalized as the solution to a constrained optimization problem of the lithography rule values:

[0146]

[0147] where, denotes the expected lithography cost after changing the parameter . denotes the expected lithography cost under the regular distribution parameter, denotes the amount of change in the parameter, denotes the gradient of the loss function with respect to the parameter.

[0148] The constraint is:

[0149]

[0150] where, denotes the distance between the distribution with as the parameter and the distribution with as the parameter, denotes an arbitrarily small amount.

[0151] In the above equation, is the expected lithography cost under the regular distribution parameter. A common natural distance metric is the Kullback-Leibler (KL) divergence between two probability distributions. In the case of , the natural distance metric is:

[0152]

[0153] where, denotes the KL divergence between the parameter distribution and , and is the Fisher information matrix for the given regular parameter family, which is expressed as

[0154]

[0155] where, denotes the probability distribution, and E[·] denotes the average over the regular vector m.

[0156] By substituting the direction of the natural gradient into the Lagrange multiplier method of the constrained optimization problem, the update step is obtained:

[0157] .

[0158] where, denotes the scaling factor.

[0159] The natural gradient can be determined in the same direction as , which is calculated as:

[0160]

[0161] where, denotes the natural gradient of the loss function with respect to the regular distribution parameters denotes the inverse of the Fisher information matrix.

[0162] Therefore, the parameters can be updated as:

[0163]

[0164] where, denotes the update step size for the regular distribution parameters.

[0165] B2 Adaptation function based on relative ranking: The mask optimization problem is a highly non-convex problem, and the optimization path can be different for different regular vectors on the same mask segment. In addition, different design pattern segments have different geometric complexity, resulting in large differences in lithography performance after mask optimization between different segments. These changes in mask optimization can distort the gradient estimation due to fluctuating lithography cost.

[0166] To solve this problem, the RuleLearner of the present application converts the lithography cost into an utility value , and ranks the individuals according to the cost, where the i-th highest ranked individual is when ranked in ascending order of cost, and is the regular parameter with the lowest cost, is the parameter with the highest cost. This method corrects the gradient estimation in the formula to maintain scale invariance and rank invariance between different pattern segments:

[0167]

[0168] Based on the past experience of the covariance matrix adaptation evolution strategy (CMA-ES), the utility value

[0169]

[0170] B3 Natural exponential expansion: In a d-dimensional multivariate Gaussian distribution, the parameter , where is the mean vector, is the covariance matrix. When updating , its positive definiteness needs to be maintained, which cannot be guaranteed by the prior method. In addition, the dimension of the distribution parameter is​ , while the dimension of the Fisher information matrix in the previous context is . Therefore, the complexity of computing its inverse matrix reaches , which is very costly. To address these challenges, the present invention introduces an exponential mapping method for the covariance matrix and employs natural coordinate to update the distribution vector. Specifically, instead of directly computing the new mean and covariance parameters , the present invention decomposes the covariance matrix into the form , by which the update is performed on the tangent space of the parameter manifold. The update formula is as follows:

[0171]

[0172] where and represent the direction of natural gradient update, represents the updated distribution represents one of the quantities represented by the distribution: the mean (in Gaussian distribution, the distribution parameters are the mean and variance), represents the standard deviation of the updated distribution .

[0173] In this formula, and represent the direction of natural gradient update. By this method, the positive definiteness of the matrix is automatically maintained, and the natural gradient update can be efficiently performed. In addition, this update method can significantly reduce the complexity and avoid directly dealing with the large-scale Fisher information matrix inversion.

[0174] This natural exponential expansion method not only solves the problem of computational complexity, but also ensures the stability and efficiency of distribution update. In the regular parameter optimization process, this method can achieve faster and more stable distribution adjustment.

[0175] The present invention also utilizes the natural gradient coordinate system, in which the Fisher matrix is an identity matrix with respect to the orthogonal basis . In the new coordinate system, the current search distribution is encoded as . In this coordinate system, the logarithmic expression of the probability density is:

[0176]

[0177] where denotes the trace operation on matrix Q, denotes the rule parameters to be optimized, denotes the mean of the distribution of m.

[0178] By this method, the generation of samples follows the formula: for sample where the calculation of the natural gradient of the loss function on becomes more efficient, denotes the identity matrix:

[0179]

[0180] where, denotes the gradient of the loss function on the distribution parameters of the current distribution state, (0, 0) represents the current distribution state, and n denotes the nth sampled sample, and N denotes the total number of samples for each cycle to update the distribution parameters, denotes the gradient on the orthogonal basis .

[0181] Approximately, the gradient of can be derived as:

[0182]

[0183] where, denotes the natural gradient of , denotes the gradient of the loss function on the orthogonal basis W.

[0184] To further improve efficiency, the transformation matrix is decomposed into a step size and a normalized matrix that satisfies . The gradients of and are respectively:

[0185]

[0186]

[0187] where, denotes the total dimension of the parameters (the number of elements in the m vector).

[0188] Finally, the natural gradient can be calculated with the complexity of . After the above discussion, the overall algorithm can be summarized as:

[0189]

[0190] B4: Adaptive sampling on unseen graphic fragments. This sampling-based rule-based distribution update, taking into account geometric characteristics, aims to improve performance in complex metallic graphics. However, due to computational limitations, mask optimization using this rule is fragment-based, and optimizing the entire mask for complex VLSI designs on a single system is not feasible.

[0191] In photolithography, the capabilities of an optical projection system are determined by factors such as light wavelength and numerical aperture, which limit linewidth and depth of focus. As design nodes shrink, diffraction effects become more pronounced. Therefore, mask optimization of the layout tiles also needs to consider the influence of adjacent tiles. To this end, this invention divides the layout into overlapping tiles, as mentioned in the literature, and calculates the width and height of each tile based on the following considerations:

[0192]

[0193] in, and These represent the width and height of the complete layout, respectively. and Indicates the width and height of the clipping tile. Terminology and The stride represents the width and height directions and is a predefined constant. These represent the number of blocks.

[0194] The rule distribution is updated after lithography engine simulation and natural evolution optimization. To effectively generalize across various graphic complexities, graphic fragment sampling must be adaptive based on the distribution characteristics. If the rules favor denser SRAFs, coarser segmentation, and more aggressive fragment movements, the sampled fragments may appear sparse and simple. Therefore, the next iteration should prioritize complex fragments with higher metal area ratios. Conversely, if the rules involve smaller SRAFs and more complex model-based OPC processes, subsequent sampling should focus on sparse and simple fragments to maintain a balance.

[0195] C. General Mask Optimization Process

[0196] In this invention, the proposed rules are applied to an industrial-grade mask optimization process, which includes rule-based SRAF insertion and model-based OPC.

[0197] The key elements for SRAF generation are position and size information. Compared to simple single-level rule SRAF insertion methods, the rules considered in this invention are able to adapt to the geometric relationships inherent in complex primary patterns. Since the auxiliary features (SRAFs) are critical for isolated patterns, in the rule set of this invention, the sparseness near each primary pattern boundary is first examined. The optimized parameters are then used to characterize this isolation and determine whether SRAFs can be inserted and how many levels of SRAFs should be inserted. Depending on the specifics of each case, the position rule parameters are used to define the distance of SRAFs from the primary pattern edges. Next, the configuration of SRAFs is generated by the shape rule parameters The generated SRAFs will be fixed in the subsequent model-based OPC process. It is worth noting that RuleLearner can be easily extended to accommodate more complex rules without much effort.

[0198] After SRAFs are generated, model-based OPC modifies the complex primary patterns to compensate for imaging distortions. In the new complex segments, the edges of primary patterns are adaptively segmented according to the influence of neighboring primary patterns: the influence of neighboring primary patterns can have an impact on the lithography results, especially within a threshold distance , if the edge length is less than , the corner case is ignored; otherwise, the edge will be split into a corner segment with length

[0199] and the remaining part is evenly segmented into segments with length In each iteration, the deviation of wafer image from the target edge will guide the shift of segments: if the wafer image shrinks, segments will expand by a fixed step according to the segment type; if a protrusion occurs, the segments will shrink accordingly. It is worth mentioning that, in addition to the mask optimization process designed in this invention, the lithography-aware natural evolution strategy in RuleLearner can be extended to consider other lithography-related rule parameters in specific mask optimization processes. This iteration process will stop when the cost is less than a predefined threshold or the maximum number of iterations is reached.

[0200] IV. Experiments

[0201] A. Experimental Setup

[0202] The RuleLearner framework of the present invention is implemented in PyTorch, and all experiments are conducted on a Linux system with an Nvidia GeForce RTX 3090 GPU. The present invention employs a Calibre-compatible lithography simulator to perform the lithography process, which contains an optical and compact photoresist model from industry, with a reference threshold of 0.25. The relevant Calibre OPC scripts are from industry partners. Following the state-of-the-art OPC methods such as AdaOPC and LithoBench, the present invention crops a large number of layout tiles from a 45nm design synthesized by the IC design tool OpenROAD. Given a GDS-II layout file, the present invention uses KLayout to crop the layout into 1024nm x 1024nm tiles with a stride of 256nm x 256nm and extracts the first metal layer. Each tile contains a portion of the entire design pattern, and the local geometric characteristics can be simple (metal area ratio less than 0.25) or complex (metal area ratio greater than 0.35). The present invention selects 5 simple tiles and 5 complex tiles to test the performance of the optimized rules. The detailed pattern area (nm2) and area ratio of each benchmark are listed in Table I.

[0203] Similar to previous work, the performance evaluation is based on the final wafer image, and the evaluation metrics used include the process variation band area (PVB, unit: square nanometer), the sum of edge placement error (EPESum, unit: nanometer), the maximum edge placement error (EPEMax, unit: nanometer), and the runtime (RT, unit: second). The detection points of edge placement error are uniformly distributed on the target edge, and the distance between each detection point is 40 nanometers. The total number of detection points in each test case is listed in the last column of Table II. PVB is the exclusive or area between wafer contours under different process conditions. The defocus value is selected from the set {-10nm, 0nm, 10nm}, where the nominal focal length corresponds to 0nm, and the exposure dose value is selected from the set {0.95, 1.0, 1.05}, with the nominal dose value set to 1.0.

[0204] Table 1, Statistical information of benchmark tiles

[0205]

[0206] B. Comparison with different model-based OPC methods

[0207] In the first experiment, the present application evaluated the mask quality of RuleLearner and compared it with other model-based OPC methods. The results were compared with the industrial tool Calibre and two state-of-the-art model-based OPC methods that can adapt to metal layer patterns. AccOPC performs mask optimization by inserting single-level SRAFs, and CAMO is a reinforcement learning based OPC method that considers geometric correlations using graph-based mask encoding and performs the serialization of fragment movements by recurrent neural networks (RNN).

[0208] As shown in Table II, RuleLearner demonstrated superior mask optimization performance. Benefiting from more sophisticated SRAF insertion decisions and fragment control mechanisms, the present application reduced the PVB area by 8%, the EPE Sum by 18%, and the maximum EPE value by 15% compared to AccOPC. RuleLearner showed better scalability in handling complex 2D patterns compared to CAMO, which uses a reinforcement learning strategy, reducing the PVB area by 6%, the EPE Sum by 8%, and the maximum EPE length by 21%. Even compared to the commercial tool Calibre, RuleLearner improved by 6%, 3%, and 7% in PVB area, EPE Sum, and maximum EPE, respectively.

[0209] Mask optimization is a complex iterative process that occurs in a highly non-convex space, and different methods require different numbers of iterations to reach optimal results. In addition to mask quality metrics, the inherent adaptive flexibility of RuleLearner accelerated the convergence speed of mask optimization. RuleLearner achieved 1.55x, 2.23x, and 1.51x speedup compared to Calibre, AccOPC, and CAMO, respectively.

[0210] The present application also evaluated the mask segmentation times of different methods, which represent the average number of rectangular segmentations required to accurately reproduce the optimized mask. The results showed that RuleLearner had slightly higher mask segmentation times than other model-based OPC engines, because RuleLearner introduced more complex edge segmentation and SRAF insertion processes to achieve higher wafer image quality.

[0211] Table II and Comparison of Different OPC Engines

[0212]

[0213] C. Comparison with Different Rule Optimization Methods

[0214] In the second experiment, the quality of the rules generated by RuleLearner is evaluated by assessing the quality of the final optimized mask.

[0215] To the best of the inventors' knowledge, this is the first model-based OPC framework that simultaneously considers rule optimization in both SRAF generation and OPC stages. Therefore, the inventors compare the developed RuleLearner with different parameter optimization methods and test the generated rules using the same rule-based SRAF and model-based OPC. The different rule value candidates are manually sampled by experts and the best candidate is chosen based on the experience of parameter tuning. This approach mimics the behavior of experts when choosing the right rule values. Covariance Matrix Adaptation Evolution Strategy (CMA-ES) focuses on adjusting the covariance matrix of the sampling distribution, enabling the algorithm to learn the shape of the objective function landscape. GA-Rule employs a genetic algorithm to optimize SRAF insertion rules. For fair comparison, the inventors re-implement this algorithm and optimize both SRAF and OPC rules. The number of iterations for the above methods is set to 50.

[0216] As shown in Table III, RuleLearner optimizes the rules and achieves the best mask quality in these test cases. Compared to the expert-tuned rules, RuleLearner of the present invention reduces the PVB area by 9%, the EPE Sum by 17%, and the maximum EPE by 11%. Compared to CMA-ES, RuleLearner significantly improves the efficiency of updating the rule values through natural gradient and adaptive evaluation, reducing the PVB area by 7%, the EPE Sum by 12%, and the maximum EPE by 10%. In addition, RuleLearner guides the optimization process using a customized ILT engine, improving the PVB area and EPE Sum by 13% and 24%, respectively, and reducing the maximum EPE by 19% compared to GA-rule.

[0217] The rules generated by RuleLearner also achieve the fastest mask optimization convergence time. Compared to the other listed methods, the mask optimization runtime of the present invention is improved by 58%, 43%, and 47%, respectively. This advantage can be more significant when implemented on large-scale mask fragment sets.

[0218] The number of mask splits for different rule optimization methods is comparable. The difference mainly comes from the chosen fragment length and SRAF insertion decisions.

[0219] Table III and comparison of different rule optimization algorithms

[0220] D. Visualization of lithographic performance

[0221] To visually demonstrate the lithographic performance of the RuleLearner-optimized mask, the present invention compares the optimized mask profile with the final imaging results on a wafer, as shown in FIG. 6. The imaging results of each pattern are generated based on the mask created by the best rule set. The dashed line in the figure represents the design target pattern, the black solid line represents the nominal focus imaging profile of the wafer image, and the red area is the process variation band (PVBand). The present invention shows the lithographic results of three representative metal patterns, which have different geometric complexities and significantly different edge characteristics. Figure 3

[0222] From Figure 3 it can be seen that RuleLearner is able to generate high-quality masks on different patterns, showing a small process variation band (PVBand) and a very high matching degree between the wafer imaging profile and the target pattern. This indicates that RuleLearner can optimize the mask to achieve excellent lithographic performance under different geometric complexity conditions.

[0223] E. Ablation study on CTM engine guidance and adaptive adjustment

[0224] The present invention conducted an ablation study to evaluate the role of the CTM engine in the optimization effect, and the results are shown in Table V. In the first row, the rule values are randomly initialized, and then the same natural evolution optimization process is used. The results clearly show that the rules initialized using the customized CTM engine are superior to the randomly initialized rules, with improvements in PVB area, EPE total length, maximum EPE length, and optimization time.

[0225] In addition, the adaptive adjustment function in the RuleLearner framework was also verified, and the results are shown in the second row of Table IV. Applying adaptive adjustment to RuleLearner can effectively prevent premature convergence and numerical instability during rule optimization, ultimately leading to better mask optimization results.

[0226] Table IV Ablation experiment

[0227]

[0228] V. Conclusion

[0229] ​The present invention focuses on optimizing the rule values in the complex SRAF generation and model-based OPC flow. In the present invention, the present invention proposes RuleLearner, an innovative automated framework that leverages ILT engines to extract latent features and uses a natural evolutionary strategy to optimize rule values. To the best of the present invention's knowledge, this is the first work that can extract rules from small portions of complex designs and generalize them to other portions. Experimental results show that the present invention's method exhibits superior practical performance on more complex metal layers. The present invention believes that the rule extraction strategy implemented in RuleLearner will play a key role in the evolution of modern lithography mask optimization.

[0230] CITATIONS:

[0231] [2] B. Jiang, H. Zhang, J. Yang, and E. F. Young, “A fast machine learning based mask printability predictor for OPC acceleration,” in IEEE / ACM Asia and South Pacific Design Automation Conference (ASPDAC), 2019, pp. 412-419.

[0232] [4] X. Liang, H. Yang, K. Liu, B. Yu, and Y. Ma, “CAMO: Correlationaware mask optimization with modulated reinforcement learning,” arXiv preprint arXiv:2404.00980, 2024.

[0233]

[19] Y. Xu, B. Zhang, C. Wang, W. Wilkinson, and J. Bolton, “The performance improvement of SRAF placement rules using GA optimization,” in Proceedings of SPIE, vol. 9985, 2016.

[0234]

[31] N. Hansen and A. Ostermeier, “Completely derandomized selfadaptation in evolution strategies,” Evolutionary computation, vol. 9, no. 2, pp. 159-195, 2001.

[0235] Based on the above embodiments, the embodiments of the present application further provide a computer program, which, when executed on a computer, causes the computer to perform the method provided by the above embodiments.

[0236] Based on the above embodiments, the embodiments of the present application further provide a computer storage medium, which stores a computer program, and the computer program, when executed by a computer, causes the computer to perform the method provided by the above embodiments.

[0237] The storage medium can be any available medium that can be accessed by a computer. By way of example, and not limitation, such computer-readable media can comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store desired program code in the form of instructions or data structures and that can be accessed by a computer. Computer-readable media can further include, but are not limited to, wireline-based media, such as a telephone line, a cable line, etc., wireless-based media, such as acoustic, RF, infrared, etc., etc.

[0238] Based on the above embodiments, the embodiments of the present application further provide a chip, which is used for reading a computer program stored in a memory, and implements the method provided by the above embodiments.

[0239] Based on the above embodiments, the embodiments of the present application provide a computer program product, which, when executed on an electronic device, implements the method provided by the above embodiments.

[0240] Those skilled in the art will appreciate that embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including, but not limited to, disk memory, CD-ROM, optical memory, etc.) containing computer-usable program code.

[0241] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0242] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instructions which implement the function specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0243] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 one or more flow or blocks

[0244] Obviously, numerous modifications and variations of the present application are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims and their equivalents, the application can be practiced otherwise than as specifically described.

Claims

1. A mask optimization method, characterized in that, include Determine the rule set; The sampled metal layer fragments are processed by the ILT engine to generate a quasi-optimized continuous transmission mask (CTM) for initializing regular distribution values; Optimize the regular distribution value using a photolithographically-aware exponential natural evolution strategy; The optimized regular distribution values ​​were used to test the generation sub-resolution auxiliary features (SRAF) of graphic fragments and the model-based optical proximity correction (OPC). Optimize the regular distribution value using a photolithographically-aware exponential natural evolution strategy, including The regular distribution parameters are optimized using a photolithography-aware natural gradient method, where the regular distribution parameters... The updates are performed in the direction of minimizing the expected lithography cost, while imposing constraints on the information gain at each step; where natural gradient updates are performed through natural exponential expansion. An adaptive function based on relative ranking is shaped to correct the gradient estimate in order to maintain scale invariance and ranking invariance among different graph fragments; wherein, adaptive sampling is performed on the test graph fragment.

2. The mask optimization method according to claim 1, characterized in that, The optimization objectives of the ILT engine include wafer images and target images under nominal conditions. The deviation between them, and the area of ​​the process variation zone; a given regular distribution vector. loss function It is a mask The function is expressed by the following formula: In the formula, Indicates a printed image. Represents the target image. This indicates the mask used to generate the printed image. These are weighting coefficients. Indicates the outermost contour. Indicates the innermost contour; Iterative updates of unconstrained intermediate variables To minimize the loss function The quasi-optimized continuous transmission mask (CTM) is generated, as expressed by the following formula: In the formula, Represents unconstrained intermediate variables. This indicates that the step size parameter is updated iteratively; where, , Indicates from arrive The gradient parameters of the transformation.

3. The mask optimization method according to claim 2, characterized in that, in, Methods for iterative updating of continuous transmission mask (CTM), including After average pooling downsampling, fast optimization is performed at low resolution r_L; Perform double upsampling for detailed optimization at a higher resolution r_H; The optimization is represented by the following formula: The constraints are: ； In the formula, This represents the final optimized continuous projection mask (CTM). Indicates passage The initial high-resolution continuous transmission mask obtained by upsampling. This represents a continuous transmission mask (CTM) obtained through iterative updates at low resolution. This represents a continuous transmission mask (CTM) obtained through iterative updates at high resolution. The function indicates that the mask is interpolated using nearest neighbor interpolation. proportional factor Perform upsampling.

4. The mask optimization method according to claim 1, characterized in that, The natural gradient is formalized as a solution to a constrained lithography regular value optimization problem: In the formula, This indicates that the regular distribution parameters have changed. The expected cost of subsequent photolithography This represents the expected lithography cost under the regular distribution parameters. This indicates the amount of parameter change. This represents the gradient of the loss function with respect to the parameters; The constraints are: In the formula, Indicates The distribution of parameters and with The distance between the distributions of the parameters. Represents arbitrarily small quantities; exist In this case, the natural distance metric is: In the formula, Represents the regular distribution parameters and KL divergence, Given a family of rule parameters, the Fisher information matrix is ​​represented as: In the formula, Let E[·] represent the probability distribution, and let E[·] represent the probability distribution of the regular vector. average By substituting the direction of the natural gradient into the Lagrange multiplier method of the constrained optimization problem, the update step is expressed as: In the formula, Indicates the scale factor; The natural gradient is determined to be... For the same direction, the calculation is as follows: In the formula, The loss function represents the parameters of the regular distribution. The natural gradient Represents the inverse of the Fisher information matrix; Update parameters: In the formula, This indicates the update step size for the parameters of the regular distribution.

5. The mask optimization method according to claim 4, characterized in that, in, The natural gradient is updated using the natural exponential expansion, and the update formula is as follows: In the formula, and These represent the directions of the natural gradient update. Represents the updated regular distribution parameters The representative mean, Represents the updated regular distribution parameters The standard deviation represents; In the natural gradient coordinate system, the Fisher matrix Relative to orthogonal basis It is the identity matrix, and the current search distribution. Encoded as The logarithmic expression for the probability density is: In the formula, This represents the trace operation on matrix Q. This represents the rule parameters that need to be optimized. express The mean of the distribution; For the sample In the formula, Represents the loss function. Represents the identity matrix; Sample generation follows the formula: right Calculation of the natural gradient: In the formula, The loss function representing the current distribution state with respect to the parameters of the regular distribution. The gradient of , where (0,0) represents the current distribution state, n represents the nth sample, and N represents the total number of samples taken in each cycle to update the distribution parameters. Indicates the orthogonal basis gradient, Represents the utility function; right Calculation of the natural gradient: In the formula, express The natural gradient This represents the gradient of the loss function with respect to the orthogonal basis W; Transformation matrix Decompose into step size and satisfaction normalized matrix ; gradient Represented as: gradient Represented as: In the formula, Indicates the total dimension of the parameters; Natural gradient The complexity is calculated.

6. The mask optimization method according to claim 4, characterized in that, Converting photolithography costs into utility value Individuals are ranked according to their cost, where the i-th highest-ranking individual is the one whose cost is in ascending order. ,and It is the rule parameter with the lowest cost. It is the parameter with the highest cost, expressed as: In the formula, This represents the number of population samples taken per cycle to update the distribution parameters. Indicates the parameters of the regular distribution Next, the nth parameter takes The posterior distribution of; Select utility value The utility function is derived from the following formula. express: In the formula, n represents the nth sample.

7. The mask optimization method according to claim 6, characterized in that, in, In adaptive sampling on the test graphic fragment, the layout is divided into overlapping tiles, and the width and height of each tile are calculated: In the formula, and These represent the width and height of the complete layout, respectively. and This indicates the width and height of the clipping tile. and Indicates the stride length in the width and height directions. These represent the number of blocks.

8. An electronic device, the electronic device comprising: One or more processors, a memory, and one or more programs; wherein the one or more programs are stored in the memory, and the one or more programs include instructions that, when executed by the electronic device, cause the electronic device to perform the method of any one of claims 1-7.

9. A computer-readable storage medium comprising a computer program that, when executed on an electronic device, causes the electronic device to perform the method of any one of claims 1-7.

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