Method and system for detecting defects in a photolithography mask from an aerial image and computer implemented method

Abstract

This invention relates to a method (60) for detecting defects (58) in a photomask (14), the method comprising: i. acquiring a spatial image (54) of the photomask (14); ii. obtaining a base design (64) of the photomask (14); iii. generating a rational design (62) of the acquired spatial image (54) by solving an optimization problem that minimizes the deviation between a simulated spatial image (72) of the rational design (62) and the acquired spatial image (54); and iv. detecting defects (58) in the photomask (14) by comparing the base design (64) and the rational design (62). The invention also relates to a corresponding defect detection system.

Description

[Technical Field]

[0001] This invention relates to methods and systems for quality control and quality assurance of photomasks, and more specifically, to a method and a corresponding system for defect detection in spatial images of photomasks. These methods and systems can be used for quantitative measurement, process monitoring, defect detection, and defect verification of photomasks. [Previous Technology]

[0002] Semiconductor manufacturing involves precision operations (such as etching) on ​​materials like silicon or oxides at a very fine scale, down to the nanometer level. Therefore, a quality management process that includes quality assurance and quality control is crucial to ensuring the high quality standards of the manufactured wafers. Quality assurance refers to a series of activities that ensure product quality by preventing any defects that may occur during the development process. Quality control refers to a system for inspecting the final quality of the product. Quality control is part of the quality assurance process.

[0003] A wafer made of a thin silicon sheet serves as a substrate for a microelectronic device, which contains semiconductor structures built into and on the wafer. The semiconductor structure is constructed layer by layer using repeated process steps involving repetitive chemical, mechanical, thermal, and optical processes. The size, shape, and placement of the semiconductor structure and pattern are influenced by a variety of factors. One of the most critical steps is lithography.

[0004] Lithography is a process used to create patterns on a substrate. The pattern to be printed on the surface of the substrate is generated by computer-aided design (CAD). According to the design, a lithography mask is generated for each layer, containing an enlarged image of the computer-generated pattern to be etched into the substrate. The lithography mask can be further adjusted, for example, by optical proximity correction techniques. In the printing process, an illumination image projected from the lithography mask is focused onto a photoresist film formed on the substrate. A semiconductor chip that powers a mobile phone or tablet computer includes, for example, approximately 80 to 120 patterned layers.

[0005] As the integrated density of the semiconductor industry continues to increase, lithography masks must image increasingly smaller structures onto wafers. The aspect ratios and number of layers in integrated circuits are constantly increasing, and structures are increasingly moving towards a third (vertical) dimension. Currently, the height of memory stacks exceeds tens of micrometers. In contrast, feature sizes are becoming smaller. The minimum feature size, or critical size, is below 10 nm, such as 7 nm or 5 nm, and will approach feature sizes below 3 nm in the near future. While the complexity and size of semiconductor structures are evolving into a third dimension, the lateral dimensions of integrated semiconductor structures are becoming smaller. Producing small structure sizes imaged onto wafers requires lithography masks or templates for nanoimprint lithography with small structural or patterned elements. Therefore, the manufacturing processes for lithography masks and templates used in nanoimprint lithography are becoming increasingly complex, time-consuming, and ultimately more expensive. With the advent of EUV lithography scanners, the nature of the mask has changed from transmissive patterning to reflective patterning.

[0006] Due to the minute structural dimensions of the pattern elements in lithography masks or templates, it is impossible to rule out errors during mask or template production. For example, defects may arise from lithography mask degradation or particulate contamination. Of the various defects occurring in semiconductor structure manufacturing processes, lithography-related defects account for nearly half of all defects. Therefore, lithography mask inspection, verification, and measurement play a crucial role in monitoring system defects in semiconductor process control. Defects detected in the quality assurance process can be used for root cause analysis, such as modifying or repairing the lithography mask. These defects can also serve as feedback to improve process parameters in the manufacturing process, such as exposure time and focus variation.

[0007] Every defect in a lithography mask can cause malfunctions in the produced wafer, or even serious damage to the wafer. Therefore, every defect must be detected and repaired, if possible and necessary. Thus, reliable and rapid defect detection methods are crucial for lithography masks.

[0008] To analyze large amounts of data requiring extensive measurements, machine learning methods can be used. Machine learning is a field of artificial intelligence. Machine learning methods typically build parametric machine learning models based on training data consisting of a large number of samples. After training, the method can generalize the knowledge gained from the training data to new samples that have not been encountered before, thereby making predictions on new data. There are many machine learning methods, such as linear regression, k-means, support vector machines, decision trees, random forests, neural networks, or deep learning methods.

[0009] Deep learning is a type of machine learning that uses artificial neural networks with many hidden layers between the input and output layers. Due to this complex internal structure, the network is able to progressively extract higher-level features from the raw input data. Each layer learns to transform its input data into a slightly more abstract and complex representation, thereby acquiring low-order and high-order knowledge from the training data. Hidden layers can have different sizes and tasks, such as convolutional layers or pooling layers.

[0010] Methods for automatically detecting defects in lithography masks include defect detection algorithms, which are typically based on a die-to-die or die-to-database principle. The die-to-die principle compares an imaging dataset of a lithography mask with a reference dataset, which includes identical structures obtained from different portions of the same lithography mask or from different lithography masks. Detected discrepancies are considered defects. However, this method requires usability and time-consuming scanning of the two corresponding portions of the lithography mask to accurately determine their relative positions. Furthermore, the method fails if duplicate defects are found.

[0011] The bare-chip to database principle compares an imaging dataset of a lithography mask with a reference dataset in a database (e.g., a simulated image, a design, a CAD file, or a model of a lithography mask) to detect deviations from ideal data. Due to significant differences, unexpected patterns are detected in the image dataset. It can handle recurring defects. However, comparing an image dataset (especially a spatial image) with a reference dataset is very difficult due to differences in image acquisition parameters, styles, and quality.

[0012] Therefore, one object of the present invention is to improve the accuracy of lithography mask defect detection based on a bare-die to database method. Specifically, the object is to improve the comparison accuracy between a spatial image and a corresponding design of a lithography mask. Another object of the present invention is to reduce the computation time for defect detection in lithography masks.

[0013] These objectives are achieved by means of the invention specified in the independent claim, and advantageous embodiments and further developments of the invention are described in detail in the appendix claim. [Summary of the Invention]

[0014] Specific embodiments of the present invention relate to a method and system for detecting defects in spatial images of lithography masks.

[0015] A first embodiment relates to a method for detecting defects in a lithography mask, the method comprising: i. acquiring a spatial image of the lithography mask using an optical system; ii. obtaining a base design of the lithography mask; iii. generating a rational design of the acquired spatial image by solving an optimization problem that minimizes the deviation between a simulated spatial image of the rational design and the acquired spatial image, wherein the simulated spatial image simulates the application of the optical system in the rational design; and iv. detecting defects in the lithography mask by comparing the base design and the rational design.

[0016] Another specific embodiment relates to a method for detecting defects in a lithography mask, the method comprising: i. acquiring a spatial image of the lithography mask using an optical system; ii. obtaining a base design of the lithography mask; iii. generating a rational design by solving an optimization problem such that the deviation between a simulated spatial image of the rational design and the acquired spatial image is less than the deviation between a simulated spatial image of the base design and the acquired spatial image, wherein the simulated spatial image of the rational design simulates the application of the optical system in the rational design, and wherein the simulated spatial image of the base design simulates the application of the optical system in the base design; and iv. detecting defects in the lithography mask by comparing the base design and the rational design.

[0017] The term "defect" refers to a local deviation of an integrated circuit pattern in a photomask or on a wafer from a predefined norm of the integrated circuit pattern. For example, a defect in an integrated circuit pattern of a semiconductor structure can cause a failure of a related semiconductor device. For example, based on the detected defect, the photolithography process can be improved, or the photomask or wafer can be repaired or discarded. The norm of the integrated circuit pattern can be defined by one or more corresponding references or reference sets (e.g., design data sets, simulation data sets, or acquired defect-free data sets).

[0018] The lithography mask can have an aspect ratio between 1:1 and 1:4, preferably between 1:1 and 1:2, and most preferably 1:1 or 1:2. The lithography mask can have a nearly rectangular shape. The lithography mask is preferably 5 to 7 inches long and wide, and most preferably 6 inches long and wide. Alternatively, the lithography mask can be 5 to 7 inches long and 10 to 14 inches wide, preferably 6 inches long and 12 inches wide.

[0019] An optimization problem involves an objective function to be maximized or minimized. The optimization problem may also include constraints. Solving an optimization problem means applying some mathematical optimization method. A mathematical optimization method calculates a point where the objective function value is expected to be better than the objective function values ​​at several other points. Solving an optimization problem may, for example, mean calculating a global optimum or a local optimum of the objective function. Mathematical optimization methods may include calculating an analytical solution or applying an iterative method, such as gradient descent, a simplex method, a variational method, a combinatorial optimization method, etc. Iterative methods may use an initial solution and iteratively adjust the initial solution until a convergence criterion is met or a predetermined number of iterations is reached. Mathematical optimization methods may also include calculating the output of a model trained to optimize an objective function for a given input; for example, applying a machine learning model trained to minimize a loss function to the input. Thus, the model is trained to directly predict the solution to an optimization problem.

[0020] A spatial image indicates the radiation intensity distribution of a lithography system within a wafer plane under a given lithography mask. Therefore, the spatial image is used to simulate the structure of the wafer surface when a wafer is printed using a lithography mask in a lithography system. A wafer plane refers to a plane within the resist on the top of the wafer in a lithography system. A spatial image can also be generated by applying a mask inspection system or an optical mask verification system to a lithography mask. A spatial image can also be simulated using a lithography mask design and a spatial image simulation method.

[0021] A spatial image can refer to a spatial image of a complete lithography mask, or it can refer to a spatial image of a cross-section of a lithography mask. A design can refer to the design of a complete lithography mask, or it can refer to the design of a cross-section of a lithography mask.

[0022] An optical system refers to a system that uses light to inspect a lithography mask or wafer, or to generate an integrated circuit pattern on a wafer. Optical systems include, for example, inspection systems, optical mask authentication systems, lithography systems, and metrology systems.

[0023] A lithography system refers to a system used to manufacture integrated circuits. For this purpose, during the lithography process, light shines on a lithography mask, and an integrated circuit pattern is transferred onto a substrate via the lithography mask through a photosensitive chemical process.

[0024] An inspection system is an optical system that detects defects in a photomask by acquiring a spatial image of the photomask.

[0025] An optical mask verification system is a system used to acquire a spatial image of a portion of a mask (particularly using a potential defect detected by an inspection system). The optical mask verification system simulates the settings of a lithography system, such as illumination and imaging parameters, to examine the impact of a potential defect on a printed wafer, to verify whether the lithography mask is defect-free or whether a repair attempt is successful.

[0026] Parameters describing an optical system include, for example, - illumination parameters describing the illumination settings of the lithography system, including the distribution and intensity of different illumination angles, such as a ring illumination setting, a dipole illumination setting, a quasar illumination setting, etc.; - imaging parameters, such as the numerical aperture of the lithography system and the magnification, occlusion, aberration, apodization, or distortion of the lithography system; - design parameters, such as parameters describing the material of the lithography mask, such as layer thickness, refractive index of different layers, etc.

[0027] A design of a lithography mask refers to a representation of a lithography mask or a cross-section thereof. This design may include, for example, a computer-readable file (e.g., a CAD file or a GDS file), a technical drawing, or a set of polygons representing the structure of the lithography mask or a cross-section thereof. A design of a lithography mask may include material information, such as the complex refractive index, dielectric constant, permeability, or derived representations thereof of the materials contained in the lithography mask. A design of a lithography mask may include parameters describing the structural dimensions of the lithography mask, such as the thickness of each layer in the multilayer structure of an EUV mask, the thickness of the absorption layer, or the dimensions of the absorption structure.

[0028] A design of a lithography mask can include parameters describing the location of structures within the lithography mask, such as the location of an absorbing structure or the location of layers in a multilayer structure. A design of a lithography mask can include parameters describing the shape of structures within the lithography mask, such as the shape of an absorbing structure (e.g., sidewall angles or fillets). A design of a lithography mask can include an image, such as a 2D image or a 3D image (e.g., a single pixel volume or multiple 2D slices of a volume), representing the properties of the lithography mask. The image can contain one, two, or more channels. The image can include image elements, such as pixels or voxels. The properties of the lithography mask can include material properties, such as refractive index, dielectric constant, permeability, or derivative representations thereof. A design of a lithography mask can include a description of the internal structure of the lithography mask, such as curves, contours, polygons, splines, non-uniform rational B-splines (NURBS), Bézier curves, etc.

[0029] A design can refer to the design of a complete lithography mask, or it can refer to the design of a cross section of a lithography mask.

[0030] A “low-level design” for a lithography mask refers to a design that describes the structure of the lithography mask, such as a model, like a CAD model. A low-level design can be provided for a lithography mask, or can be derived from an image of the lithography mask, for example, using image processing or machine learning methods.

[0031] A “rational design” for a spatial image refers to a design that can be used by a spatial image simulation method to simulate a spatial image. If the spatial image is a rational result obtained by a spatial image simulation method applied to a rational design, then the design is a rational design for the spatial image. A rational design can be a base design of a lithography mask that can be used to generate a spatial image, for example, using a spatial image measurement system for a set of parameters.

[0032] A simulated spatial image can be simulated using a spatial image simulation method. The spatial image simulation method simulates the generation of a spatial image of a lithography mask based on the design of the lithography mask in the optical system. The spatial image simulation method can use physically based models, such as lithography mask models and / or electromagnetic wave propagation models through the lithography mask. The spatial image simulation method can also use non-physical models, such as machine learning models trained using training data.

[0033] In a preferred example, the underlying design and the overall design are represented in a vector format. A design in a vector format represents the structure of the design using continuous coordinates and the connections between them. Compared to a raster image, which has limited precision due to pixel size, the advantage of using a vector format is that the structure in the design can be represented very accurately due to the continuous coordinates. Furthermore, measurements of the structure, such as the distances between structures, can be determined more precisely.

[0034] In another example, the underlying design and rational design are represented by a non-binary image. A non-binary image can be used to represent structures in a lithography mask with sub-pixel precision. Thus, structural boundaries can, for example, be located between pixels. Therefore, the accuracy of the method is improved.

[0035] According to one aspect of the present invention, the underlying design of the lithography mask is generated from the acquired spatial image, for example, using image processing or machine learning. Therefore, this defect detection method can be used even without the underlying design of the lithography mask.

[0036] In a preferred example, in step iv, defects in the lithography mask are detected by comparing the underlying design with the rational design in a mathematical space (e.g., a Fourier space, a polygonal space, or a wavelet space). For this purpose, the underlying design and the rational design can be transformed into a mathematical space. Alternatively, the underlying design and the rational design can be directly represented in a mathematical space. A distance measurement can be defined in the mathematical space to measure a distance between the underlying design and the rational design. In this way, the comparison process can be simplified and the accuracy of the distance measurement can be improved. Furthermore, the comparison can be targeted at or limited to specific details in the design that can be particularly well represented in a particular mathematical space.

[0037] According to an example of the present invention, solving the optimization problem in step iii includes applying a machine learning model to the acquired spatial imagery, wherein the machine learning model is trained to map the acquired spatial imagery to a rational design of the acquired spatial imagery. This machine learning model is capable of accurately and quickly calculating the rational design from the acquired spatial imagery. The accuracy is determined by the selected training data. Since only one forward pass is required for the trained machine learning model during inference, the computation time is very fast.

[0038] According to an example of the present invention, solving the optimization problem in step iii includes generating a reasonable design by modifying the underlying design to minimize the deviation between the acquired spatial image and a simulated spatial image, wherein the simulated spatial image is obtained by applying a spatial image simulation method to a design. The design that minimizes the deviation between the corresponding simulated spatial image and the acquired spatial image is the reasonable design obtained by solving the optimization problem. By solving an optimization problem in this way to generate a reasonable design, the accuracy of the reasonable design is improved by minimizing the deviation between the simulated spatial image and the acquired spatial image. In this way, a reasonable design that can be used to approximately generate the acquired spatial image can be obtained.

[0039] In a preferred example, a parametric representation of the underlying design is optimized by solving an optimization problem. This allows for a flexible representation of the underlying design, which can be modified to achieve a reasonable design. In one example, the parametric representation describes the structural boundaries of the underlying design, such as geometry, contours, or polygons. Thus, a particularly simple, flexible, and low-complexity parametric representation can be obtained, which is well-suited for optimization. Specifically, the parametric representation can include contours represented graphically by nodes and edges. The positions of the nodes and edges can be optimized by solving an optimization problem.

[0040] According to one aspect of the first embodiment, the optimization problem includes modified parameters describing the underlying design, and the optimization problem imposes a sparsity constraint on these parameters. Thus, the accuracy of the optimization parameters and the accuracy of the rational design are improved. Furthermore, the rational design is less susceptible to noise in the acquired spatial imagery.

[0041] In one example, the space image simulation method includes using a physical model to generate a space image from a rational design. Using a physical model allows for high-precision simulations that follow physical laws.

[0042] According to one example, this space image simulation method includes a machine learning model. In this way, the space image simulation method is improved because the machine learning model can directly and automatically learn important correlations from the training data without manually writing rule-based and often error-prone programs. Furthermore, this machine learning model can also be used to improve the accuracy of other methods used in the space image simulation method. Therefore, it allows for the generation of space imagery based on a reasonable design using methods with lower complexity and shorter computation time.

[0043] In a preferred example, this spatial image simulation method includes a physical model and a machine learning model for generating a spatial image from a rational design. The machine learning model can then be applied to the physical model. In this way, the machine learning model can improve the accuracy of the physical model's output. Meanwhile, since the results will be further processed by the machine learning model, a less complex physical model can be used.

[0044] In one example, this spatial image simulation method includes a machine learning model trained to map a design to a spatial image. Because the machine learning model learns important correlations directly from the training data, this spatial image simulation method has high accuracy. Furthermore, because only one forward pass is required during inference, the computation time is short.

[0045] According to a preferred embodiment of the invention (hereinafter referred to as not quite rigorous (NQR)), a spatial image simulation method in an optical system (particularly in an inspection system, a lithography system, or an optical mask identification system) generates a spatial image from a rational design by irradiating a corresponding lithography mask with incident electromagnetic waves, and includes: a) approximating the propagation of the incident electromagnetic waves in a first region of a lithography mask comprising multiple structures; b) analyzing or numerically simulating the propagation of the electromagnetic waves simulated in step a) in a second region of the lithography mask; c) simulating a representation of a rationally designed electromagnetic near field by propagating the electromagnetic waves simulated in step b) to a near-field plane; and d) generating a spatial image from a rational design by applying a simulation of an imaging process of the optical system to the representation of the electromagnetic near field. For example, a first region of an EUV lithography mask may contain absorbing and non-absorbing structures, while a second region of an EUV lithography mask may contain multilayer structures, as described below. Compared with standard spatial image simulation methods, this spatial image simulation method can generate spatial images with higher accuracy and shorter computation time, thereby improving the accuracy of defect detection.

[0046] The calculation methods for the electromagnetic near field differ between the first and second zones of the lithography mask. In the first zone, several assumptions can be made in the lithography setup, thereby simplifying and rapidly calculating the propagation of electromagnetic waves within the first zone. The electromagnetic wave propagation in the first zone is calculated using a wave propagation method that considers the inhomogeneity of the medium within the first zone of the lithography mask. In the second zone, the propagation of electromagnetic waves is calculated analytically or numerically. In this way, a highly accurate approximation of the electromagnetic wave propagation within the lithography mask can be obtained, requiring several orders of magnitude less computation time than rigorous simulation methods. Therefore, simulating the electromagnetic near field and spatial imaging becomes feasible in industrial applications.

[0047] Preferably, the lithography mask is designed to describe the lithography mask at least partially in a dimension orthogonal to a base plane of the lithography mask. The lithography mask design can include one or more different regions or portions thereof, such as a first region and / or a second region. The one or more different regions can be arranged at different depths relative to the surface normal.

[0048] The first region of the lithography mask includes multiple structures. These structures can be arranged according to a design that determines the pattern imprinted on the wafer during the printing process. This design can include structures and unstructures, particularly absorbing and non-absorbing structures. The second region of the lithography mask can contain a mask carrier, which can include one or more layers of one or more materials. These structures and unstructures can be deposited on the mask carrier. The mask carrier can include a substrate layer. The second region can be configured to transmit incident electromagnetic waves (for transmissive lithography masks) or to reflect incident electromagnetic waves (for reflective lithography masks). The first region can be directly adjacent to the second region of the lithography mask. Therefore, the first and second regions can have a common boundary, such as a boundary plane. The mask carrier in the lithography mask can be defined by the boundary plane and a base plane. The boundary plane can be a surface plane of the mask carrier. The base plane is preferably parallel to the boundary plane. The base plane can define the second region from the outside. It can form an interface between the photomask carrier and the exterior of the lithography mask, through which electromagnetic waves propagate. The structure of the first region of the lithography mask can be defined by a boundary plane and a structural plane. The structural plane can define the first region of the lithography mask from the exterior. The structural plane can contain a portion of the structural surface facing away from the boundary plane. Preferably, the structural plane is parallel to the boundary plane. The first region of the lithography mask can extend between the structural plane and the boundary plane and is defined by these planes. The second region of the lithography mask can extend between the boundary plane and the base plane. It can be defined by the boundary plane and the base plane. The second region can contain a stack of homogeneous parallel layers. Homogeneity means that the material properties remain unchanged within a single layer. Other configurations of the lithography mask containing a first region and a second region can also be used.

[0049] An electromagnetic near-field represents the distribution of electromagnetic waves in a near-field plane. The near-field plane can be located next to a structural plane of a photomask that defines a first region of the photomask from the outside. Preferably, the near-field plane is parallel to the structural plane of the photomask. The near-field plane can be located anywhere between the structural plane and a wafer plane; for example, the near-field plane can be located at a distance of 0 to 1000 nm from the structural plane, preferably at a distance of 0 to 100 nm, more preferably at a distance of 0 to 50 nm, even more preferably at a distance of 0 to 20 nm, and most preferably at a distance of 0 to 10 nm. In a preferred embodiment of the invention, the near-field plane and the structural plane are the same. In principle, the near-field plane can also be located within the first region, the second region, on the structural plane, on the base plane, or outside the lithography mask, or on the side of the base plane of the lithography mask, for example, when electromagnetic waves propagate back to the lithography mask after passing through the first region.

[0050] A representation of an electromagnetic (near) field can refer to a (complex) electric field E or a (complex) scattered electric field, where E represents the incident electric field. A complex electromagnetic field can be represented by, for example, real and imaginary parts, or amplitude and phase. A representation of an electromagnetic field can refer to a (complex) magnetic field H or a (complex) scattered magnetic field, where H represents the incident magnetic field. A representation of an electromagnetic field can include measurements derived from the electromagnetic field, such as diffraction order, spectrum, far field, or intensity field. A representation of an electromagnetic field within a lithography mask can refer to the electromagnetic field within the lithography mask, a region of the electromagnetic field within the lithography mask, or an electromagnetic field near the lithography mask (e.g., a near field). A representation of an electromagnetic field can include representations of electromagnetic fields in different spatial directions. For example, the representation of electromagnetic fields can include 2D or 3D images with one, two or more channels, such that the 2D or 3D images include a representation of the electromagnetic field in each spatial direction, for example, complex electric fields in the x and y directions or the x, y and z directions produce 2D or 3D images with four or six channels.

[0051] According to a first example of a specific embodiment, the propagation of the incident electromagnetic wave in the first region of the photomask in step a) is approximated using a Helmholtz equation. This simplifies the approximation process, thereby reducing complexity and computation time.

[0052] According to a second example of a specific embodiment, the propagation of the incident electromagnetic wave in the first region of the lithography mask in step a) is approximated using a machine learning model. By using a machine learning model, computation time can be significantly reduced because, after training, a single fast forward pass is sufficient to calculate the propagation of the incident electromagnetic wave.

[0053] According to a state example of the first specific embodiment, a forward Helmholtz equation is used to approximate the Helmholtz equation. This simplifies the approximation process, thereby reducing complexity and computation time.

[0054] The forward Helmholtz equations can be solved using the one-beam propagation method. This simplifies the approximation process, thereby reducing complexity and computation time.

[0055] In a preferred embodiment, the forward Helmholtz equations are solved using a wave propagation method that approximates the propagation of electromagnetic waves through a non-uniform medium. By using the wave propagation method, the forward Helmholtz equations can be solved quickly, thereby shortening the computation time of this method. Furthermore, by considering the inhomogeneities of the first region of the lithography mask (e.g., inhomogeneities caused by differences in materials in the absorbing and non-absorbing structures), wave propagation can be simulated with high accuracy.

[0056] According to one aspect of the first embodiment, after step i, one or more regions of interest containing potential defect candidates are identified in the acquired spatial image, and steps ii. to iv. apply only to these one or more regions of interest. In this way, the computation time of the defect detection method can be significantly reduced because most of a lithography mask typically does not contain any defects, and a fast defect detection method with a low false alarm rate can exclude them from further investigation. Alternatively, the region of interest can be defined using user input. In subsequent processing steps, the detected potential defects can be examined using the method according to the invention. In this way, the accuracy of the defect detection method can also be improved.

[0057] A computer implementation method for training a machine learning model according to a second embodiment of the present invention includes training any machine learning model according to an example or pattern of the first embodiment of the present invention.

[0058] A system for detecting defects in a photomask according to a third embodiment of the present invention includes: an optical system for acquiring a spatial image of the photomask; and a data analysis device including at least one memory and at least one processor, configured to perform the steps of a method for detecting defects in a photomask according to any example or form of the first embodiment.

[0059] The present invention described through examples and specific embodiments is not limited to the specific embodiments and examples, but can be implemented by those skilled in the art through various combinations or modifications of the examples and specific embodiments.

Implementation Method

[0078] Next, advantageous exemplary embodiments of the invention will be described and schematically illustrated in the accompanying drawings. Throughout the drawings and description, the same reference numerals are used to describe the same features or components. Dashed lines indicate optional features.

[0079] The methods and systems described herein can be used with a variety of lithography systems, such as the transmissive lithography system 10 or the reflective lithography system 10'.

[0080] Figure 1 illustrates an exemplary transmissive lithography system 10, such as a DUV lithography system. The main components are a radiation source 12, which may be a deep ultraviolet (DUV) excimer laser source; imaging optics, which, for example, define partial coherence and are capable of including optics for shaping radiation from the radiation source 12; a lithography mask 14; an illumination optics 16 illuminating the lithography mask 14; and a projection optics 17 projecting an image designed by the lithography mask onto a wafer plane 18. An adjustable filter or aperture at the pupil plane of the projection optics 17 limits the range of beam angles striking the wafer plane 18, where the maximum possible angle defines the numerical aperture NA of the projection optics as NA = n sin(Gmax), where n is the refractive index of the medium between the substrate and the last element of the projection optics 17, and Gmax is the maximum angle of the beam emitted from the projection optics 17 that can still strike the wafer plane 18.

[0081] In this document, the terms “radiation” or “beam” are used to cover all types of electromagnetic radiation, including ultraviolet radiation (e.g., a wavelength having a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultraviolet radiation, e.g., a wavelength having a wavelength in the range of about 3-100 nm).

[0082] The illumination optics 16 may include optical components for shaping, adjusting, and / or projecting radiation from the radiation source 12 before the radiation passes through the lithography mask 14. The projection optics 17 may include optical components for shaping, adjusting, and / or projecting radiation after it has passed through the lithography mask 14. The illumination optics 16 does not include the light source 12, and the projection optics does not include the lithography mask 14.

[0083] The illumination optics 16 and the projection optics 17 may include various types of optical systems, including, for example, refractive optics, reflective optics, aperture and catadioptric optics. The illumination optics 16 and the projection optics 17 may also include components that operate according to any of these design types for guiding, shaping or controlling the projected beam of radiation, either jointly or individually.

[0084] Figure 2 illustrates an exemplary reflective lithography system 10', such as an extreme ultraviolet (EUV) lithography system. The main components are a radiation source 12, which may be a laser plasma source; an illumination optics 16, which, for example, defines partial coherence and may include optics for shaping radiation from the radiation source 12; a lithography mask 14; and a projection optics 17 that projects an image of the lithography mask design onto a wafer plane 18. An adjustable filter or aperture at the pupil plane of the projection optics 17 limits the range of beam angles striking the wafer plane 18, where the maximum possible angle defines the numerical aperture NA of the projection optics as NA = n sin(Gmax), where n is the refractive index of the medium between the substrate and the last element of the projection optics 17, and Gmax is the maximum angle of the beam emitted from the projection optics 17 that can still strike the wafer plane 18.

[0085] Figure 3 illustrates a die-to-database method for defect detection, which compares a spatial image 54 of a lithography mask 14 with a design 56 of the corresponding lithography mask 14 to detect defects 58. However, the spatial image 54 and the design 56 have very different appearances, making direct comparison difficult, and the detected defects 58 are not very reliable.

[0086] To improve the reliability of defect detection, FIG4 shows a flowchart of a method 60 for detecting defects in a lithography mask according to a first embodiment of the present invention. The method includes: in step M1, acquiring a spatial image of the lithography mask using an optical system; in step M2, obtaining a base design of the lithography mask; in step M3, generating a rational design of the acquired spatial image by solving an optimization problem that minimizes the deviation between a simulated spatial image of the rational design and the acquired spatial image, wherein the simulated spatial image simulates the application of the optical system in the rational design; and in step M4, detecting defects in the lithography mask by comparing the base design and the rational design. After optimization, the deviation between the simulated spatial image of the rational design and the acquired spatial image is smaller than the deviation between the simulated spatial image of the base design and the acquired spatial image.

[0087] Figure 5 illustrates an exemplary application of this method. In step M1, a spatial image 54 of a lithography mask 14 is obtained. For example, a spatial image measurement system can be used to acquire the spatial image 54, which simulates the intensity distribution in a wafer plane when an incident electromagnetic wave irradiates a lithography mask. A metrology system can also be used to acquire the spatial image 54.

[0088] In step M2, a low-level design 64 of the lithography mask 14 is obtained. For example, the low-level design 64 can be loaded from a database, obtained from a designer of the lithography mask, or derived from a spatial image of the lithography mask using image processing methods.

[0089] A rational design 62 is used to generate the acquired spatial image 54, such that the acquired spatial image 54 is a rational result of applying a spatial image simulation method to the rational design 62 in step M3. Therefore, the rational design 62 is generated in such a way that the acquired spatial image 54 can be generated from the rational design 62 by a certain spatial image acquisition method or system and a set of corresponding parameters.

[0090] In step M4, the rational design 62 is compared with a base design 64 of the lithography mask 14. For example, a difference 66 between the base design and the rational design can be calculated, for example, if the base design and the rational design are represented as images (e.g., non-binary images), then a difference image is used; or for example, if the base design and the rational design are represented as vectors, then a difference vector is used. The location of the difference 66 corresponds to the location of the defect 58 in the lithography mask. A machine learning model can be used to detect the defect 58 directly from the base design 64 and the rational design 62, or from a representation indicating the difference between the base design and the rational design (e.g., from a difference image or difference vector).

[0091] According to an example of the first embodiment, in step iv, defects 58 in the lithography mask 14 are detected by comparing the underlying design 64 and the rational design 62 in a mathematical space. For this purpose, the underlying design 64 and the rational design 62 can be transformed to different mathematical spaces, such as a Fourier space, a polygonal space, a wavelet space, a CAD space, etc. Alternatively, the underlying design 64 and the rational design 62 can be directly represented in the mathematical space without transformation. A distance measurement can be defined in the mathematical space to measure the deviation between the underlying design and the rational design.

[0092] Various methods for quantitatively measuring the distance between contours or shapes in a mathematical space are known in the literature. For example, in a polygonal or CAD space, the deviation between two contours, one a base design 64 and the other a rational design 62, can be measured in different ways. For example, a bending energy measurement measures the energy required to transform one contour into another. For example, the deviation between two contours can be measured by calculating the maximum, average, or median distance between each contour point of the first contour and the nearest contour point of the second contour. Alternatively, a Hausdorff distance can be used to measure the deviation between two contours. Alternatively, regions defined by the contours, such as overlapping or non-overlapping regions, can be compared to measure the contour deviation. If the base design 64 and the rational design 62 are represented by images (e.g., non-binary images), the overlapping or non-overlapping regions of the structures can be used to measure the deviation. The distance between contours or shapes can be measured in Fourier space, for example, by calculating a difference vector of Fourier coefficients. Distance measurements in other mathematical spaces can be defined relative to a base (e.g., a wavelet base or a master component base). For example, the difference vector of coefficients relative to the base can be used as a distance measurement. The underlying design 64 and / or the rational design can also be converted into a profile or region and compared using any of the above measurements.

[0093] In a preferred example of the first embodiment, the underlying design 64 and the rational design 62 are represented in the same way, for example, they share the same format, representation type, color or grayscale value space, etc. By using the same representation, the comparability of the designs can be ensured, thereby obtaining improved comparison and defect detection results. Each design type has different advantages, thereby achieving a more accurate comparison.

[0094] For example, the underlying design 64 and the rational design 62 can be represented using a vector format. The vector format is based on the mathematical principles of coordinate geometry, where a shape is defined as a set of points in a two- or three-dimensional Cartesian coordinate system. Since almost all shapes consist of an infinite number of points, the vector format defines a finite set of geometric primitives that can be specified using a finite set of salient points (called vertices). For example, a square can be explicitly defined by the positions of three of its four corners, and the software can interpolate the boundary lines and interior space based on these corner positions. Since a square is a regular shape, it can also be defined by the position of one corner, a size, and a rotation angle. The basic geometric primitives of vector formats include points, line segments, polygons, two-dimensional or three-dimensional parametric shapes (e.g., circles, ellipses, squares, spheres, hyperellipses, etc.), parametric curves (where polylines or polygons are parametrically enhanced to define a nonlinear interpolation between vertices, such as circular arcs, cubic splines, Bézier curves, etc.), and three-dimensional surfaces that are usually defined as a set of connected polygons or parametric surfaces, such as polygonal meshes or non-uniform rational basis splines (NURBS).

[0095] In another example, the underlying design 64 and the rational design 62 are represented by non-binary images. The underlying design 64 and the rational design 62 can also be represented by images sharing the same color or grayscale value space or range. In another example, the underlying design 64 and the rational design 62 are represented in a CAD format, such as gdsll, oasis, svg, dxf, etc. The underlying design 64 and the rational design 62 can be represented by geometric structures, such as polygons, circles, ellipses, outlines, etc. Geometric structures can be described using coordinates (e.g., corner points or center points, length, angles, directions, axes, etc.). The underlying design 64 and the rational design 62 can be represented by outlines depicting the design structure, such as straight lines, curves, or graphics containing nodes and edges. In another example, the underlying design 64 and the rational design 62 are represented by Fourier descriptors.

[0096] The process of obtaining the underlying design and rational design from spatial images will be described in detail below.

[0097] The underlying design is preferably provided together with the lithography mask. According to an example of the first specific embodiment, the underlying design of the lithography mask can also be generated from an image of the lithography mask (especially a spatial image, especially a defect-free spatial image) or a golden reference image of the lithography mask. For this purpose, image processing methods, such as image segmentation, pattern matching, thresholding, contour extraction, edge detection, etc., can be used. These methods can, for example, be used to find structures or patterns in the image (especially the spatial image) that correspond to structures (e.g., repeating structures) in the underlying design. These structures can, for example, be represented by polygons in the underlying design; for example, a circle can represent a memory hole, etc. In addition to image processing methods, machine learning methods can also be used to map an image (especially a spatial image) to an underlying design. The machine learning methods can be trained using paired images of the lithography mask (especially spatial images) and their corresponding underlying designs (in the desired representation) as training data. Deep learning methods (e.g., CNN, U-Net, GAN) and models containing attention mechanisms (e.g., transformer, diffusion model, etc.) can achieve particularly good results.

[0098] A reasonable design that can acquire spatial images of a place through different methods.

[0099] According to an example illustrated in Figures 6a and 6b, solving the optimization problem in step iii involves applying a machine learning model 70 to the acquired spatial image 54, wherein the machine learning model 70 is trained to map the acquired spatial image 54 to a rational design 62 (e.g., represented as a non-binary image) of the acquired spatial image 54. Therefore, the acquired spatial image 54 is a rational result of applying a spatial image simulation method to the rational design 62. Here, solving the optimization problem involves applying a mathematical optimization method, which includes computing the output of a model trained to optimize an objective function for a given input. As shown in Figure 6a, the machine learning model 70 can be trained using a training image 68, which includes a lithography mask 14 and correspondingly designed pairs of acquired spatial images 54 or simulated spatial images.

[0100] For example, the training images may include the acquired spatial image 54 or the simulated spatial image and its underlying design 64. Preferably, at least a portion of the design in the training image 68 contains uncommon structures. Uncommon structures are generally not part of a design, such as defects, design deviations that may not be classified as defects (e.g., subtle variations in the design structure, such as variations in the thickness of the structure, rounded corners, etc.), or structures of unexpected types (e.g., if the training images contain designs that only contain lines and space, uncommon structures may include holes, intersections, auxiliary features, complex polygons, etc.).

[0101] Designs containing uncommon structures can be obtained by selecting a base design that already contains an acquired or simulated spatial image of the uncommon structure, or by manually introducing the uncommon structure into a design. A spatial image simulation method can be used to simulate a spatial image corresponding to the modified design from the modified design. In this way, the location, size, intensity, type, frequency, etc., of uncommon structures such as defects can be controlled. Therefore, pairs of designs containing uncommon structures and their corresponding spatial images can be obtained as training data 68 for the machine learning model 70.

[0102] The objective function optimized during training can include the deviation between the predicted rational design 62 and the corresponding design. The objective function can also include the deviation between the acquired spatial image and the simulated spatial image obtained by applying a spatial image simulation method to the corresponding design. The machine learning model can use the underlying design 64 as additional input. In this case, the machine learning model can be trained using the spatial image and the perturbed underlying design as input, and output with the unperturbed underlying design. The perturbation can model, for example, line edge roughness or local defects.

[0103] Machine learning models can include neural networks, particularly deep neural networks, such as a CNN or U-Net. These neural networks can include one or more attention mechanisms to learn connections between different regions of an acquired spatial image 54, thereby improving the results. Machine learning models can also include random forests, support vector machines, decision trees, clustering methods, etc.

[0104] As shown in Figure 6b, during the inference process, the machine learning model 70 directly maps an acquired spatial image 54 to a reasonable design 62 of the acquired spatial image 54. Since this only requires one forward pass, this mapping process can be completed very quickly. By using, for example, a GPU or TPU that supports parallelization to implement the machine learning model 70, the runtime of learning and inference can be significantly shortened.

[0105] According to the examples shown in Figures 7a to 7c, solving the optimization problem in step iii includes minimizing the deviation between the acquired spatial image 54 and a simulated spatial image 72, wherein the simulated spatial image 72 is obtained by applying a spatial image simulation method 74 to the design (in particular a modification to the underlying design 64).

[0106] According to a sample, as shown in Figure 7a, a parameterized representation 76 of the underlying design 64 is optimized by an optimization problem. The parameterized representation 76 of the underlying design 64 can be continuously modified until the optimization problem is solved. The solution to the optimization problem corresponds to a rational design 62. For example, the parameterized representation 76 can describe the structural boundaries of the underlying design 64. For example, a parameterized representation 76 can include lines, edges, contours, or geometric shapes, such as polygons, circles, ellipses, etc. These parameterized representations 76 can contain, for example, the positions of optimized control points. In one example, as shown in Figure 7a, the parameterized representation 76 includes a contour 78 represented by a graphic containing nodes and edges. These contours (or nodes and edges) can be derived, for example, from an underlying CAD design. The positions of the nodes can be optimized by solving an optimization problem. The contour can also be represented by an analytical function describing a curve. The corresponding optimization problem can include an active contour or serpentine objective function term for aligning the contour of the parameterized representation 76 with the contour in the acquired spatial image 54.

[0107] As shown in Figure 7b, the parameters of the parameter representation 76 of the underlying design 64 are modified in the optimization problem. An initial parameter representation 77 is a parameter representation 76 corresponding to the underlying design 64. This initial parameter representation 77 is optimized 80 to obtain an optimized parameter representation 82 to minimize the deviation between the acquired spatial image 54 and a simulated spatial image 72 of the optimized parameter representation 82. The optimized parameter representation 82 is the parameter representation 76. In an iterative optimization process, a simulated spatial image 72 is obtained from a modified parameter representation of the underlying design using a spatial image simulation method 74. The optimized parameter representation 82 corresponds to a reasonable design 62 of the acquired spatial image 54. In this case, the parameter representation 76 (including the initial parameter representation 77 and the optimized parameter representation 82) is an over-parameterized parameter representation 83 because more parameters than necessary are used to represent the design (to be explained with reference to Figure 8).

[0108] Based on the deviation 81 between the optimized parameterized representation 82 (corresponding to the rational design 62) and the initial parameterized representation 77 (corresponding to the underlying design 64), a defect 58 can be detected. An index can be defined to measure this deviation 81, for example, as shown in Figure 7c, the distance between a node in the initial parameterized representation 77 and the corresponding shifted node in the optimized parameterized representation 82, or differences in dimensions, lengths, areas, etc. To detect defects, a threshold can be defined, for example, a confidence interval derived from statistical data. Alternatively, the q-value of the statistical data can be used. Alternatively, a machine learning model can be used to distinguish between defects and non-defects.

[0109] In one example, the optimization problem includes parameters describing a deviation 81 from the underlying design. Therefore, by solving the optimization problem, only the deviation 81 of the parameters is optimized, such as a translation of a node, edge, or contour, a modification of a contour length, a modification of an area, a dimensional modification of certain lines or structures, etc. The optimization problem can impose a sparsity constraint on these parameters describing the deviation 81 of the underlying design. Thus, the solution to the optimization problem, i.e., the rational design 62, often contains only a small number of deviations 81 of the parameters, for example, as shown in Figure 7c, only local modifications of points, nodes, or edges.

[0110] According to a sample of the first embodiment illustrated in FIG8, the parametric representation 76 of the underlying design 64 is adjusted, for example, to make it more flexible so as to be modified by overparameterization. The parametric representation 76 of the underlying design 64 is overparameterized. For this purpose, for example, the contour can be resampled to place additional nodes that can be translated during the optimization process along the contour. Overparameterization increases computation time, but also increases the accuracy of the rational design. FIG8 shows a parametric representation 76 derived from a CAD design by placing nodes at the corner points of the structure and connecting these nodes by the contour edges. To improve the flexibility of the parametric representation 76, more nodes are added along the contour edges, resulting in an overparameterized parametric representation 83. For example, a method similar to DeepSnake (e.g., conference paper "Deep snake for real-time instance segmentation." Proceedings of the IEEE / CVF conference on computer vision and pattern recognition. 2020") can be used to overparameterize the parameterized representation 76. By solving an optimization problem, the overparameterized parameterized representation 83 is optimized to fit the acquired spatial image 54, thereby producing an optimized parameterized representation 82, i.e., a rational design 62. Defects 58 can be detected by the deviation between the parameterized representation 76 of the underlying design 64 and the optimized parameterized representation 82 (rational design 62). Specifically, defects 58 can be detected based on the deviation between the optimized parameterized representation 82 (rational design 62) and the overparameterized parameterized representation 83 or the initial parameterized representation 77. The parameterized representation 76 in Figure 7, which includes the initial parameterized representation 77 and the optimized parameterized representation 82, is also overparameterized.

[0111] There are various spatial image simulation methods 74 that can be used to obtain simulated spatial images based on a reasonable design. A spatial image simulation method simulates an optical system (especially a detection system, a photomask identification system, a lithography system or a metrology system) applied to a lithography photomask corresponding to the design, and mathematically calculates a spatial image of a design for a lithography photomask.

[0112] In a preferred example, the spatial image simulation method includes using a physical model to generate a spatial image from a rational design. This method can yield accurate results, but is typically time-consuming. Among these methods are rigorous simulation methods well-known to those skilled in the art, such as the finite-difference time domain (FDTD) method or rigorous coupled-wave analysis (RCWA). Because these methods require long computation times, fast approximation methods, such as the thin element approximation (TEA), can be used. The thin element approximation (TEA) assumes that the thickness of the structure on the lithography mask is very small relative to the wavelength, while the width of the structure on the lithography mask is very large relative to the wavelength. However, as the radiation wavelengths used in lithography processes become shorter and the structures on the patterning apparatus become smaller and extend along the vertical direction, these assumptions no longer hold, and the three-dimensional effects of the mask must be considered. Therefore, the results of the TEA method are less accurate than those of rigorous simulations, but are much faster.

[0113] In order to obtain fast and accurate results, simulation methods based on physical models but not relying on thin mask assumptions can be used.

[0114] According to one example, a not-quite-rigorous (NQR) spatial image simulation method can be used to simulate a spatial image obtained by an optical system from a design (especially a well-designed one). This method can accurately and quickly simulate the spatial image from the design when incident electromagnetic waves irradiate a corresponding lithography mask in an optical system. In order to simulate the interaction between electromagnetic waves and a lithography mask, the propagation of electromagnetic waves in different layers of the lithography mask (these layers include different materials with different refractive indices) must be considered.

[0115] Figure 9a illustrates a less rigorous spatial image simulation method 200, which generates a designed spatial image of a lithography mask when it is irradiated by incident electromagnetic waves by simulating the application of an optical system (particularly a mask inspection system, optical mask identification system, or specific lithography system) to a lithography mask. The method includes: a) in a step N1, approximately simulating the propagation of incident electromagnetic waves in a first region of a lithography mask comprising multiple structures; b) in a step N2, analyzing or numerically simulating the propagation of electromagnetic waves simulated in step a) in a second region of the lithography mask; c) in a step N3, simulating a representation of an electromagnetic near field of the lithography mask by propagating the electromagnetic waves simulated in step b) to a near field plane; and d) in a step N4, generating a spatial image of the lithography mask by applying a simulation of an imaging process of the optical system to the representation of the electromagnetic near field.

[0116] The less stringent method 200 for generating a spatial image can be applied to the transmissive photomask 14' shown in FIG9b and the reflective photomask 14'' shown in FIG9d.

[0117] An electromagnetic near-field represents the distribution of electromagnetic wave 222 on a near-field plane 252. The near-field plane 252 can be located next to a structural plane 230 of the photomask 14. Preferably, the near-field plane 252 is parallel to the structural plane 230 or the base plane 234 of the photomask 14. The near-field plane 252 can generally be located at any position between the structural plane 230 and the wafer plane 18, for example, at a distance from the structural plane 230 between 0 and 1000 nm, preferably between 0 and 100 nm, more preferably between 0 and 50 nm, even more preferably between 0 and 20 nm, and most preferably between 0 and 10 nm. In a preferred embodiment of the invention, the near-field plane 252 and the structural plane 230 are the same.

[0118] According to a specific embodiment, the lithography mask 14 includes a mask carrier 248 and a grating 224. The grating 224 includes an absorption structure 226 and a non-absorption structure 228, and a design 292 is formed on at least a portion of the mask carrier 248. The lithography mask 14 includes a first region 225 extending between a structural plane 230 and a boundary plane 232 of the lithography mask 14, and a second region 227 extending between the boundary plane 232 and a base plane 234 of the lithography mask 14. The first region 225 includes the grating 224, and the second region 227 includes the mask carrier 248.

[0119] Figure 9b illustrates the propagation of an incoming electromagnetic wave 222 through a transmissive lithography mask 14' (e.g., a DUV lithography mask). The lithography mask 14' includes a first region 225 and a second region 227. The first region 225 contains a grating 224, and the second region 227 contains a mask carrier 248. The grating 224 is composed of an absorbing structure 226 and a non-absorbing structure 228. The absorbing structure 226 is made of one or more materials that absorb electromagnetic waves 222, such as titanium nitride or tantalum nitride. The non-absorbing structure 228 is made of one or more materials that absorb electromagnetic waves 222 to a lesser degree than the absorbing material. For example, the non-absorbing structure 228 can include a vacuum. Therefore, the grating 224 is a non-uniform medium. The absorbing structure 226 and the non-absorbing structure 228 are deposited on a mask carrier 248. The mask carrier 248 can include a substrate layer 246. The photomask carrier 248 in the lithography mask 14' is defined by a boundary plane 232 and a base plane 234 preferably parallel to the boundary plane 232. The boundary plane 232 can be a surface plane of the photomask carrier 248. The base plane 234 is a boundary plane through which electromagnetic waves 222 enter the grating 224. The incoming electromagnetic waves 222 impact the base plane 234. The base plane 234 can form an interface between the photomask carrier 248 and the exterior of the lithography mask 14', through which the electromagnetic waves 222 propagate. The absorbing structure 226 in the grating 224 of the lithography mask 14' is defined by the boundary plane 232 and a structural plane 230. The structural plane 230 is a boundary plane containing a portion of the surface of the absorbing structure 226 facing away from the boundary plane 232. Preferably, the structural plane 230 is parallel to the boundary plane 232. The first region 225 of the lithography mask 14' extends between the structural plane 230 and the boundary plane 232 and is defined by these planes. The second region 227 of the lithography mask 14' extends between the boundary plane 232 and the base plane 234, and is defined by the boundary plane 232 and the base plane 234.

[0120] For the transmissive lithography mask 14', according to the example, simulated electromagnetic wave 222 is incident on the base plane 34, propagates from the base plane 234 to the boundary plane 232 in the second region 227 of the lithography mask 14', and propagates from the boundary plane 232 to the structural plane 230 in the first region 225 of the lithography mask 14.

[0121] Figure 9c shows a flowchart of a less rigorous method 200 for generating a spatial image of a transmissive lithography mask 14' as shown in Figure 9b. In step P1, simulated electromagnetic waves 222 are incident on the lithography mask (e.g., base plane 234) and propagate within a second region 227 of the lithography mask, for example, from the base plane 234 to the boundary plane 232. In step P2, simulated electromagnetic waves 222 propagate within a first region 225 of the lithography mask 14', for example, from the boundary plane 232 to the structural plane 230. Then, in step P3, a representation of the electromagnetic near field of the lithography mask 14' in a near-field plane 252 is obtained. Finally, in step P4, a spatial image is generated from the near-field representation by applying a simulation of an imaging process of an optical system to the electromagnetic near-field representation.

[0122] For the reflective lithography mask 14'', according to an example shown in FIG9d, the mask carrier 248 includes a multilayer 238 in the form of a stack of optical thin films 240 for reflecting electromagnetic waves 222, and simulates electromagnetic waves 222 incident on the structural plane 230, propagating from the structural plane 230 to the boundary plane 232 within the first region 225 of the lithography mask 14'', being reflected within the multilayer 238 in the second portion 227 of the lithography mask 14'', and propagating from the boundary plane 232 to the structural plane 230 within the first region 225 of the lithography mask 14''. Thus, a less stringent method 200 for generating a spatial image can be applied to the reflective lithography mask 14'', such as an EUV lithography mask.

[0123] Figure 9d illustrates the propagation of an incoming electromagnetic wave 222 through a reflective lithography mask 14'' (e.g., an EUV lithography mask). The lithography mask 14'' includes a first region 225 and a second region 227. The first region 225 contains a grating 224, and the second region 227 contains a mask carrier 248. The grating 224 contains absorbing structures 226 and non-absorbing structures 228, which are designed on at least a portion of the mask carrier 248 for printing onto a wafer. The absorbing structures 226 are made of one or more materials that absorb electromagnetic waves 222, such as titanium nitride or tantalum nitride. The non-absorbing structures 228 are made of one or more materials that absorb electromagnetic waves 222 to a lesser degree than the absorbing materials. For example, the non-absorbing structures 228 can include a vacuum. Therefore, the absorbing structures 226 and the non-absorbing structures 228 form a non-uniform medium. The absorbing structures 226 and the non-absorbing structures 228 are deposited on a mask carrier 248. The photomask carrier 248 includes a multilayer 238, which is a stack of optical thin films 240 for reflecting electromagnetic waves 222. The photomask carrier 248 may include a cover layer 242 and / or a substrate layer 246. The reflection of electromagnetic waves 222 by the stack of optical thin films 240 corresponds to a reflection of electromagnetic waves 222 at the effective mirror surface 244. The photomask carrier 248 in the lithography photomask 14'' is defined by a boundary plane 232 and a base plane 234 preferably parallel to the boundary plane 232. The boundary plane 232 may be a surface plane of the photomask carrier 248. The absorption structure 228 in the grating 224 of the lithography photomask 14'' is defined by the boundary plane 232 and a structural plane 230. The structural plane 230 is a boundary plane containing a portion of the surface of the absorption structure 226 facing away from the boundary plane 232. Preferably, the structural plane 230 is parallel to the boundary plane 232.

[0124] The structural plane 230 is a boundary plane through which the electromagnetic wave 222 enters the first region 225 (e.g., grating 224). The incoming electromagnetic wave 222 strikes the structural plane 230. The structural plane 230 can form an interface between the lithography mask 14'' and the outside of the lithography mask 14'', through which the electromagnetic wave 222 propagates. The first region 225 of the lithography mask 14'' extends between the structural plane 230 and the boundary plane 232, and is defined by these planes. The second region 227 of the lithography mask 14'' extends between the boundary plane 232 and the base plane 234, and is defined by the boundary plane 232 and the base plane 234.

[0125] Figure 9e shows a flowchart of an example of a less rigorous method 200 for generating a spatial image of a reflective photomask 14'' as shown in Figure 9d. The photomask carrier 248 comprises a multilayer 238 in the form of a stack of optical thin films 240 for reflecting electromagnetic waves 222, and in step Q1, simulated electromagnetic waves 222 are incident on the photomask (e.g., on the structural plane 230), propagating within a first region 225 of the photomask 14'' (e.g., from the structural plane 230 to the boundary plane 232); reflected within the multilayer 238 in a second region 227 of the photomask 14'' in step Q2; and propagated within the first region 225 of the photomask 14'' in step Q3 (e.g., from the boundary plane 232 to the structural plane 230). Then, in step Q4, a representation of the electromagnetic near field of the photomask 14'' in a near-field plane 252 is obtained. Finally, in step Q5, a spatial image is generated from the near-field representation by applying a simulation of an imaging process of an optical system to the electromagnetic near-field representation.

[0126] In addition to directly solving the Maxwell equations in the first region 225, different approximation methods can be used to reduce the computation time of this method. According to one example, in step a), the propagation of the incident electromagnetic wave in the first region 225 of the lithography mask 14 is approximated using the Helmholtz equations (especially a forward Helmholtz equation).

[0127] In the lithography setting, the following assumptions can be made: 1) The refractive indices of different materials of the lithography mask 14 are similar; for example, the refractive index of structure 226 (especially the absorbing structure) is close to the refractive index of the exterior of structure 226 (especially the non-absorbing structure, such as vacuum). 2) The refractive index distribution of the first region 225 is a piecewise constant, and there is no need to model a transition. 3) The main propagation direction 250 of the incoming electromagnetic wave 222 is nearly perpendicular to a main surface of the lithography mask (especially the base plane 234). These assumptions allow for a simplified approximation of the propagation of the electromagnetic wave 222 within the first region 225.

[0128] Based on the Maxwell equation for time harmonics, the following equation for the electric field E of an electromagnetic wave 222 can be derived: (1) where ω is the angular frequency, c is the speed of light, and ϵ(r,ω) is the dielectric function characterizing a particular material. These relationships are related to the refractive index n(r,ω) of the material. The right side of the equation is coupled with the electric field component, which makes the equation difficult to solve. Therefore, it is best to ignore the right side. Ignoring the right side is still valid if the following two assumptions are met: the optical system under consideration does not exhibit a unique response dependent on the incident polarization, and there is no cross-coupling between the polarization components. For short-wavelength lithography setups, such as for DUV or EUV lithography masks, there are two reasons to ignore polarization and phonon effects, so these assumptions are valid. First, in the first region 225, the difference in refractive index between the different materials inside and outside the structure 226 is small. Second, the height a of the structure 226 is greater than the wavelength λ, i.e. Therefore, the right side of equation (1) can be ignored, thus yielding the Helmholtz equation.

[0129] The Helmholtz equation can be further simplified. Using the following relationship between the wavenumber magnitude and the wavelength λ, where and are the wave vector and wavelength in vacuum, respectively, the Helmholtz equation can be rewritten as follows.

[0130] This equation can be rewritten using the transverse Helmholtz operator as follows: , where.

[0131] This equation can be rewritten as follows.

[0132] Here, the square root Helmholtz operator is introduced, which is defined as a power series. Furthermore, it is assumed that the commutator is negligible, which physically means that back reflections in a non-uniform medium can be ignored. Thus, the forward Helmholtz equation can be expressed as follows.

[0133] Ordinary partial differential equations can be solved by multiplying by an integrating factor:

[0134] The exponential operator can be approximated by an integral operator, as shown in Appendix A of Soeren Schmidt's doctoral dissertation, "Efficient Wave Optical Simulation for Modeling Micro-Optical Elements," published at the University of Jena. The aforementioned doctoral dissertation is cited in its entirety, and its disclosure is included in this description. By approximating with an integral operator, we can obtain: .

[0135] This method is called the angular spectrum of plane wave decomposition (ASPW), as shown in Equation 1.8 of the aforementioned doctoral dissertation. It assumes that the electromagnetic wave propagates in a homogeneous medium with a refractive index of n. However, this does not apply to the first region 225 of the lithography mask 14, because it contains structure 226 and unstructured components.

[0136] Therefore, it is necessary to extend ASPW to non-uniform media to describe the propagation of electromagnetic wave 222 in the first region 225 of lithography mask 14.

[0137] To explain the non-homogeneous medium, the propagation constant from a subsequent plane z to a given plane was calculated based on the refractive index distribution described in Section 1.4 of the aforementioned doctoral dissertation.

[0138] Therefore, according to an example, the forward Helmholtz equation can be solved using a wave propagation method. The wave propagation method is a generalization of ASPW in non-uniform media, which describes the wave propagation steps in a plane along the z-direction perpendicular to the base plane, as shown in the formula, (2) where E represents the electric field component of the electromagnetic field, λ represents the wave vector, and locally obeys the dispersion relation, (3) where λ represents the wave number of a light with wavelength λ in vacuum, λ represents the refractive index distribution, and λ represents the spatial Fourier transform. The magnitude of the wave vector k is inversely proportional to the wavelength λ, and the direction of the wave vector is perpendicular to the wavefront. By using this wave propagation method, the propagation of electromagnetic waves in a non-uniform medium can be simulated, thereby accurately approximating the propagation of electromagnetic waves in the first region of the lithography mask.

[0139] In one specific embodiment, the first region 225 of the lithography mask 14 includes a structure 226 and a non-structure 228, forming a non-uniform medium. For example, the grating 224 includes an absorbing structure and a non-absorbing structure. The propagation simulation of electromagnetic wave 222 within the first region 225 takes into account this non-uniformity of the material within the first region 225. Simultaneously, some simplifying assumptions can be utilized in the lithography setup. Furthermore, the propagation simulation of electromagnetic wave 222 within the second region 227 employs analytical or numerical calculations. In this way, the propagation of electromagnetic wave 222 within the lithography mask 14 can be simulated accurately and quickly.

[0140] Alternatively, the forward Helmholtz equations can be solved using a beam propagation method. The beam propagation method has been described, for example, in Chapter 1.3 of Dr. Soeren Schmidt's dissertation "Efficient wave-optical simulations for the modeling of micro-optical elements" mentioned above.

[0141] In one embodiment, the propagation of the incident electromagnetic wave in the first region of the lithography mask in step a) is approximated using a machine learning model. This machine learning model can, for example, include a neural network, such as a deep learning model. For instance, this machine learning model can include a U-Net or a neural network with at least one attention mechanism, such as a Transformer machine learning model. This machine learning model can take a model of the lithography mask (e.g., a design pattern) as input and map the input to an electromagnetic field as output. The machine learning model can be trained using training data, for example, obtained from the simulation described above. By using a machine learning model, computation time can be significantly reduced because, after training, a single fast forward pass is sufficient to calculate the propagation of the incident electromagnetic wave.

[0142] Since the dispersion relation in equation (3) depends on the spatial variable (x,y), the wave propagation method in equation (2) cannot be implemented using the Fast Fourier Transform (FFT). In order to use FFT and reduce computation time, the wave propagation method in equation (2) can be reformulated using the characteristic function.

[0143] In one example, the first region 225 of the photomask 14 is decomposed into different materials by defining a characteristic function for each material, the characteristic function indicating the presence of the material in different locations of the first region 225 of the photomask 14, wherein at least one characteristic function is non-binary.

[0144] The first region 225 of the lithography mask 14 can be decomposed into a finite number of M pairwise disjoint and uniform subregions with a refractive index of . Then, the refractive index distribution within a given layer can be rewritten using characteristic functions. The characteristic function of material m is a mapping from a spatial domain to a range of values, representing the existence of material m at each position (x, y) in the spatial domain. For example, represents a bivariate characteristic function with a range of values ​​of , where represents the refractive index of material m. D can be a subset of real numbers ( ), or a subset of complex numbers ( ).

[0145] Figure 9f shows a flowchart of a less rigorous method for generating a spatial image according to an example, including an additional feature function step R1, then simulating a representation of an electromagnetic near field in a step R2, and applying a simulation of an imaging process of an optical system to the resulting electromagnetic near field representation in a step R3.

[0146] Step R1 includes: determining a quantity M of materials M of the structure 226 constituting the design 292 of the photomask 14 in the first region 225; defining a characteristic function for each material m, which indicates that the material exists in a subset of an x / y plane at position (x,y) of the photomask 14. Where the x / y plane is orthogonal to the z direction perpendicular to the base plane 34; within each identified material, simulating the propagation of electromagnetic wave 222 as a weighted sum in a propagation step: , (4) where represents the inverse Fourier transform. The use of the characteristic function enables the wave propagation method in equation (2) to be implemented based on an FFT, thereby saving computation time. The integrator in equation (4) converges linearly with the step size.

[0147] However, discretization of commonly used binary characteristic functions presents problems. Since binary characteristic functions are discontinuous, the Shannon-Nyquist theorem requires a very high sampling frequency (at least twice the maximum frequency of the signal), thus requiring a very high resolution of the sampling grid. Specifically, if the edges of structure 226 are not aligned with the sampling grid, sampling will be inaccurate. Furthermore, the resolution of the sampling grid depends on the size of the smallest feature. This high resolution of the sampling grid, in turn, leads to excessively long computation times for generating spatial images.

[0148] Therefore, according to the one-state sample of this example, the characteristic function is band-limited. A band-limited characteristic function is an characteristic function with a finite frequency that satisfies the following condition.

[0149] According to the Shannon-Nyquist theorem, on the one hand, the sampling frequency required for discretization of a band-limited characteristic function depends on its maximum frequency. On the other hand, a given sampling frequency for discretization of a band-limited characteristic function directly implies its maximum frequency.

[0150] By using a band-limited eigenfunction, the maximum frequency of the eigenfunction can be limited. Thus, according to the Shannon-Nyquist theorem, the required sampling frequency is reduced, allowing the use of a lower-resolution single-sampling grid to discretize the eigenfunction (compared to the case of a binary eigenfunction). This reduces the computation time required to generate spatial images. Furthermore, the resolution of the sampling grid is independent of the feature size of the features in the lithography mask design. In contrast, for a binary eigenfunction, the resolution of the sampling grid depends on the minimum feature size of the lithography mask design.

[0151] One reason for using the discretized band-limited characteristic function is as follows: Assuming that the electromagnetic field E contains only long-wavelength energy in the x / y plane perpendicular to the base plane 234 of the photomask 14, then a linear space-invariant low-pass filter P applied to the electromagnetic field E has no effect, that is:

[0152] Equivalently, P can be written as a convolution in the time domain, and the above equation means: .

[0153] If the filter P is applied to the product of E and a function Θ with energy of a shorter wavelength, the following situation will occur: .

[0154] Therefore, if a low-pass filter is applied to the product of a slow-changing function E and a fast-changing function Θ, the result is approximately the product of the slow-changing function E and the filtered fast-changing function.

[0155] Apply this result to the propagator of the wave propagation method in (4), where O represents the linear ASPW propagator (5) and assuming that the scale of change of the electromagnetic field is larger than the scale of change of the characteristic function, then:

[0156] Therefore, by applying filter P to the characteristic function, the low-frequency part of the propagator of the field E in the wave propagation method in (4) can be obtained.

[0157] By extending the concept of feature functions to non-binary feature functions, it is possible to solve sub-pixel design features and achieve an acceleration of about 100 times.

[0158] In addition to band-limited characteristic functions, it is also advantageous to use other non-binary characteristic functions to describe the presence of a specific material at different locations of the lithography mask 14.

[0159] For example, it is advantageous to use continuous characteristic functions or complex-valued characteristic functions. This allows for a more flexible description of the material distribution within the lithography mask, thereby obtaining more accurate approximations.

[0160] According to the example, the range of values ​​for at least one characteristic function includes at least one value. Therefore, the at least one characteristic function is not a binary characteristic function because it maps to at least one non-binary value. In this way, different materials m can exist at the same location (x,y), thereby enabling more flexible modeling of the refractive index distribution in the lithography mask 14, thus obtaining a more general description of the material distribution in the lithography mask. By using characteristic functions with overlapping supports, the accuracy of wave propagation methods can be improved. The support of a real-valued function is a subset of the function domain containing elements not mapped to zero. On the one hand, when different materials exist at the same location in the lithography mask, the presence of different materials at the same location in the lithography mask can be used to model the material distribution. On the other hand, assuming that different materials exist at the same location, it can be used as a mathematical means to improve the accuracy of electromagnetic near-field and spatial imaging, even if this material distribution does not match the actual material distribution. In this way, more accurate electromagnetic near-field and spatial imaging can be calculated.

[0161] According to the example, the characteristic function forms an affine combination at each position in the first region of the lithography mask. This means that at: Specifically, the characteristic function can form a convex combination at each position in the lithography mask. This constraint ensures that the amount of material present at each position in the characteristic function domain is the same and equal to 1. Therefore, the material distribution within the lithography mask 14 can be accurately described, thereby accurately approximating the propagation of electromagnetic wave 222 within the lithography mask 14.

[0162] According to one aspect of the example, obtaining the characteristic function involves decomposing the design of the photomask 14 into elements (e.g., using mathematical functions describing the contour or area of ​​structure 226, such as polygons, spline functions, curve elements, etc.), representing element 294 with a characteristic function (particularly a binary characteristic function), and applying a low-pass filter to the characteristic function. For example, element 294 can be represented by a characteristic function that takes a non-zero value (e.g., 1 inside element 294 and 0 outside element 294). For example, each element 294 can be decomposed into one or more triangles, which can be represented by a characteristic function. The Fourier transform of the polygon can be obtained as described in Appendix A of the doctoral dissertation "Photolithography Simulation by Heinrich Kirchauer at the Technical University of Wien". The aforementioned doctoral dissertation is hereby cited in its entirety, the disclosure of which is included in this description. By applying a low-pass filter to the characteristic function, a band-limited characteristic function 68 can be obtained. Therefore, the wave propagation method in Equation (4) can be simulated using the coarse sampling grid as described above, thereby reducing computation time.

[0163] In one example, a low-pass filter is applied to the characteristic function. Specifically, applying a low-pass filter to the characteristic function can involve applying a spatial analytic Fourier transform to the characteristic function, followed by an inverse Fourier transform. The analytic Fourier transform can only be computed for the spatial frequencies in the discrete domain of the inverse Fourier transform. According to the Shannon-Nyquist theorem, this subsampling of the spatial domain limits the maximum frequency of the characteristic function. Therefore, discretization corresponds to a low-pass filter for the characteristic function. The result is that the design of lithography masks can be represented by band-limited characteristic functions that can be discretized using a sampling grid with a resolution much lower than that of binary characteristic functions, thereby reducing computation time.

[0164] According to one example, the analytical Fourier transform used in the wave propagation method in formula (4) can be approximated by a fast Fourier transform (FFT), and / or an analytical inverse Fourier transform can be approximated by a fast inverse Fourier transform. In this way, the computation time is reduced.

[0165] FFT implies periodic boundary conditions. However, since the angle of the incident electromagnetic wave is arbitrary, this assumption no longer holds. Approximation methods usually ignore this inaccuracy. Even assuming that the photomask design 292 is periodic, the arbitrary illumination angle of the incident electromagnetic wave 222 (e.g., the normal 254 relative to the structure plane 230) means that, according to the Floquet Theorem, the solution to equation (4) is only quasi-periodic, that is, periodic with an additional phase shift α:

[0166] Therefore, according to one example, the wave propagation method approximates an analytical Fourier transform by a fast Fourier transform, and by assuming quasi-periodic boundary conditions at one or more pairs of opposite boundaries perpendicular to the base plane 234 of the photomask 14 (i.e., in the x / y plane) in the propagation step of equation (4), the wave propagation method takes into account the angle ϕ of the incident electromagnetic wave 222 (e.g., the angle relative to the normal 254 of the structure plane 230). By assuming quasi-periodic boundary conditions, the accuracy of the simulated electromagnetic near field is improved.

[0167] Let E be quasi-periodic in the x and y coordinates. Then, according to Flocai's theorem, E can be rewritten as the periodic part E' in the x and y coordinates multiplied by a non-periodic phase shift, as shown below:

[0168] Then the Fourier transform of the periodic part E' can be written as.

[0169] This shows that.

[0170] Use. We obtain... .

[0171] It can be concluded that a phase shift α in an input field that is linear in the x and y coordinates can be adapted by re-evaluating the dispersion relation in equation (3), as shown below:

[0172] Therefore, according to an example, the dispersion relation in (3) can be re-expressed using the Flokai theorem. The terms in the inverse Fourier transform are periodic and can be computed using the standard FFT.

[0173] Specifically, the dispersion relation of the electromagnetic wave 222 in the first region 225 depends on the angle ϕ of the incident electromagnetic wave 222.

[0174] Specifically, the dispersion relation in the first region 225 is modified by a phase shift in a coordinate system parallel to the base plane 234 of the photomask 14.

[0175] Figure 9g illustrates the dependence of the phase shift vector α on the angle ϕ of the incoming electromagnetic wave 222. The angle ϕ can be measured relative to the normal 254 of the structural plane. The electromagnetic wave 222 propagates along the direction of the wave vector 256. Let and represent the boundaries of the unit lattice in the x-direction, i.e., the smallest aperiodic subset in the periodic design 292. Then, the phase difference between and can be expressed by ϕ as follows:

[0176] It is possible to similarly calculate the dependence on the angle of the incoming electromagnetic wave 222.

[0177] In one example, the photomask 14 is a transmissive photomask.

[0178] In another example, the photomask 14 is a reflective photomask, and the second region 227 includes a multilayer 238 in the form of a stack of optical thin films 240 for reflecting electromagnetic waves 222.

[0179] For the reflective photomask 14, the propagation of the electromagnetic wave 222 simulated in step a) within the second region 227 of the photomask 14 can be simulated in an analytical or numerical manner by including an analytical description of the propagation of the electromagnetic wave within the photomask carrier 248 and analytical calculation of the reflection of the electromagnetic wave 222 at the multilayer 238.

[0180] Therefore, according to an example, simulating the reflection of the simulated electromagnetic wave 222 within the multilayer 238 obtained in step a) involves analyzing and calculating the reflection coefficient at a boundary, for example, at the boundary plane 232 between the second region 227 and the first region 225 of the photomask 14. This reflection coefficient describes the propagation of the electromagnetic wave 222 within the stack of optical thin films 240 of the multilayer 238. The propagation within the stack of optical thin films 240 of the multilayer 238 corresponds to reflection at an effective mirror 244 at a specific distance from the boundary plane 232.

[0181] Specifically, the reflection coefficients at the interior of structure 26 and at the outer boundary 232 of structure 226 in the first region 225 of the lithography mask 14 can be calculated separately. For example, the reflection coefficients of each medium of the absorbing and non-absorbing structures of the grating 224 at the location of the boundary plane 232 can be calculated separately. In this way, the accuracy of the generated spatial image is improved.

[0182] In one example, the propagation of the simulated electromagnetic wave 222 within the second region 227 of the photomask 14 includes applying a reflection coefficient to the electromagnetic wave 222 incident on the boundary 232.

[0183] Specifically, simulating the reflection of electromagnetic wave 222 within the multilayer 238 involves replacing the phase term in equation (4) with the analytical reflection coefficient at the boundary plane: , where indicates the scalar electric field at the boundary plane of the structural plane 230 directly facing the lithography mask 14, and indicates the scalar electric field at the boundary plane of the base plane 234 directly facing the lithography mask 14. Thus, a computer-implemented method for generating a spatial image of a lithography mask design can be applied to reflective lithography masks. Furthermore, the accuracy of this method is also improved.

[0184] As shown in the article "Optical properties of a thin-film stack illuminated by a focused field" by S. Kim, Y. Kim and I. Park, Journal of the Optical Society of America A, Vol. 17, No. 8, August 2000, Equations 33 to 41, the analytical reflectance coefficient of each of the N optical thin films 240 in the multilayer 238 can be calculated for s-polarized waves and p-polarized waves as follows: , where are the elements of the characteristic matrix A, where , are , where is used for s-polarized waves and p-polarized waves.

[0185] Here, represents the vacuum permittivity, represents the vacuum permeability, represents the refractive index of the (j+1)th optical thin film 240, and represents the thickness of the (j+1)th optical thin film 240. The foregoing article is hereby quoted in its entirety, and its disclosure is included in this description.

[0186] In another example, the reflection of electromagnetic waves by the multilayer structure 238 can be numerically calculated as follows: In a first step, the electric field at the boundary plane 232 is decomposed into its Fourier modes. In a second step, for each Fourier mode, the reflected electric field can be calculated using, for example, the transfer matrix method (see Section 2.2 of the article "Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures", Achim Schädle, Lin Zschiedrich, Sven Burger, Roland Klose, Frank Schmidt, arXiv:math / 0602179v1). In a third step, the reflected Fourier modes are superimposed to obtain the reflected electromagnetic wave. Alternatively, a machine learning model can be trained to numerically simulate the propagation of electromagnetic waves within the second region of the lithography mask.

[0187] Figures 9h a) to d) illustrate the steps of an example of a less stringent method 200 for generating spatial images. The design 292 of the lithography mask 14 includes elements 294 composed of the rectangular polygons shown in Figure 9h a). In a feature function step R1, element 294 is represented by a feature function (e.g., a binary feature function obtained by any of the methods described above). For example, element 294 is represented by a binary feature function whose value is 1 inside element 294 and 0 outside element 294. Then, a spatial analytic Fourier transform is performed on the feature function, followed by an inverse FFT transform, to obtain a band-limited feature function 268. Here, the analytic Fourier transform is calculated only for the spatial frequencies in the discrete domain of the inverse FFT. According to the Shannon-Nyquist theorem, this subsampling of the spatial domain limits the maximum frequency of the feature function. Therefore, discretization corresponds to a low-pass filter for the feature function. The result is a band-limited discretized representation of the design 292 of the lithography mask 14, namely the band-limited characteristic function 268 sampled on a low-resolution sampling grid as shown in FIG. 9h b). Based on the band-limited characteristic function 268, FIG. 9h c) shows an electromagnetic near field 220 represented in amplitude form, which is simulated by propagating simulated electromagnetic waves to a near-field plane. Finally, a simulation 290 of the imaging process of the lithography systems 10, 10' within a projection region 19 between the near-field plane 252 and a wafer plane 18 is applied to the representation of the electromagnetic near field 220 to calculate a spatial image 264 as shown in FIG. 9h d). The imaging process can include resampling the electromagnetic near field 220 onto a higher-resolution grid. By applying steps R1 and R3 to calculate the spatial image 264, an accurate spatial image 264 can be simulated for the design 292 of the lithography mask 14 with lower computation time due to the low resolution of the sampling grid. Therefore, the computation time for obtaining the spatial image 264 is reduced compared to applying a rigorous simulation method (such as RCWA) to rigorously simulate the design 292 of the lithography mask 14 (which requires a high-resolution sampling grid) to simulate the spatial image 264. Further details of a less rigorous method for generating spatial imagery are described in International Patent Application PCT / EP2023 / 087651 and German Patent Application 102022135019.3, which are incorporated herein by reference in their entirety.

[0189] According to one example, the spatial image simulation method 74 includes a machine learning model. The spatial image simulation method 74 can be configured as shown in Figure 10. The input is a design, specifically a parametric representation 76 of the design. The output is a simulated spatial image 72. A transformation method 88 can optionally convert the parametric representation 76 into a normalized representation, which will be used as input to subsequent methods 90 and 92. For example, a parametric representation 76 containing contours, nodes and edges, geometry, etc., can be converted into an image or a vector format, etc.

[0190] A simulation method 90 is used to generate a spatial image based on an input design or a standardized representation of the design. The simulation method 90 may include, for example, a physical model of electromagnetic wave propagation within a lithography mask, or a machine learning model simulating electromagnetic wave propagation within a lithography mask. The simulation method 90 may include, for example, a TEA method, a rigorous simulation method, a physics-based machine learning model, an NQR method, or any other simulation method capable of calculating a spatial image from a design of a lithography mask.

[0191] The spatial image simulation method 74 can include a machine learning model 92. This machine learning model includes at least one parameter, preferably multiple parameters, such as 100,000 parameters. The values ​​of these parameters are determined during training of the machine learning model using training data. The machine learning model 92 can be applied to the results of the simulation method 90. In this way, the machine learning model 92 can improve the accuracy of the simulation method 90. Therefore, simulations with lower accuracy or lower complexity can be used because these simulations are followed by a machine learning model that can improve the accuracy of the simulation results. Furthermore, since the input to the machine learning model already comes from a physics-based simulation, a machine learning model 92 with lower complexity can be used. In this way, computation time can be reduced and accuracy improved.

[0192] The training of the spatial image simulation method 74 can be performed in different ways. In a first example, the simulation method 90 is first adjusted or trained. If the simulation method 90 does not contain a machine learning model, the parameters of the simulation method 90 can be adjusted, for example, using training data or prior knowledge. If the simulation method contains a machine learning model, the machine learning model can be trained using training data (e.g., including pairs of design drawings and corresponding spatial images). In subsequent steps, the subsequent machine learning model 92 can be trained using pairs including the output of the simulation method 90 and the corresponding spatial image 54 as training data.

[0193] In a second example, simulation method 90 and machine learning method 92 can be jointly trained. In this case, it is beneficial to weight the effects of the parameters of simulation method 90 and machine learning method 92 in the objective function.

[0194] In one example, this spatial image simulation method includes a machine learning model trained to map a design to a spatial image. The design can be optimized by solving an optimization problem (e.g., iteratively). The deviation between the spatial image and the acquired spatial image can be used as the objective function of the optimization problem.

[0195] In another example, a machine learning model can be used to predict a reasonable design 62 of the lithography mask 14 based on the acquired spatial image 54, thereby directly solving the optimization problem with less computation time. For this purpose, for example, a machine learning model such as an encoder-decoder architecture (e.g., U-Nets or Vision Transformer architecture) can be used. Since the mapping from the acquired spatial image 54 to a reasonable design 62 is not injective, i.e., the acquired spatial image 54 can map to different reasonable designs 62, a machine learning model can be used to predict a distribution of the reasonable designs 62. To obtain a distribution of potential reasonable designs 62, for example, a diffusion model can be used.

[0196] An optimization problem that minimizes the deviation between the acquired spatial image and the simulated spatial image can be expressed as follows, wherein the simulated spatial image is obtained using a spatial image simulation method.

[0197] Let p denote a set of parameters in a parameterized representation of a rationally designed system. p may also include design parameters such as a mask thickness and refractive index. These parameters can be optimized iteratively. Let q denote a set of parameters describing the optical system to be simulated, particularly illumination parameters, imaging parameters, and / or the design parameters as described above. For this purpose, an optimization problem can be solved as follows: is the objective parameter vector used to minimize a difference measure χ between an acquired spatial image and a simulated spatial image. χ is an objective function or loss function defining the optimal conditions. It can be associated with a noise model of the acquired spatial image; for example, a norm can be used for Gaussian independent and identically distributed noise. The objective function may also include other normalization terms as a function of p, such as sparsity constraints, to obtain a well-defined objective function.

[0198] Several methods are known to be able to calculate the intensity of simulated spatial images using partially coherent imaging of the incoming electromagnetic near field corresponding to different illumination angles: for example, the Hopkins method, the Abbe method, and the local Hopkins method. An incoming electromagnetic near field can be calculated using a near-field simulation method (e.g., RCWA, FDTD, TEA, or NQR).

[0199] The Hopkins method is based on the observation that even a small change in the incident angle of light waves can be expected to result in only very small deviations in the intensity, phase, and polarization of the light waves. Therefore, a change in the illumination angle will only cause a frequency shift in the diffraction spectrum corresponding to the lithography mask. Thus, for all illumination angles, the same mask spectrum will be used, where an incoming electromagnetic near field will be shifted according to the illumination angle: , where represents the intensity of the simulated spatial image, is a complex imaging pupil function, is an angle-weighted intensity distribution (e.g., with respect to illumination intensity), is the dielectric constant, is the speed of light in the assumed vacuum, is the illumination angle, and is the number of illumination angles in the illumination angle distribution in the pupil plane.

[0200] This method is simple and fast. This assumption always holds for simulations using thin masks or Kirchhoff methods (such as TEA). However, the Hopkins method is not accurate enough when the thickness of the structure on the lithography mask cannot be ignored and a rigorous electromagnetic field simulation of mask diffraction at different illumination angles is required.

[0201] In this case, the Abbe method can be used to adapt the non-constant diffraction spectrum of the lithography mask because the Abbe method assumes that the diffraction spectrum is related to the illumination angle: where represents the intensity of the simulated spatial image. However, the Abbe method is computationally expensive because an electromagnetic near-field must be simulated for each individual illumination angle. Therefore, the Abbe method may not be suitable for, for example, a whole-wafer die-to-die or die-to-database defect detection method.

[0202] To obtain a fast and accurate method for simulating spatial images of lithography masks, a local Hopkins method, such as the local Hopkins method disclosed in US 2007 / 0253637 A1, can be used. The local Hopkins method is a combination of the Hopkins method and the Abbe method, based on the local assumption that the diffraction spectrum of the lithography mask is constant. For this purpose, the source map is divided into multiple segments. For each segment, the diffraction spectrum is assumed to be constant, such that only one diffraction spectrum needs to be simulated for each segment. Therefore, using the local Hopkins method, a smaller number of spectra can be simulated for a selected subset of illumination angles. For the remaining illumination angles, the simulated spectrum is shifted according to the illumination angle: , where and . where represents the intensity of the simulated spatial image. The local Hopkins method requires careful selection of the segments and the illumination angles within the segments to simulate the diffraction spectrum, for example, as described in US 2007 / 0253637 A1.

[0203] The Hopkins, Abbe, or Local Hopkins methods allow for the calculation of the gradient of the objective function with respect to the parameter vector p. Therefore, by using one of these methods to calculate a simulated spatial image based on the electromagnetic near field in the objective function, the parameter p can be optimized in an iterative manner (e.g., by gradient descent). It should be noted that, unlike design parameters (e.g., bias, fillet, etc.), the optimization of optical parameters (e.g., Zernike aberrations) that can be modeled by changing the pupil function does not require a new simulation of the electromagnetic near field, but only a re-evaluation of the aforementioned Hopkins, Abbe, or Local Hopkins imaging equations, thereby simplifying and accelerating the optimization of the parameter vector p.

[0204] Optimization problems can include a sparsity constraint imposed on one or more parameters of the parameter vector p. Such a sparsity constraint is particularly useful if many parameters in the solution to the optimization problem are typically zero (e.g., when these parameters indicate a modification to the underlying design). A sparsity constraint can be implemented, for example, by norm normalization. To maintain the differentiability of the objective function, a Huber loss function can be used to approximate the norm, which penalizes smaller parameter deviations quadratically, larger parameter deviations linearly, and is differentiable; or a different optimizer can be used, such as the fast iterative shrinkage-thresholding algorithm (FISTA). Sparsity constraints can also be implemented, for example, by L_p normalization, where...

[0205] The resulting optimization problem can be solved using different methods. For example, the downslope simplex method or gradient descent method can be used. The conjugate gradient method or the restricted memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is preferred. Furthermore, evolutionary algorithms such as simulated annealing can also be used to solve the optimization problem.

[0206] In one example of the first specific embodiment, after step i, one or more regions of interest containing potential defect candidates are identified in the acquired spatial image, and steps ii to iv are applied only to these one or more regions of interest. This can significantly reduce computation time. Regions of interest can be determined using fast but less accurate defect detection algorithms. Examples of such methods include low-complexity machine learning models, pattern recognition or template matching methods, filtering methods, wavelet transforms, image processing methods, etc. These methods are less accurate, for example, because they can only roughly locate defects (e.g., using boundary blocks) or due to an increased false positive rate. It is preferable to use a fast method with a low false positive rate to determine regions of interest so that defect-free areas can be reliably excluded from further inspection. Alternatively or additionally, regions of interest can be user-defined. Alternatively or additionally, regions of interest can be derived from the underlying design, for example, by extracting or excluding specific areas of a lithography mask or design from defect detection; for example, depending on the type of structure, the nature of the structure, or the spatial nature of the area.

[0207] Different preliminary defect detection methods can be used to obtain defect candidates in different regions of interest. For example, different characteristics of the structure can be examined in different regions of the lithography mask. For example, different minimum defect sizes can be defined based on the structural dimensions in a region of interest of the lithography mask, and larger defect candidates can be detected only in the corresponding regions for further examination by the method of the present invention. Alternatively, a defect detection method according to the present invention with parameters specific to the region of interest can be used in different regions of interest. For example, the parameters in step iii. or step iv. of the method can be varied, such as formulas for optimization problems or rules for distinguishing between defects and non-defects. In this way, the sensitivity and specificity of the method can be adjusted according to the different requirements of different regions of interest of the lithography mask.

[0208] A second embodiment of the present invention relates to a computer-implemented method for training a machine learning model based on any example in the first embodiment, namely, for mapping an acquired spatial image to a reasonably designed machine learning model of the acquired spatial image 54, such that the acquired spatial image is a reasonable result of a reasonably designed spatial image simulation method, or a machine learning model that is part of one of the spatial image simulation methods described above.

[0209] According to a third embodiment of the present invention, a system 94 for detecting defects 58 in a photomask 14 illustrated in FIG11 includes: an optical system 96 for acquiring a spatial image 54 of the photomask 14; and a data analysis device 98 including at least one memory 100 and at least one processor 102, configured to perform the steps of a method 60 for detecting defects in a photomask 14 according to any example or form of the first embodiment.

[0210] The optical system 96 for acquiring a spatial image 54 of the lithography mask 14 may include a detection system, an optical mask identification system, a lithography system, a metrology system, or a spatial image measurement system. The optical system 96 for acquiring a spatial image 54 of the lithography mask 14 provides the spatial image 54 to a data analysis device 98. The data analysis device 98 includes a processor 102, for example, implemented as a CPU or GPU. The processor 102 can receive the spatial image 54 via an interface 104. The processor 102 can load code from memory 100, for example, code for executing a method for detecting defects 58 according to the first embodiment described above. The processor 102 can execute the code.

[0211] Throughout this specification, references to "a particular embodiment," "an example," or "a pattern" mean that a particular feature, structure, or characteristic described in connection with a particular embodiment, example, or pattern is included in at least one particular embodiment, example, or pattern. Therefore, the phrases "according to a particular embodiment," "according to an example," or "according to a pattern" appearing in various places throughout the specification do not necessarily refer to the same particular embodiment, example, or pattern, but may do so. Furthermore, in more than one particular embodiment, as will be apparent to those skilled in the art based on this disclosure, particular features or characteristics can be combined in any suitable manner.

[0212] Furthermore, as those skilled in the art will understand, although some specific embodiments, examples or styles described herein include some other features not included in other specific embodiments, examples or styles, combinations of features of different specific embodiments, examples or styles are meant to be within the scope of the claim and form different specific embodiments.

[0213] In summary, the present invention relates to a method 60 for detecting defects 58 in a photolithography mask 14, the method comprising: i. acquiring a spatial image 54 of the photolithography mask 14 using an optical system 96; ii. obtaining a base design 64 of the photolithography mask 14; iii. generating a rational design 62 of the acquired spatial image 54 by solving an optimization problem that minimizes the deviation between a simulated spatial image 72 of the rational design 62 and the acquired spatial image 54, wherein the simulated spatial image 72 simulates the application of the optical system in the rational design 62; and iv. detecting defects 58 in the photolithography mask 14 by comparing the base design 64 and the rational design 62. The present invention also relates to a corresponding defect detection system. [Simplified Explanation of the Diagram]

[0060] Figure 1 illustrates an exemplary transmission lithography system, such as a deep ultraviolet (DUV) lithography system;

[0061] Figure 2 illustrates an exemplary reflective lithography system, such as an extreme ultraviolet (EUV) lithography system;

[0062] Figure 3 illustrates a die-to-database method for defect detection, which compares a spatial image of a lithography mask with a design of a corresponding lithography mask to detect defects;

[0063] Figure 4 shows a flowchart of a method for detecting defects in a photomask according to a first specific embodiment of the present invention;

[0064] Figure 5 illustrates an exemplary application of the method in Figure 4;

[0065] Figures 6a and 6b illustrate the application of a machine learning model to solve the optimization problem, which is trained to map an acquired spatial image onto a reasonable design of the acquired spatial image.

[0066] Figures 7a to 7c illustrate the solution to the optimization problem, including minimizing the deviation between a rationally designed simulated spatial image and the acquired spatial image, wherein the rational design is obtained by modifying the underlying design of the acquired spatial image;

[0067] Figure 8 illustrates the adjustment of a parametric representation of a design to increase its flexibility in defect detection;

[0068] Figure 9a shows a flowchart of a less rigorous method for generating a spatial image of a photomask;

[0069] Figure 9b illustrates the propagation of an incoming electromagnetic wave through a transmissive photomask;

[0070] Figure 9c shows a flowchart of a less rigorous method for generating a spatial image of a transmissive lithography mask;

[0071] Figure 9d illustrates the propagation of an incoming electromagnetic wave through a reflective photomask;

[0072] Figure 9e shows a flowchart of a less rigorous method for generating a spatial image of a reflective lithography mask;

[0073] Figure 9f shows a flowchart of an example of a less rigorous method for generating a spatial image of a lithography mask, including an additional feature function step;

[0074] Figure 9g illustrates the dependence of the phase shift vector α on the angle φ of the incoming electromagnetic wave;

[0075] Figures 9h a) to d) illustrate the steps of a less rigorous method for generating a spatial image of a photomask according to an example;

[0076] Figure 10 illustrates a spatial image simulation method that includes a machine learning model; and

[0077] Figure 11 illustrates a system for detecting defects in a photomask according to a third embodiment of the present invention.

Claims

1. A method (60) for detecting defects (58) in a photomask (14), the method comprising: i. Acquire a spatial image (54) of the photomask (14) using an optical system (96); ii. Obtain a base design (64) of the photomask (14); iii. Generate a rational design (62) of the acquired spatial image (54) by solving an optimization problem that minimizes the deviation between a simulated spatial image (72) of the rational design (62) and the acquired spatial image (54), wherein the simulated spatial image (72) simulates the application of the optical system in the rational design (62); and iv. Detect defects (58) in the photomask (14) by comparing the base design (64) and the rational design (62).

2. The method as described in Request 1, wherein the underlying design (64) and the rational design (62) are represented in a vector format.

3. The method as described in claim 1, wherein the underlying design (64) and the rational design (62) are represented by non-binary images.

4. The method as described in claim 1, wherein the underlying design (64) of the photomask (14) is generated based on the acquired spatial image (54).

5. The method as described in claim 1, wherein the defect (58) in the photomask (14) is detected in step iv by comparing the underlying design (64) with the rational design (62) in a mathematical space.

6. The method as described in claim 1, wherein solving the optimization problem in step iii includes applying a machine learning model (70) to the acquired spatial image (54), wherein the machine learning model (70) is trained to map the acquired spatial image (54) to a reasonable design (62) of the acquired spatial image (54).

7. The method as described in claim 1, wherein solving the optimization problem in step iii includes minimizing the deviation between a simulated spatial image (72) of the rational design (62) and the acquired spatial image (54), wherein the rational design (62) is obtained by modifying the underlying design (64) of the acquired spatial image (54).

8. The method as described in claim 1, wherein the simulated spatial image (72) of the rational design (62) is obtained by applying a spatial image simulation method (74) to the rational design (62).

9. The method as described in claim 8, wherein the spatial image simulation method (74) includes using a physical model for generating a spatial image from the rational design (62).

10. The method as described in claim 8, wherein the spatial image simulation method (74) includes a machine learning model (92) trained to map a design to a spatial image.

11. The method as described in claim 8, wherein the spatial image simulation method (74) includes using a physical model for generating the spatial image from the rational design (62), wherein a machine learning model (92) is subsequently applied to the generated spatial image to improve its accuracy.

12. The method as claimed in claim 8, wherein the spatial image simulation method (74) generates a spatial image from the rational design (62) by irradiating the corresponding photomask (14) with incident electromagnetic waves (222) in an optical system, and includes: a) Approximately simulate the propagation of the incident electromagnetic wave (222) within a first region (225) of the photomask (14) comprising multiple structures (226); b) Analyze or numerically simulate the propagation of the electromagnetic wave (222) simulated in step a) within a second region (227) of the photomask (14); c) Simulate a representation of an electromagnetic near field (220) of the rational design (62) by propagating the electromagnetic wave simulated in step b) to the near field plane (252); and d) By applying a simulation of an imaging process of the optical system to the representation of the electromagnetic near field (220), a spatial image is generated from the rational design (62).

13. The method as described in claim 12, wherein the propagation of the incident electromagnetic wave in step a) within the first region (225) of the photomask (14) is approximated by a Helmholtz equation.

14. The method as described in claim 12, wherein the propagation of the incident electromagnetic wave in step a) within the first region (225) of the photomask (14) is approximated using a machine learning model.

15. The method as described in claim 13, wherein a forward Helmholtz equation is used to approximate the Helmholtz equation.

16. The method as described in claim 14, wherein the forward Helmholtz equation is solved using a beam propagation method.

17. The method as described in claim 15, wherein the forward Helmholtz equation is solved using a wave propagation method that approximates the propagation of electromagnetic wave (222) through a non-uniform medium.

18. The method as described in claim 1, wherein a parameter representation (76) of the underlying design (64) is optimized by the optimization problem.

19. The method as described in request item 18, wherein the parameter represents (76) the structural boundary describing the underlying design (64).

20. The method as described in claim 18, wherein the parameter representation (76) includes a contour (78) represented by a graph containing nodes and edges, the positions of which are optimized by solving the optimization problem.

21. The method as described in claim 1, wherein the optimization problem includes parameters describing a modification of the underlying design (64), and wherein the optimization problem imposes a sparsity constraint on these parameters.

22. The method as described in claim 1, wherein, After step i, one or more regions of interest containing possible defect candidates are identified in the acquired spatial image (54), and steps ii to iv apply only to the one or more regions of interest.

23. A computer implementation method for training a machine learning model, which is applied when performing the method as described in claim 6 or 11, or is included in a spatial image simulation method (74) as described in claim 10.

24. A system (94) for detecting defects (58) in a photomask (14), the system (94) comprising: i. an optical system (96) for acquiring a spatial image (54) of the photomask (14); and ii. a data analysis device (98) including at least one memory (100) and at least one processor (102), the optical system (96) and the data analysis device (98) being configured to perform the steps of a method (60) for detecting defects (58) in a photomask (14) as described in any one of claims 1 to 22.

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