Computer-implemented method for detecting defects in a photolithography mask using a machine learning model

Abstract

The invention relates to a computer-implemented method (54) for simulating an aerial image (124) of a model (56, 56') of a photolithography mask (14) illuminated by incident electromagnetic waves (22), the method comprising: obtaining the model of the photolithography mask (14), wherein the model (56, 56') describes the photolithography mask (14) at least partially in one dimension orthogonal to the mask support plane (32); simulating the propagation of the incident electromagnetic waves (22) through the model of the photolithography mask (14) using a machine learning model, wherein the machine learning model maps the model (56, 56') of the photolithography mask (14) to a representation of an electromagnetic field (68) generated by the electromagnetic waves (22) incident on the photolithography mask (14);Obtaining the aerial image (124) of the model (56, 56') of the photolithography mask (14) by applying a simulation of an imaging process. The invention also relates to corresponding computer programs, computer-readable media and systems.

Description

Field of the invention

[0001] The invention relates to a computer-implemented method, a computer-readable medium, a computer program product, and corresponding systems for detecting defects in photolithography masks. The method, the computer-readable medium, the computer program product, and the systems can be used for quantitative metrology, for evaluating defect relevance in photolithography masks, for improving photolithography masks, for system simulation, or for process control, process monitoring, or process improvement. Background of the invention

[0002] A wafer made of a thin silicon disc serves as the substrate for microelectronic components containing semiconductor structures embedded in and built onto the wafer. The semiconductor structures are constructed layer by layer through repeated processing steps that include repeated chemical, mechanical, thermal, and optical processes. The dimensions, shapes, and placement of the semiconductor structures and patterns are subject to multiple influences. One of the most important steps is the photolithography process.

[0003] Photolithography is a process for creating patterns on the substrate of a wafer. The patterns to be printed on the substrate surface are typically created using computer-aided design (CAD). Based on the design, a photolithography mask is created for each layer. This mask contains a magnified image of the computer-generated pattern to be etched into the substrate. The photolithography mask can be further customized, for example, using optical proximity correction techniques. During the printing process, an illuminated image projected by the photolithography mask is focused onto a photoresist thin film formed on the substrate. A semiconductor chip that powers mobile phones or tablets, for example, comprises approximately 80 to 120 patterned layers. In the past, when photolithography required less precision, the circuit layout corresponded to the mask pattern, which in turn corresponded to the wafer pattern.

[0004] Due to the growing integration density in the semiconductor industry, photolithography masks must image ever smaller features on wafers. The aspect ratio and number of layers of integrated circuits are constantly increasing, and features are growing into the third (vertical) dimension. The current height of memory stacks is more than a dozen micrometers. In contrast, feature sizes are becoming smaller. The minimum feature size, or critical dimension, is below 20 nm, or even below 10 nm, for example, 7 nm or 5 nm, and will approach feature sizes below 3 nm in the near future. As the complexity and dimensions of semiconductor features grow into the third dimension, the lateral dimensions of integrated semiconductor features are becoming smaller. Producing the small feature dimensions imaged onto the wafer requires photolithography masks or stencils for nanoimprint photolithography with ever smaller features or pattern elements.The manufacturing process of photolithography masks and stencils for nanoimprint photolithography is therefore becoming increasingly complex, time-consuming, and ultimately expensive. With the advent of EUV photolithography scanners, the nature of the masks changed from transmission-based patterning to reflection-based patterning.

[0005] Today, the minimum feature size on the mask has reached subwavelength dimensions. Consequently, the so-called optical proximity effect occurs, caused by the non-uniformity of energy intensity due to optical diffraction during the exposure process. This results in images formed on the substrate not faithfully reproducing the patterns on the photolithography mask.

[0006] Therefore, many applications require an aerial image of the photolithography mask, which simulates the radiation intensity distribution at the substrate level. This aerial image enables analysis of the semiconductor structures printed on the substrate during the printing process. However, creating an aerial image is time-consuming and expensive. Therefore, methods for simulating aerial images based on a model of a photolithography mask have become important.

[0007] These methods include time-consuming rigorous simulations, such as FDTD (Finite Difference Time Domain) or RCWA (Rigorous Coupled Wave Analysis), as well as fast approximations, such as the Thin Element Approximation (TEA). Due to the high computational burden for full-chip applications, rigorous simulation is not typically used in commercial computer photolithography software. The Thin Element Approximation (TEA) assumes that the thickness of the features on the photolithography mask is very small compared to the wavelength and that the widths of the features on the photolithography mask are very large compared to the wavelength.However, as lithographic processes use radiation of increasingly shorter wavelengths, and the structures on the pattern creation device become smaller and grow into the vertical dimension, these assumptions no longer apply. The interaction of the incident radiation with the absorber structures results in 3D mask effects that must be accounted for in the simulation. Therefore, the TEA for shortwave radiation produces inaccurate aerial images.

[0008] A typical 3D mask effect is mask shadowing. The CRA (Chief Ray Angle) specifies the angle between the optical axis and the normal vector of the mask surface. Current EUV projection systems, for example, use a CRA of 6°. Due to the height of the absorber structures and the non-telecentric illumination at the mask level, which modulates the recorded intensity from the shadowed mask area through the reflective optics onto the wafer, mask shadowing occurs. At the wafer level, this leads to asymmetric shadowing, image shift and size bias depending on the feature orientation, as well as a shift in the process window.

[0009] Another mask 3D effect is phase shifts caused by diffraction at the absorber structures. These phase effects produce imaging effects that are very similar to phase deformations caused by wave aberrations in the projection systems.

[0010] Another mask 3D effect can be attributed to the reflective nature of EUV photolithography masks. The majority of the reflected light originates from the multilayer, which is designed to provide high reflectivity over a sufficiently wide range of incident angles. However, there is also some reflected light from the top of the absorber, resulting in double images.

[0011] These mask 3D effects cannot be ignored during the lithography process. However, rigorous simulation methods, such as FDTD (Finite Difference Time Domain) or RWCA (Rigorous Coupled Wave Analysis), which consider mask 3D effects, are computationally infeasible.

[0012] Therefore, there is a need for an accurate and fast simulation method for aerial images of photolithography masks.

[0013] A known method for simulating an aerial image of a photolithography mask is disclosed in US 10,209,615 B2. The method uses a thin mask image as input and simulates the near-field image using a neural network.

[0014] Similar methods can also be found in the article "Fast mask near-field calculation using fully convolution network" by Lin, Jiaxin et al., 2020 International Workshop on Advanced Patterning Solutions (IWAPS). IEEE, 2020. pp. 1-4, and in the article "Accurate prediction of EUV lithographic images and 3D mask effects using generative networks" by Awad, Abdalaziz, et al., Journal of Micro / Nanopatterning, Materials, and Metrology, 2021, Vol. 20, No. 4, pp. 1-15.

[0015] Due to the use of a neural network, no rigorous simulations are required, thus reducing computation time. The neural network is trained using thin mask images and corresponding near-field images, which can be obtained using rigorous simulation methods. However, thin mask images represent the mask as a 2D image, thus ignoring its thickness and topographical structures and any mask 3D effects, especially in the case of short wavelengths. In addition, training the neural network requires large amounts of training data. This training data must be generated using rigorous simulation methods with very long runtimes. Therefore, training the neural network is still very time-consuming, even if the inference is fast.Furthermore, the loss function penalizes deviations of the near-field images obtained using the neural network from the simulated near-fields, although the predictions of the neural network may not be consistent with the underlying physical principles that generated the simulated near-fields.

[0016] The article "MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations" by J. Lim and D. Psaltis, published in APL Photonics on January 10, 2022, describes neural networks that use the residual of physical differential equations as a loss function to simulate physical relationships—in this case, Maxwell's equations. Such neural networks are also described in US 2022 / 0414429 A1.

[0017] The article "Modeling of Near- and Far-field Diffraction from EUV Absorbers Using Physics-informed Neural Networks" by V. Medvedev, A. Erdmann, and A. Rosskopf, published in the Photonics & Electromagnetics Research Symposium in July 2023, describes neural networks that use the residual of the Helmholtz differential equation as a loss function to simulate the scattering of EUV light from absorber structures of photomasks. However, periodic boundary conditions are used here, which are not suitable for describing electromagnetic waves with arbitrary incidence directions.

[0018] One object of the invention is therefore to obtain a simulation method for aerial images with increased accuracy. Another object of the invention is to obtain a simulation method for aerial images that requires little computing time. Another object of the invention is to reduce the required memory. Another object of the invention is to reduce the required user effort and the required expert knowledge. Another object of the invention is to obtain an aerial image simulation method that is applicable to transmission-based and reflection-based photolithography masks. Another object of the invention is to improve the design of a photolithography mask without actually having to print a wafer. Another object of the invention is to detect defects in photolithography masks with high accuracy and low computing times.A further object of the invention is to evaluate the relevance of defects detected in photolithography masks with high accuracy and low computation times. Another object of the invention is to detect edge placement errors of structures on photolithography masks.

[0019] These objects are achieved by the invention specified in the independent claims. Advantageous embodiments and further developments of the invention are specified in the dependent claims. Brief description of the invention

[0020] Embodiments of the invention relate to computer-implemented methods, a computer-readable medium, a computer program product and corresponding systems for detecting defects in photolithography masks.

[0021] A first embodiment of the invention relates to a computer-implemented method for detecting defects in a photolithography mask, wherein the photolithography mask comprises a mask support and a grating, wherein the grating comprises absorber structures and non-absorber structures forming a pattern on at least a portion of the mask support, wherein the photolithography mask further comprises an absorber portion extending between an absorber plane and a mask support plane of the photolithography mask, and a mask support portion extending between the mask support plane and a base plane of the photolithography mask, wherein the photolithography mask is illuminated by incident electromagnetic waves, the method comprising: obtaining an aerial image of the photolithography mask; simulating an aerial image of a model of the photolithography mask by the following steps: obtaining the model of the photolithography mask,wherein the model describes the photolithography mask at least partially in a dimension orthogonal to the mask support plane; simulating the propagation of the incident electromagnetic waves through the model of the photolithography mask using a machine learning model comprising a convolutional neural network, wherein the machine learning model maps the model of the photolithography mask to a representation of an electromagnetic field generated by the electromagnetic waves incident on the model of the photolithography mask, wherein Floquet-Bloch boundary conditions are used on at least one pair of opposite edges of the model of the photolithography mask that are orthogonal to the mask support plane,and wherein the Floquet-Bloch boundary conditions are implemented using circular padding in the convolutions at the at least one pair of opposing edges and multiplying the padded values ​​by a phase shift induced by an angle of incidence of the electromagnetic waves; obtaining the aerial image of the model of the photolithography mask by applying a simulation of an imaging process of a photolithography system or an optical measurement system within a projection section to the representation of the electromagnetic field in a near-field plane close to the absorber plane, the projection section extending between the near-field plane and a wafer plane; and detecting defects in the photolithography mask by comparing the obtained aerial image with the simulated aerial image,Wherein the defects include edge placement errors, and wherein the edge placement errors are detected by registering the obtained aerial image with the simulated aerial image. The simulated aerial image can be used for defect detection and verification of photolithography masks, for example, for defect detection in a photolithography mask, for improving the design of a photolithography mask, or for detecting edge placement errors of structures on a photolithography mask. Based on the calculated aerial image, various important metrics for photolithography can then be evaluated, for example, critical dimension (CD), normalized image log-slope (NILS), or by calculating a defocus stack of aerial images, also the best focus and / or depth of field.

[0022] According to one embodiment of the invention, the simulated electromagnetic waves are incident on the base plane, propagated within the mask support portion of the photolithography mask from the base plane to the mask support plane, and propagated within the absorber portion of the photolithography mask from the mask support plane to the absorber plane. Thus, the computer-implemented method for detecting defects in photolithography masks can be applied to transmission-based photolithography masks, e.g., DUV photolithography masks.

[0023] According to one embodiment of the invention, the mask support comprises a multilayer in the form of a stack of optical thin films for reflecting the electromagnetic waves. The simulated electromagnetic waves are incident on the absorber plane, are propagated within the absorber portion of the photolithography mask from the absorber plane to the mask support plane, are reflected within the multilayer in the mask support portion of the photolithography mask, and are propagated within the absorber portion of the photolithography mask from the mask support plane to the absorber plane. In this way, the computer-implemented defect detection method can be applied to reflection-based photolithography masks, e.g., EUV photolithography masks.

[0024] According to one embodiment of the invention, the mask carrier comprises a glass substrate, and the simulated electromagnetic waves are incident on the absorber plane, are partially reflected by the absorber plane, and are partially propagated within the absorber portion of the photolithography mask from the absorber plane to the mask carrier plane, are reflected at the mask carrier plane of the photolithography mask, and are propagated within the absorber portion of the photolithography mask from the mask carrier plane to the absorber plane. In this way, the computer-implemented defect detection method can be applied to reflection-based measurements of DUV photolithography masks.

[0025] In each of the aforementioned embodiments, further reflections and interference effects at other material interfaces are also taken into account.

[0026] The photolithography mask may have an aspect ratio between 1:1 and 1:4, preferably between 1:1 and 1:2, most preferably 1:1 or 1:2. The photolithography mask may have a nearly rectangular shape. The photolithography mask may preferably be 5 to 7 inches long and wide, most preferably 6 inches long and wide. Alternatively, the photolithography mask may be 5 to 7 inches long and 10 to 14 inches wide, preferably 6 inches long and 12 inches wide.

[0027] A "model" of the photolithography mask refers to a representation of the photolithography mask or a portion thereof. The model may, for example, comprise a computer-readable file, such as a CAD file or a GDS file, or a technical drawing, a set of polygons representing the structures of the photolithography mask or a portion thereof. A model of a photolithography mask may include material information, such as complex refractive indices of the materials included in the photolithography mask, electrical permittivities, magnetic permeabilities, or derived representations. A model of a photolithography mask may include parameters describing dimensions of structures in the photolithography mask, such as the thicknesses of the layers in the multilayer of an EUV mask or the thickness of absorber layers, or the dimension of the absorber structures.A model of a photolithography mask may include parameters that describe the location of structures in the photolithography mask, e.g., the location of absorber structures or layers in the multilayer. A model of a photolithography mask may include parameters that describe the shape of structures in the photolithography mask, e.g., the shape of the absorber structures such as sidewall angles or corner rounding, etc. A model of a photolithography mask may include an image, e.g., a 2D image or a 3D image (e.g., a volume of voxels or a number of 2D slices of a volume), that represents properties of the photolithography mask. The image may contain one, two, or more channels. The image may include image elements, e.g., pixels or voxels. The properties of the photolithography mask may include material properties, e.g., refractive indices, electrical permittivities, magnetic permeabilities, or derived representations.The model of a photolithography mask can include descriptions of the structures within the photolithography mask, e.g., in the form of curves, contours, polygons, splines, NURBS, Bezier curves, etc.

[0028] The model of the photolithography mask describes the photolithography mask at least partially in a dimension orthogonal to the mask support plane. For example, the model describes the structure of the photolithography mask at least partially in a dimension orthogonal to the mask support plane. For example, the model of the photolithography mask comprises the absorber section and / or the mask support section. Since both the absorber section and the mask support section extend along a dimension orthogonal to the mask support plane of the photolithography mask, the model describes the photolithography mask at least partially in this dimension. Preferably, the model also describes the photolithography mask in other dimensions, e.g., in one or more dimensions parallel to the mask support plane. The model of the photolithography mask may comprise one or more different sections of the photolithography mask.The one or more different sections may be arranged at different depths relative to the normal of the mask support plane. For example, the model of the photolithography mask may include sections of the absorber section and / or the mask support section and / or the absorber plane and / or the mask support plane. In particular, the model may include sections of the absorber section and the mask support section.

[0029] The electromagnetic near-field and the aerial image are calculated using a model of the photolithography mask that explicitly describes the photolithography mask at least partially in a dimension orthogonal to the mask support plane. For example, the model comprises an absorber section and / or a mask support section. In this way, the configuration of the mask in the vertical dimension, including mask 3D effects such as non-telecentricities, shifts of the best focus position, and image blurring, are taken into account in the simulation, rather than using a thin mask image as a model of the photolithography mask. Thus, the propagation of electromagnetic waves within the photolithography mask is simulated with increased accuracy, especially for short wavelengths such as those used with EUV photolithography masks.At the same time, a machine learning model is used to simulate the propagation of electromagnetic waves within the photolithography mask. This eliminates the need for time-consuming, rigorous simulations of the physical processes. Instead, during training, the machine learning model learns to map the photolithography mask model to a representation of an electromagnetic field generated by the electromagnetic waves incident on the photolithography mask model. During inference, i.e., during the application of the method after training, no time-consuming simulations are required, since the machine learning model has already derived this knowledge during the training phase. Therefore, the method for simulating an aerial photograph of a photolithography mask model according to this invention is much faster than rigorous simulations.

[0030] A representation of an electromagnetic field generated by electromagnetic waves incident on the photolithography mask can refer to the (complex) electric field E or the (complex) scattered electric field E sc = E - E inc refer, where E inc the incident electric field. A complex electromagnetic field can be represented, for example, in terms of the real and imaginary parts or the amplitude and phase, etc. A representation of an electromagnetic field can refer to the (complex) magnetic field H or the (complex) scattered magnetic field H sc = H - H inc where H inc the incident magnetic field. A representation of an electromagnetic field may include the envelope of the total or scattered electric field or the total or scattered magnetic field, for example, the envelope of the total electric field Ee−ikinc⋅r, where the rapidly varying electric field is represented by the rapidly varying component e -ikinc·r is demodulated, where k incdenotes the incident wavevector and r the spatial coordinate vector. The envelope is finally multiplied by the phase term to obtain the electric field E. A representation of an electromagnetic field may include the measurements derived from the electromagnetic field, e.g., diffraction orders, the spectrum, the far field or the intensity field, etc. The representation of the electromagnetic field in the photolithography mask may refer to the electromagnetic field in the photolithography mask, to a portion of the electromagnetic field in the photolithography mask, to an electromagnetic field adjacent to the photolithography mask, e.g., a near field, etc. A representation of an electromagnetic field may include representations of the electromagnetic field for different spatial directions.For example, a representation of an electromagnetic field may comprise a 2D or 3D image containing one, two, or more channels, such that the 2D or 3D image comprises a representation of the electromagnetic field in each spatial direction, e.g., the complex electric field in the x and y or x, y, and z directions, resulting in a 2D or 3D image with four or six channels.

[0031] An optical metrology system refers to a system that measures an aerial image of at least a portion of a photolithography mask or quantities that can be derived from the aerial image, for example, critical dimension (CD), normalized image log slope (NILS), edge placement, defects, etc.

[0032] In the case of a photolithography system, a wafer plane refers to a plane in the photoresist on a wafer when the wafer is placed into the photolithography system. In the case of an optical metrology system, a wafer plane refers to a plane where the camera sensors are located.

[0033] According to a preferred embodiment of the invention, the machine learning model was trained using a loss function comprising one or more partial differential equations (PDEs) describing properties of the representation of the electromagnetic field in the photolithography mask. Machine learning models with loss functions comprising PDEs describing physical principles are also called physics-informed neural networks. Preferably, the one or more partial differential equations are derived from the Maxwell equations or from a Helmholtz equation. These can be used to model the propagation of electromagnetic waves within an inhomogeneous medium comprising different materials.In this way, the trained machine learning model produces predictions consistent with the underlying physical principles of the electromagnetic field in the photolithography mask. Additionally, user effort is greatly reduced because training data can be generated automatically by simply creating models of photolithography masks. The corresponding representations of the electromagnetic fields are not required as part of the training data—instead, the loss function of the machine learning model can include the residual of the one or more PDEs used to evaluate the simulated electromagnetic fields.

[0034] According to one example, the model of the photolithography mask comprises an image in the form of a cross-sectional image comprising properties of a cross-section of the photolithography mask. In this way, the different sections of the photolithography mask arranged along a dimension orthogonal to the mask support plane of the photolithography mask can be represented in the model used as input to the machine learning model. Thus, the propagation of the electromagnetic waves in the different sections of the photolithography mask can be learned by the machine learning model, thereby increasing the accuracy of its predictions. To represent a 3D model of a photolithography mask, an image comprising multiple cross-sectional images can be processed by the machine learning model, either simultaneously or sequentially.

[0035] According to one example, the model of the photolithography mask comprises an image in the form of a voxel volume that includes properties of a portion of the photolithography mask, e.g., a voxel volume that includes material properties of the portion of the photolithography mask. In this way, a 3D model of a photolithography mask can be represented and used as input to the machine learning model. By using voxel-based models, spatial correlations between adjacent slices of the photolithography mask can be modeled, and the electromagnetic field can be simulated for the full 3D model of the photolithography mask by the machine learning model. In this way, the accuracy of the predictions of the machine learning model, the simulated near fields, and the aerial images is improved.

[0036] In one example, the model of the photolithography mask contains properties of the materials in the photolithography mask. The properties of the materials can include, for example, refractive indices, electrical permittivities, magnetic permeabilities, or derived representations, e.g., normalized values ​​such as relative permittivity values ​​normalized to the range [0,1] using the minimum and maximum relative permittivity values. The refractive indices can be specified as complex numbers such that the imaginary part contains the attenuation, while the real part accounts for refraction. In particular, in the case of absorbing materials in the photolithography mask, complex numbers are advantageous for modeling the attenuation of the electromagnetic waves.

[0037] In one example, the photolithography mask model contains characteristic functions of the materials in the photolithography mask. For each material in the photolithography mask, the material distribution can be modeled as a characteristic function that contains the value 1 at locations encompassing the material and the value 0 at other locations.

[0038] In one example, the photolithography mask model comprises two or more channels. For example, each channel may include a characteristic function indicating the presence of a specific material at each location. Alternatively, the photolithography mask model may include refractive indices of the material at each location, and the refractive indices may be specified using complex numbers. For this purpose, the real part and the imaginary part may each constitute a channel of the photolithography mask model. Alternatively, the real part and the imaginary part may be encoded sequentially or alternately in a single channel; other encoding techniques may be used, or machine learning models that handle complex inputs, such as a complex-valued neural network, may be used.

[0039] The machine learning model can be configured and trained in various ways. For example, the machine learning model may include a random forest, a decision tree, a regression model (e.g., a linear regression model, a polynomial regression model, a ridge regression model, etc.), a support vector regression model, a neural network, a neural operator, etc. It takes a model of the photolithography mask as input, which is mapped to a representation of an electromagnetic field generated by the electromagnetic waves incident on the photolithography mask as output. Preferably, the machine learning model is implemented using parallel processing, e.g., using GPUs or TPUs, since many of the operations in a machine learning model can usually be performed in parallel. This reduces the computation time during training and inference.

[0040] In a preferred embodiment, the machine learning model comprises a neural network, in particular a deep neural network, a neural operator, a convolutional neural network, a conditional generative adversarial neural network, a transformer, etc. Neural networks, in particular deep neural networks, are modeled after the human brain and have been successfully used in the recent past to solve difficult modeling tasks. Instead of requiring a user to define modeling rules, the neural network learns from the training data in a data-driven manner. Thus, the trained model is optimally and automatically adapted to the modeling task. Neural networks, in particular deep neural networks, are capable of learning complex relationships between the input and output of a machine learning model.Using a neural network to simulate the propagation of electromagnetic waves in the photolithography mask therefore improves the accuracy of the simulated near-fields and aerial images. Additionally, neural networks are very fast during inference, as a single forward pass is sufficient to obtain the output. Since the operations within each layer during the forward pass can usually be performed in parallel, the neural network can be implemented using parallel processing, e.g., using GPUs or TPUs, thus obtaining very short computation times. Furthermore, using a neural network reduces user effort, as no expert knowledge of the relationships within the data is required to train the neural network.

[0041] In one example, the machine learning model includes a neural operator. A neural operator is a neural network that learns a mapping between infinite-dimensional function spaces (rather than functions between finite-dimensional vector spaces). Neural operators can thus be used to approximate the solution operators posed by PDEs. Instead of learning a mapping from the input space to the output space from training data comprising pairs of models of photolithography masks and simulated electromagnetic fields, a neural operator is trained to solve a PDE by including the residual of the PDE in the loss function. In this way, the predictions generated by the neural operator are consistent with the physical principles underlying the predictions, thus improving the accuracy of the predictions.Furthermore, user effort is reduced because the training data only includes models of photolithography masks, not the corresponding simulated representations of the electromagnetic fields. These are not required because the PDE is included in the loss function used to evaluate the neural operator's predictions. Since rigorous simulations of electromagnetic fields require a long computation time, the computation time for training the neural operator is greatly reduced. Furthermore, the effort required to generate training data for a neural operator is reduced compared to standard neural network training procedures.

[0042] The machine learning model includes a convolutional neural network (CNN). A convolutional neural network is a neural network that includes at least one convolutional layer. Convolutional neural networks use one or more convolutional layers that apply different types of learned filters to the input. Because the filters are learned from training data in a data-driven manner, they are optimally adapted to the input data. By applying cascades of learned filters to the input, the CNN is able to extract low- and high-level features that are specifically tailored to the modeling task. This improves the accuracy of the CNN's predictions and thus of the simulated near-field and aerial imagery.

[0043] Floquet-Bloch boundary conditions are applied to at least one pair of opposing edges of the photolithography mask model that are orthogonal to the mask support plane. For example, in the case of a model in the form of a 2D cross-sectional image with a horizontal mask support plane, opposing edges that are orthogonal to the mask support plane are the vertical edges of the 2D cross-sectional image. For example, in the case of a model in the form of a 3D image with a horizontal mask support section, opposing edges that are orthogonal to the mask support plane include the vertical edge planes in the y / z direction and in the x / z direction.In the case where a) the structures within the photolithography mask model are periodic and b) the incident plane electromagnetic wave exhibits a phase shift with respect to opposite edges of the photolithography mask model due to its angle of incidence, the electric field generated by the incident electromagnetic wave exhibits the same phase shift at the opposite edges and is thus quasi-periodic, according to the Floquet-Bloch theorem. Thus, the arbitrary illumination angle of the incident electromagnetic waves implies that the simulated electromagnetic field is quasi-periodic according to the Floquet-Bloch theorem, i.e., periodic with an additional phase shift α.To accurately model the quasi-periodic electromagnetic field, Floquet-Bloch boundary conditions are implemented using circular padding in the CNN convolutions of at least one pair of opposing edges orthogonal to the mask support plane, and multiplying the padded values ​​by a phase shift induced by the angle of incidence of the electromagnetic waves on the photolithography mask. The angle of incidence can be measured with respect to the normal of the absorber plane. This improves the accuracy of the predicted near-fields and aerial images.

[0044] According to one example, the machine learning model comprises a neural network with an encoder-decoder architecture. An encoder-decoder architecture can comprise an encoder and a decoder, an encoder alone, or a decoder alone. By using an encoder, the input can be mapped to a low-dimensional feature space (called a bottleneck). The decoder then maps the features in the low-dimensional feature space to the desired output. By using a low-dimensional feature space, the information in the input is compressed so that only the most meaningful information for the learning task is preserved. This improves the accuracy of the neural network's predictions. Additionally, computation time is reduced because only the training of the neural network requires time, but inference is fast.

[0045] In one example, the machine learning model includes a neural network with a U-Net architecture. A U-Net architecture is an encoder-decoder architecture with additional skip connections between layers of the encoder and layers of the decoder. A U-Net is an image-to-image model that transforms one type of image (the model of the photolithography mask) into another image (the representation of the electromagnetic field). The additional skip connections can be used to directly access information in the encoder before it is mapped to the latent space. This allows specific details of the input to be preserved, which can be used by the decoder. This improves the accuracy of the near-field and aerial images simulated by the neural network. Additionally, computation time is reduced because only the training of the neural network requires time, but inference is fast.

[0046] In one example, the machine learning model includes a neural network with at least one attention mechanism, such as a Transformer machine learning model. An attention mechanism is a part of a neural network used to consider a local or global context of portions of the input.

[0047] The term "attention mechanism" refers to a computational procedure that is part of a machine learning procedure that transforms input data into output data. The computational procedure is used to detect relationships between pieces of the input data that are relevant to the transformation. To detect relationships between pieces of the input data, the attention mechanism may transform an element of the input data into a new representation, exploiting one or more other elements of the input data and their similarity to the element. The transformation may comprise a similarity function and an aggregation function, where the similarity function assesses the similarity of an element to one or more other elements in the input data, and the aggregation function maps the element and the one or more other elements and their similarities to the new representation of the element.The aggregation function may generate the new representation of the element using a weighted combination of the one or more other elements in the input data, wherein the weights depend on the similarities of the element to the one or more other elements. An attention mechanism may comprise at least one trainable parameter, preferably for preprocessing elements of the input data, such that the similarity function is applied to the preprocessed elements. The at least one trainable parameter may define at least one projection matrix used to preprocess the elements of the input data. Throughout the aforementioned definition of the term "attention mechanism," representations of the elements of the input data may be processed instead of elements of the input data.

[0048] Unlike convolutional or fully connected layers, for example, in CNNs, the weights applied to the elements of the input data depend on the input data, more precisely on the similarity of each element to the other elements of the input data, rather than being fixed after training. Furthermore, unlike convolutional operations, the attention mechanism does not require a fixed sequence of elements in the input data. Instead, context windows with dynamic or global sizes can be implemented, rather than using fixed-size context windows as in the case of convolutions. Finally, unlike fully connected layers, the attention mechanism does not require a fixed number of elements in the input data, but can be applied to input data sets of arbitrary size.An attention mechanism can thus be understood as a convolution with input-dependent weights and a context window of arbitrary size. For example, the context window can contain the entire input data. In the case of an imaging dataset, the input data can, for example, contain a sequence of locations that forms a partition of the imaging dataset.

[0049] By using at least one attention mechanism in the machine learning model, dependencies of the electromagnetic field on different sections of the photolithography mask model, e.g., on material properties in different sections, can be taken into account. By using attention mechanisms, this spatial context is not limited to the local receptive field of a convolution, but can encompass larger contexts or even the entire input data, i.e., the global context. By using at least one attention mechanism, the accuracy of the predicted electromagnetic field is thus improved.

[0050] In a preferred embodiment, the machine learning model calculates the representation of the electromagnetic field generated by the electromagnetic waves incident on the photolithography mask model for any given angle of incidence of the electromagnetic waves. For this purpose, the machine learning model may comprise multiple machine learning submodels, each trained for a specific angle of incidence. Alternatively, a single machine learning model may be trained that can handle different angles of incidence of the electromagnetic waves. In this way, user effort can be reduced, since only a single machine learning model needs to be trained to simulate the propagation of electromagnetic waves with arbitrary angles of incidence in the photolithography mask.

[0051] In one example, the angle of incidence is an input parameter of the machine learning model. An input parameter refers to a parameter that is not learned, but is, for example, specified by a user and used by the machine learning model in some way. Using the input parameter, the machine learning model can be parameterized to the angle of incidence. During training, the training data can include the corresponding angle of incidence so that the machine learning model can learn relationships between the angle of incidence and the other parameters of the machine learning model. Using the angle of incidence as an input parameter simplifies the learning task and improves the accuracy of the simulated near-fields and aerial images.

[0052] In one example, the machine learning model comprises a neural network, and the angle of incidence of the electromagnetic waves is used as an input parameter in one of the layers of the neural network. For example, the angle of incidence can be used as an additional parameter in the input layer. Alternatively, the angle of incidence can be used as an additional parameter in any of the other layers of the neural network. If the neural network comprises an encoder-decoder architecture, the angle of incidence of the electromagnetic waves can be used as an input parameter in one of the layers of the encoder. The angle of incidence can, for example, be used as an input parameter in a first, second, or third convolutional layer of the encoder. The angle of incidence can also be used as an input parameter in the bottleneck.This improves the neural network's adaptability to different angles of incidence and simplifies the optimization of the loss function during training. This improves the accuracy of the neural network's predictions and thus of the simulated near-fields and aerial images.

[0053] A computer-implemented method for training a machine learning model for simulating the propagation of electromagnetic waves through a model of a photolithography mask, as described in any of the above embodiments, may comprise: generating models of photolithography masks and, optionally, angles of incidence of the electromagnetic waves incident on the photolithography masks as training data, wherein the photolithography masks comprise a mask support and a grating, the grating comprises absorber structures and non-absorber structures forming a pattern on at least a portion of the mask support, the photolithography masks further comprise an absorber portion extending between an absorber plane and a mask support plane of the photolithography mask, and a mask support portion extending between the mask support plane and a base plane of the photolithography mask,wherein each model of a photolithography mask describes the photolithography mask at least partially in a dimension orthogonal to the mask support plane; and iteratively presenting one or more models of photolithography masks and, optionally, angles of incidence based on the training data to the machine learning model, evaluating the loss function, and modifying the parameters of the machine learning model.

[0054] For example, the loss function includes one or more partial differential equations that describe properties of the representation of the electromagnetic field within the photolithography mask. This ensures that the predictions of the machine learning model are consistent with the physical principles underlying the learning task. Thus, the accuracy of the predictions is improved. Furthermore, the generation of training data is simplified because the predicted representations of the electromagnetic field can be evaluated by calculating the residual of the one or more PDEs, rather than comparing the representations of the predicted electromagnetic fields with computationally expensive, rigorously simulated representations of electromagnetic fields.The training data can even be generated automatically, since only different models of photolithography masks and angles of incidence are needed, which can be generated automatically using simple rule-based randomized algorithms.

[0055] In one example, the one or more partial differential equations are derived from the Maxwell equations or from a Helmholtz equation. This improves the accuracy of the predictions because the Maxwell equations, or their simplified version in the form of a Helmholtz equation, describe the propagation of electromagnetic waves within an inhomogeneous medium.

[0056] The simulated aerial image of the photolithography mask model can be used in different ways.

[0057] For example, based on an accurate aerial image of a photolithography mask model, the photolithography mask design can be improved and mask 3D effects can be mitigated, e.g., by modifying the photolithography mask design layout or by modifying the absorber material and / or absorber thickness within the photolithography mask grid or by applying optical proximity correction techniques to the photolithography mask, e.g., by adding sub-resolution assist structures, etc.

[0058] For example, the simulated aerial image can be used for defect detection. Using an acquired aerial image of a photolithography mask, the photolithography mask can be inspected for defects by simulating an aerial image of the photolithography mask based on a model of the photolithography mask, using the illumination and imaging settings of the optical inspection tool to obtain the acquired aerial image, and comparing the acquired aerial image with the simulated aerial image.

[0059] For example, the relevance of defects in an acquired charged particle beam image of a photolithography mask can be evaluated by simulating an aerial image using the acquired charged particle beam image of the photolithography mask as a model of the photolithography mask, using the illumination and imaging settings of the optical inspection tool to obtain the acquired aerial image, and evaluating the relevance of the defects using the simulated aerial image. Specifically, the defects in the charged particle beam image can be compared with the corresponding locations in the simulated aerial image. This saves considerable time and resources because there is no need to acquire an aerial image of the photolithography mask.

[0060] For example, the simulated aerial image can be used to create a digital twin of a machine that utilizes captured aerial images of photolithography masks. The machine's digital twin is a digital simulation of the machine that uses a method for simulating an aerial image of a photolithography mask model, as described above, to simulate the capture of an aerial image within the machine. The machine's digital twin can be used for many different purposes, such as specifying the functionality and requirements of the machine, demonstrating the machine's functionality to the customer prior to building or shipping the machine, or accelerating the development of parts of the machine, such as the user interface, etc.

[0061] In these applications, instead of capturing an aerial image of the photolithography mask, a simulation of the aerial image of a model of the photolithography mask is used, which significantly reduces the computation time.

[0062] Therefore, the computer-implemented method for detecting defects in a photolithography mask comprises: obtaining an aerial image of the photolithography mask; simulating an aerial image of a model of the photolithography mask using a method for simulating an aerial image as described above; and detecting defects in the photolithography mask by comparing the obtained aerial image with the simulated aerial image. In this way, simulated aerial images can be used to detect defects in a photolithography mask with high accuracy and low computation times. This allows the design of the photolithography mask to be improved without actually printing a wafer.

[0063] Simulating the imaging process can involve resampling the simulated electromagnetic near-field. This can increase the resolution of the simulated aerial image without significantly increasing runtime. This allows aerial images to be obtained with a resolution comparable to that of rigorous simulation methods.

[0064] A computer-readable medium according to an embodiment of the invention stores a computer program that can be executed by a computer device, the computer program comprising code for carrying out a method according to one of the previously described embodiments of the invention.

[0065] A computer program product according to an embodiment of the invention comprises instructions which, when the program is executed by a computer, cause the computer to carry out a method according to one of the previously described embodiments of the invention.

[0066] A system for simulating an aerial image of a model of a photolithography mask according to an embodiment of the invention comprises a data analysis device comprising at least one memory and at least one processor configured to perform the steps of a computer-implemented method for simulating an aerial image of a model of a photolithography mask described above.

[0067] A system for detecting defects in a photolithography mask according to an embodiment of the invention comprises: a subsystem for obtaining an aerial image of the photolithography mask, a data analysis device comprising at least one memory and at least one processor configured to perform the steps of a computer-implemented method for detecting defects in a photolithography mask described above.

[0068] The invention described by examples and embodiments is not limited to the embodiments and examples, but can be implemented by a person skilled in the art through various combinations or modifications thereof. Short description of the drawings Fig. Figure 1 illustrates an exemplary transmission-based photolithography system, e.g., a deep ultraviolet (DUV) photolithography system. Fig. Figure 2 illustrates the propagation of incoming electromagnetic waves through a transmission-based photolithography mask; Fig. 3 illustrates an exemplary reflection-based photolithography system, e.g., an extreme ultraviolet (EUV) light photolithography system; Fig. Figure 4 illustrates the propagation of incoming electromagnetic waves through a reflection-based photolithography mask; Fig.Figure 5a shows the amplitude of a simulated electromagnetic near field of a photolithography mask using the RCWA method or Rigorous Coupled-Wave Analysis method; Fig. Figure 5b shows the amplitude of a simulated electromagnetic near field of a photolithography mask using the thin element approximation (TEA) method; Fig. 6 shows a flowchart of a computer-implemented method for simulating an aerial image of a model of a photolithography mask; Fig. 7a shows a model of an EUV photolithography mask in the form of a cross-sectional image including features of a cross-section of the photolithography mask; Fig. Figure 7b shows a model of an EUV photolithography mask in the form of a voxel volume containing properties of the photolithography mask; Fig.8a, b illustrate the application of the Floquet-Bloch theorem to opposite edges of a photolithography mask orthogonal to the mask support plane; Fig. Figure 9 shows a neural network with a U-Net architecture that maps a model of a photolithography mask in the form of a cross-sectional image to a representation of an electromagnetic field within the cross-section; Fig. Figure 10 shows a neural network with a U-Net architecture that maps a model of a photolithography mask in the form of a voxel volume to a representation of an electromagnetic field; Fig. 11a-d illustrate a single neural network generating representations of electromagnetic fields for arbitrary angles of incidence Θ of the electromagnetic waves onto the photolithography mask; Fig.12a-c illustrate the variation of the shape of the absorber structures within the absorber section of the photolithography mask and the simulated representations of the electromagnetic fields; Fig. 13 illustrates an exemplary electromagnetic field simulated using a model of a photolithography mask in the form of a voxel volume that includes refractive indices of the materials within the photolithography mask; Fig. 14 illustrates a flowchart of a computer-implemented method for training a machine learning model to simulate the propagation of electromagnetic waves through a model of a photolithography mask; Fig. 15 illustrates the training progress for training a Fig. 9 U-Nets shown; Fig. 16 illustrates a computer-implemented method for detecting defects in a photolithography mask; Fig. Figure 17 illustrates a computer-implemented method for evaluating the relevance of defects in a photolithography mask; Fig. 18 illustrates a system for simulating an aerial image of a model of a photolithography mask according to an embodiment of the invention; Fig. 19 illustrates a system for detecting defects in a photolithography mask according to an embodiment of the invention; and Fig. 20 illustrates a system for evaluating the relevance of defects in a photolithography mask according to an embodiment of the invention. Detailed description

[0069] Advantageous exemplary embodiments of the invention are described below and schematically shown in the figures. In the figures and in the description, the same reference numerals are used throughout to describe identical features or components. Dashed lines indicate optional features.

[0070] The methods and systems described herein may be used with a variety of photolithography systems, e.g., transmission-based photolithography systems 10 or reflection-based photolithography systems 10'.

[0071] Fig.1 illustrates an exemplary transmission-based photolithography system 10, e.g., a DUV photolithography system. Main components include a radiation source 12, which may be a deep ultraviolet (DUV) excimer laser source; imaging optics, which may define, for example, partial coherence, and which may include optics that shape the radiation from the radiation source 12; a photolithography mask 14; illumination optics 16 that illuminates the photolithography mask 14; and projection optics 17 that projects an image of the photolithography mask pattern 92 onto a wafer plane 18.An adjustable filter or aperture in the pupil plane of the projection optics 17 can limit the range of beam angles impinging on the wafer plane 18, with the largest possible angle defining the numerical aperture of the projection optics 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 largest angle of the beam emerging from the projection optics 17 that can still impinge on the wafer plane 18.

[0072] In this document, the terms ‘radiation’ or ‘ray’ refer to all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultraviolet radiation, e.g. with a wavelength in the range of approximately 3-100 nm).

[0073] 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 photolithography mask 14. The projection optics 17 may include optical components for shaping, adjusting, and / or projecting the radiation after the radiation has passed through the photolithography mask 14. The illumination optics 16 excludes the light source 12, and the projection optics excludes the photolithography mask 14.

[0074] The illumination optics 16 and the projection optics 17 may comprise various types of optical systems, including, for example, refractive optics, reflective optics, apertures, 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 to collectively or individually direct, shape, or steer the projection beam of radiation.

[0075] Fig.2 illustrates the propagation of incoming electromagnetic waves 22 through a transmission-based photolithography mask 14, e.g., a DUV photolithography mask. The photolithography mask 14 includes a mask carrier 48 and a grating 24. The mask carrier 48 is disposed in a mask carrier portion 27 of the photolithography mask 14, and the grating 24 is disposed in an absorber portion 25 of the photolithography mask 14. The grating 24 is formed by a combination of absorber structures 26 and non-absorber structures 28. The absorber structures 26 are made of one or more materials that absorb electromagnetic waves 22, e.g., titanium nitride or tantalum nitride, etc. The non-absorber structures 28 are made of one or more materials that absorb electromagnetic waves 22 to a lesser extent than the absorber material. For example, the non-absorber structures 28 may comprise vacuum. Thus, the grating 24 is an inhomogeneous medium.The absorber structures 26 and the non-absorber structures 28 are deposited on a mask carrier 48. The mask carrier 48 can comprise a substrate layer 46. The mask carrier 48 in the photolithography mask 14 is delimited by a mask carrier plane 32 and a base plane 34, which preferably runs parallel to the mask carrier plane 32. The mask carrier plane 32 is a surface plane of the mask carrier 48. The base plane 34 is an edge plane through which the electromagnetic waves 22 enter the grating 24. The incoming electromagnetic wave 22 strikes the base plane 34. The base plane 34 forms an interface between the mask carrier 48 and the outside of the photolithography mask 14, through which the electromagnetic waves 22 propagate. The absorber structures 26 in the grating 24 of the photolithography mask 14 are delimited by the mask carrier plane 32 and an absorber plane 30.The absorber plane 30 is an edge plane that contains the part of the surface of the absorber structures 26 that faces away from the mask support plane 32. The absorber plane 30 is preferably parallel to the mask support plane 32. The absorber section 25 of the photolithography mask 14 extends between the absorber plane 30 and the mask support plane 32 and is delimited by these planes. The mask support section 27 of the photolithography mask 14 extends between the mask support plane 32 and the base plane 34 and is delimited by the mask support plane 32 and the base plane 34.

[0076] In transmission-based photolithography masks 14, the simulated electromagnetic waves 22 are incident on the base plane 34 and propagate within the mask support section 27 of the photolithography mask 14 from the base plane 34 to the mask support plane 32 and within the absorber section 25 of the photolithography mask 14 from the mask support plane 32 to the absorber plane 30.

[0077] Fig.3 illustrates an exemplary reflection-based photolithography system 10', e.g., a lithography system for extreme ultraviolet (EUV) light. Main components are a radiation source 12, which may be a laser plasma light source, illumination optics 16, which may, for example, define partial coherence and may include optics that shape the radiation from the radiation source 12, a photolithography mask 14, and projection optics 17, which projects an image of the photolithography mask pattern 92 onto a wafer plane 18.An adjustable filter or aperture in the pupil plane of the projection optics 17 can limit the range of beam angles impinging on the wafer plane 18, with the largest possible angle defining the numerical aperture of the projection optics 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 largest angle of the beam emerging from the projection optics 17 that can still impinge on the wafer plane 18.

[0078] Fig.4 illustrates the propagation of incoming electromagnetic waves 22 through a reflection-based photolithography mask 14, e.g., an EUV photolithography mask. The photolithography mask 14 includes a mask carrier 48 and a grating 24. The mask carrier 48 is disposed in a mask carrier portion 27 of the photolithography mask 14, and the grating 24 is disposed in an absorber portion 25 of the photolithography mask 14. The grating 24 includes absorber structures 26 and non-absorber structures 28 that form a pattern 92 on at least a portion of the mask carrier 48 to be printed onto a wafer. The absorber structures 26 are made of one or more materials that absorb electromagnetic waves 22, e.g., titanium nitride or tantalum nitride, etc. The non-absorber structures 28 are made of one or more materials that absorb electromagnetic waves 22 to a lesser extent than the absorber material.For example, the non-absorber structures 28 may comprise a vacuum. Thus, the absorber structures 26 and the non-absorber structures 28 form an inhomogeneous medium. The absorber structures 26 and the non-absorber structures 28 are deposited on a mask carrier 48. The mask carrier 48 comprises a multilayer 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic waves 22. The mask carrier 48 may have a cover layer 42 and / or a substrate layer 46. The mask carrier 48 in the photolithography mask 14 is delimited by a mask carrier plane 32 and a base plane 34, which preferably runs parallel to the mask carrier plane 32. The mask carrier plane 32 is a surface plane of the mask carrier 48. The absorber structures 28 in the grating 24 of the photolithography mask 14 are delimited by the mask carrier plane 32 and an absorber plane 30.The absorber plane 30 is an edge plane that contains the part of the surface of the absorber structures 26 that faces away from the mask support plane 32. Preferably, the absorber plane 30 is parallel to the mask support plane 32.

[0079] The absorber plane 30 is an edge plane through which the electromagnetic waves 22 enter the grating 24. The incoming electromagnetic waves 22 strike the absorber plane 30. The absorber plane 30 forms an interface between the mask 14 and the outside of the photolithography mask 14, through which the electromagnetic waves 22 propagate. The absorber section 25 of the photolithography mask 14 extends between the absorber plane 30 and the mask support plane 32 and is delimited by these planes. The mask support section 27 of the photolithography mask 14 extends between the mask support plane 32 and the base plane 34 and is delimited by the mask support plane 32 and the base plane 34.

[0080] For reflection-based photolithography masks 14, the mask carrier 48 comprises a multilayer 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic waves 22, wherein the simulated electromagnetic waves 22 are incident on the absorber plane 30, propagate within the absorber section 25 of the photolithography mask 14 from the absorber plane 30 to the mask carrier plane 32, are reflected within the multilayer 38 in the mask carrier section 27 of the photolithography mask 14, and propagate within the absorber section 25 of the photolithography mask 14 from the mask carrier plane 32 to the absorber plane 30.

[0081] An electromagnetic near-field 20 indicates the distribution of electromagnetic waves 22 in a near-field plane 52 near the absorber plane 30 of the photolithography mask 14. Preferably, the near-field plane 52 is parallel to the absorber plane 30 or to the base plane 34 of the photolithography mask 14. The near-field plane 52 can generally be located anywhere between the absorber plane 30 and the wafer plane 18 or outside the photolithography mask. The term "near" refers to a distance between 0 and 1000 nm, preferably a distance between 0 and 100 nm, more preferably a distance between 0 and 50 nm, even more preferably a distance between 0 and 20 nm, and most preferably a distance between 0 and 10 nm. In a preferred embodiment of the invention, the near-field plane 52 and the absorber plane 30 are identical.

[0082] An aerial photograph shows the radiation intensity distribution in the wafer plane 18.

[0083] Known methods for simulating electromagnetic near fields 20 or aerial photographs either require enormous computing times or are not sufficiently accurate.

[0084] In order to simulate the interaction of electromagnetic waves 22 with a photolithography mask 14, the propagation of the electromagnetic waves 22 within the different layers of the photolithography mask 14, which have different materials with different refractive indices, must be taken into account.

[0085] To simulate electromagnetic near fields 20, rigorous simulation techniques such as FDTD (Finite Difference Time Domain), the finite element method (FEM) or the RCWA method (Rigorous Coupled Wave Analysis) are often used. For example, Fig.Figure 5a shows the amplitude of a simulated electromagnetic near field 20 of a photolithography mask 14 using the RCWA method, or Rigorous Coupled Wave Analysis. However, these methods are extremely computationally intensive, making them unsuitable for full-chip applications. A full mask simulation could even take several years.

[0086] Fig.Figure 5b shows the amplitude of a simulated electromagnetic near-field 20 of an EUV photolithography mask 14 for oblique, coherent illumination at EUV wavelength using the thin-element approximation (TEA) method. The thin-element approximation (TEA) method is an efficient algorithm for analyzing diffractive optical elements. The TEA method assumes that the thickness of the structures on the photolithography mask 14 is very small compared to the wavelength of the incoming light and that the widths of the structures on the photolithography mask 14 are very large compared to the wavelength. However, since photolithography processes use radiation of increasingly shorter wavelengths and the structures on the photolithography mask 14 become increasingly smaller, the assumptions of the TEA method may fail. In this case, the photolithography mask 14 can no longer be approximated by a photolithography mask with a flat surface.Instead, the interaction of the radiation at a wavelength below the height of the structures on the photolithography mask 14, which leads to so-called mask 3D effects, must be taken into account. Therefore, a method for simulating an aerial image of a photolithography mask 14 is needed that is fast and accurate even for short wavelengths.

[0087] To reduce computation time, attempts have been made to simulate the near fields of photolithography masks using neural networks. Neural networks can require long training times but are very fast in the inference phase. However, these approaches also require a flat photolithography mask (thin mask) and thus cannot account for mask 3D effects, especially for short wavelengths. One object of the invention is therefore to simulate aerial images by accurately simulating the propagation of electromagnetic waves incident on a photolithography mask with short computation times.

[0088] To achieve these objectives, a computer-implemented method for simulating an aerial image of a model of a photolithography mask according to an embodiment of the invention is described. Fig.6 shows a corresponding flowchart. The computer-implemented method 54 for simulating an aerial image of a model of a photolithography mask 14, wherein the photolithography mask 14 comprises a mask carrier 48 and a grating 24, wherein the grating 24 comprises absorber structures 26 and non-absorber structures 28 forming a pattern 92 on at least a portion of the mask carrier 48, the photolithography mask 14 further comprises an absorber portion 25 extending between an absorber plane 30 and a mask carrier plane 32 of the photolithography mask 14, and a mask carrier portion 27 extending between the mask carrier plane 32 and a base plane 34 of the photolithography mask 14, wherein the photolithography mask 14 is illuminated by incident electromagnetic waves 22, comprises: obtaining the model of the photolithography mask 14, wherein the model 56,56' describes the photolithography mask 14 at least partially in a dimension orthogonal to the mask support plane 32, in a step S1; simulating the propagation of the incident electromagnetic waves 22 through the model of the photolithography mask 14 using a machine learning model, wherein the machine learning model maps the model of the photolithography mask 14 to a representation of an electromagnetic field generated by the electromagnetic waves 22 incident on the model of the photolithography mask 14, in a step S2; and obtaining the aerial image of the model of the photolithography mask 14 by applying a simulation of an imaging process of a photolithography system 10, 10' or an optical measuring system within a projection section 19 to the representation of the electromagnetic field 68 in a near-field plane 52 close to the absorber plane 30,wherein the projection section 19 extends between the near-field plane 52 and a wafer plane 18. The electromagnetic field in the near-field plane 52 near the absorber plane 30 is referred to as the near-field 20.

[0089] Since the grating 24 of the photolithography mask 14 comprises absorber structures 26 and non-absorber structures 28 that form an inhomogeneous medium, the simulation of the propagation of the electromagnetic waves 22 within the absorber section 25 takes into account the inhomogeneity of the grating material. Instead of using a thin mask model to simulate the electromagnetic field incident on the photolithography mask 14, the methods described here use a model of the photolithography mask 14 that explicitly includes the different sections of the photolithography mask 14, i.e., at least the absorber section 25 and the mask support section 27. In this way, the machine learning model can use the model of the photolithography mask 14 as input, which includes, for example, the material distribution within the various structures of the photolithography mask 14.The machine learning model can thus learn the relationships between the structures of different materials and the representation of the electromagnetic field in the photolithography mask 14. Thus, the simulated representation of the electromagnetic field within the photolithography mask is of increased accuracy. Furthermore, machine learning models may require longer computation times during training, but they are typically very fast during inference because only a single forward pass is required. Thus, using a machine learning model that uses an accurate model of the photolithography mask 14, highly accurate representations of electromagnetic fields, and thus near fields and aerial images, can be simulated with short computation times.

[0090] The computer-implemented method 54 for simulating an aerial image of a model of a photolithography mask can be applied to transmission-based photolithography masks and reflection-based photolithography masks.

[0091] In a preferred embodiment, the machine learning model was trained using a loss function comprising one or more partial differential equations describing properties of the electromagnetic field representation in the photolithography mask 14. By directly incorporating the partial differential equations into the loss function during training of the machine learning model, predictions of the machine learning model conform to the underlying physical principles. Thus, the accuracy of the predictions and the simulated aerial images is increased.

[0092] In one example, the one or more partial differential equations are derived from the Maxwell equations or from a Helmholtz equation. The Maxwell equations can be rewritten in a simplified way as a Helmholtz equation in the photolithography context. Thus, by using the Maxwell equations or a Helmholtz equation in the loss function during training of the machine learning model, the machine learning model learns to approximate the PDEs that describe the propagation of the electromagnetic waves within the various sections of the photolithography mask. In this way, the underlying physical principles are explicitly learned by the machine learning model, thereby increasing the accuracy of its predictions and thus of the simulated near fields or aerial images. Furthermore, it is sufficient to include models of photolithography masks in the training data.No time-consuming rigorous simulations of electromagnetic fields corresponding to the photolithography mask models in the training data are required, as the residuals of the PDEs can be used to evaluate the quality of the electromagnetic fields simulated by the machine learning model. Thus, the effort and time required to generate training data are greatly reduced.

[0093] The propagation of electromagnetic waves within a medium can be described using Maxwell's equations. ∈ 0 be the vacuum electrical permittivity and ∈(r, ω) a dielectric function that characterizes the relative electrical permittivity of a specific material within the photolithography mask.

[0094] These relationships are related to the refractive index n(r,ω) of a material via ∈(r,ω) = n(r,ω) 2, where vacuum permeability is assumed for simplicity. Based on these material relationships and in the absence of free charges and currents, the time-harmonic Maxwell equations are given as ∇×E(r,ω)=iωμ0H(r,ω), ∇⋅H(r,ω)=0,∇×H(r,ω)=−iωε0ε(r,ω)E(r,ω), ∇⋅(ε(r,ω)E(r,ω))=0, where E is the electric field strength, H is the magnetic field strength, r is the spatial coordinate vector and ω is the angular frequency.

[0095] Based on Maxwell's equations, the following equation can be derived for the electric field E of an electromagnetic wave: ΔE(r,ω)+ω2c2ε(r,ω)E(r,ω)=−∇(∇ε(r,ω)ε(r,ω)⋅E(r,ω)), where c is the speed of light. The right-hand side couples the components of the electric field, making it difficult to find solutions to this equation. Therefore, the right-hand side is preferably neglected. Neglecting the right-hand side remains valid if the following two assumptions are met: the optical system under consideration does not show a pronounced response depending on the incident polarization, and there is no cross-coupling between individual polarization components. For the lithography environment at short wavelengths, e.g., for EUV photolithography masks, there are two reasons for neglecting polarization and phonon effects, so these assumptions are permissible. First, the contrasts in the refractive index with respect to the different materials of the absorber structures and the non-absorber structures are small.Secondly, the height α of the absorber structures and the non-absorber structures in the grating is larger than the wavelength λ, i.e. aλ≥2. Therefore, the right-hand side of equation (2) can be neglected, resulting in the following Helmholtz equation ΔE(r,ω)+ω2c2ε(r,ω)E(r,ω)=0.

[0096] Fig. 7a and Fig.7b illustrate various models 56, 56' of an EUV photolithography mask that describe the photolithography mask 14 in a dimension orthogonal to the mask support plane 32. Thus, in contrast to thin mask images, the models 56, 56' may include a vertical dimension, i.e., a dimension orthogonal to the mask support plane 32. In particular, the models 56, 56' include different portions of the photolithography mask, i.e., an absorber portion 25 and a mask support portion 27. The models 56, 56' include a grating 24 within the absorber portion 25 and a multilayer 38 within the mask support portion 27. The models 56, 56' in Fig. 7a and Fig. 7b also describe the photolithography mask 14 in one or two directions parallel to the mask support plane 32. In Fig.7a, the model 56 of the photolithography mask 14 includes an image in the form of a cross-sectional image 78 that includes characteristics of a cross-section of the photolithography mask 14. In Fig. 7b, the model 56' of the photolithography mask 14 includes an image in the form of a voxel volume 79 comprising properties of the photolithography mask. In this way, the different sections of the photolithography mask, including the absorber section 25 and the mask support section 27, are represented in the model 56, 56', which is used as input to the machine learning model. Using accurate descriptions of the structures within the photolithography mask, particularly in the vertical direction (z-direction), improves the accuracy of the machine learning model's predictions.

[0097] The propagation of electromagnetic waves within the photolithography mask depends on the materials within the photolithography mask. For example, the Maxwell equations in (1) and the Helmholtz equation in (3) contain dielectric functions ∈(r,ω) that characterize specific materials within the photolithography mask. These functions are related to the refractive indices n(r, ω). By using material properties within the photolithography mask as input to the machine learning model, the machine learning model can learn to map specific material distributions to representations of electromagnetic fields generated by the electromagnetic waves 22 incident on the photolithography mask 14. This can improve the accuracy of the simulated near fields 20 and aerial images.

[0098] In both Fig.7a and 7b, the model 56, 56' of the photolithography mask 14 contains properties of the materials within the photolithography mask 14. The model 56, 56' of the photolithography mask 14 can, for example, contain refractive indices of the materials within the photolithography mask 14. Light propagation in absorbing materials can, for example, be described using a complex-valued refractive index. Thus, the refractive indices can be represented by complex numbers. The imaginary part then handles attenuation, while the real part takes refraction into account. The machine learning model then maps models 56, 56' of the photolithography mask 14 containing refractive indices, e.g., in the form of complex numbers, represented as a 2D or 3D image comprising two channels, to representations of an electromagnetic field generated by the electromagnetic waves 22 incident on the photolithography mask 14.The output of the machine learning model is a representation of the electromagnetic field, for example, a 2D or 3D image comprising two or more channels, e.g., the real and imaginary parts of the electric field or the scattered electric field, or the amplitude and phase of the electric field or the magnetic field, etc. The scattered electric field was found to be easier for the machine learning model to learn due to its lower complexity. However, other representations can also be used as output of the machine learning model. Each representation can be obtained for one or more spatial dimensions of the electromagnetic field, e.g., an x-, y-, and z-component of the real and imaginary parts of the electric field, thus resulting in six channels.

[0099] Other machine learning models may be used for mapping a model 56, 56' of a photolithography mask 14 to a representation of an electromagnetic field generated by the electromagnetic waves 22 incident on the photolithography mask 14.

[0100] In one example, the machine learning model includes a neural operator. A neural operator is a neural network that learns a mapping between infinite-dimensional function spaces (rather than functions between finite-dimensional vector spaces). Neural operator methods represent the solution map of parametric PDEs as a Hilbert-Schmidt integral operator whose kernel is parameterized and learned from paired observations, either using local message passing on a graph discretization of the physical domain or using global Fourier approximations in the frequency domain, as described, for example, in "Learning the solution of parametric partial differential equations with physics-informed DeepONets, Sifan Wang, Hanwen Wang, Paris Perdikaris, arXiv:2103.10974v1." Neural operator methods are resolution-independent.Thus, the model can be queried at an arbitrary input location. To achieve resolution independence, the neural operator can, for example, comprise two subnetworks to achieve abstraction from the discretization of the input and output. A so-called branch net can be used to map the input to a latency representation, and a so-called trunk net can be used to extract latency representations at given coordinates where the output functions are evaluated. Thus, the solution of PDEs, such as the Maxwell equations in (1) or the Helmholtz equation in (3), can be obtained by training a neural operator, thereby improving the accuracy of the predicted near-fields and aerial images and reducing computation time.

[0101] The machine learning model includes a convolutional neural network (CNN). CNNs are a class of neural networks that use convolutions in at least one of their layers. A convolution represents a filtering operation with a filter of a specific size, called the receptive field, applied to the output of the previous layer. During training, the filters are learned from training data to optimally solve the given task by minimizing the loss function. Using a CNN improves the accuracy of the simulated near fields and the aerial image.

[0102] The illumination angle of the incident electromagnetic waves 22 can vary, and a representation of the electromagnetic field often needs to be calculated for a wide variety of incident electromagnetic waves in order to simulate the respective near fields 20, for example, in the context of partially coherent imaging simulations. However, the arbitrary illumination angle of the incident electromagnetic waves 22, which can be measured with respect to the normal of the absorber plane 30, implies that the calculated electromagnetic field is not periodic, but rather, according to Floquet's theorem, only quasi-periodic, i.e., periodic with an additional phase shift. Fig. Figures 8a and b illustrate the application of the Floquet-Bloch theorem to opposite edges of the photolithography mask model that are orthogonal to the mask support plane 32. Fig.Figure 8a illustrates a phase shift between opposite edges 60, 62 of the model of the photolithography mask 14 that are orthogonal to the mask support plane 32. The phase shift is caused by an incident plane electromagnetic wave 22 with an arbitrary angle of incidence. The arbitrary angle of incidence results in a phase shift between the left edge 60 and the right edge 62. To accurately model a plane wave with an arbitrary angle of incidence, Floquet-Bloch boundary conditions are used on at least one pair of opposite edges of the model of the photolithography mask that are orthogonal to the mask support plane. As in Fig.As illustrated in Figure 8b, the Floquet-Bloch boundary conditions can be implemented by using circular padding in the convolutions at at least one pair of opposite boundaries and multiplying the padded values ​​by a phase shift induced by an angle of incidence of the electromagnetic waves. A function E is said to satisfy the 'Floquet-Bloch boundary conditions' or to be 'quasiperiodic' if it is constant over a distance L > 0 with an additional phase factor Δ φ is periodic: E(x+L)=E(x)exp−iΔϕ.

[0103] If such a function is applied to a discrete grid with N E lattice points, the edges can be filled in a convolution by circular padding with an additional phase factor exp -iΔϕ be implemented, where the number of samples that need to be copied is N K- 1. is.

[0104] N K < N E denotes the number of samples in the convolution kernel K : (E∗K)n=∑m=0Nk−1En−m+⌊NK / 2⌋Km En=En+NEexpiΔϕ (n<0) En=En−NEexp−iΔϕ (n≥NE)

[0105] By implementing this type of padding in one or more layers of a neural network, for example in the output layer, it becomes possible to correctly implement the Floquet-Bloch boundary conditions for these layers.

[0106] According to one aspect of the invention, the machine learning model comprises a neural network with an encoder-decoder architecture. An encoder-decoder architecture is a special case of a CNN. It can be used to map an input of a specific size to an output of a specific size, e.g., a model of a photolithography mask to a representation of an electromagnetic field. The encoder-decoder architecture involves a two-stage process in which the input data is first encoded by an encoder into a fixed-length numerical representation, which is then decoded by a decoder to produce an output conforming to the desired format. The encoder maps the input to a latency representation, whereas the decoder maps the latency representation to the output. The spatial resolution of the inputs of the various layers typically decreases in the encoder and increases in the decoder.The layer with the smallest spatial resolution is called the "bottleneck." The output of the bottleneck can be thought of as the most abstracted representation of the input in a latency space or feature space. An encoder-decoder architecture can comprise an encoder and a decoder, an encoder alone, or a decoder alone. Due to the lower dimensionality of the feature space, only the most relevant information in the feature space is preserved, e.g., noise or rare structures are removed. The input is thus represented in an abstract way, and the abstract feature vector is then mapped to the output. Due to this structure, the training of the neural network can be carried out very efficiently, and the near-field and aerial images simulated this way are more accurate.

[0107] In a preferred example, illustrated in Fig. 9 and Fig.10, the machine learning model includes a neural network 74 with a U-Net architecture. In Fig. 9, the neural network 74 maps a model 56 of a photolithography mask 14 in the form of a cross-sectional image 78, comprising properties of a cross-section of the photolithography mask, to a representation of an electromagnetic field 68 within the cross-section. In Fig.10, the neural network 74 maps a model 56' of a photolithography mask 14 in the form of a voxel volume 79 to a representation of an electromagnetic field 68 within the volume. In both figures, the U-Net comprises an encoder 64 that extracts relevant features from the input (the model 56, 56' of the photolithography mask) in a latency space (bottleneck 76), and a decoder 66 that generates the output (the representation of the electromagnetic field 68 within the photolithography mask) from the extracted features in the latency space. The latency space thus contains a compressed representation of the essential input data. Fig. 9 and Fig.10, the representation of the electromagnetic field 68 corresponds to the real and imaginary parts of the scattered complex electric field. The total complex electric field 69 can be obtained by adding the incident complex electric field 67 to the scattered complex electric field.

[0108] A potential implementation of the transformations between the different layers of the U-Net is given in the following table. The abbreviation 'c' refers to the number of channels before and after the transformation. Other transformations can also be used, e.g., different convolution sizes or different channel counts. Transformations in Fig. 9 (2D input) Transformations in Fig. 10 (3D input) A 3x3 fold (c: 1→16) + weight standard + 3x3 fold (c: 16→16) 3x3x3 fold (c: 1→16) + weight norm + 3x3x3 fold (c: 16→16) B 2x2 average pooling 2x2x2 average pooling C 3x3 fold (c→2c) + weight standard + 3x3 fold (2c→2c) 3x3x3 fold (c→2c) + weight norm + 3x3x3 fold (2c→2c) D 3x3 fold (c→2c) + weight standard + 3x3 fold (2c→c) 3x3x3 fold (c→2c) + weight norm + 3x3x3 fold (2c→c) E Transpose folding with step size of 2 Transpose folding with step size of 2 F 3x3 fold (2c→c) + weight standard + 3x3 fold (c→c) 3x3x3 fold (2c→c) + weight norm + 3x3x3 fold (c→c) G 3x3 fold (c: 16→2) 3x3x3 folding (c: 16→2) H Skip connection Skip connection I Input parameter neural network Input parameter neural network

[0109] The skip connections H are used to directly access information in the encoder 64 from the decoder 66. In this way, details contained in the input can be used by the decoder 66, rather than relying solely on the information contained in the features in the latency space. The layer sizes resulting from the transformations are shown in Fig. 9 and Fig. 10. For the convolutional blocks of the U-Net, for example, the activation function for continuously differentiable exponential linear units (CELU) can be used.

[0110] To obtain a single machine learning model that can be used for arbitrary angles of incidence of the electromagnetic waves, the angle of incidence Θ is used as an input parameter of the machine learning model, in particular as an input parameter of the neural network 74. The angle of incidence can be used as a parameter in any of the layers of the neural network 74, e.g., in the input layer, in a layer of the encoder 64, in the bottleneck 76, or in a layer of the decoder 66. In this way, no retraining of the machine learning model is required, and a single machine learning model can be used for any angle of incidence Θ.

[0111] The angle of incidence can be encoded as an input parameter in various ways.

[0112] For example, the angle of incidence can be encoded as a scalar value. Alternatively, the angle of incidence can be encoded using an additional input channel that includes a representation of the electromagnetic field of the incident plane wave.

[0113] Alternatively, an input parameter neural network (I) can be added to the machine learning model, which maps an input comprising the angle of incidence to a feature map as output. Thus, the angle of incidence is encoded as a feature map by the input parameter neural network (I). In a case of a machine learning model in the form of a neural network, the feature map can be used, for example, as an input parameter of any of the layers of the neural network. For example, the feature map can be concatenated with any of the layers of the neural network or used as an additional channel. In the case of an encoder-decoder architecture, the feature map can be used, for example, as an input parameter of the bottleneck or any of the encoder layers.The input parameter neural network can be designed, for example, as a multilayer perceptron (MLP), which comprises, for example, fully connected layers, as in . Fig. 9 and Fig.10. The CELU activation function can be used, for example, for the fully connected layers of the input parameter neural network. The output of the input parameter neural network can be transformed to match the machine learning model, e.g., to match the size of the neural network layer to which it is concatenated, e.g., by transforming the output to the same size as the layer to which it is added. The information from the input parameter neural network is thus propagated through the machine learning model, e.g., through the layers of the machine learning model, in particular, through the decoder layers. In this way, the output of the machine learning model can be controlled by selecting a scalar value.

[0114] Alternatively, the angle of incidence can be used as an input parameter by defining an angle-dependent convolution, i.e., an angle-dependent kernel and bias, for convolving any intermediate results of the machine learning model. For example, in the case of a neural network, the angle-dependent convolution can be applied to the output of any of the neural network layers, thereby introducing the angle of incidence as a parameter into the respective layer of the neural network. In the case of an encoder-decoder neural network, the angle-dependent convolution can be preferentially applied to the bottleneck or to any of the encoder layers. In one example, the angle-dependent convolution can be trained end-to-end with the machine learning model, in particular with the neural network.The encoded angle of incidence can be used as an input parameter into the machine learning model, for example, by adding it to any of the layers of the neural network, e.g., as a scalar value, as an additional channel, by concatenation, or by applying an angle-dependent convolution to any of the layers of the neural network.

[0115] Fig. 11a to 11d illustrate a single neural network 74 that generates representations of electromagnetic fields for arbitrary angles of incidence Θ of the electromagnetic waves 22 onto the photolithography mask 14. The angle of incidence Θ can, for example, be measured with respect to the normal of the surface of the absorber plane 30. Values ​​for Θ are considered within the range [0°, 45°], although other ranges may also be considered.

[0116] Fig.Figure 11a shows a cross-sectional image 78 of the absorber section 25, which contains various absorber materials, tantalum boron oxide (TaBO) and tantalum boron nitride (TaBN), within a support made of ruthenium (Ru). The cross-sectional image 78 contains 224 x 256 pixels, corresponding to a physical size of 112 x 128 nm. The physical width of the absorber section 25 is 27 nm. A single neural network 74 is trained to map a cross-sectional image 78 to a corresponding representation of an electromagnetic field 68 for various angles of incidence Θ. The angle of incidence Θ is specified as a parameter of the neural network 74, as in Fig. 9 and Fig. 10, in particular as a parameter of the second convolutional layer. Fig.11b, c and d show the simulated representations of the electromagnetic fields 68 in the form of the amplitude of the complex total electric field within the region of the photolithography mask corresponding to the cross-sectional image 78 in Fig. 11a, for different angles of incidence Θ= 7°, Θ = 19° and Θ= 32°.

[0117] The absorber structures 26 within the absorber section 25 vary not only in material but also in shape. Fig. 12a-c illustrate the variation of the shape of the absorber structures 26 within the absorber section 25 of the photolithography mask 14 and the simulated representations of the electromagnetic fields 68. In Fig.12a, the absorber structures 26 within the absorber section 25 vary in width and sidewall angles 80, 80'. The sidewall angles 80, 80' denote the angles of the sidewalls 84 of the absorber structures 26 with respect to the absorber plane 30. The sidewall angles 80, 80' can both vary within a range of [79.66°, 100.66°], although other ranges are also possible. The width of the absorber structures 26 can vary within a range of [27 nm, 54 nm]. Thus, the sidewalls 84 can vary within the sidewall variation region 82. For example, beveled absorber structures or trapezoidal absorber structures can be represented in this way.Using training data comprising models of photolithography masks containing absorber sections 25 with absorber structures 26 of various shapes, the neural network 74 can be trained to generate representations of electromagnetic fields 68 for various absorber structure shapes. The training data can be automatically generated by defining ranges for the sidewall angles and width of the absorber structures 26, randomly selecting values ​​from these ranges, and creating the corresponding cross-sectional image as a training sample. The size of the cross-sectional image 78 used as input to the machine learning model is 448 x 672 pixels, corresponding to a physical size of 224 x 336 nm. Fig.12b and c show representations of electromagnetic fields 68 in the form of the amplitude of the complex total electric field, which were simulated for different shapes of the absorber structures 26 for an angle of incidence Θ = 6° of the electromagnetic waves.

[0118] Fig. 13 illustrates an exemplary representation of an electromagnetic field 68 in the form of the amplitude of the complex total electric field, which was simulated using a model of a photolithography mask in the form of a voxel volume 79, comprising properties of a portion of the photolithography mask, in particular refractive indices of the materials within the portion of the photolithography mask. The voxel volume 79 is used as input to the trained machine learning model, for example, the U-Net in Fig.10. The resulting representation of the electromagnetic field 68, generated by the incident electromagnetic waves on the photolithography mask, is highly accurate because it is consistent with the underlying physical principles applied during training of the machine learning model. From the simulated representation of the electromagnetic field, a near-field can be obtained in a near-field plane, and from the near-field, an aerial image in the wafer plane can be obtained by simulating an imaging process of a photolithography system or optical measurement system within the projection section extending between the near-field plane and the wafer plane.The computation time required to simulate the electromagnetic field inside the photolithography mask during inference is several orders of magnitude faster than simulating the electromagnetic field using a rigorous method such as RWCA.

[0119] Fig.14 illustrates a flowchart of a computer-implemented method 98 for training a machine learning model for simulating the propagation of electromagnetic waves through a model of a photolithography mask, as used in any of the above embodiments. The method comprises: generating models of photolithography masks and, optionally, angles of incidence of the electromagnetic waves incident on the photolithography masks as training data, wherein the photolithography masks comprise a mask support and a grating, the grating comprises absorber structures and non-absorber structures forming a pattern on at least a portion of the mask support, the photolithography masks further comprise an absorber portion extending between an absorber plane and a mask support plane of the photolithography mask, and a mask support portion,which extends between the mask support plane and a base plane of the photolithography mask, wherein each model describes the photolithography mask at least partially in a dimension orthogonal to the mask support plane, in a step T1; iteratively presenting one or more models of photolithography masks and, optionally, angles of incidence based on the training data for the machine learning model in a step T2; and evaluating the loss function and modifying the parameters, in particular the weights, of the machine learning model in a step T3.

[0120] The angle of incidence Θ can be used as an additional input parameter to the machine learning model, as described above. Alternatively, the machine learning model can include different sub-machine learning models for different angles of incidence Θ. The training data sample whose angle of incidence Θ is closest to the angle of incidence of the training data sample is then used as input to the machine learning model.

[0121] The training data 72 for training the neural network 74 in Fig. 9 or Fig. 10 contain, for example: - a model 56, 56' of a photolithography mask 14, e.g. a 2D cross-sectional image or a voxel volume, with two channels, comprising refractive indices in the form of complex numbers, - optionally, an angle of incidence of the incident electromagnetic waves, e.g. within the range [0, 45°].

[0122] The models 56, 56' of the photolithography masks can, for example, be generated according to specific rules that define the structure of photolithography masks, e.g., by randomly selecting parameters within predefined ranges, as in Fig. 12a. The parameters may, for example, define material properties, locations, dimensions, and sidewall angles of absorber structures in the absorber section 25 and / or materials, locations, and dimensions of layers within a multilayer in the mask support section 27.

[0123] In a preferred embodiment, the loss function comprises one or more partial differential equations (PDEs) that describe properties of the representation of the electromagnetic field within the photolithography mask. In a preferred example, the one or more partial differential equations are derived from the Maxwell equations in (1) or from the Helmholtz equation in (3). During a training step, one or more training patterns are presented to the machine learning model as inputs. The machine learning model simulates the electromagnetic field within the photolithography mask. For the simulated electromagnetic field, the one or more PDEs, e.g., the Maxwell equations in (1) or the Helmholtz equation in (3), are evaluated, and the residual is calculated. In the case of a perfect simulation, the evaluation of the PDEs should result in a residual of 0.The residual can thus be used to modify the parameters of the machine learning model, e.g., the weights of the various layers of the neural network. To modify the parameters, learning algorithms are used, e.g., a backpropagation algorithm or one of its derivatives. Because the loss function contains one or more PDEs, the learned mapping of a model of a photolithography mask to a representation of an electromagnetic field generated by incident electromagnetic waves on the photolithography mask is consistent with the underlying physical principles. Additionally, no time-consuming rigorous simulation of electromagnetic fields, corresponding to the models of the photolithography masks, is required to generate the training data.

[0124] Since the PDEs in the loss function contain derivatives of the electromagnetic field, which is given in the form of a 2D or 3D image, these derivatives must be evaluated on a discretized grid of the photolithography mask model. To evaluate derivatives on a discretized grid, approximation schemes such as finite differences or finite elements can be used. Circular padding to implement Floquet-Bloch boundary conditions, as described above, can be used here.

[0125] According to one aspect of the invention, the loss function is evaluated using an approximation scheme of derivatives of the electromagnetic field representation that takes into account the physical quantities of the image elements, in particular approximation schemes based on finite differences or finite elements. In the context of Maxwell's equations, approximation schemes for regular grids using finite differences can be used, as described, for example, in the Supplementary Material, Section 2, of "MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations," Joowon Lim, Demetri Psaltis; APL Photonics, January 1, 2022; 7 (1): 011301. In the case of finite element methods, approximation schemes are described in "Finite Element Methods for Maxwell's Equations," Peter Monk, Oxford Science Publications, 2003.

[0126] An example of a potential loss function is described below. Starting from the Helmholtz equation in (3), ΔE(r,ω)+ω2c2ε(r,ω)E(r,ω)=0 the electric field E(r,ω)=Einc(r,ω)+Esc(r,ω). into an incident electric field E inc and a scattered electric field E sc · disassembled.

[0127] E(r, ω) = E inc (r, ω) + E sc (r, ω). The incident electric field E inc (r, ω) satisfies the Helmholtz equation for the permittivity of the background material ∈ b , so that ΔEinc(r,ω)=−ω2c2εbEinc(r,ω).

[0128] This results ΔESC(r, ω)−ω2c2ϵbEinc(r, ω)+ω2c2ϵ(r, ω)E(r, ω)=0.

[0129] The loss function can be defined as the root mean square of the residual of equation (4), evaluated at each pixel within the computation domain: L=1N∑j=1N‖ΔESC(rj, ω)−ω2c2ϵbEinc(rj, ω)+ω2c2ϵ(rj, ω)E(rj, ω)‖2 where r j = 1,.., N denotes the coordinates of the N pixels. The derivatives are approximated by a higher-order finite-difference approximation, using the circular padding described above to appropriately account for the Floquet-Bloch boundary conditions for oblique angles of incidence. To implement the loss function within a real-valued machine learning framework, the components of the complex electric field E=R[E]+iJ[E] and material parameters ϵ=R[ϵ]+iJ[ϵ] further into their real part R[⋅] and imaginary part J[⋅] and calculate the different contributions to the residual in (4) separately. In particular, this leads to the alternative (real-valued) representation of the loss function in (5) LR(r, ω)=ΔR[Esc(r, ω)]−ω2c2ϵbR[Einc(r, ω)]+ω2c2(R[ϵ(r, ω)]R[ϵ(r, ω)]−J[ϵ(r, ω)]J[E(r, ω)]) LJ(r, ω)=ΔJ[Esc(r, ω)]−ω2c2ϵbJ[Einc(r, ω)]+ω2c2(J[ϵ(r, ω)]R[ϵ(r, ω)]−J[ϵ(r, ω)]R[E(r, ω)]) L=1N∑j=1N‖LR(rj, ω)‖2+‖LJ(rj, ω)‖2

[0130] Different variants of this loss function are possible, e.g., other norms can be used, instead of the mean square error in the loss function, the full Maxwell equations in (1) can be used instead of the Helmholtz equation in (3), the residual can be evaluated at other coordinates (e.g., only at a subset of the pixels), etc.

[0131] To train the neural network, physical and mathematical parameters must be selected, e.g., the wavelength of the incident electromagnetic waves, polarization, the boundary condition, specific approximation schemes for the derivatives in the PDEs, etc. In addition, hyperparameters of the machine learning model must be selected, e.g., the U-Net in Fig. 9 or Fig.10. In the case of a neural network, these hyperparameters may include, among other things, the depth of the neural network, the filter size of the convolutional layers, the number of input and output channels in each step, the order and number of convolutional, normalization, and other layers, the normalization, the upsampling scheme, the learning rate, the learning rate decay, the number of epochs, the batch sizes, the optimizers, etc. The hyperparameters of the machine learning model may be automatically selected using hyperparameter optimization techniques known to those of ordinary skill in the art. These techniques can be used to automatically find optimal hyperparameter combinations for the given task.

[0132] The training process of the U-Net can be carried out as follows: in a first step, training and validation datasets are generated that contain models of photolithography masks in the form of material distributions, including refractive indices of the various materials within the photolithography masks. The refractive indices can be represented by complex numbers. Alternatively, the training and validation datasets can be randomly generated by applying rules that define photolithography mask structures, e.g., ranges for the width, sidewall angles, and distances of absorber structures, ranges for the number and thickness of layers within the multilayers, etc. Additionally, each training pattern contains an angle of incidence Θ of the incident electromagnetic waves.Within a training epoch, a batch of training samples is presented to the U-Net as input, and the electromagnetic fields are computed by the U-Net in a forward pass. The one or more electromagnetic fields are padded depending on the selected boundary condition, e.g., using zero padding or circular Floquet-Bloch padding for quasi-periodic boundary conditions. For example, a discretization scheme based on Yee grids can then be used to approximate first- and second-order derivatives. Higher-order finite-difference approximation schemes, e.g., second- or fourth-order, can also be employed for this purpose. After discretization, the derivatives are unpadded to the original size, and the physics-based loss function is evaluated.For this purpose, the original Helmholtz PDE in (3) is modified in two ways: first, the total electric field E = E. inc + E sc into an incident electric field E inc and a scattered electric field E sc Second, the PDE is decomposed into two parts, the real and imaginary parts, to support the material distribution in the form of complex refractive indices within a real-valued neural network architecture. The physics-based loss function is then calculated by computing the residual of the PDE. Finally, backpropagation or a variant of it is used to modify the weights of the U-Net based on the value of the loss function.

[0133] Fig. 15 illustrates the training progress for the training of the Fig.9 according to the previously described training process. The training of the U-Net was performed using the following parameters: the depth of the U-Net was set to 8, the filter size of the convolutional layers was set to 16, the learning rate was set to 0.0001, the learning rate decay was set to 0.5 every 1000 epochs, the batch size was selected as 4, the CELU activation function was used for the convolutional blocks of the U-Net and for fully connected layers of the input parameter neural network, and the Adam optimizer with an initial learning rate within [0.0001, 0.0005] was used. Fig. Figure 15 shows the number of epochs on the horizontal axis 102 and the value of the loss function on the vertical axis 100 for the training dataset 104 and for the validation dataset 106. The graph shows that the value of the loss function is rapidly reduced for both the training data and the validation data and converges to a loss function value close to 0 within 1500 epochs.

[0134] Fig. 16 illustrates a computer-implemented method 108 for detecting defects in a photolithography mask according to an embodiment of the invention, the computer-implemented method 108 comprising: obtaining an aerial image of the photolithography mask in a step M1; simulating an aerial image of a model of the photolithography mask using a computer-implemented method 54 for simulating an aerial image of a model of a photolithography mask according to one of the embodiments described above, in a step M2; and detecting defects in the photolithography mask by comparing the obtained aerial image with the simulated aerial image in a step M3.

[0135] Fig. 17 illustrates a computer-implemented method 110 for evaluating the relevance of defects in a photolithography mask according to an embodiment of the invention, the computer-implemented method 110 comprising: providing a charged particle beam image of the photolithography mask having one or more defects in step N1; simulating an aerial image of a model of the photolithography mask using a computer-implemented method 54 for simulating an aerial image of a model of a photolithography mask according to one of the embodiments described above, wherein the charged particle beam image is used as a model of the photolithography mask, in a step N2; evaluating the relevance of the one or more defects in the photolithography mask using the simulated aerial image, in a step N3. A defect is evaluated as relevant if it is printed on the wafer during the printing process.In contrast, defects that cannot be printed onto the wafer are assessed as irrelevant. The image of the charged particle beam is obtained using a charged particle beam device, for example, a helium ion microscope (HIM), a crossed-beam device, including FIB and SEM, or any charged particle imaging device. The assessment step N3 may include comparing the simulated aerial image with the charged particle beam image. For example, the one or more locations of the one or more defects in the charged particle beam image may be compared with the corresponding one or more locations in the simulated aerial image. If a defect is not visible in the simulated aerial image, it can be concluded that it is not printed onto the wafer and is therefore irrelevant.If a defect is visible in the simulated aerial image, it can be concluded that it is printed on the wafer and is therefore relevant. The simulated aerial image can also be compared with a reference image, e.g. another simulated or captured aerial image of the photolithography mask, to assess the relevance of the one or more defects. For example, if the simulated aerial image is very similar to the reference image at the location of a defect, the defect can be assessed as not relevant. If the simulated aerial image deviates from the reference image at the location of a defect, the defect can be assessed as relevant. The assessment step N3 can additionally or alternatively comprise the calculation of a critical dimension (CD). The calculated CD can be compared with a predefined CD.For example, if the calculated CD is lower than the predefined CD at one or more locations, these locations can be assessed as relevant defects.

[0136] Fig. 18 illustrates a system 112 for simulating an aerial image of a photolithography mask model according to an embodiment of the invention, wherein the system 112 comprises: a data analysis device 114 comprising at least one memory 118 and at least one processor 116 configured to perform a computer-implemented method for simulating an aerial image of a photolithography mask model according to any of the embodiments described above. The processor 116 may be implemented, for example, as a CPU, GPU, or TPU.

[0137] Fig. 19 illustrates a system 120 for detecting defects in a photolithography mask according to one embodiment of the invention, the system 120 comprising: a subsystem 122 for obtaining an aerial image 124 of the photolithography mask; a data analysis device 114 comprising at least one memory 118 and at least one processor 116 configured to perform the steps of the computer-implemented method 108 for detecting defects in a photolithography mask according to any of the above-described embodiments of the invention. The subsystem 122 for obtaining an aerial image 124 of the photolithography mask may comprise an aerial image acquisition system. Alternatively, the subsystem 122 may comprise a database or other storage comprising an aerial image 124 of the photolithography mask, and the subsystem 122 may be configured to load the aerial image 124 from the database or memory.The subsystem 122 for obtaining an aerial image 124 of the photolithography mask 14 can provide an aerial image 124 to the data analysis device 114. The data analysis device 114 includes a processor 116, implemented, for example, as a CPU or GPU. The processor 116 can receive the aerial image 124 via an interface 120. The processor 116 can load program code from a memory 118, for example, program code for executing a computer-implemented method for detecting defects according to one of the embodiments of the invention described above. The processor 116 can execute the program code.

[0138] Fig.20 illustrates a system 126 for assessing the relevance of defects in a photolithography mask according to one embodiment of the invention, the system 126 comprising: a subsystem 128 for obtaining a charged particle beam image 130 of the photolithography mask; a data analysis device 114 comprising at least one memory 118 and at least one processor 116 configured to perform the steps of the computer-implemented method 110 for assessing the relevance of defects in a photolithography mask according to any of the above-described embodiments of the invention. The subsystem 128 for obtaining a charged particle beam image 130 of the photolithography mask may comprise a charged particle beam device, for example, a helium ion microscope (HIM), a crossed beam device, including FIB and SEM, or any charged particle imaging device.Alternatively, the subsystem 128 may include a database or any other storage containing a charged particle beam image 130 of the photolithography mask, and the subsystem 128 may be configured to load the charged particle beam image 130 from the database or storage. The subsystem 128 for obtaining a charged particle beam image 130 of the photolithography mask 14 may provide a charged particle beam image 130 to the data analysis device 114. The data analysis device 114 includes a processor 116, implemented, for example, as a CPU or GPU. The processor 116 may receive the charged particle beam image 130 via an interface 120. The processor 116 may load program code from a memory 118, for example, program code for a computer-implemented method for evaluating the relevance of defects in a photolithography mask according to an embodiment of the invention, as described above. The processor 116 can execute the program code.

[0139] Any of the systems described above may include a user interface, e.g., for showing loss plots, accuracy metrics, training progress, or intermediate predictions to the user, or for receiving input from the user, e.g., machine learning model parameters such as the learning rate, the angle of incidence Θ, or physical parameters. Any of the systems described above may include a database for loading and / or saving training data, validation data, intermediate results, pre-trained machine learning models, e.g., for reuse in another application, etc.

[0140] In summary, the invention relates to a computer-implemented method 54 for simulating an aerial image 124 of a model of a photolithography mask 14 illuminated by incident electromagnetic waves 22, the method comprising: obtaining the model of the photolithography mask 14, wherein the model 56, 56' describes the photolithography mask 14 at least partially in a dimension orthogonal to the mask support plane 32; simulating the propagation of the incident electromagnetic waves 22 through the model of the photolithography mask 14 using a machine learning model, wherein the machine learning model maps the model 56, 56' of the photolithography mask 14 to a representation of an electromagnetic field 68 generated by the electromagnetic waves 22 incident on the photolithography mask 14;and obtaining the aerial image 124 of the model of the photolithography mask 14 by applying a simulation of an imaging process of a photolithography system or an optical measurement system. The invention also relates to corresponding computer programs, computer-readable media, and systems. List of reference symbols 10, 10' photolithography system 12 Radiation source 14 Photolithography mask 16 Lighting optics 17 Projection optics 18 wafer level 19 Projection section 20 Near field 22 electromagnetic wave 24 grids 25 absorber section 26 absorber structures 27 Mask wearer section 28 non-absorber structures 30 absorber level 32 mask wearer level 34 Base Level 38 multi-layer 40 optical thin film 42 Top layer 46 Substrate layer 48 mask wearers 50 Main propagation direction 52 Near-field plane 54 Computer-implemented method 56, 56' model 58 rand 60 Left margin 62 Right margin 64 encoders 66 decoders 67 complex incident electromagnetic field 68 Representation of an electromagnetic field 69 complex electromagnetic total field 70 angle of incidence 72 training data 74 Neural Network 76 Bottleneck 78 Cross-sectional image 79 voxel volumes 80, 80' side wall angle 82 Sidewall variation area 84 side wall 98 Computer-implemented procedure 100 Vertical axis 102 Horizontal axis 104 training data sets 106 Validation dataset 108 Computer-implemented method 110 Computer-implemented method 112 System 114 Data analysis device 116 memory 118 processor 120 systems 122 Subsystem 124 aerial photograph 126 systems 128 subsystems 130 Charged particle beam image

Claims

[1] A computer-implemented method (54) for detecting defects in a photolithography mask (14), wherein the photolithography mask (14) comprises a mask carrier (48) and a grating (24), wherein the grating (24) comprises absorber structures (26) and non-absorber structures (28) forming a pattern (92) on at least a portion of the mask carrier (48), wherein the photolithography mask (14) further comprises an absorber portion (25) extending between an absorber plane (30) and a mask carrier plane (32) of the photolithography mask (14), and a mask carrier portion (27) extending between the mask carrier plane (32) and a base plane (34) of the photolithography mask (14), wherein the photolithography mask (14) is illuminated by incident electromagnetic waves (22), the method comprising: - obtaining an aerial image (124) of the photolithography mask (14); - Simulating an aerial image (124) of a model of the photolithography mask (14) by the following steps: ◯ Obtaining the model (56, 56') of the photolithography mask (14), wherein the model (56, 56') describes the photolithography mask (14) at least partially in a dimension orthogonal to the mask carrier plane (32); ◯ Simulating the propagation of the incident electromagnetic waves (22) through the model (56, 56') of the photolithography mask (14) using a machine learning model comprising a convolutional neural network, wherein the machine learning model maps the model (56, 56') of the photolithography mask (14) to a representation of an electromagnetic field (68) generated by the electromagnetic waves (22) incident on the photolithography mask, wherein Floquet-Bloch boundary conditions are used on at least one pair of opposite edges of the model (56, 56') of the photolithography mask (14) that are orthogonal to the mask support plane (32), and wherein the Floquet-Bloch boundary conditions are calculated using circular padding in the convolutions at the at least one pair of opposite edges and multiplying the padded values ​​by a phase shift,which is induced by an angle of incidence (70) of the electromagnetic waves (22); , o Obtaining the aerial image (124) of the model (56, 56') of the photolithography mask (14) by applying a simulation of an imaging process of a photolithography system (10, 10') or an optical measuring system within a projection section (19) to the representation of the electromagnetic field (68) in a near-field plane (52) close to the absorber plane (30), wherein the projection section (19) extends between the near-field plane (30) and a wafer plane (18); and - detecting defects in the photolithography mask (14) by comparing the obtained aerial image with the simulated aerial image, wherein the defects comprise edge placement errors and wherein the edge placement errors are detected by registering the obtained aerial image with the simulated aerial image. [2] The method of claim 1, wherein the machine learning model was trained using a loss function comprising one or more partial differential equations describing properties of the representation of the electromagnetic field (68) in the photolithography mask (14). [3] The method of claim 2, wherein the one or more partial differential equations are derived from the Maxwell equations or from a Helmholtz equation. [4] The method of any preceding claim, wherein the model (56, 56') of the photolithography mask (14) comprises an image in the form of a cross-sectional image (78) comprising characteristics of a cross-section of the photolithography mask (14). [5] The method of any preceding claim, wherein the model (56, 56') of the photolithography mask (14) comprises an image in the form of a voxel volume (79) comprising properties of a portion of the photolithography mask (14). [6] A method according to any one of the preceding claims, wherein the model (56, 56') of the photolithography mask (14) contains properties of the materials within the photolithography mask (14). [7] A method according to any one of the preceding claims, wherein the model (56, 56') of the photolithography mask (14) contains refractive indices of the materials within the photolithography mask (14). [8] Method according to one of the preceding claims, wherein the model (56, 56') of the photolithography mask (14) comprises characteristic functions of the materials in the photolithography mask (14). [9] Method according to one of the preceding claims, wherein the machine learning model comprises a neural network (74). [10] A method according to any one of the preceding claims, wherein the machine learning model comprises a neural operator. [11] A method according to any one of the preceding claims, wherein the machine learning model comprises a neural network (74) with an encoder-decoder architecture. [12] The method of any preceding claim, wherein the machine learning model comprises a neural network (74) having a U-Net architecture. [13] Method according to one of the preceding claims, wherein the machine learning model comprises a neural network (74) with at least one attention mechanism. [14] The method of any preceding claim, wherein the machine learning model calculates the representation of the electromagnetic field (68) generated by the electromagnetic waves (22) incident on the model of the photolithography mask (14) for any given angle of incidence (70) of the electromagnetic waves (22). [15] The method of claim 14, wherein the angle of incidence (70) is an input parameter of the machine learning model. [16] The method of claim 14 or 15, wherein the angle of incidence (70) of the electromagnetic waves (22) is used as an input parameter in one of the layers of the neural network (74). [17] The method of claim 16, wherein the neural network (74) comprises an encoder-decoder architecture and wherein the angle of incidence (70) of the electromagnetic waves (22) is used as an input parameter in the encoder (64) of the neural network (74). [18] A computer-readable medium having stored thereon a computer program executable by a computer device, the computer program comprising code for carrying out a method according to any one of the preceding claims. [19] A computer program product comprising instructions which, when executed by a computer, cause the computer to perform a method according to any one of claims 1 to 17. [20] A system (112) for simulating an aerial image (124) of a model of a photolithography mask (14), the system (112) comprising a data analysis device (114) comprising at least one memory (116) and at least one processor (118) configured to perform the steps of a computer-implemented method according to any one of claims 1 to 17. [21] A system (120) for detecting defects in a photolithography mask (14), the system (120) comprising: - a subsystem (122) for obtaining an aerial image (124) of the photolithography mask (14); - a data analysis device (114) comprising at least one memory (116) and at least one processor (118) configured to perform the steps of the computer-implemented method according to any one of claims 1 to 17.

Images