Computer-implemented method for generating spatial image of lithographic mask, computer-readable medium, computer program product and corresponding system

Abstract

The invention relates to a computer-implemented method (54, 54 ', 54' ', 54 ''') for generating a spatial image (64) of a model of a lithographic mask (14) under irradiation of an incident electromagnetic wave (22), the method comprising: (a) approximately simulating propagation of the incident electromagnetic wave (22) within a first portion (25) of the lithographic mask (14) comprising a plurality of structures; (b) analytically or numerically simulating the propagation of the simulated electromagnetic waves (22) from step (a) within the second portion (27) of the lithographic mask (14); (c) simulating a representation of an electromagnetic near field (20) of the model of the lithographic mask (14) by propagating the simulated electromagnetic waves (22) from step (b) to a near field plane (52); and (d) generating a spatial image (64) of the lithographic mask (14).

Description

[0001] Related applications

[0002] This application claims priority from German Patent Application No. 10 2022 135019.3, filed on December 29, 2022, the entire contents of which are incorporated herein by reference. Technical Field

[0003] The present invention relates to a computer-implemented method, computer-readable medium, computer program product, and corresponding system for generating an aerial image of a lithography mask. The method, computer-readable medium, computer program product, and system can be used for quantitative metrology, defect detection in lithography masks, assessment of defect correlation in lithography masks, lithography mask improvement, system simulation, or process control, process monitoring, or process improvement. Background Art

[0004] Wafers, made of thin slices of silicon, serve as substrates for microelectronic devices, including semiconductor structures built within and on the wafers. Semiconductor structures are built layer by layer using repetitive processing steps involving repeated chemical, mechanical, thermal, and optical processes. The size, shape, and layout of semiconductor structures and patterns are influenced by many factors. One of the most critical steps is photolithography.

[0005] Photolithography is a process for producing patterns on the substrate of a wafer. The pattern to be printed on the substrate surface is usually generated by computer-aided design (CAD). According to the design, a photolithography mask is generated for each layer, which contains a magnified image of the computer-generated pattern to be etched into the substrate. The photolithography mask can be further adjusted, for example, by optical proximity correction technology. During the printing process, the illumination image projected from the photolithography mask is focused onto the photoresist film formed on the substrate. For example, the semiconductor chip that powers a mobile phone or tablet computer contains about 80 to 120 patterned layers. In the past, when the requirements for precision in photolithography were lower, the circuit layout was equivalent to the mask pattern, and the mask pattern was equivalent to the wafer pattern.

[0006] Due to the increasing integration density in the semiconductor industry, photomask have to image smaller and smaller structures on wafers. The aspect ratio and the number of layers of integrated circuits is constantly increasing and the structures are developing towards the third (vertical) dimension. The current height of memory stacks already exceeds several micrometers. In contrast, the feature size becomes smaller. The minimum feature size or critical dimension is below 10 nm, e.g. 7 nm or 5 nm, and will approach feature sizes below 3 nm in the near future. When the complexity and the dimension of semiconductor structures grow to the third dimension, the lateral dimensions of integrated semiconductor structures become smaller. Producing small structure sizes imaged on wafers requires photomasks or templates for nanoimprint lithography having smaller structures or pattern elements. Therefore, the production process of photomasks and templates for nanoimprint lithography becomes more and more complex and, thus, more time-consuming and eventually also more expensive. With the advent of EUV lithography scanners, the nature of the mask is changing from a transmissive to a reflective based patterning.

[0007] Today, the minimum feature size on a mask has reached sub-wavelength dimensions. Therefore, so-called optical proximity effects due to an inhomogeneous energy intensity in the exposure process caused by optical diffraction occur. Thus, the image formed on the substrate does not faithfully reproduce the pattern on the photomask.

[0008] Therefore, many applications require aerial images of the photomask, which simulate the radiation intensity distribution on the substrate level. In this way, the aerial image allows analyzing the semiconductor structure to be printed on the substrate in the printing process. However, the generation of aerial images is both time-consuming and expensive. Therefore, methods for generating aerial images based on a model of the photomask become important.

[0009] In these methods, there are time-consuming rigorous simulations (e.g. finite difference time domain (FDTD) or rigorous coupled wave analysis (RWCA)) and fast approximations (e.g. thin element approximation (TEA)). Due to the large computational load for full chip applications, rigorous simulations are usually not used in commercial computational lithography software. The thin element approximation (TEA) assumes that the thickness of the structures on the photomask is very small compared to the wavelength and the width of the structures on the photomask is very large compared to the wavelength. However, with the lithography process using shorter and shorter wavelengths of radiation and the structures on the patterning device becoming smaller and developing to the vertical dimension, these assumptions are no longer valid. The interaction of the incident radiation with the absorber structures leads to mask 3D effects, which have to be taken into account by the simulation. Therefore, the aerial image generated by the TEA is not accurate for short wavelengths of radiation.

[0010] For example, a typical mask 3D effect is mask shadowing. The chief ray angle refers to the angle between the optical axis and the surface normal vector of the mask. Current EUV projection systems for example employ a CRA of 6°. Mask shadowing occurs due to the height of the absorber structure and the non-telecentric illumination at the mask level, which modulates the intensity captured from the shadow mask through the reflective optics onto the wafer. At the wafer level, this results in asymmetric shadowing, image shifts and size deviations (depending on the feature direction) as well as a shift of the process window.

[0011] Another mask 3D effect is the phase shift caused by diffraction at the absorber structure. These phase effects produce imaging effects which are very similar to the phase distortions caused by the wave aberrations of the projection system.

[0012] Another mask 3D effect can be attributed to the reflective properties of the EUV lithography mask. The main part of the reflected light originates from the multilayer which is designed to provide high reflectivity over a sufficiently large range of incidence angles. However, there is also some reflected light from the top of the absorber which leads to ghosting.

[0013] These mask 3D effects should not be neglected in the course of a lithography process. However, a rigorous simulation method which takes mask 3D effects into account, such as the finite difference time domain (FDTD) or the rigorous coupled wave analysis (RWCA), is computationally infeasible.

[0014] Therefore, there is a need for an accurate, fast simulation method of the aerial image of a lithography mask which takes at least some mask 3D effects into account.

[0015] A known method for generating an aerial image of a lithography mask is disclosed in WO 2019 / 214909 Al. The method comprises generating one or more electromagnetic field determinations expressions based on Maxwell’s equations and the quantum Schrödinger equation. The method comprises determining an electromagnetic field by propagating an electromagnetic wave through a region of interest of a mask stack based on Maxwell’s equations and the quantum Schrödinger equation. SUMMARY

[0016] It is an object of the present invention to obtain an alternative method for generating aerial images. It is another object of the present invention to generate aerial images with high accuracy. In particular, it is an object of the present invention to take into account mask 3D effects during the generation of aerial images. It is another object of the present invention to generate aerial images requiring low computation time. It is another object of the present invention to obtain a method for aerial image generation applicable to transmissive and reflective photomasks. It is another object of the present invention to make the resolution of aerial images independent of the smallest design feature. It is another object of the present invention to make the resolution of sub-pixel design features possible. It is another object of the present invention to allow a more flexible representation of photomasks. It is another object of the present invention to improve photomask design without the need to actually print a wafer. It is another object of the present invention to detect defects or placement errors in a photomask or to measure structures on a photomask with high accuracy and low computation time. It is another object of the present invention to evaluate the relevance of defects detected in a photomask with high accuracy and low computation time.

[0017] These objects are achieved by the present invention as specified in the independent claims. Advantageous embodiments and further developments of the present invention are specified in the dependent claims. SUMMARY

[0019] Embodiments of the present invention relate to a computer-implemented method, a computer-readable medium, a computer program product and a corresponding system for generating an aerial image of a photomask or for detecting defects and evaluating the relevance of defects in a photomask.

[0020] Embodiments of the present invention comprise a computer-implemented method for generating an aerial image of a model of a photomask under illumination with incident electromagnetic waves. The method comprises: a) approximately simulating the propagation of the incident electromagnetic waves within a first portion of the photomask comprising a plurality of structures; b) analytically or numerically simulating the propagation of the simulated electromagnetic waves from step a) within a second portion of the photomask; c) simulating a representation of an electromagnetic near-field within the model of the photomask by propagating the simulated electromagnetic waves from step b) to a near-field plane; and d) generating an aerial image of the photomask by applying a simulation of an imaging process of a lithography system or a metrology system to the representation of the electromagnetic near-field.

[0021] This method can be used for various purposes, e.g. for improving a model of a photomask (e.g. a design pattern), for repairing a photomask, for deciding on the quality of a photomask, for performing a measurement of a photomask, for detecting or evaluating defects in a photomask, for selecting an illumination setting of a lithography system, for source mask optimization or inverse lithography.

[0022] According to embodiments of the present application, in particular with respect to a transmission-based photomask, such as a DUV photomask, the simulated electromagnetic wave is incident on a substrate plane of the photomask, propagates through the second portion of the photomask and subsequently through the first portion of the photomask to the near-field plane.

[0023] According to embodiments of the present application, in particular with respect to a reflection-based photomask, such as an EUV photomask, the second portion comprises a plurality of layers in the form of an optical thin-film stack for reflecting electromagnetic waves, and the simulated electromagnetic wave is incident on a structure plane, propagates through the first portion of the photomask, is reflected within the plurality of layers in the second portion of the photomask, and propagates back through the first portion of the photomask to the structure plane and the near-field plane. Thus, according to embodiments of the present application, the electromagnetic wave propagates through the first portion, is then reflected by the second portion, and propagates again through the first portion of the photomask to the near-field plane.

[0024] According to embodiments of the present application with respect to reflection-based measurements of a DUV photomask, the second portion comprises a mask carrier comprising a glass substrate, and the simulated electromagnetic wave is incident on a structure plane which externally bounds the first portion, the electromagnetic wave is partially reflected by the structure plane and partially propagates through the first portion of the photomask, is partially reflected by the mask carrier of the photomask and propagates through the first portion of the photomask.

[0025] In each of the foregoing embodiments, further reflection and interference effects at other material interfaces can also be considered.

[0026] The term "photomask" refers to a general patterning device which can be used to impart a patterned cross-section to an incident beam of radiation corresponding to a pattern to be established in a target portion of a substrate.

[0027] The photomask can have an aspect ratio between 1 : 1 and 1 : 4, preferably between 1 : 1 and 1 : 2, and most preferably 1 : 1 or 1 : 2. The photomask can have a shape close to a rectangle. The photomask can preferably be 5 to 7 inches long and wide, and most preferably 6 inches long and wide. Alternatively, the photomask can be 5 to 7 inches long and 10 to 14 inches wide, preferably 6 inches long and 12 inches wide.

[0028] A "model" of a lithographic mask refers to a representation of the lithographic mask or a portion thereof. The model may, for example, comprise a computer readable file (e.g., a CAD file or a GDS file), or a set of polygons representing the structure of the lithographic mask or a portion thereof. The model of the lithographic mask can comprise material information, e.g., the complex refractive index, the permittivity, the permeability, or a derived representation of the materials comprised in the lithographic mask. The model of the lithographic mask can comprise parameters describing the dimensions of structures in the lithographic mask, e.g., the thickness of a layer in a multilayer of an EUV mask or the thickness of an absorber layer or the dimensions of an absorber structure. The model of the lithographic mask can comprise parameters describing the position of structures in the lithographic mask, e.g., the position of an absorber structure or a layer in a multilayer. The model of the lithographic mask can comprise parameters describing the shape of structures in the lithographic mask, e.g., the shape of an absorber structure, such as a sidewall angle or a roundness. The model of the lithographic mask can comprise an image, e.g., a 2D image or a 3D image (e.g., a volume of voxels or a plurality of 2D slices of a volume), representing a property of the lithographic mask. The image can comprise one, two, or more channels. The image can comprise image elements, e.g., pixels or voxels. The properties of the lithographic mask can comprise material properties, e.g., the refractive index, the permittivity, the permeability, or a derived representation. The model of the lithographic mask can comprise a description of structures within the lithographic mask, e.g., in the form of curves, contour lines, polygons, splines, NURBS, Bezier curves, etc.

[0029] The model of the lithographic mask preferably describes the lithographic mask at least partially in a dimension orthogonal to the plane of the substrate of the lithographic mask. The model of the lithographic mask can comprise one or more different portions of the lithographic mask or a portion thereof, e.g., a first portion and / or a second portion. The one or more different portions can be configured at different depths relative to the surface normal.

[0030] The first portion of the photomask comprises a plurality of structures. These structures can be configured in a design pattern or model, which determines the pattern that is imprinted on a wafer in a printing process. The design pattern or model can comprise structures and non-structures, in particular absorber structures and non-absorber structures. The second portion of the photomask can comprise a mask carrier, which can comprise one or more layers of one or more materials. The structures and non-structures can be deposited on the mask carrier. The mask carrier can comprise a substrate layer. The second portion can be configured to be transmissive for incident electromagnetic waves (for a transmissive-based photomask), or it can be configured to be reflective for incident electromagnetic waves (for a reflective-based photomask). The first portion can be directly adjacent to the second portion of the photomask. Thus, the first portion and the second portion can have a common boundary, for example a boundary plane. The mask carrier in the photomask can be bounded by the boundary plane and a base plane. The boundary plane can be a surface plane of the mask carrier. The base plane is preferably parallel to the boundary plane. The base plane can externally bound the second portion. It can form an interface between the mask carrier and the outside of the photomask, through which the electromagnetic waves propagate. The structures in the first portion of the photomask can be bounded by the boundary plane and a structure plane. The structure plane can externally bound the first portion of the photomask. The structure plane can comprise portions of the surface of the structures facing away from the boundary plane. Preferably, the structure plane is parallel to the boundary plane. The first portion of the photomask can extend between the structure plane and the boundary plane and can be bounded by these planes. The second portion of the photomask can extend between the boundary plane and the base plane. It can be bounded by the boundary plane and the base plane. The second portion can comprise a stack of uniform parallel layers. Uniform means that the material properties do not change within the layers. Other configurations of the photomask comprising the first portion and the second portion can also be used.

[0031] The electromagnetic near-field represents the distribution of electromagnetic waves in a near-field plane. The near-field plane can be located next to the structure plane of the photomask, which externally bounds the first portion of the photomask. Preferably, the near-field plane is parallel to the structure plane of the photomask. The near-field plane can be located at any position between the structure plane and the wafer plane, for example, the near-field plane can be located at a distance of between 0 and 1000 nm from the structure plane, preferably at a distance of between 0 and 100 nm, more preferably at a distance of between 0 and 50 nm, even more preferably at a distance of between 0 and 20 nm, and most preferably at a distance of between 0 and 10 nm. In preferred embodiments of the present invention, the near-field plane and the structure plane are identical. In principle, the near-field plane can also be located within the first portion, within the second portion, on the structure plane, on the base plane, or outside the photomask on the side of the base plane of the photomask, for example in the case where the electromagnetic waves propagate back into the photomask after propagating through the first portion.

[0032] The representation of the electromagnetic (near-)field can refer to the (complex) electric field E or the (complex) scattered electric field wherein represents the incident electric field. For example, the complex electromagnetic field can be represented by real and imaginary parts, or amplitude and phase. The representation of the electromagnetic field can refer to the (complex) magnetic field H or the (complex) scattered magnetic field ,in Represents the incident magnetic field. The representation of the electromagnetic field can include the envelope of the total electric field or the scattered electric field or the total magnetic field or the scattered magnetic field, such as the total electric field envelope , where the rapidly changing electric field consists of the rapidly changing component demodulation, Represents the incident wave vector, and r represents the spatial coordinate vector. Finally, the envelope is multiplied by the phase term to obtain the electric field E. The representation of the electromagnetic field may include measurements derived from the electromagnetic field, such as diffraction orders, spectrum, far field or intensity field, etc. The representation of the electromagnetic field within the lithography mask may refer to the electromagnetic field within the lithography mask, a portion of the electromagnetic field within the lithography mask, the electromagnetic field next to the lithography mask, such as the near field, etc. The representation of the electromagnetic field may include representations of the electromagnetic field in different spatial directions. For example, the representation of the electromagnetic field may include a 2D or 3D image with one, two or more channels, such that the 2D or 3D image includes a representation of the electromagnetic field in each spatial direction, such as the complex electric field in the x and y directions, or the x, y and z directions, resulting in a 2D or 3D image with four or six channels.

[0033] The aerial image represents the radiation intensity distribution in the wafer plane and is generated from the representation of the electromagnetic near field by applying a simulation of the imaging process of a lithography system or a metrology system to the representation of the electromagnetic near field.

[0034] An optical metrology system refers to a system that measures the aerial image of at least one portion of a lithographic mask or quantities derivable from the aerial image, such as critical dimension (CD), normalized image logarithmic slope (NILS), edge position, defects, etc.

[0035] In the case of a lithography system, the wafer plane refers to the plane within the resist on top of the wafer if the wafer is placed in the lithography system. In the case of an optical metrology system, the wafer plane refers to the plane where the camera sensor is located.

[0036] The electromagnetic near-field is calculated differently within the first portion of the lithography mask and the second portion of the lithography mask. Within the first portion, several assumptions can be made in the lithography setup which allow for a simplified and fast calculation of the propagation of the electromagnetic waves within the first portion. The propagation of the electromagnetic waves within the first portion is calculated by a wave propagation method which takes into account the inhomogeneity of the medium within the first portion of the lithography mask. In the second portion, the propagation of the electromagnetic waves is calculated analytically or numerically. In this way, a highly accurate approximation of the electromagnetic wave propagation within the lithography mask can be obtained with a required calculation time which is several orders of magnitude lower than a rigorous simulation method. Thus, the simulation of the electromagnetic near-field and the aerial image becomes feasible for industrial applications.

[0037] According to a first example of a specific embodiment, the Helmholtz equation is used to approximately simulate the propagation of the incident electromagnetic wave within the first portion of the lithography mask in step a). In this way, the approximation is simplified, thereby reducing complexity and calculation time.

[0038] According to a second example of a specific embodiment, a machine learning model is used to approximately simulate the propagation of the incident electromagnetic wave within the first portion of the lithography mask in step a). By using a machine learning model, the calculation time can be significantly reduced, since after training, a single and fast forward pass is sufficient to calculate the propagation of the incident electromagnetic wave.

[0039] According to an aspect of the first example, a forward Helmholtz equation is used to approximate the Helmholtz equation. In this way, the approximation is simplified, thereby reducing complexity and calculation time.

[0040] The forward Helmholtz equation can be solved using a beam propagation method. In this way, the approximation is simplified, thereby reducing complexity and calculation time.

[0041] In a preferred specific embodiment, the forward Helmholtz equation is solved using a wave propagation method which approximately describes the propagation of electromagnetic waves through inhomogeneous media. With the wave propagation method, the forward Helmholtz equation can be solved quickly, thereby reducing the calculation time of this method. Furthermore, by taking into account the inhomogeneity of the first portion of the lithography mask, for example due to different materials in the absorber structures and the non-absorber structures, the wave propagation can be simulated with high accuracy.

[0042] According to an aspect of a preferred embodiment, the first portion of the lithography mask is decomposed into different materials by defining a characteristic function for each material, which indicates the presence of the material at different locations in the first portion of the lithography mask, wherein at least one characteristic function is non-binary. This allows a more general representation of the material distribution within the first portion of the lithography mask, such that intermediate material properties can be represented as a weighted average of discrete materials. This can occur, for example, when using an effective material representation to approximate the interaction of sharp contrasts with electromagnetic waves. This can be used as a mathematical approach to describe the material distribution, thereby improving the simulation results and thereby obtaining more accurate aerial images.

[0043] In an example, the characteristic functions form an affine combination at each location in the first portion of the lithography mask. An affine combination of functions is a linear combination such that the sum of all functions at each location equals one. This mathematically ensures that the sum of all materials at each location equals one.

[0044] In an example, the characteristic functions are bandwidth limited. This allows to select a sampling grid with a lower resolution than required for binary characteristic functions. In this way, the computation time of the method is reduced.

[0045] In an example, a low-pass filter is applied to the characteristic functions. This allows a fast computation of the bandwidth limited characteristic functions.

[0046] Applying a low-pass filter can include applying a spatial Fourier transform to the characteristic functions followed by an inverse Fourier transform. In this way, the low-pass filter can be applied quickly, reducing the computation time of the method.

[0047] In an example, the wave propagation method approximates the analytical Fourier transform by a fast Fourier transform and / or the analytical inverse Fourier transform by a fast inverse Fourier transform, thereby reducing the computation time of this method.

[0048] According to an aspect, the wave propagation method approximates the analytical Fourier transform by a fast Fourier transform and the wave propagation method takes the angle of the incident electromagnetic wave into account by assuming quasi-periodic boundary conditions in the fast Fourier transform at one or more pairs of opposite boundaries perpendicular to the base plane of the lithography mask. In this way, the accuracy of the approximation of the electromagnetic wave propagation is improved.

[0049] In an example, the electromagnetic wave within the first portion has a dispersion relation that depends on the angle of the incident electromagnetic wave. Preferably, the dispersion relation within the first portion is modified by a phase shift in the coordinates parallel to the base plane of the lithography mask. In this way, the accuracy of the approximation of the electromagnetic wave propagation is improved.

[0050] The lithography mask can be a transmissive lithography mask or a reflective lithography mask.

[0051] In the case of a reflection-based lithography mask, the second portion comprises a multilayer in the form of a stack of optical thin films for reflecting electromagnetic waves. According to an aspect, simulating the reflection of the simulated electromagnetic wave within the multilayer comprises analytically or numerically calculating a reflection coefficient at the boundary between the second portion and the first portion of the lithography mask, the reflection coefficient describing the propagation of the electromagnetic wave within the stack of optical thin films of the multilayer. Rather than iteratively propagating the electromagnetic wave through each layer of the multilayer, the reflection within the multilayer can be simulated by calculating the reflection coefficient at the boundary only once. In this way, a high-precision simulation of the propagation of the electromagnetic wave within the multilayer can be obtained with a significant reduction in computation time.

[0052] According to an aspect, the reflection coefficient at the boundary is calculated separately within and outside the structure in the first portion of the lithography mask. In this way, the accuracy of the propagation of the electromagnetic wave is improved.

[0053] In an example, the simulation of the propagation of the simulated electromagnetic wave within the second portion of the lithography mask comprises applying the reflection coefficient to the electromagnetic wave incident on the boundary. In this way, a fast and accurate simulation within the second portion is obtained.

[0054] According to a preferred embodiment of the invention, the method for generating a spatial image of a model of a lithography mask further comprises adjusting at least one parameter of the method to minimize a difference between one or more reference spatial images of the lithography mask and a corresponding one or more generated spatial images of the model of the lithography mask, wherein the at least one parameter is from the group comprising a mask parameter and an optical parameter. The reference spatial image can for example be a spatial image obtained using a certain spatial image capturing system, or a simulated spatial image, for example a target spatial image indicative of a desired result of the spatial image generation method. The mask parameter describes a characteristic of the lithography mask, for example a deviation of a structure size on the lithography mask relative to its size in the model or design, a corner rounding, a sidewall angle, a layer height, a refractive index, etc. The optical parameter describes an optical device characteristic of the system used to acquire the reference spatial image, for example an illumination setting (NA, pupil geometry, intensity distribution, coherence, etc.), an imaging setting (NA, pupil geometry, obscuration, aberrations, apodization, defocus, distortion, magnification, etc.), a sensor model (intensity integration over the active sensor area, pixel size, dark current, shot noise, stray light, etc.), motion blur caused by scanning, field dependent effects (aberrations of illumination and imaging, etc.). By adjusting at least one parameter of the method based on the reference spatial image, the accuracy of the generated spatial image can be improved. This calibration step can further allow various applications requiring a high-precision simulation of a spatial image, comparable to the acquired spatial image, for example in defect detection in a die-to-database method.

[0055] In a preferred embodiment (which can be combined with the previous preferred embodiments), the method for generating a spatial image of a model of a lithography mask further comprises registering one or more reference spatial images of the lithography mask to the generated spatial image of the model of the lithography mask, and reporting at least one registration parameter. The registration parameter comprises any (isomorphic) transformation to align the reference spatial image with the generated spatial image, such as a translation, shift, rotation, scaling, shear, non-linear transformation, etc. This way, an accurate comparison of the reference spatial image and the simulated spatial image can be made for various applications, such as for defect detection in the database method for a die.

[0056] According to an aspect of the preferred embodiments, the one or more reference spatial images comprise a focal stack of the lithography mask. In this way, calibration and registration parameters can be found that accommodate different defocus values.

[0057] According to an embodiment, a computer-implemented method for improving a model of a lithography mask (e.g. a design pattern), for repairing a lithography mask, for determining a quality of a lithography mask, for performing a measurement of a lithography mask, for detecting or assessing a defect in a lithography mask, or for selecting an illumination setting in a lithography system, comprises: generating a spatial image of a model of a lithography mask using the method according to any of the above embodiments, examples or aspects; analyzing the generated spatial image accordingly; using the results of the analysis, improving a design of the lithography mask, repairing the lithography mask, determining a quality of the lithography mask, detecting or assessing a defect in the lithography mask, or selecting an illumination setting in the lithography system.

[0058] According to an embodiment, a computer-implemented method for training a machine learning model that maps a model of a lithography mask to a spatial image of the lithography mask, comprises: generating spatial images of models of a plurality of lithography masks using the method according to any of the above embodiments, examples or aspects; and training the machine learning model using training data comprising the generated spatial images. In this way, a particularly fast and efficient training of the machine learning model can be achieved, as a large portion of the training data can be obtained by simulation rather than time-consuming acquisition.

[0059] According to an embodiment, a computer-implemented method for training a machine learning model for defect detection in acquired spatial images of lithography masks, comprises: generating a plurality of model pairs of lithography masks, each model pair comprising a defect-free model (i.e. a model without defects) of a lithography mask, and a defective model (i.e. a model comprising one or more defects) of the same lithography mask; generating a pair of spatial images from the model pairs by applying the method according to any of the above embodiments, examples or aspects to the defect-free model and the defective model of each model pair; and training the machine learning model using training data comprising the pairs of spatial images.

[0060] The computer readable medium according to embodiments of the present application has stored thereon a computer program, which can be executed by a computing device, the computer program comprising program code for executing a method for generating a aerial image of a photomask according to the above described embodiments, examples or aspects.

[0061] The computer program product according to embodiments of the present application comprises instructions, which, when executed by a computer, cause the computer to perform a method for generating a aerial image of a photomask according to the above described embodiments, examples or aspects.

[0062] The system for generating a aerial image of a model of a photomask according to embodiments of the present application 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 generating a aerial image of a photomask according to the above described embodiments, examples or aspects.

[0063] According to an embodiment of the present application, a system for improving a model of a photomask, for repairing a photomask, for determining a quality of a photomask, for measuring a photomask, for detecting or evaluating defects in a photomask, or for selecting an illumination setting of a lithography system 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 generating a aerial image of a photomask according to the above described embodiments, examples or aspects. Optionally, the system can further comprise a subsystem for obtaining a aerial image of a photomask, which can be used for analyzing the generated aerial image of a photomask, e.g. for comparison.

[0064] According to an embodiment of the present application, a system for detecting defects in a photomask comprises a subsystem for obtaining a aerial image of a photomask; 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 generating a aerial image of a photomask according to the above described embodiments, examples or aspects.

[0065] According to a tenth embodiment of the present application, a system for evaluating a relevance of defects in a photomask comprises a subsystem for obtaining a charged particle beam image of a photomask; a data analysis device comprising at least one memory and at least one processor configured to perform the steps of a computer implemented method according to the fourth embodiment of the present application.

[0066] The application described by way of example and embodiments is not limited to the specific embodiments and examples described, but can be practiced by a person skilled in the art with various combinations or modifications of the described embodiments and examples. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 An exemplary transmission-based lithography system, such as a deep-ultraviolet (DUV) lithography system, is shown;

[0068] Figure 2 Propagation of an incident electromagnetic wave through a transmission-based lithography mask is shown;

[0069] Figure 3 An exemplary reflection-based lithography system, such as an extreme-ultraviolet (EUV) lithography system, is shown;

[0070] Figure 4 Propagation of an incident electromagnetic wave through a reflection-based lithography mask is shown;

[0071] Figure 5a Amplitude of the simulated electromagnetic near-field of a lithography mask generated using the rigorous coupled wave analysis (RCWA) method is shown;

[0072] Figure 5b Amplitude of the electromagnetic near-field of the same lithography mask simulated using the thin element approximation (TEA) method is shown;

[0073] Figure 6 A flowchart of a computer-implemented method according to embodiments of the present invention is shown;

[0074] Figure 7 A flowchart of a computer-implemented method for generating a spatial image of a model of a lithography mask according to an example of a transmission-based lithography mask is shown;

[0075] Figure 8 A flowchart of a computer-implemented method for generating a spatial image of a model of a lithography mask according to another example of a reflection-based lithography mask is shown;

[0076] Figure 9 A flowchart of a computer-implemented method for generating a spatial image of a model of a lithography mask according to an example is shown;

[0077] Figure 10a 、 10b Influence of the sampling grid resolution during the sampling of the characteristic function on the approximation error is shown;

[0078] Figure 11 Comparison of the convergence speed of wave propagation methods implemented using discrete binary characteristic functions or discrete bandwidth-limited characteristic functions is shown;

[0079] Figure 12 Dependence of the phase shift vector a on the angle of the incident electromagnetic wave is shown;

[0080] Figures 13a-13dSteps of a computer implemented method for generating aerial images of a model of a lithography mask according to embodiments of the application are shown;

[0081] Figures 14a-14c A comparison of aerial images of a model of a lithography mask obtained by three different simulation methods is shown;

[0082] Figure 15 The use of a calibration step in a defect detection method for detecting defects in aerial images of a lithography mask is shown;

[0083] Figure 16 The calibration step in Figure 15 is shown, comprising an optimization of mask and / or optical parameters;

[0084] Figure 17 The effect of the calibration step of Figure 15 is shown;

[0085] Figure 18 The calibration step in Figure 15 is shown, comprising a registration;

[0086] Figure 19 The calibration step in Figure 15 is shown, comprising a combination of a previous mask and / or optical parameter optimization and a subsequent registration;

[0087] Figure 20 The calibration step of Figure 15 is shown, comprising a joint mask and / or optical parameter optimization and registration;

[0088] Figure 21 A training method of a machine learning model for generating aerial images of a model of a lithography mask is shown;

[0089] Figure 22 A sample model used as training data to train a machine learning model of Figure 21 is shown;

[0090] Figure 23 A training method of a machine learning model for defect detection in a lithography mask is shown;

[0091] Figure 24 The generation of training data for training a machine learning model for defect detection in Figure 23 is shown;

[0092] Figure 25 A method for improving a model of a lithography mask, for repairing a lithography mask, for deciding on a quality of a lithography mask, for performing a measurement of a lithography mask, for detecting or assessing defects in a lithography mask, or for selecting an illumination setting in a lithography system is shown;

[0093] Figure 26 A computer-implemented method for detecting defects in a photomask according to embodiments of the application is shown;

[0094] Figure 27 A computer-implemented method for assessing the relevance of defects in a photomask according to embodiments of the application is shown;

[0095] Figure 28 A system for generating aerial images of a model of a photomask according to embodiments of the application is shown;

[0096] Figure 29 A system for detecting defects in a photomask according to embodiments of the application is shown; and

[0097] Figure 30 A system for assessing the relevance of defects in a photomask according to embodiments of the application is shown. DETAILED DESCRIPTION

[0098] In the following, advantageous exemplary embodiments of the application are described and schematically shown in the accompanying drawings. In all drawings and descriptions, the same reference symbols are used to describe the same features or components.

[0099] The methods and systems herein can be used with a variety of photolithography systems, such as a transmission-based photolithography system 10 or a reflection-based photolithography system 10'.

[0100] Figure 1 An exemplary transmission-based photolithography system 10, such as a DUV photolithography system, is shown. The main components are a radiation source 12 (which can be a deep ultraviolet (DUV) excimer laser source), imaging optics (which, for example, define a partial coherence and which can include optics that shape the radiation from the radiation source 12), a photomask 14, illumination optics 16 that illuminate the photomask 14, and projection optics 17 that project an image of a model 92 (e.g., a design pattern) of the photomask onto a wafer plane 18. An adjustable filter or aperture at the pupil plane of the projection optics 17 can limit the range of beam angles that impinge on the wafer plane 18, with the maximum 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 maximum angle of a beam that exits the projection optics 17 and still impinges on the wafer plane 18.

[0101] In the present document, the terms "radiation" or "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 3- 100 nm).

[0102] The illumination optics 16 can include optical components for shaping, adjusting, and / or projecting the radiation from the radiation source 12 before the radiation passes through the photomask 14. The projection optics 17 can include optical components for shaping, adjusting, and / or projecting the radiation after the radiation passes through the photomask 14. The illumination optics 16 do not include the light source 12, and the projection optics do not include the photomask 14.

[0103] The illumination optics 16 and the projection optics 17 can include 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 can also include components operating according to any of these design types for collectively or individually directing, shaping, or controlling the projection beam of radiation.

[0104] According to Figure 2 In the example shown, the simulated electromagnetic wave is incident on the base plane 34, propagates from the base plane 34 to the boundary plane 32 within the second portion 27 of the photomask 14, and propagates from the boundary plane 32 to the structure plane 30 within the first portion 25 of the photomask 14. In this way, the computer-implemented method for generating a spatial image 64 can be applied to a transmission-based photomask, such as a DUV photomask.

[0105] Figure 2Propagation of an incident electromagnetic wave 22 through a transmission-based photomask 14 (e.g., a DUV photomask) is shown. The photomask 14 includes a first portion 25 and a second portion 27. The first portion 25 includes a grating 24, and the second portion 27 includes a mask carrier 48. The grating 24 is formed from a combination of absorber structures 26 and non-absorber structures 28. The absorber structures 26 are made from one or more materials that absorb the electromagnetic wave 22, such as titanium nitride or tantalum nitride, for example. The non-absorber structures 28 are made from one or more materials that have a lower degree of absorption of the electromagnetic wave 22 than the absorber material. For example, the non-absorber structures 28 can include a vacuum. Thus, the grating 24 is a non-uniform medium. The absorber structures 26 and the non-absorber structures 28 are deposited on the mask carrier 48. The mask carrier 48 can include a substrate layer 46. The mask carrier 48 in the photomask 14 is bounded by a boundary plane 32 and a base plane 34 that is preferably parallel to the boundary plane 32. The boundary plane 32 is a surface plane of the mask carrier 48. The base plane 34 is a boundary plane through which the electromagnetic wave 22 enters the grating 24. The incident electromagnetic wave 22 impinges on the base plane 34. The base plane 34 forms an interface between the mask carrier 48 and the outside of the photomask 14 through which the electromagnetic wave 22 propagates. The absorber structures 26 in the grating 24 of the photomask 14 are bounded by the boundary plane 32 and a structure plane 30. The structure plane 30 is a boundary plane that includes a portion of a surface of the absorber structures 26 that is facing away from the boundary plane 32. Preferably, the structure plane 30 is parallel to the boundary plane 32. The first portion 25 of the photomask 14 extends between the structure plane 30 and the boundary plane 32 and is bounded by these planes. The second portion 27 of the photomask 14 extends between the boundary plane 32 and the base plane 34 and is bounded by the boundary plane 32 and the base plane 34.

[0106] For the transmission-based photomask 14, according to an example, the simulated electromagnetic wave 22 is incident on the base plane 34, propagates from the base plane 34 to the boundary plane 32 within the second portion 27 of the photomask 14, and propagates from the boundary plane 32 to the structure plane 30 within the first portion 25 of the photomask 14.

[0107] Figure 3An exemplary reflective-based lithography system 10' is shown, for example an extreme ultraviolet (EUV) lithography system. The main components are a radiation source 12 (which can be a laser plasma source), illumination optics 16 (which for example define the partial coherence and which can include optics that shape the radiation from the radiation source 12), a lithography mask 14, and projection optics 17 (which project an image of the model 92 (design pattern) of the lithography mask onto a wafer plane 18). An adjustable filter or aperture at the pupil plane of the projection optics 17 can limit the range of beam angles that are incident on the wafer plane 18, with the maximum 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 maximum angle of the beam that exits the projection optics 17 and still can illuminate the wafer plane 18.

[0108] According to Figure 4 According to the example shown, the mask carrier 48 includes a multilayer 38 in the form of an optical thin film stack for reflecting electromagnetic waves, and the simulated electromagnetic waves are incident on the structure plane 30, propagate from the structure plane 30 to the boundary plane 32 within the first portion 25 of the lithography mask 14, reflect within the multilayer 38 in the second portion 27 of the lithography mask 14, and propagate from the boundary plane 32 to the structure plane 30 within the first portion 25 of the lithography mask 14. In this way, the computer-implemented method for generating a aerial image 64 can be applied to reflective-based lithography masks, for example EUV lithography masks.

[0109] Figure 4The propagation of an incident electromagnetic wave 22 through a reflection-based photomask 14, e.g., an EUV photomask, is shown. The photomask 14 comprises a first portion 25 and a second portion 27. The first portion 25 comprises a grating 24 and the second portion 27 comprises a mask carrier 48. The grating 24 comprises absorber structures 26 and non-absorber structures 28 which form a pattern 92 to be printed onto a wafer on at least a portion of the mask carrier 48. The absorber structures 26 are made of one or more materials which absorb the electromagnetic wave 22, e.g., titanium nitride or tantalum nitride, etc. The non-absorber structures 28 are made of one or more materials which have a lower degree of absorption of the electromagnetic wave 22 than the absorber material. For example, the non-absorber structures 28 can comprise a vacuum. Thus, the absorber structures 26 and the non-absorber structures 28 form a non-uniform medium. The absorber structures 26 and the non-absorber structures 28 are deposited on the mask carrier 48. The mask carrier 48 comprises a plurality of layers 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic wave 22. The mask carrier 48 can comprise a cover layer 42 and / or a substrate layer 46. The reflection of the electromagnetic wave 22 by the stack of optical thin films 40 corresponds to a reflection of the electromagnetic wave 22 at an effective mirror plane 44. The mask carrier 48 in the photomask 14 is bounded by a boundary plane 32 and a base plane 34 which is preferably parallel to the boundary plane 32. The boundary plane 32 is a surface plane of the mask carrier 48. The absorber structures 28 in the grating 24 of the photomask 14 are bounded by the boundary plane 32 and a structure plane 30. The structure plane 30 is a boundary plane which comprises a portion of a surface of the absorber structures 26 which faces away from the boundary plane 32. Preferably, the structure plane 30 is parallel to the boundary plane 32.

[0110] The structure plane 30 is a boundary plane through which the electromagnetic wave 22 enters the first portion 25, e.g., the grating 24. The entering electromagnetic wave 22 hits the structure plane 30. The structure plane 30 forms an interface between the mask 14 and the outside of the photomask 14 through which the electromagnetic wave 22 propagates. The first portion 25 of the photomask 14 extends between the structure plane 30 and the boundary plane 32 and is bounded by these planes. The second portion 27 of the photomask 14 extends between the boundary plane 32 and the base plane 34 and is bounded by the boundary plane 32 and the base plane 34.

[0111] For a reflection-based photomask 14, according to an example, the mask carrier 48 comprises a plurality of layers 38 in the form of a stack of optical thin films 40 for reflecting the electromagnetic wave 22, and wherein the simulated electromagnetic wave 22 is incident on the structure plane 30, propagates within the first portion 25 of the photomask 14 from the structure plane 30 to the boundary plane 32, is reflected within the plurality of layers 38 in the second portion 27 of the photomask 14, and propagates within the first portion 25 of the photomask 14 from the boundary plane 32 to the structure plane 30.

[0112] The electromagnetic near-field 20 indicates the distribution of the electromagnetic wave 22 in a near-field plane 52. The near-field plane can be located adjacent to the structure plane 30 of the photolithography mask. Preferably, the near-field plane 52 is parallel to the structure plane 30 or the substrate plane 34 of the photolithography mask 14. In general, the near-field plane 52 can be located anywhere between the structure plane 30 and the wafer plane 18, for example, at a distance between 0 and 1000 nm, preferably between 0 and 100 nm, more preferably between 0 and 50 nm, even more preferably between 0 and 20 nm, and most preferably between 0 and 10 nm from the structure plane 30. In a preferred embodiment of the present invention, the near-field plane 52 and the structure plane 30 are identical.

[0113] Known methods for simulating the electromagnetic near field 20 or the aerial image 64 generally require too much computing time or are not sufficiently accurate.

[0114] In order to simulate the interaction of the electromagnetic wave 22 with the photolithographic mask 14 , it is necessary to consider the propagation of the electromagnetic wave 22 within the different layers of the photolithographic mask 14 comprising different materials having different refractive indices.

[0115] To simulate the electromagnetic near field 20 or the aerial image 64, rigorous simulation techniques such as finite-difference time-domain (FDTD), finite element method (FEM), or rigorous coupled wave analysis (RCWA) methods are often used. For example, Figure 5a The amplitude of the simulated electromagnetic near field 20 of a lithographic mask 14 using the rigorous coupled wave analysis (RCWA) method is shown. However, these methods are computationally expensive, making these techniques impractical for full-chip applications. Full mask simulations can take years.

[0116] Figure 5b The thin element approximation (TEA) method is shown. Figure 5a Amplitude of the simulated electromagnetic near field 20 for the same photolithography mask 14. The thin element approximation is an effective method 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 incident light, and that the width of the structures on the photolithography mask 14 is very large compared to the wavelength. However, as the photolithography process uses radiation with shorter and shorter wavelengths, and the structures on the photolithography mask 14 become smaller and smaller, the assumptions of the TEA method may not hold. In this case, the photolithography mask 14 can no longer be approximated by a photolithography mask with a flat surface. Instead, it must be taken into account that the interaction of radiation with a wavelength lower than the height of the structures on the photolithography mask 14 leads to the so-called mask 3D effect. Therefore, a method is needed that can quickly and accurately produce an aerial image 64 of the photolithography mask 14 even for short wavelengths.

[0117] To achieve these goals, refer toFigure 6 A computer-implemented method for generating a spatial image of a model of a lithography mask according to a specific embodiment of the present application is described.

[0118] According to a specific embodiment, a computer-implemented method 54 for generating a spatial image of a model of a lithography mask under illumination of incident electromagnetic waves comprises: a) approximatively simulating the propagation of incident electromagnetic waves within a first portion of the lithography mask, the first portion comprising a plurality of structures, in a first portion simulation step 56; b) analytically or numerically simulating the propagation of the simulated electromagnetic waves from step a) within a second portion of the lithography mask, in a second portion simulation step 58; c) simulating a representation of the electromagnetic near-field within the model of the lithography mask by a near-field plane to which the simulated electromagnetic waves from step b) propagate, in a near-field generation step 60; and d) generating a spatial image of the lithography mask by applying a simulation of an imaging process of a lithography system or a metrology system to the representation of the electromagnetic near-field, in a spatial image generation step 63.

[0119] According to a specific embodiment, the lithography mask 14 comprises a mask carrier 48 and a grating 24 comprising absorber structures 26 and non-absorber structures 28 forming a model 92 or a design pattern on at least a portion of the mask carrier 48. The lithography mask 14 comprises a first portion 25 extending between a structure plane 30 and a boundary plane 32 of the lithography mask 14 and a second portion 27 extending between the boundary plane 32 and a substrate plane 34 of the lithography mask 14. The first portion 25 comprises the grating 24 and the second portion 27 comprises the mask carrier 48.

[0120] According to an example, the simulated imaging process comprises a resampling of the simulated electromagnetic near-field 20. Thus, the resolution of the simulated spatial image 64 can be increased without a significant increase of the computation time. In this way, a spatial image 64 can be obtained which has a resolution comparable to the resolution obtained by a rigorous simulation method.

[0121] The computer-implemented method 54 according to a specific embodiment can be applied to a transmission-based lithography mask 14 and to a reflection-based lithography mask 14.

[0122] Figure 7 It is shown that the computer-implemented method 54 according to a specific embodiment can be applied to a lithography mask 14 having a grating 24 with a pitch of 100 nm and a grating 24 with a pitch of 50 nm. Figure 2A flow chart of a computer-implemented method 54' for generating an aerial image 64 of a model of a lithography mask 14 based on a transmission-based lithography mask 14 is shown. In the computer-implemented method 54, a simulated electromagnetic wave 22 is incident on the lithography mask, for example, on the substrate plane 34, propagates within the second portion 27 of the lithography mask (e.g., from the substrate plane 34 to the boundary plane 32) in a second partial simulation step 58, and propagates within the first portion 25 of the lithography mask 14 (e.g., from the boundary plane 32 to the structure plane 30) in a first partial simulation step 56. Next, a representation of the electromagnetic near-field 20 of the model of the lithography mask 14 in the near-field plane 52 is obtained in a near-field generation step 60. Finally, in an aerial image generation step 63, an aerial image 64 is generated from the representation of the near-field 20 by applying a simulation of the imaging process of the lithography system or metrology system to the representation of the electromagnetic near-field.

[0123] Figure 8 Shown in Figure 4 A flow chart of a computer-implemented method 54 ″ for generating an aerial image 64 of a model of a lithography mask 14 in the case of a reflection-based lithography mask 14 is shown. In the computer-implemented method 54 ″, a mask carrier 48 includes a stack of multiple layers 38 in the form of optical films 40 for reflecting electromagnetic waves 22. The simulated electromagnetic wave 22 is incident on the lithography mask (e.g., on a structure plane 30), propagates within a first portion 25 of the lithography mask 14 (e.g., from the structure plane 30 to the boundary plane 32) in a first partial simulation step 56, is reflected within the multiple layers 38 in a second portion 27 of the lithography mask 14 in a second partial simulation step 58, and propagates within the first portion 25 of the lithography mask 14 (e.g., from the boundary plane 32 to the structure plane 30) in the first partial simulation step 56. Subsequently, a representation of the electromagnetic near-field 20 of the model of the lithography mask 14 in the near-field plane 52 is obtained in a near-field generation step 60. Finally, an aerial image 64 is generated from the representation of the near field 20 by applying a simulation of the imaging process of the lithography system or metrology system to the representation of the electromagnetic near field in an aerial image generation step 63 .

[0124] Instead of solving Maxwell's equations directly in the first portion 25, different approximations can be used to reduce the computational time of the method. According to one example, the propagation of the incident electromagnetic wave in step a) within the first portion 25 of the lithographic mask 14 is approximately simulated using the Helmholtz equation, in particular the forward Helmholtz equation.

[0125] In a lithography setting, the following assumptions can be made: 1) the refractive indices are similar for the different materials of the lithography mask 14, e.g. the refractive index of the structures 26 (in particular the absorber structures) is close to the refractive index outside the structures 26 (in particular the non-absorber structures), e.g. vacuum. 2) the refractive index profile of the first portion 25 is piecewise constant, without the need to model the transitions. 3) the main propagation direction 50 of the incident electromagnetic wave 22 is close to perpendicular with respect to the main surface of the lithography mask, in particular the substrate plane 34. These assumptions can simplify the approximation of the propagation of the electromagnetic wave 22 within the first portion 25.

[0126] Based on the Maxwell equations, the following equation can be derived for the electric field E of the electromagnetic wave 22:

[0127] (1)

[0128] where ω is the angular frequency, c is the speed of light, and is the dielectric function characterizing a particular material. These relations are connected to the refractive index n(r, ω) of the material via The right side couples the electric field components, which makes it difficult to find a solution of this equation. Therefore, it is preferable to neglect the right side. Neglecting the right side is still valid if the following two assumptions are met: the optical system under consideration does not have different responses for different incident polarizations, and there is no cross-coupling between the individual polarization components. For a lithography setting with short wavelengths, e.g. DUV or EUV lithography masks, there are two reasons to neglect the polarization and phonon effects, and therefore these assumptions are valid. First, the contrast of the refractive indices with respect to the different materials inside and outside the structures 26 of the first portion 25 is low. Second, the height a of the structures 26 is larger than the wavelength λ, i.e. . Therefore, the right side of equation (1) can be neglected, resulting in the Helmholtz equation

[0129]

[0130] The Helmholtz equation can be further simplified. Using the following relation for the size of the wave number

[0131]

[0132] and its relation to the wavelength λ

[0133]

[0134] where and are the wave vector and the wavelength in vacuum, respectively, the Helmholtz equation can be rewritten as

[0135]

[0136] ​This equation can be rewritten using the transverse Helmholtz operator as follows:

[0137]

[0138] where

[0139]

[0140] This equation can be rewritten as

[0141]

[0142] Here the square root Helmholtz operator is introduced and formally defined in terms of a power series. Furthermore, it is assumed that the commutator can be neglected, which physically means that back reflections within the inhomogeneous medium are ignored. Then, the forward Helmholtz equation is defined as

[0143]

[0144] Ordinary partial differential equations can be solved by multiplying with an integration factor:

[0145]

[0146] Exponential operators can be approximated by integral operators as shown in Appendix A of the doctoral thesis "Efficient wave-optical simulations for the modeling of micro-optical elements" by Soeren Schmidt, Jena University. The entire content of the aforementioned doctoral thesis is incorporated herein by reference, including the disclosure thereof. The result of the approximation by an integral operator is

[0147]

[0148] This method is called angular spectrum plane wave (ASPW) decomposition as shown in equation 1.8 of the aforementioned doctoral thesis. It assumes that the electromagnetic wave propagates within a homogeneous medium with refractive index n. However, this is not true for the first part 25 of the lithography mask 14 which contains the structure 26 and the non-structured part.

[0149] Therefore, it is necessary to extend the ASPW to inhomogeneous media to describe the propagation of the electromagnetic wave 22 within the first part 25 of the lithography mask 14.

[0150] In order to take into account the inhomogeneous medium, the propagation constant

[0151] ​

[0152] Thus, according to an example, the forward Helmholtz equation can be solved using a wave propagation method. The wave propagation method is a generalization of the ASPW to non-uniform media and describes the wave propagation step in the z-direction perpendicular to the plane of the substrate

[0153] (2)

[0154] where E denotes the electric field component of the electromagnetic field and is the wave vector, which locally obeys the dispersion relation

[0155] (3)

[0156] where denotes the wave number of light with wavelength in vacuum, denotes the refractive index distribution, denotes the spatial Fourier transform. The magnitude of the wave vector k is inversely proportional to the wavelength λ, and the direction of the wave vector is perpendicular to the wave front. By using this wave propagation method, the propagation of electromagnetic waves in non-uniform media can be modeled, resulting in an accurate approximation of the propagation of electromagnetic waves in the first portion of the lithography mask.

[0157] In a specific embodiment, the first portion 25 of the lithography mask 14 comprises structures 26 and non-structures 28 forming a non-uniform medium, e.g. the grating 24 comprises absorber structures and non-absorber structures. The simulation of the propagation of the electromagnetic wave 22 in the first portion 25 takes into account this non-uniformity of the material in the first portion 25. At the same time, several simplifying assumptions can be made in the lithography setup. In addition, the simulation of the propagation of the electromagnetic wave 22 in the second portion 27 is calculated analytically or numerically. In this way, an accurate and fast simulation of the propagation of the electromagnetic wave 22 in the lithography mask 14 is obtained.

[0158] Alternatively, the forward Helmholtz equation can be solved using a beam propagation method. The beam propagation method is described, for example, in chapter 1.3 of the above-mentioned PhD thesis of Dr. Soeren Schmidt, "Efficient wave-optical simulations for the modeling of micro-optical elements".

[0159] ​In an example, the propagation of the incident electromagnetic wave in the first portion of the photomask in step a) is approximated using a machine learning model. The machine learning model can for example comprise a neural network, e.g. a deep learning model. For example, the machine learning model can comprise a U-Net or a neural network with at least one attention mechanism, e.g. a Transformer machine learning model. The machine learning model can use a model of the photomask, e.g. a design pattern, as input and map the input to an electromagnetic field as output. The machine learning model can be trained using training data, e.g. obtained from the simulation described above. By using a machine learning model, the computation time can be greatly reduced, since after training a single fast forward computation is sufficient to compute the propagation of the incident electromagnetic wave.

[0160] Since the dispersion relation in (3) depends on the spatial variables (x, y), the wave propagation method in (2) cannot be implemented using a Fast Fourier Transform (FFT). To use the FFT and reduce the computation time, the wave propagation method in (2) can be re-expressed using a characteristic function.

[0161] In an example, the first portion 25 of the photomask 14 is decomposed into different materials by defining a characteristic function for each material that indicates the presence of the material in different locations in the first portion 25 of the photomask 14, wherein at least one characteristic function is non-binary.

[0162] The first portion 25 of the photomask 14 can be decomposed into a finite number M of pairs of disjoint and homogeneous sub-regions with refractive indices Then, the refractive index distribution in a given layer can be rewritten using a characteristic function The characteristic function of a material m is a mapping from the spatial domain to the value domain D that represents the presence of the material m in each location (x, y) of the spatial domain. For example

[0163]

[0164] represents a binary characteristic function, the numerical range where represents the refractive index of the material m. D can for example be a subset of real ( or complex ( numbers.

[0165] Figure 9 A flowchart of a computer-implemented method 54’’’ for generating a spatial image 64 of a model of a photomask 14 according to an example is shown, comprising an additional characteristic function step 61.

[0166] An additional feature function step 61 comprises: identifying the number M of materials of structures 26 in a first portion 25 of a model 92 (e.g. design pattern) forming the lithography mask 14; for each material m Defining a feature function which indicates whether this material is present at a position (x,y) of the lithography mask 14 within a subset of an x / y plane orthogonal to the z direction, the z direction being perpendicular to the substrate plane 34; simulating the propagation of the electromagnetic wave 22 as a weighted sum of propagation steps within each identified material:

[0167] (4)

[0168] wherein denotes the inverse Fourier transform. The use of a feature function can allow for an FFT-based implementation of the wave propagation method in (2), saving computation time. The integrator in (4) converges linearly with step size.

[0169] However, the discretization of commonly used binary feature functions is problematic. Since binary feature functions are discontinuous, the Shannon-Nyquist theorem requires a very high sampling frequency (at least twice the maximum frequency of the signal), and thus a very high resolution of the sampling grid. In particular, if the edges of the structures 26 are not aligned with the sampling grid, the sampling is inaccurate. Moreover, the resolution of the sampling grid depends on the size of the smallest feature. The high resolution of the sampling grid in turn leads to long computation times for generating the aerial image 64.

[0170] According to an aspect of this example, the feature function is therefore bandwidth limited. A bandwidth limited feature function is a feature function which exists for a finite frequency bandwidth.

[0171] .

[0172] According to the Shannon-Nyquist theorem, on the one hand, the sampling frequency required for the discretization of a bandwidth limited feature function depends on its maximum frequency. On the other hand, a given sampling frequency for the discretization of a bandwidth limited feature function directly implies its maximum frequency.

[0173] ​By using a bandwidth-limited characteristic function, the maximum frequency of the characteristic function can be limited. In this way, the required sampling frequency is reduced according to the Shannon-Nyquist theorem, and thus a lower resolution sampling grid can be used to discretize the characteristic function (compared to the case of a binary characteristic function). In this way, the computation time required to generate the aerial image 64 can be reduced. Furthermore, the resolution of the sampling grid is independent of the feature size of the features in the model of the photomask (e.g. the design pattern). In contrast, for a binary characteristic function, the sampling grid resolution depends on the smallest feature of the model of the photomask.

[0174] The reason for using a discretized bandwidth-limited characteristic function is given in the following: Assuming that the electromagnetic field E contains only energy at long wavelengths in the x / y plane perpendicular to the base plane 34 of the photomask 14, then a linear spatially invariant low-pass filter P will have no effect when applied to the electromagnetic field E, i.e.

[0175]

[0176] Equivalently, P can be written as a time-domain convolution, which means that:

[0177]

[0178] If the filter P is applied to the product of E and a function Θ with energy at shorter wavelengths, then the following holds:

[0179]

[0180] Thus, if a low-pass filter is applied to the product of a slowly varying function E and a rapidly varying function Θ, the result is approximately the product of the slowly varying function E and the filtered rapidly varying function .

[0181] Applying this result to the propagator

[0182]

[0183] where O denotes the linear ASPW propagator

[0184] (5)

[0185] Assuming that the variation range of the electromagnetic field is larger than the characteristic function , then the following holds:

[0186]

[0187] Thus, by applying the filter P to the characteristic function, a propagator for the low-frequency part of the field E in the wave propagation method in (4) is obtained.

[0188] Figure 10a The effect of the sampling grid resolution on the approximation error 66 during the sampling of the binary characteristic function 62 is shown. In each column, different sampling spacings between 30.18 nm and 2 nm are indicated. In the top row, the sampled binary characteristic functions 62 of a model 92 (e.g., a design pattern) of a lithography mask 14 are shown for different sampling spacings (pixel size in nm). Using the indicated binary characteristic functions 62, an aerial image 64 is generated by a computer-implemented method 54''' for generating an aerial image 64 of a model of a lithography mask 14, which includes the characteristic function step 61 described above. In the middle row, the generated aerial images 64 corresponding to the respective binary characteristic functions 62 in the top row are shown. In the bottom row, the approximation error 66 is shown as the difference between the generated aerial image 64 and a rigorously simulated aerial image 64 of the same base model 92 (e.g., a design pattern) of the lithography mask 14. From the results, it can be concluded that for the binary characteristic function 62, an approximation error 66 below 1% requires a sampling spacing below 2 nm. Therefore, for the binary characteristic function 62, a high resolution of the sampling grid is required, resulting in a high computation time.

[0189] Figure 10b The effect of the sampling grid resolution on the approximation error 66 during sampling of a bandwidth-limited characteristic function 68 is shown. Different sampling pitches between 30.18 nm and 2 nm are indicated in each column. In the top row, the sampled bandwidth-limited characteristic function 68 of a model 92 (e.g., a design pattern) of a lithography mask 14 is shown for different sampling pitches (pixel size in nm). Using the indicated bandwidth-limited characteristic function 68, an aerial image 64 is generated by the computer-implemented method 54''' for generating an aerial image 64 of a model of a lithography mask 14, including the characteristic function step 61 described above. In the middle row, the generated aerial image 64 corresponding to the corresponding bandwidth-limited characteristic function 68 in the top row is shown. In the bottom row, the approximation error 66 is shown as the difference between the generated aerial image 64 and a rigorously simulated aerial image 64 of the same underlying model 92 (e.g., a design pattern) of the lithography mask 14. The results indicate that, for the bandwidth-limited characteristic function 68, an approximation error of less than 1% requires a sampling pitch of even less than 8 nm. Therefore, for a bandwidth-limited characteristic function 68 , a coarse resolution is sufficient to obtain an accurate aerial image 64 in less computation time.

[0190] Figure 11A comparison of the convergence speed of the wave propagation method in equation (4) using the discrete binary characteristic function 62 or the discrete bandwidth limited characteristic function 68 is shown. The sampling interval is shown on the top horizontal axis 78. The number of corresponding pixels in one dimension of the sampling grid is shown on the bottom horizontal axis 80. The approximation error 66 is indicated on the vertical axis 82. The graphs show the root mean square error 70 and the maximum error 72 for the binary characteristic function 62, and the root mean square error 74 and the maximum error 76 for the bandwidth limited characteristic function 68. From these graphs it can be concluded that the convergence speed of the bandwidth limited characteristic function 68 is exponential compared to the convergence speed of the binary characteristic function 62.

[0191] By generalizing the concept of characteristic functions to non-binary characteristic functions, the design features of sub-pixels can be addressed and an acceleration factor of about 100 can be achieved.

[0192] In addition to the bandwidth limited characteristic function 68, it is also advantageous to use other non-binary characteristic functions to describe the presence of a specific material in different positions of the lithography mask 14 at . It is also advantageous to use a continuous characteristic function or a complex valued characteristic function.

[0193] For example, it is advantageous to use a continuous characteristic function or a complex valued characteristic function. In this way, the material distribution within the lithography mask can be described in a more flexible way, resulting in a more accurate approximation.

[0194] According to an aspect of this example, the numerical range of the at least one characteristic function contains at least one number . Thus, the at least one characteristic function is not a binary characteristic function, as it maps to at least one non-binary value. In this way, different materials m can exist at the same position (x, y), allowing for a more flexible modeling of the refractive index distribution in the lithography mask 14, resulting in a more comprehensive description of the material distribution in the lithography mask. By using characteristic functions with overlapping support, the accuracy of the wave propagation method can be improved. The support of a real valued function is a subset of the domain of the function containing the elements that are not mapped to zero. On the one hand, the presence of different materials in the same position of the lithography mask can be used to model the distribution of materials in the case that different materials exist at the same position. On the other hand, the assumption that different materials exist at the same position can be used as a mathematical means to improve the accuracy of the electromagnetic near field and the spatial image, even if this material distribution does not correspond to the real material distribution. In this way, a more accurate electromagnetic near field and spatial image can be calculated.

[0195] According to an aspect of this example, the characteristic functions form an affine combination at each position of the lithography mask at . In particular, the characteristic functions can form an affine combination at each position of the lithography mask at The convex combination of each position of the photomask forms a position of the characteristic function. This restriction ensures that the amount of material present in each position of the characteristic function domain is the same and equal to one. Thus, an exact description of the material distribution within the photomask 14 is obtained, resulting in an exact approximation of the propagation of the electromagnetic wave 22 within the photomask 14.

[0196] According to an aspect of this example, obtaining the characteristic function comprises decomposing the model 92 (e.g. design pattern) of the photomask 14 into elements 94 (e.g. using mathematical functions describing the contour or area of the structure 26, such as polygons, spline lines, curve elements, etc.), representing the elements 94 by the characteristic function, in particular by the binary characteristic function, and applying a low-pass filter to the characteristic function. The elements 94 can for example be represented by a characteristic function that takes a non-zero value (e.g. one) inside the element 94 and zero outside the element 94. For example, each element 94 can be decomposed into one or more triangles, and the triangles can be represented by the characteristic function. The Fourier transform of a polygon can be obtained as described in Appendix A of the PhD thesis "Lithography simulation at Vienna University of Technology" by Heinrich Kirchauer. The aforementioned PhD thesis is incorporated herein in its entirety and the disclosure thereof is included in the present specification. By applying a low-pass filter to the characteristic function, a bandwidth-limited characteristic function 68 is obtained. Thus, the wave propagation method in (4) can be simulated using the coarse sampling grid described above, reducing the computation time.

[0197] In particular, applying a low-pass filter to the characteristic function can comprise applying a spatially analytical Fourier transform to the characteristic function, followed by an inverse fast Fourier transform. The analytical Fourier transform can only be computed for spatial frequencies in the discrete domain of the inverse FFT. According to the Shannon-Nyquist theorem, this subsampling in spatial domain limits the maximum frequency of the characteristic function. Thus, the discretization corresponds to a low-pass filter of the characteristic function. As a result, the model (e.g. design pattern) of the photomask is represented by a bandwidth-limited characteristic function, which can be discretized using a sampling grid with a resolution much lower than the binary characteristic function, reducing the computation time.

[0198] According to the example, the analytical Fourier transform used in the wave propagation method in (4) is approximated by a fast Fourier transform (FFT). In this way, the computation time is reduced.

[0199] The FFT implies periodic boundary conditions. However, due to the arbitrary angle of the incident electromagnetic wave, this assumption is no longer valid. This inaccuracy is often neglected by the approximation method. Even if the model 92 (e.g. design pattern) of the mask is assumed to be periodic, the arbitrary angle of incidence of the electromagnetic wave 22 (e.g. with respect to the normal 84 of the structure plane 30) implies that the solution of (4) is only quasi-periodic according to the Floquet theorem, which means periodic with an additional phase shift a:

[0200]

[0201] Thus, according to an example, the wave propagation method takes into account the angle of the incident electromagnetic wave 22 (e.g., an angle relative to the normal 84 of the structure plane 30) by assuming quasi-periodic boundary conditions in the propagator step in equation (4) at one or more pairs of opposing boundaries that are perpendicular to the substrate plane 34 of the photolithographic mask 14 (i.e., in the x / y plane). By assuming quasi-periodic boundary conditions, the accuracy of the simulated electromagnetic near-field is improved.

[0202] make is quasi-periodic in the x and y coordinates. Then, according to Floquet's theorem, E can be rewritten as the periodic part E' in x and y multiplied by the non-periodic phase shift , as shown below:

[0203]

[0204] The Fourier transform of the periodic part E' can be written as

[0205]

[0206] It follows

[0207]

[0208] use

[0209]

[0210] get

[0211]

[0212] It follows that the phase shift α in the input field, which is linear in the x and y coordinates, can be incorporated by reformulating the dispersion relation in equation (3) as follows:

[0213]

[0214] Thus, according to an example, the dispersion relation in (3) can be reformulated using Floquet's theorem. The terms in the inverse Fourier transform are periodic and can be computed using a standard FFT.

[0215] Specifically, the dispersion relation of the electromagnetic wave 22 in the first portion 25 depends on the angle of the incident electromagnetic wave 22 .

[0216] Figure 12 The angle between the phase shift vector α and the incident electromagnetic wave 22 is shown. the relationship of the angle may be measured relative to a normal 84 of the structure plane Electromagnetic waves 22 propagate in the direction of the wave vector 86. Let and denote the boundary of a unit cell in the x-direction, i.e. the smallest non-periodic subset of the periodic model 92 (e.g. design pattern). Then, using the relationship

[0217]

[0218] and the phase difference between may be expressed as follows:

[0219]

[0220] The dependence of on the angle of the incident electromagnetic wave 22 can be calculated analogously.

[0221] For a reflection-based lithography mask 14, analytically or numerically simulating the propagation of the simulated electromagnetic wave 22 from step a) within the second portion 27 of the lithography mask 14 can comprise using an analytical description of the electromagnetic wave propagation within the mask carrier 48 and analytically calculating the reflection of the electromagnetic wave 22 at the multilayer 38.

[0222] According to an example, therefore, simulating the reflection of the simulated electromagnetic wave 22 within the multilayer 38 comprises analytically calculating a reflection coefficient at the boundary between the second portion 27 and the first portion 25 of the lithography mask 14 (e.g. at the boundary plane 32), which describes the propagation of the electromagnetic wave 22 within the stack of optical films 40 of the multilayer 38. The propagation within the stack of optical films 40 of the multilayer 38 corresponds to a reflection at the effective mirror plane 44 at a distance from the boundary plane 32.

[0223] In particular, the reflection coefficient at the boundary 32 can be calculated separately inside and outside the structure 26 in the first portion 25 of the lithography mask 14. For example, the reflection coefficient can be calculated individually for each medium of the absorber structures and the non-absorber structures of the grating 24 at the location of the boundary plane 32. In this way, the accuracy of the resulting aerial image 64 is improved.

[0224] In an example, simulating the propagation of the simulated electromagnetic wave 22 within the second portion 27 of the lithography mask 14 comprises applying the reflection coefficient to the electromagnetic wave 22 incident on the boundary 32.

[0225] In particular, simulating the reflection of the electromagnetic wave 22 within the multilayer 38 comprises using an analytical reflection coefficient at the boundary plane ​Substitute (4) for the phase term :

[0226]

[0227] wherein denotes the scalar electric field at the boundary plane pointing to the structure plane 30 of the photolithography mask 14, and denotes the scalar electric field at the boundary plane pointing to the base plane 34 of the photolithography mask 14. In this way, the computer-implemented method for generating a spatial image of a model of a photolithography mask can be applied to reflection-based photolithography masks. Furthermore, the accuracy of the method is improved.

[0228] As shown in equations 33 to 41 in the article "Optical properties of thin-film stacks illuminated by a focused field" by S. Kim, Y. Kim, and I. Park, Journal of the Optical Society of America A, Vol. 17, No. 8, August 2000, for s-polarized and p-polarized waves, the analytical reflection coefficient of each of the N optical thin films 40 of the multilayer 38 can be calculated as follows :

[0229]

[0230] wherein is an element of the characteristic matrix A

[0231]

[0232] Herein, is given by

[0233] ,

[0234] wherein

[0235]

[0236] Herein, denotes the vacuum permittivity, denotes the vacuum permeability, denotes the refractive index of the j+1th optical thin film 40, and denotes the thickness of the j+1th optical thin film 40. Reference is made herein throughout to the aforementioned article, the disclosure of which is included in the specification of the present invention.

[0237] In another example, the reflection of an electromagnetic wave by multiple layers 38 can be numerically calculated as follows: In a first step, the electric field at the boundary plane 32 is decomposed into its Fourier modes. In a second step, for each Fourier mode, the reflected electromagnetic field can be calculated using, for example, the transfer matrix method (described in Section 2.2 of "Domain Decomposition Methods for Maxwell's Equations: Scattering from Periodic Structures," by Achim Schädle, Lin Zschiedrich, Sven Burger, Roland Klose, and Frank Schmidt, available at arXiv:math / 0602179v1). In a third step, the superposition of the reflected Fourier modes yields the reflected electromagnetic wave. Alternatively, a machine learning model can be trained to numerically simulate the propagation of the electromagnetic wave within the second portion of the photolithographic mask.

[0238] Figures 13a to 13d The steps of a computer-implemented method 54, 54', 54", 54''' for generating an aerial image 64 of a model of a photolithographic mask 14 according to a specific embodiment of the present invention are shown. The model 92 of the photolithographic mask 14 comprises Figure 13a The element 94 is composed of polygons in the form of rectangles as shown. In the characteristic function step 61, the element 94 is represented by a characteristic function (for example, a binary characteristic function 62) obtained by any of the above methods. For example, the element 94 is represented by a binary characteristic function 62 having a value of 1 inside the element 94 and a value of 0 outside. Then, a spatially analytical Fourier transform is applied to the characteristic function, followed by an inverse FFT for the inverse transform, resulting in a bandwidth-limited characteristic function 68. Here, the analytical Fourier transform is calculated only for the spatial frequencies of the discretized domain of the inverse FFT. According to the Shannon-Nyquist theorem, this subsampling of the spatial domain limits the maximum frequency of the characteristic function. Therefore, the discretization corresponds to a low-pass filter of the characteristic function. The result is a bandwidth-limited discrete representation of the model 92 of the photolithography mask 14, that is, in Figure 13b The bandwidth-limited characteristic function 68 is sampled on a low-resolution sampling grid. Based on the bandwidth-limited characteristic function 68, Figure 13c A representation of the electromagnetic near field 20 in the form of its amplitude is shown in FIG. , which is simulated by propagating a simulated electromagnetic wave into the near field plane. Finally, by applying the simulation of the imaging process of the lithography system 10, 10' in the projection portion 19 between the near field plane 52 and the wafer plane 18 to the representation of the electromagnetic near field 20, the Figure 13dAn aerial image 64 is shown. The imaging process may include resampling the electromagnetic near-field 20 to a higher resolution grid. By applying the characteristic function step 61 and the aerial image generation step 63 to calculate the aerial image 64, due to the low resolution of the sampling grid, an accurate aerial image 64 can be simulated for the model 92 of the lithography mask 14 at a low computation time. Therefore, by applying a rigorous simulation method (e.g., RCWA) to the model 92 of the lithography mask 14 (e.g., a design pattern) in the rigorous simulation step 95 (which requires a high resolution sampling grid), the computation time used to obtain the aerial image 64 can be reduced compared to the simulation of the aerial image 64.

[0239] Figures 14a to 14c A comparison of aerial images 64 of a model of a lithography mask 14 obtained by three different simulation methods is shown. All three figures show the intensity distribution of the aerial images 64 obtained. Figure 14a An aerial image 64 simulated using the TEA method described above is shown. Figure 14b An aerial image 64 produced using the computer-implemented method described above for producing an aerial image 64 of a model of a photolithographic mask 14 is shown. Figure 14c An aerial image 64 simulated by a rigorous simulation method, the RCWA method described above, is shown. The results show that the computer-implemented method 54, 54', 54", 54'" for generating an aerial image 64 of a model of a lithographic mask 14 approximates the real aerial image 64 with a very low error rate and is therefore more accurate than the TEA method.

[0240] The computation time of the above method can be reduced even further using acceleration methods known to those skilled in the art, such as graphics processing units (GPUs), distributed GPUs, field programmable gate arrays (FPGAs), etc.

[0241] Figure 14a The calculation time of the TEA method is 0.4 seconds. Figure 14b The computation time for the computer-implemented method 54, 54', 54", 54'' of the aerial image 64 in FIG. 5 is between 0.01 and 0.4 seconds, depending on the grid structure. If the edges of the model 92 (e.g., the design pattern) are aligned with the FFT grid, the computation time is approximately 0.01 seconds. However, in general, the model 92 (e.g., the design pattern) is composed of elements 94 that are not aligned with the grid used in the FFT. In this case, a Fourier transform is applied to the elements 94 to map the characteristic function to the FFT grid, which requires a computation time of approximately 0.4 seconds. Figure 14cThe computation time of the rigorous simulation method (RCWA method) in the above-mentioned examples is between 10 seconds and 1000 seconds, depending on the smallest feature size, e.g. the defect size. Therefore, the computer-implemented method according to embodiments of the present application produces an approximation of the aerial image 64 with a very low error rate in a computation time comparable to ordinary spatial image approximation methods. Compared to the rigorous simulation method, the method according to embodiments of the present application is about two to four orders of magnitude faster at an error level of only a few percent.

[0242] In many applications, the produced aerial image is compared to a reference aerial image. For example, the reference aerial image can be an acquired aerial image (referred to as die-to-die mode) or a simulated aerial image (referred to as die-to-database mode). In order to obtain meaningful results from such a comparison, the produced aerial image has to be highly accurate and precisely replicate the reference aerial image, e.g. the reference aerial image capturing conditions. For this reason, it is advantageous to calibrate the produced aerial image.

[0243] In Figure 15 In the example shown, a defect 93 is detected by comparing an acquired aerial image 83 (here an aerial image acquired using a certain metrology system 85) with a simulated aerial image 64. The acquired aerial image 83 contains the defect 93. According to embodiments, examples or aspects of the present application described above, the simulated aerial image 64 is produced using the method 54, 54', 54", 54"' for producing an aerial image of a model 92 of a lithography mask. In this way, the simulated aerial image 64 can be produced accurately and with low computation time. The acquired aerial image 83 is compared to the simulated aerial image 64, e.g. by subtracting the images, resulting in a comparison result 90. From the comparison result 90, a defect map 91 is produced indicating the position of the defect 93, e.g. by thresholding, adaptive thresholding or by applying a machine learning method for defect detection.

[0244] In order to be able to compare the aerial image 83 acquired by the metrology system 85 with the simulated aerial image 64 using the method 54, 54', 54", 54"' according to the present application, it is important to take into account the characteristics of the metrology system 85 and the lithography mask when producing the aerial image 64 from the model 92 of the lithography mask. The characteristics of the metrology system 85 to be taken into account include, for example, illumination settings, imaging settings, sensor model, motion blur due to scanning, field dependent effects, defocus, distortion, aberrations, apodization, etc. The characteristics of the mask to be taken into account include, for example, bias, round corners, side wall angles, layer height, refractive index, etc. as described in detail before. In order to take these characteristics into account, the method for producing an aerial image of a model of a lithography mask can include adjustable mask parameters and / or optical parameters as described above.

[0245] Therefore, in preferred embodiments, the method 54, 54', 54", 54"' for generating aerial images 64 from models 92 of lithography masks further comprises adjusting at least one parameter of the method to minimize the difference between one or more reference aerial images 88 of one or more lithography masks and corresponding generated aerial images 64 of corresponding models 92 of the one or more lithography masks, wherein the at least one parameter is from the group comprising mask parameters and optical parameters. These parameter adjustments can be performed in a calibration step 89, which is shown in detail in Figure 16 .

[0246] As Figure 16 illustrated, the calibration step 89 comprises adjusting parameters p of the method 54 for generating aerial images of models of lithography masks by minimizing the difference between one or more reference aerial images 88 and corresponding generated aerial images 64. The one or more reference aerial images 88 can for example comprise aerial images acquired by a metrology system 85 for one or more lithography masks. The corresponding generated aerial images 64 can for example be generated using an embodiment of the method for generating aerial images using corresponding models 92 of the one or more lithography masks. Other methods for generating aerial images from models of lithography masks can also be used. However, the method according to the present application has the advantage of being fast, accurate and having a computable gradient, which greatly simplifies the optimization of the method parameters. The one or more reference images 88 can comprise acquired aerial images of models 92 of lithography masks, which will be used for inspecting defects 93.

[0247] The parameters p of the method 54 comprising mask parameters and / or optical parameters can for example be optimized 81 in an iterative manner 79 comprising one or more iterations as illustrated in Figure 16 . To this end, the following optimization problem can be solved:

[0248]

[0249] is the target parameter vector that minimizes a difference measure x between the acquired aerial images and the generated aerial images . x is the objective function or loss function that defines the optimization condition. It can be linked to a noise model of the acquired aerial images, for example the -norm can be used for Gaussian i.i.d noise. The objective function can also comprise additional normalization terms as a function of p, in particular in order to obtain a well-conditioned objective function.

[0250] There are different methods known for computing the generated aerial images the Hopkins method, the Abbe method, and the partial Hopkins method.

[0251] The Hopkins method relies on the observation that for a small change in the angle of incidence of an optical wave, the intensity, phase, and polarization of the expected optical wave will only deviate very slightly. Therefore, a change in the angle of illumination results in a frequency shift of the corresponding diffraction spectrum of the lithography mask only. Thus, the incident electromagnetic near-field of the same mask spectrum for all angles of illumination and is shifted according to the angle of illumination:

[0252]

[0253] wherein denotes the intensity of the aerial image, is a complex imaging pupil function, is an illumination angle weight distribution (e.g., with respect to the illumination intensity), is a permittivity, is the speed of light in vacuum, is the angle of illumination, is the number of angles of illumination in the distribution of angles of illumination in the pupil plane.

[0254] This method is both simple and fast. For simulations using thin masks or the Kirchhoff method (e.g., TEA), this assumption is always fulfilled. However, if the thickness of the structures on the lithography mask can no longer be neglected and a strict electromagnetic field simulation of the mask diffraction is required for different angles of illumination, the Hopkins method is not accurate enough.

[0255] In this case, the Abbe method can be used to adapt to the non-constant diffraction spectrum of the lithography mask, since the Abbe method assumes a diffraction spectrum :

[0256]

[0257] However, the Abbe method is very computationally expensive, since the electromagnetic near-field must be simulated for each single angle of illumination. Therefore, the Abbe method can not be suitable for simulations of, for example, full-chip optical proximity correction (OPC) or verification systems.

[0258] To obtain a fast and accurate simulation method of aerial images of a lithographic mask, a local Hopkins method can be used, for example as disclosed in US 2007 / 0253637 Al. The local Hopkins method is a combination of the Hopkins method and the Abbe method based on a constant diffraction spectrum of the local assumed lithographic mask. To this end, the source map is divided into a number of segments. For each segment, the diffraction spectrum is assumed to be constant, so that only a single diffraction spectrum needs to be simulated for each segment. Thus, using the local Hopkins method, a smaller number of spectra can be simulated for a subset of the selected illumination angles For the remaining illumination angles, the simulated spectrum is moved according to the illumination angle

[0259]

[0260] where , and .

[0261] The local Hopkins method requires careful selection of the segments and the illumination angles within the segments and the simulation of their diffraction spectrum, for example as described in US 2007 / 0253637 Al.

[0262] The Hopkins method, the Abbe method or the local Hopkins method allow to compute the gradient with respect to the parameter vector p. Thus, by using one of these methods to compute the spatial image generated by the electromagnetic near-field in the above objective function x, the parameters p can be optimized using an iterative approach, for example by gradient descent. Note that the optimization of optical parameters, for example Zernike aberrations, which can be modeled by a change of the pupil function, as opposed to mask parameters, for example bias, round corners, etc., does not require a new simulation of the electromagnetic near-field, but only a re-evaluation of the Hopkins method, the Abbe method or the local Hopkins method, thus simplifying and speeding up the optimization of the parameter vector p.

[0263] Figure 17The benefit of mask and optical parameters of the method for generating a aerial image of a model of a lithographic mask is shown. In the center of the top row, the obtained aerial image 83 for a given lithographic mask is shown. The obtained aerial image 83 can be an aerial image acquired using a metrology system. On the left side of the top row, the generated aerial image without calibration 87 is shown, which is obtained by applying the method for generating an aerial image of a model of a lithographic mask according to the present application to a model of a lithographic mask without performing the calibration step 89. On the right side of the top row, the generated aerial image with calibration 64 is shown, which is obtained by applying the method for generating an aerial image of a model of a lithographic mask according to the present application to a model of a lithographic mask and performing the calibration step 89. In the bottom row, the comparison result 90 (i.e. the difference image) is presented, which clearly shows the benefit of the additional calibration step 89.

[0264] For mask analysis, metrology systems typically compare the position of a structure in a reference aerial image (e.g. an acquired aerial image) to the corresponding position in a generated aerial image. For this, a high precision reference aerial image is required, e.g. sub-pixel accuracy of about below 1 nm. However, the structure position cannot be directly estimated from the reference aerial image due to the low-pass filtering effect of the numerical aperture and due to mask 3D effects or optical proximity effects leading to a displacement of the structure in the aerial image. Furthermore, the reference aerial image of a metrology system is often affected by aberrations and further image error sources, e.g. distortions, apodization, noise (e.g. shot noise, read-out noise, etc.).

[0265] Therefore, according to Figure 18 The method for generating an aerial image of a model of a lithographic mask 92 shown in the example further comprises registering 99 one or more obtained aerial images 83 (e.g. acquired aerial images) of a lithographic mask to the corresponding generated aerial image 64 of the model 92 of the lithographic mask and reporting at least one registration parameter. The registration 99 can also be understood as the calibration step 89 or as part of the calibration step 89. In this way, the displacement of the structure is reduced and the defect detection is improved. In Figure 18 In the middle, the obtained aerial image 83 is the aerial image of a lithographic mask acquired using a metrology system. It is registered 99 to the generated aerial image 64 according to the base model 92 of the lithographic mask. The registration result 97 is shown on the right side.

[0266] For registering the aerial images, various registration methods can be used. For example, a (sub-pixel) shift of one of the aerial images can be optimized, e.g. by minimizing an error norm (e.g. least squares error norm) between the two aerial images. The error norm can be calculated for each pixel of the aerial images, e.g. by calculating the difference between the two aerial images for each pixel and squaring the result. The error norm can then be calculated by summing the squared differences for all pixels of the aerial images. or by maximizing a similarity measure (e.g. cross-correlation of aerial images). Note that when using a sinusoidal interpolation (Fourier shift) and neglecting boundary effects, the sub-pixel shift of a simulated (noiseless) reference image of the Nyquist sampling is exact. Non-linear registration methods can also be used. For example, a continuous optimization method can be used for registration, e.g. in a variational approach. Further constraints can be imposed on the registration result. For example, different registration methods with sub-pixel accuracy are described in Manuel Guizar-Sicairos, Samuel T. Thurman, and James R. Fienup, "Efficient subpixel image registration algorithms", Opt. Lett. 33, pp. 156 - 158, 2008.

[0267] In addition to the registration parameters, critical dimension (CD) parameters can also be estimated by an optimization method. The CD can be a global parameter (constant over the whole model) or a local parameter (locally varying over the model) of the mask model and the loss function.

[0268] According to an example, one or more of the obtained aerial images 83 contain a focus stack of the lithography mask. A focus stack contains multiple aerial images of the same measurement site obtained at different focus settings. The focus stack can be used in different ways. For example, a best focus aerial image can be interpolated from the focus stack and this best focus aerial image can be registered to the generated aerial image 64 of the model 92 of the lithography mask during registration. Alternatively, all aerial images in the focus stack can be considered during registration, e.g. by computing the norm of the complete stack of difference images during optimization.

[0269] In an example, the calibration step 89 contains an optimization 81 of mask and optical parameters and a registration 99. In this example, the optimization 81 is performed before the registration 99. Figure 19 In the example shown, the optimization 81 can be performed before the registration 99. The optimized mask and optical parameters are then used to generate the aerial image 64. In this example, the registration 99 is performed after the optimization 81. Figure 20 In another example shown, the mask parameters and / or the optical parameters and / or the registration parameters are jointly optimized in an iterative 79 manner using a single or multiple iterations. In each optimization step, the mask and optical parameters of the generated aerial image 64 are optimized 81, followed by registering 99 the optimized aerial image to the obtained aerial image 83. Finally, the optimized mask and optical parameters and the complete registration are reported. By jointly optimizing the parameters, a higher accuracy result is obtained. For example, in this way parameter drifts are taken into account as the parameters are optimized based on the latest measurements.

[0270] The generation of aerial images of a model of a lithography mask according to the above described embodiments, examples or aspects of the application can facilitate the training of a machine learning model.

[0271] Due to the large size of a lithography mask relative to the wavelength, e.g. the size of an EUV lithography mask is about Thus, the generation of aerial images of a complete lithography mask is very time consuming - even using the above described method of the application. Therefore, according to embodiments of the application, a machine learning model is trained to emulate the mapping from a model of a lithography mask to an aerial image. The machine learning model is even faster than the method according to the application. In Figure 21 In the shown example, the computer implemented method for training a machine learning model 128 mapping a model 92 of a lithography mask to an aerial image 64 of the lithography mask comprises generating aerial images 64 of a plurality of models 92 of lithography masks using the method for generating aerial images of a model of a lithography mask 54 according to any of the above described embodiments, examples or aspects; and training the machine learning model 128 using training data comprising the generated aerial images 64. By using the method for generating aerial images of a model of a lithography mask 54 according to any of the above described embodiments, examples or aspects, aerial images 64 of various models 92 of lithography masks can be generated accurately and quickly. Using the models 92 and the generated aerial images 64 as training data, the machine learning model 128 can be efficiently trained which emulates the method for generating aerial images of a model of a lithography mask. The trained machine learning model 128 can then be used to quickly generate aerial images 64 of a model 92 of a lithography mask. The machine learning model can for example comprise a neural network, e.g. a deep learning model. Due to the single forward pass only required during inference, the trained machine learning model 128 requires very low computation time. Figure 22 A sample model 130 of a lithography mask is shown which can be used to generate training data. In addition to the generated training data, other training data can be used, e.g. acquired aerial images or emulated aerial images obtained by different methods. For example, transfer learning can be used to consider additional training data after pre-training on the generated training data.

[0272] Due to their high quality results and inference speed, machine learning models are also particularly promising for other lithography applications, e.g. for defect detection in lithography masks. However, these methods require a large amount of real and annotated training data which is usually not available from acquired aerial images.

[0273] Thus, the method for generating aerial images of a model of a lithography mask according to the above described embodiments, examples or aspects of the application can also be advantageously used to train machine learning models for other applications, e.g. for defect detection in lithography masks. Figure 23The machine learning model 136 for defect detection receives as input the reference aerial image 132 and the acquired aerial image 134 and maps it to an aerial image with marked defects 138 as output. The reference aerial image 132 can for example be an acquired aerial image (die-to-die defect detection) or a simulated aerial image (die-to-database defect detection).

[0274] As shown in Fig. 1 1, the computer-implemented method for training a machine learning model 136 for defect detection in an acquired aerial image 134 of a photomask comprises generating model pairs 140 for a plurality of photomasks, each model pair 140 comprising a defect-free model 142 of a photomask (i.e. a model without defects) and a defect model 144 of the same photomask (i.e. a model comprising one or more defects 93), generating from the model pairs 140 aerial image pairs 146 comprising a defect-free aerial image 148 and a defective aerial image 150 of a photomask by applying the aforementioned method for generating an aerial image of a model of a photomask according to embodiments, examples or aspects of the present invention to the defect-free model 142 and the defect model 144 of each model pair 140, and training the machine learning model 136 using training data comprising the aerial image pairs 146. Figure 24

[0275] Optionally, noise 152 can be added to the aerial image pairs 146, thereby generating noisy aerial image pairs 154 comprising a noisy defect-free aerial image 156 and a noisy defective aerial image 158. The noise can comprise a realization of all parameters that vary randomly (e.g. shot noise, readout noise, aberrations, line edge roughness, etc.). By adding noise to the aerial image pairs 146, the training data becomes more realistic, in particular in case of die-to-die defect detection, which improves the training results.

[0276] In addition to the generated training data, other training data can be used, for example acquired aerial images or simulated aerial images obtained by different methods, for example defect-free acquired aerial images or simulated aerial images. For example, transfer learning can be used to take into account additional training data after pre-training on the generated training data.

[0277] The generated aerial images of models of photomasks can be used for different purposes.

[0278] For example, based on the accurate aerial images, the model of the respective photomask (e.g. the design pattern) can be improved and mask 3D effects can be mitigated, for example by modifying the model of the photomask. Or, the material or thickness within the first portion can be modified, for example the absorber material and / or the absorber thickness within the grating of the photomask. Or, optical proximity correction techniques can be applied to the photomask, for example by adding sub-resolution assist features, etc. ​

[0279] For example, the generated aerial image can be used for defect detection. Given an acquired or simulated aerial image of a lithography mask, defects of the lithography mask can be checked by generating an aerial image of the lithography mask based on a model of the lithography mask, and by comparing the acquired or simulated aerial image with the generated aerial image.

[0280] For example, the relevance of defects in a charged-particle beam image of an acquired lithography mask can be assessed by generating an aerial image using the charged-particle beam image as a model of the lithography mask, and by comparing the defects in the charged-particle beam image with corresponding locations in the generated aerial image. This saves a lot of time and resources, as it is not necessary to acquire an aerial image of the lithography mask.

[0281] For example, the generated aerial image can be used to generate a digital twin of a machine, which uses the acquired aerial image of the lithography mask. The digital twin of the machine is a digital simulation of the machine, which simulates the capture of aerial images within the machine using the method of generating an aerial image of a model used to generate the lithography mask. The digital twin of the machine can be used for many different purposes, for example to specify the functions and requirements of the machine, to show the functions of the machine to a customer before the machine is manufactured or delivered, or to accelerate the development of machine components (e.g. user interfaces, etc.).

[0282] In these applications, the generated aerial image using a model of the lithography mask is used instead of acquiring an aerial image of the lithography mask, thereby greatly reducing the computation time.

[0283] According to Figure 25The computer-implemented method 103 for improving a model 92 of a photomask 14 (e.g. a design pattern), for repairing a photomask 14, for deciding on a quality of a photomask 14, for measuring a photomask 14, for detecting or assessing a relevance of a defect in a photomask 14, or for selecting an illumination setting in a lithography system of the specific embodiments, examples or aspects described above comprises: generating a aerial image 64 of a model of a photomask 14 using the method 54 for generating an aerial image of a model of a photomask of any of the specific embodiments, examples or aspects described above; analyzing the generated aerial image 64 accordingly in an analysis step 105; and improving a design of a photomask 14, repairing a photomask 14, deciding on a quality of a photomask 14, detecting or assessing a relevance of a defect in a photomask 14, or selecting an illumination setting in a lithography system using the results of the analysis in an application step 107. Optionally, the method can use further information of the photomask obtained in a further information step 101, e.g. an aerial image obtained using some aerial image acquisition system or metrology system or by simulation, or a SEM image of the photomask, layer size information, mask type, etc. The method can apply iterations of the method steps described above, e.g. for source mask optimization, inverse lithography or for selecting an illumination setting. Iterative optimization of mask parameters or illumination parameters can be performed, e.g. gradient-based optimization, which requires iterations of generating an aerial image and adjusting parameters.

[0284] For improving a model of a photomask, the analysis step 105 can comprise comparing the generated aerial image 64 of a model of a photomask with a taken aerial image of a photomask, e.g. an acquired aerial image or a simulated aerial image, e.g. a simulated target aerial image, e.g. for detecting defects or measuring critical dimensions. Based on the analysis results, the model of the photomask can be modified, e.g. in case of source mask optimization or inverse lithography. Based on the analysis results, a repair shape indicating a defect position and a correction thereof can be generated and used for repairing the photomask. Based on the analysis results, a quality of the photomask can be decided, e.g. by using some quality measure, e.g. a number of defects, an average number of defects per area, a maximum severity of defects, a type of defects and a frequency thereof, etc. Based on the analysis results, defects can be detected. Based on the analysis results, a relevance of defects can be assessed by checking whether the detected defects in the generated aerial image of a photomask are actually printed in the taken aerial image of a photomask.

[0285] Using the obtained aerial image for comparison with the generated aerial image in the analysis step 105 is optional. For example, defects can also be detected by analyzing the generated aerial image only. A photomask can be measured using the generated aerial image only. An illumination setting in a lithography system can be selected by generating different aerial images using different illumination settings in the analysis step 105 and selecting the illumination setting which generates the aerial image of highest quality.

[0286] The analyzing step 105 can be performed once or iteratively, e.g. during an iterative optimization. For example, during a source mask optimization, the lithography mask can be optimized by iteratively generating aerial images of a current model of the lithography mask and modifying the model based on the aerial images.

[0287] Figure 26 A computer-implemented method 96 for detecting defects in a lithography mask 14 is shown, the computer-implemented method 96 comprising: obtaining an aerial image of the lithography mask 14 in an aerial image step 98; generating an aerial image 64 of a model of the lithography mask 14 using the computer-implemented method 54, 54', 54", 54"' according to embodiments of the present application; detecting defects in the lithography mask 14 in a defect detection step 100 by comparing the obtained aerial image with the generated aerial image 64. The aerial image of the lithography mask can be obtained, e.g., by capturing the aerial image of the lithography mask using a capturing tool or by simulating or generating the aerial image using some method, e.g., the methods described above.

[0288] Figure 27A computer-implemented method 102 for assessing the relevance of defects in a lithography mask 14 according to embodiments of the present application is shown, the computer-implemented method 102 comprising: in an imaging step 104, providing a charged particle beam image of a lithography mask comprising one or more defects; generating a spatial image 64 of a model of the lithography mask using the computer-implemented method 54, 54', 54", 54"' wherein the charged particle beam image is used as a model of the lithography mask 14; in an assessing step 106, assessing the relevance of the one or more defects in the lithography mask 14 using the generated spatial image 64. A defect is assessed as relevant if it will be printed on a wafer during a printing process. Conversely, a defect is assessed as irrelevant if it will not be printed on a wafer. The charged particle beam image is obtained by a charged particle beam device, e.g. a helium ion microscope (HIM), a cross-beam device comprising a FIB and a SEM or any charged particle imaging device. The assessing step 106 can comprise comparing the generated spatial image 64 with the charged particle beam image. For example, the position(s) of the one or more defects in the charged particle beam image can be compared with the corresponding position(s) in the generated spatial image 64. If the defect is not visible in the generated spatial image 64, it can be concluded that the defect is not printed on a wafer and is therefore irrelevant. If the defect is visible in the generated spatial image 64, it can be concluded that the defect is indeed printed on a wafer and is therefore relevant. The generated spatial image 64 can also be compared with a reference image (e.g. another generated, simulated or acquired spatial image 64 of the lithography mask 14) to assess the relevance of the one or more defects. For example, if the generated spatial image 64 is very similar to the reference image in the defect position(s), the defect can be assessed as irrelevant. If the simulated spatial image 64 is different from the reference image in the defect position(s), the defect can be assessed as relevant. Additionally or alternatively, the assessing step 106 can comprise a calculation of the 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 in one or more positions, these positions can be assessed as relevant defects.

[0289] Figure 28 A system 108 for generating a spatial image 64 of a model of a lithography mask 14 according to embodiments of the present application is shown, the system 108 comprising: a data analysis device 110 comprising at least one memory 114 and at least one processor 112, the processor 112 being configured to perform the steps of the computer-implemented method for generating a spatial image according to the above described embodiments of the present application. The processor 112 can be implemented as a CPU or a GPU.

[0290] Figure 28A system 108 for improving a lithography mask, for repairing a lithography mask, for determining a quality of a lithography mask, for performing a measurement of a lithography mask, for detecting or evaluating defects in a lithography mask, or for selecting an illumination setting of a lithography system is also shown. The system 108 comprises a data analysis device 110 comprising at least one memory 114 and at least one processor 112 configured to perform the steps of the computer-implemented method for generating a aerial image of a lithography mask according to the above described embodiments, examples or aspects. Optionally, the system can also comprise a subsystem for acquiring an aerial image of a lithography mask, which can be used for analyzing the generated aerial image of a lithography mask, e.g. for comparison, as Figure 29 shown.

[0291] Figure 29 A system 116 for detecting defects in a lithography mask 14 according to embodiments of the present application is shown, the system 116 comprising a subsystem 118 for acquiring an aerial image 64 of the lithography mask 14, a data analysis device 110 comprising at least one memory 114 and at least one processor 112 configured to perform the steps of the computer-implemented method 96 for detecting defects according to embodiments of the present application. The subsystem 118 for acquiring an aerial image 64 of the lithography mask 14 can comprise an aerial image capturing system. Alternatively, the subsystem 118 can comprise a database or any other memory containing an aerial image 64 of the lithography mask 14, and the subsystem 118 can be configured to load the aerial image 64 from the database or memory. The subsystem 118 for acquiring an aerial image 64 of the lithography mask 14 can provide the aerial image 64 to the data analysis device 110. The data analysis device 110 comprises a processor 112, e.g. implemented as a CPU or a GPU. The processor 112 can receive the aerial image 64 via an interface 120. The processor 112 can load program code from the memory 114, e.g. program code for performing the above described computer-implemented method for detecting defects. The processor 112 can execute the program code.

[0292] Figure 30A system 122 for evaluating the relevance of defects in a photomask 14 according to embodiments of the present application is shown, the system 122 comprising a subsystem 124 for acquiring charged particle beam images 126 of the photomask 14 and a data analysis device 110 comprising at least one memory 114 and at least one processor 112, the processor 112 being configured to perform the steps of a computer-implemented method 102 for evaluating the relevance of defects in a photomask 14 according to embodiments of the present application. The subsystem 124 for acquiring charged particle beam images 126 of the photomask 14 can comprise a charged particle beam device, such as a helium ion microscope (HIM), a cross-beam device comprising a FIB and a SEM or any charged particle imaging device. Alternatively, the subsystem 124 can comprise a database or any other memory with charged particle beam images 126 of the photomask 14 and the subsystem 124 can be configured to load the charged particle beam images 126 from the database or the memory. The subsystem 124 for acquiring charged particle beam images 126 of the photomask 14 can provide the charged particle beam images 126 to the data analysis device 110. The data analysis device 110 comprises a processor 112, such as implemented as a CPU or a GPU. The processor 112 can receive the charged particle beam images 126 via an interface 120. The processor 112 can load program code from the memory 114, such as program code of a computer-implemented method for evaluating the relevance of defects as described above. The processor 112 can execute the program code.

[0293] Embodiments, examples and aspects of the present application can be described by the following items:

[0294] 1. A computer-implemented method for simulating the electromagnetic near-field of a model of a photomask in a near-field plane, the photomask being illuminated by incident electromagnetic waves, the photomask comprising a mask support and a grating, the grating comprising absorber structures and non-absorber structures forming a pattern on at least a portion of the mask support, the photomask comprising a grating portion extending between an absorber plane of the photomask and a mask support plane and a mask support portion extending between the mask support plane of the photomask and a substrate plane, the method comprising:

[0295] a) simulating the propagation of electromagnetic waves within the grating portion of the photomask using a wave propagation algorithm describing the propagation of electromagnetic waves through a non-uniform medium;

[0296] b) modeling the propagation of electromagnetic waves within the mask support portion of the photomask using an analytical description of the propagation of electromagnetic waves within the mask support; and

[0297] c) obtaining the electromagnetic near-field of the model of the photomask as the simulated propagated electromagnetic waves in the near-field plane next to the absorber plane of the photomask.

[0298] 2. The method according to item 1, wherein the wave propagation algorithm describes the wave propagation step in a plane

[0299]

[0300] where E denotes the electric field component of the electromagnetic field, and is the wave vector, which locally follows the dispersion relation

[0301]

[0302] where denotes the wave number of light with wavelength λ in vacuum, denotes the local refractive index, and denotes the Fourier transform.

[0303] 3. The method according to item 2, further comprising

[0304] a) identifying the material quantities M of absorber structures and non-absorber structures forming the pattern of the lithography mask;

[0305] b) defining for each material m a characteristic function indicating whether there is material at the position (x,y) of the lithography mask within a subset of the x / y plane orthogonal to the z direction;

[0306] c) simulating the propagation step of the electromagnetic wave as a weighted sum of the propagation steps within each identified material:

[0307]

[0308] where denotes the inverse Fourier transform.

[0309] 4. The method according to item 3, wherein the numerical range of at least one characteristic function contains at least one number .

[0310] 5. The method according to item 3 or 4, wherein the characteristic functions form a convex combination at each position of the lithography mask at

[0311]

[0312] 6. The method according to any one of items 3 to 5, wherein the characteristic functions are bandwidth limited.

[0313] ​​7. The method according to any one of items 3 to 6, wherein obtaining the characteristic function comprises decomposing the pattern of the lithography mask into polygons, representing the polygons by the characteristic function, in particular by a binary characteristic function, and applying a low-pass filter to the characteristic function.

[0314] 8. The method according to item 7, wherein applying the low-pass filter comprises applying a spatially resolved Fourier transform to the characteristic function, followed by an inverse fast Fourier transform.

[0315] 9. The method according to any one of items 3 to 8, wherein the analytical Fourier transform used in the wave propagation algorithm is approximated by a fast Fourier transform.

[0316] 10. The method according to item 9, wherein the wave propagation algorithm takes into account the angle of the incident electromagnetic wave with respect to the normal of the absorber plane by assuming a quasi-periodic boundary condition in the fast Fourier transform in one or more directions perpendicular to the base plane of the lithography mask .

[0317] 11. The method according to item 9 or 10, wherein the dispersion relation is reformulated using Floquet’s theorem

[0318]

[0319] 12. The method according to any one of items 9 to 11, wherein the electromagnetic wave within the grating section has a dispersion relation that depends on the angle of the incident electromagnetic wave with respect to the normal of the absorber plane in one or more directions perpendicular to the base plane of the lithography mask .

[0320] 13. The method according to any one of items 9 to 12, wherein the dispersion relation within the grating section is modified using a phase shift vector that depends on the angle as follows:

[0321]

[0322] 14. The method according to any one of the preceding items, wherein the simulated electromagnetic wave is incident on the base plane, propagates from the base plane to the mask carrier plane within the mask carrier section of the lithography mask, and propagates from the mask carrier plane to the absorber plane within the grating section of the lithography mask.

[0323] 15. The method according to any of the preceding items, wherein the mask carrier comprises a multilayer in the form of an optical thin film stack for reflecting electromagnetic waves, and wherein the simulated electromagnetic waves are incident on the absorber plane, propagate from the absorber plane to the mask carrier plane within the grating portion of the lithography mask, are reflected within the multilayer in the mask carrier portion of the lithography mask, and propagate from the mask carrier plane to the absorber plane within the grating portion of the lithography mask.

[0324] 16. The method according to item 15, wherein the reflection of the simulated electromagnetic waves within the multilayer comprises an analytical calculation of reflection coefficients at the mask carrier plane, which describe the propagation of the electromagnetic waves within the stack of optical thin films of the multilayer.

[0325] 17. The method according to item 16, wherein the reflection coefficients are calculated separately for each medium of the absorber structures and the non-absorber structures of the grating at the location of the mask carrier plane.

[0326] 18. The method according to any of items 3 to 13, wherein the mask carrier comprises a multilayer in the form of an optical thin film stack for reflecting electromagnetic waves, and

[0327] wherein the simulated electromagnetic waves are incident on the absorber plane, propagate from the absorber plane to the mask carrier plane within the grating portion of the lithography mask, are reflected within the multilayer in the mask carrier portion of the lithography mask, and propagate from the mask carrier plane to the absorber plane within the grating portion of the lithography mask, and

[0328] wherein the reflection of the simulated electromagnetic waves within the multilayer comprises an analytical reflection coefficient at the mask carrier plane replacing the phase term :

[0329]

[0330] wherein represents a scalar electric field at the mask carrier plane pointing towards the absorber plane of the lithography mask, represents a scalar electric field at the mask carrier plane pointing towards the substrate plane of the lithography mask.

[0331] 19. A computer implemented method for simulating a spatial image of a model of a lithography mask, the method comprising:

[0332] a) the method for simulating an electromagnetic near field of a model of a lithography mask according to any of items 1 to 18;

[0333] ​b) simulating the aerial image of the model of the lithography mask by applying a simulation of the imaging process of the lithography system within the projection part of the electromagnetic near-field extending between the near-field plane and the wafer plane.

[0334] 20. The method according to item 19, wherein simulating the imaging process comprises resampling the simulated electromagnetic near-field.

[0335] 21. A computer implemented method for detecting defects in a lithography mask, the method comprising:

[0336] - taking an aerial image of the lithography mask;

[0337] - simulating the aerial image of a model of the lithography mask using the method according to item 19 or 20; and

[0338] - detecting defects in the lithography mask by comparing the taken aerial image with the simulated aerial image.

[0339] 22. A computer implemented method for assessing the relevance of defects in a lithography mask, the method comprising:

[0340] - providing a charged particle beam image of a lithography mask comprising one or more defects;

[0341] - simulating the aerial image of a model of the lithography mask using the method according to item 19 or 20, wherein the charged particle beam image is used as the model of the lithography mask;

[0342] - assessing the relevance of the one or more defects in the lithography mask using the simulated aerial image.

[0343] 23. A computer readable medium having stored thereon a computer program, executable by a computing device, the computer program comprising program code for executing the method of any one of items 1 to 22.

[0344] 24. A computer program product comprising instructions which, when executed by a computer, cause the computer to carry out the method of any one of items 1 to 22.

[0345] 25. A system for simulating the electromagnetic near-field of a model of a lithography mask, the system comprising a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer implemented method of any one of items 1 to 18.

[0346] 26. A system for simulating aerial images of a model of a lithography mask, the system comprising a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer-implemented method of item 19 or 20.

[0347] 27. A system for detecting defects in a lithography mask, the system comprising:

[0348] - a subsystem for acquiring an aerial image of a lithography mask;

[0349] - a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer-implemented method of item 21.

[0350] 28. A system for assessing the relevance of defects in a lithography mask, the system comprising:

[0351] - a subsystem for obtaining a charged particle beam image of a lithography mask;

[0352] - a data analysis device comprising at least one memory and at least one processor configured to perform the steps of the computer-implemented method of item 22.

[0353] In summary, the invention relates to a computer-implemented method 54, 54', 54", 54"' for simulating an aerial image 64 of a model of a lithography mask 14 under illumination by an incident electromagnetic wave 22, the method comprising: a) approximately simulating the propagation of the incident electromagnetic wave 22 within a first portion 25 of the lithography mask 14 comprising a plurality of structures; b) analytically or numerically simulating the propagation of the simulated electromagnetic wave 22 from step a) within a second portion 27 of the lithography mask 14; c) simulating a representation of an electromagnetic near-field 20 of the model of the lithography mask 14 by propagating the simulated electromagnetic wave 22 from step b) to a near-field plane 52; and d) generating the aerial image 64 of the lithography mask 14.

[0354] List of reference signs

[0355] 10, 10' lithography system

[0356] 12 radiation source

[0357] 14 lithography mask

[0358] 16 illumination optics

[0359] 17 projection optics

[0360] 18 wafer plane

[0361] 19 projection portion

[0362] 20 near field

[0363] 22 electromagnetic wave

[0364] 24 grating

[0365] 25 first part

[0366] 26 structure

[0367] 27 second part

[0368] 28 non-structure

[0369] 30 structure plane

[0370] 32 boundary plane

[0371] 34 base plane

[0372] 38 multilayer

[0373] 40 optical film

[0374] 42 cover layer

[0375] 44 effective mirror plane

[0376] 46 substrate layer

[0377] 48 mask carrier

[0378] 50 main propagation direction

[0379] 52 near field plane

[0380] 54, 54', 54", 54"' computer-implemented method

[0381] 56 first part simulation step

[0382] 58 second part simulation step

[0383] 60 near field generation step

[0384] 61 characteristic function step

[0385] 62 binary characteristic function

[0386] 63 aerial image generation step

[0387] 64 aerial image

[0388] 66 approximation error

[0389] 68 bandwidth-limited characteristic function

[0390] 70 root mean square error

[0391] 72 maximum error

[0392] 74 root mean square error

[0393] 76 maximum error

[0394] 78 top horizontal axis

[0395] 79 iteration

[0396] 80 bottom horizontal axis

[0397] 81 optimization

[0398] 82 vertical axis

[0399] 83 acquired aerial image

[0400] 84 normal

[0401] 85 metrology system

[0402] 86 wave vector

[0403] 87 uncalibrated aerial image

[0404] 88 reference aerial image

[0405] 89 calibration step

[0406] 90 comparison result

[0407] 91 defect map

[0408] 92 model

[0409] 93 defect

[0410] 94 element

[0411] 95 rigorous simulation step

[0412] 96 computer-implemented method

[0413] 97 registration result

[0414] 98 aerial image step

[0415] 99 registration

[0416] 100 defect detection step

[0417] 101 further information step

[0418] 102 computer-implemented method

[0419] 103 computer-implemented method

[0420] 104 imaging step

[0421] 105 analyzing step

[0422] 106 evaluating step

[0423] 107application step

[0424] 108system

[0425] 110data analysis device

[0426] 112processor

[0427] 114memory

[0428] 116system

[0429] 118subsystem

[0430] 120interface

[0431] 122system

[0432] 124subsystem

[0433] 126charged particle beam image

[0434] 128machine learning model

[0435] 130trained model

[0436] 132reference space image

[0437] 134acquired space image

[0438] 136machine learning model

[0439] 138defect map

[0440] 140model pair

[0441] 142defect-free model

[0442] 144defective model

[0443] 146space image pair

[0444] 148defect-free space image

[0445] 150defective space image

[0446] 152noise

[0447] 154noise-containing space image pair

[0448] 156noise-containing defect-free space image

[0449] 158noise-containing defective space image

Claims

1. A computer-implemented method (54, 54', 54", 54''') for generating an aerial image (64) of a model of a photolithography mask (14) under illumination by an incident electromagnetic wave (22), the method comprising: a) approximately simulating the propagation of the incident electromagnetic wave (22) within a first portion (25) of the photolithography mask (14) comprising a plurality of structures; b) analytically or numerically simulating the propagation of the simulated electromagnetic wave (22) from step a) within the second portion (27) of the photolithography mask (14); c) simulating a representation of the electromagnetic near field (20) of the model of the photolithographic mask (14) by propagating the simulated electromagnetic wave (22) from step b) to a near field plane (52); and d) generating an aerial image (64) of the lithographic mask (14) by applying a simulation of an imaging process of a lithographic system (10, 10') or a metrology system to the representation of the electromagnetic near field (20).

2. The method according to claim 1, wherein The Helmholtz equation is used to approximately simulate the propagation of the incident electromagnetic wave in the first portion (25) of the photolithographic mask (14) in step a).

3. The method according to claim 1, wherein A machine learning model is used to approximately simulate the propagation of the incident electromagnetic wave within the first portion (25) of the photolithography mask (14) in step a).

4. The method according to claim 2, wherein: The Helmholtz equation is approximated using the forward Helmholtz equation.

5. The method according to claim 4, wherein: The forward Helmholtz equation is solved using the beam propagation method.

6. The method of claim 4, wherein: The forward Helmholtz equation is solved using a wave propagation method that approximately describes the propagation of electromagnetic waves (22) through inhomogeneous media.

7. The method according to claim 6, wherein: The first portion of the photolithographic mask (14) is decomposed into different materials by defining a characteristic function for each material, the characteristic function indicating the presence of the material in different locations in the first portion (25) of the photolithographic mask (14), wherein at least one characteristic function is non-binary.

8. The method of claim 37, wherein: The characteristic functions form an affine combination at each location in the first portion (25) of the lithographic mask.

9. The method according to claim 7 or 8, wherein This characteristic function is bandwidth limited.

10. The method according to any one of claims 7 to 9, wherein Apply a low-pass filter to the characteristic function.

11. The method according to claim 10, wherein: Applying the low-pass filter comprises applying a spatially resolved Fourier transform to the characteristic function, followed by an inverse Fourier transform.

12. The method according to any one of claims 6 to 11, wherein The wave propagation method approximates the analytical Fourier transform by a fast Fourier transform and / or approximates the analytical inverse Fourier transform by an inverse fast Fourier transform.

13. The method of claim 12, wherein: The wave propagation method approximates an analytical Fourier transform by a fast Fourier transform, and wherein the wave propagation method takes into account the angle of the incident electromagnetic wave (22) by assuming quasi-periodic boundary conditions in the fast Fourier transform at one or more pairs of opposing boundaries perpendicular to a substrate plane (34) of the photolithography mask (14).

14. The method of claim 13, wherein: The electromagnetic wave (22) within the first portion (25) has a dispersion relation that depends on the angle of the incident electromagnetic wave (22).

15. The method of claim 14, wherein: The dispersion relation within the first portion (25) is modified by a phase shift in coordinates parallel to the substrate plane (34) of the photolithographic mask (14).

16. A method as claimed in any one of the preceding claims, wherein The photolithography mask (14) is a transmission-based photolithography mask.

17. The method according to any one of claims 1 to 15, wherein The photolithography mask (14) is a reflection-based photolithography mask, and wherein the second portion (27) comprises a multilayer (38) in the form of a stack of optical films (40) for reflecting the electromagnetic wave (22).

18. The method of claim 17, wherein: Simulating the reflection of the electromagnetic wave (22) within the multilayer (38) includes analyzing or numerically calculating a reflection coefficient at the boundary (32) between the second portion (27) and the first portion (25) of the photolithographic mask (14), the reflection coefficient describing the propagation of the electromagnetic wave (22) within the stack of optical films (40) of the multilayer (38).

19. The method of claim 18, wherein: The reflection coefficient at the boundary (32) is calculated inside the structure (26) and outside the structure (26) in the first portion (25) of the photolithographic mask (14).

20. The method according to claim 18 or 19, wherein Simulating propagation of the simulated electromagnetic wave (22) within the second portion (27) of the photolithography mask (14) includes applying the reflection coefficient to the electromagnetic wave (22) incident on the boundary (32).

21. A method as described in any of the preceding claims, further comprising adjusting at least one parameter of the method to minimize the difference between one or more reference aerial images of one or more photolithography masks and the corresponding generated aerial images of corresponding models of one or more photolithography masks, wherein the at least one parameter is from a group including mask parameters and optical parameters.

22. The method of any preceding claim, further comprising registering one or more reference aerial images of the lithographic mask to correspondingly generated aerial images of the model of the lithographic mask and reporting at least one registration parameter.

23. The method according to claim 21 or 22, wherein The one or more reference aerial images include a focal stack of photolithography masks.

24. A computer-implemented method (96) for improving the design of a photolithography mask (14), for repairing a photolithography mask (14), for determining the quality of a photolithography mask (14), for performing measurements of a photolithography mask (14), for detecting or evaluating defects in a photolithography mask (14), or for selecting illumination settings in a photolithography system, the method comprising: - using the method of any one of the preceding claims to generate an aerial image (64) of the model of the lithographic mask (14); - analyzing the resulting aerial image (64) accordingly; - Using the analysis results to improve the design of the photolithography mask (14), repair the photolithography mask (14), determine the quality of the photolithography mask (14), detect or evaluate defects in the photolithography mask (14), or select illumination settings in a photolithography system.

25. A computer-implemented method for training a machine learning model, which maps a model of a lithography mask to an aerial image of the lithography mask, the method comprising: generating aerial images of the models of multiple lithography masks using the method described in any one of claims 1 to 23; and training the machine learning model using training data including the generated aerial images.

26. A computer-implemented method for training a machine learning model to perform defect detection in an acquired aerial image of a lithography mask, the method comprising: generating a plurality of model pairs of lithography masks, each model pair comprising a non-defective model of the lithography mask and a defective model of the same lithography mask; generating an aerial image pair from the model pair by applying the method of any one of claims 1 to 23 to the non-defective model and the defective model of each model pair; and training the machine learning model using training data comprising the aerial image pair.

27. A computer-readable medium having stored thereon a computer program executable by a computer device, the computer program comprising codes for executing the method according to any one of claims 1 to 26.

28. A computer program product comprising instructions which, when executed by a computer, cause the computer to perform the method according to any one of claims 1 to 26.

29. A system (108) for generating an aerial image (64) of a model of a lithographic mask (14), the system comprising a data analysis device (110), the data analysis device comprising at least one memory (114) and at least one processor (112), the at least one processor being configured to perform the steps of the computer-implemented method (54, 54', 54", 54''') as claimed in any one of claims 1 to 23.

30. A system for improving a model of a lithography mask (14), for repairing a lithography mask (14), for determining the quality of a lithography mask (14), for performing measurements of a lithography mask (14), for detecting or evaluating defects in a lithography mask (14), or for selecting illumination settings for a lithography system, the system comprising: - A data analysis device (110) comprising at least one memory (114) and at least one processor (112), the at least one processor being configured to perform the steps of the computer-implemented method (96) of claim 24.

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